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Models and methods in hillslope pro�le morphometry

Cox, Nicholas John

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MODELS AND METHODS IN HILLSLOPE PROFILE MORPHOMETRY

Nicholas John Cox

Graduate Society and Trevelyan College

The copyright of this thesis rests with the author.

No quotation from it should be published without

his prior written consent and information derived

from it should be acknowledged.

A th e s i s submitted f o r the degree of Doctor o f Philosophy of the U n i v e r s i t y

of Durham

March 1979

Abstract

This t h e s i s considers and evaluates mathematical models and methods of data analysis used i n the q u a n t i t a t i v e study of h i l l s l o p e p r o f i l e form.

Models of h i l l s l o p e p r o f i l e s are brought together i n a c r i t i c a l and comprehensive review. Modelling approaches are c l a s s i f i e d using f i v e dichotomies (static/dynamic, d e t e r m i n i s t i c / s t o c h a s t i c , phenomenological/representational, a n a l y t i c a l / s i m u l a t i o n , d i s c r e t e / c o n t i n u o u s ) .

P r o f i l e data to serve as examples were c o l l e c t e d using a pantometer i n a 100 km^ square centred on B i l s d a l e i n the North York Moors. Geomorphological i n t e r p r e t a t i o n s put forward f o r t h i s area include theses of profound l i t h o l o g i c a l i n f l u e n c e , p o l y c y c l i c denudation h i s t o r y , p r o g l a c i a l lake overflow channels and profound c r y o n i v a l i n f l u e n c e .

P r o f i l e dimensions, p r o f i l e shapes, angle and curvature frequency d i s t r i b u t i o n s and bedrock geology can be r e l a t e d v i a a f o u r f o l d grouping o f p r o f i l e s . The use o f q u a n t i l e -based summary measures and of a method of s p a t i a l averaging and d i f f e r e n c i n g are advocated and i l l u s t r a t e d .

A u t o c o r r e l a t i o n analysis o f h i l l s l o p e angle s e r i e s appears to be of l i m i t e d geomorphological i n t e r e s t , as a u t o c o r r e l a t i o n f u n c t i o n s t e l l a s t o r y of o v e r a l l p r o f i l e shape, which can be measured more d i r e c t l y i n other ways. Problems o f non-s t a t i o n a r i t y and estimator choice deserve g r e a t e r emphasis.

Methods of p r o f i l e analysis p r e v i o u s l y proposed by Ahnert, Ongley, P i t t y and Young are a l l u n s a t i s f a c t o r y . A method based on a d d i t i v e e r r o r p a r t i t i o n and n o n l i n e a r smoothing i s proposed as an i n t e r i m a l t e r n a t i v e , and r e s u l t s r e l a t e d t o bedrock geology.

An approach t o model f i t t i n g i s o u t l i n e d which treats s p e c i f i c a t i o n , e s t i m a t i o n and checking i n sequence. A power f u n c t i o n due t o Kirkby i s used as an example and f i t t e d t o f i e l d data f o r components. The exercise works w e l l i f regarded as a minimum d e s c r i p t i v e approach but much greater d i f f i c u l t i e s a r i s e i f process i n t e r p r e t a t i o n i s attempted.

A ckn ow l e dgeme n t s

This t h e s i s r e p o r t s on work c a r r i e d out i n the Department o f Geography at Durham U n i v e r s i t y since October 1973. I am very g r a t e f u l to Professor W. B. Fisher, Head of Department, f o r the use o f f a c i l i t i e s w i t h i n the Department, and to the Natural Environment Research Council f o r a Research Studentship during the years 1973 - 1976.

My parents have given me c o n t i n u a l support and encouragement throughout my p r o t r a c t e d education. My bro t h e r blazed more than one academic t r a i l , before me and persuaded me at a c r u c i a l p o i n t t o continue the study o f mathematics. Dr. Ewan Anderson gave me my f i r s t systematic grounding i n geomorphology, and has provided teaching and f r i e n d l y advice over a p e r i o d of twelve years.

I thank those who taught me at Cambridge: my D i r e c t o r of Studies, Mr. A. A. L. Caesar, and my supervisors ( i n ch r o n o l o g i c a l order) Dr. L. W. Hepple, Dr. M. G. Bradford, Dr. D. E. Keeble, Dr. R. J. Bennett, Professor R. J. Chorley, Dr. A. D. C l i f f and Mr. B. W. Sparks.

At Durham my supervisor Dr. Ian Evans has provided an e x c e l l e n t mixture of advice and encouragement. He has read a l l my d r a f t s w i t h a c a r e f u l and r u t h l e s s l y c r i t i c a l eye. I would also l i k e t o thank him f o r h i s own example o f scholarship and research. I am g r a t e f u l t o I a i n Bain, Jeanne Bateson, Dr. David Rhind and Dr. Mahes Visvalingam f o r help w i t h computing and i n t e r e s t i n g discussions.

I acknowledge g r a t e f u l l y the help which I have received from s e c r e t a r i a l and t e c h n i c a l s t a f f i n the Department ( e s p e c i a l l y Kathryn Anderson who typed the manuscript), and from l i b r a r y and computing s t a f f i n the U n i v e r s i t y . The warm and f r i e n d l y atmosphere among postgraduates and s t a f f i n the Department has c o n t r i b u t e d g r e a t l y t o enjoyment o f my research.

I am g r a t e f u l t o Mr. and Mrs. N e i l McAndrew f o r t h e i r warm h o s p i t a l i t y i n B i l s d a l e , and t o landowners and tenant farmers f o r access t o land.

Contents

Chapter 1 INTRODUCTION 1 1.1 Subject 2 1.2 Aims and s t r u c t u r e 4 1.3 Presentation 6

2 PRINCIPLES OP MODELLING HILLSLOPE PROFILES 8 2.1 Geomorphology and h i l l s l o p e morphometry 9 2.2 Ph i l o s o p h i c a l issues i n modelling 13 2.3 Approaches i n modelling 28 2.4 Major geomorphological problems i n modelling 48 2.5 Summary 58 2.6 Notation 59

3 A REVIEW OP CONTINUOUS MODELS OP HILLSLOPE PROFILES 60

3.1 I n t r o d u c t i o n 6 l

3.2 S t a t i c models 62

3.3 Dynamic models 72 3.4 Notation 120

4 THE FIELD AREA 122 4.1 I n t r o d u c t i o n 123 4.2 Geological background 124 4.3 Geomorphological i n t e r p r e t a t i o n s 129 4.4 Quaternary events 143 4.5 Summary 156

5 SAMPLING AND MEASUREMENT PROCEDURES 157 5.1 I n t r o d u c t i o n 158 5.2 Sampling 159 5.3 Measurement 169

5.4 Notation 171

6 DESCRIPTION OP MEASURED PROFILES 172 6.1 P r o f i l e form 173 6.2 Angle and curvature frequency d i s t r i b u t i o n s 188

6.3 Frequency d i s t r i b u t i o n s at d i f f e r e n t scales 201 6.4 Summary 208

6.5 Notation 209

7 AUTOCORRELATION ANALYSIS OF HILLSLOPE SERIES 210 7.1 The idea of a u t o c o r r e l a t i o n 211 7.2 A u t o c o r r e l a t i o n and h i l l s l o p e p r o f i l e s 214 7.3 S t a t i o n a r i t y , n o n s t a t i o n a r i t y and a u t o c o r r e l a t i o n

216 7.4 Choice o f estimator 220 7.5 Em p i r i c a l r e s u l t s 225 7.6 Summary 245 7.7 Notation 246 8 PROFILE ANALYSIS 247 8.1 The problem o f p r o f i l e analysis 248 8.2 Geomorphological considerations 252 8.3 Basic p r i n c i p l e s 255 8.4 E x i s t i n g methods 257 8.5 A d d i t i v e e r r o r p a r t i t i o n s i n p r i n c i p l e 276

8.6 Ad d i t i v e e r r o r p a r t i t i o n s i n p r a c t i c e 279 8.7 Summary and discussion 293 8.8 Notation 297

9 PITTING CONTINUOUS CURVES TO HILLSLOPES 299 9.1 The general s i t u a t i o n 300 9.2 S p e c i f i c a t i o n and es t i m a t i o n : f i r s t approximation

302 9.3 S p e c i f i c a t i o n and es t i m a t i o n : f u r t h e r

approximations 30,6 9.4 D e t e r m i n i s t i c e s t i m a t i o n of the Kirkby parameter

315 9.5 Checking 317 9.6 Results f o r f i e l d p r o f i l e s 319

9.7 Summary and discussion 346 9.8 Notation 351

10 CONCLUSIONS 353 10.1 Retrospect 354 10.2 Prospect 358

REFERENCES 36l APPENDICES 403 APPENDIX I : H i l l s l o p e p r o f i l e data 404 APPENDIX I I : B r i e f program d e s c r i p t i o n s 421

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Chapter 1

INTRODUCTION

A l a i n once s a i d there are only two kinds o f scholars: those who love ideas and those who loathe them. I n the world o f science these two a t t i t u d e s continue t o oppose each other; but bo t h , by t h e i r c o n f r o n t a t i o n , are necessary t o s c i e n t i f i c progress. One can only r e g r e t , on behal f o f those who scorn ideas, t h a t t h i s progress, t o which they c o n t r i b u t e , i n v a r i a b l y proves them wrong.

Jacques Monod, Chance and n e c e s s i t y , Ch. 8

1.1 Subject

1.2 Aims and s t r u c t u r e

1.3 Presentation

Uni'vor,

^ SCIENCE

6/VUG ) J SECTION

J' Li b r a n .

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1.1 Subject

This t h e s i s l i e s i n the f i e l d of h i l l s l o p e p r o f i l e morphometry. The term 'morphometry' i s here used i n a wide sense t o include mathematical models and methods o f sampling, measurement and data analysis used i n the study of landsurface geometry. I t i s a convenient c o n t r a c t i o n of the more c o r r e c t term 'geomorphometry' ( c f . T r i c a r t , 1947; Evans, 1972).

A h i l l s l o p e p r o f i l e i s a l i n e connecting a drainage d i v i d e at the crest of a h i l l s l o p e w i t h a drainage l i n e at i t s base. I t i s a 'wiggly l i n e ' (Scheidegger, 1970, 7) i n two dimensions (one v e r t i c a l , one h o r i z o n t a l ) corresponding t o a maximum gradient path i n three dimensions (one v e r t i c a l , two h o r i z o n t a l ) .

The geometry of a h i l l s l o p e p r o f i l e i s i t s form sensu s t r i c t o . Form sensu l a t o includes c h a r a c t e r i s t i c s of s o i l , u n derlying bedrock, vegetation and c l i m a t e . H i l l s l o p e s are modified by various processes operating on the h i l l s l o p e and at i t s f l a n k s and endpoints. F u l l understanding of h i l l s l o p e form, process and development i s t o be sought i n the i n t e r a c t i o n of form and process over time and space: t h i s i s the concern o f h i l l s l o p e geomorphology.

I n attempting d e s c r i p t i o n and explanation of h i l l s l o p e form, h i l l s l o p e morphometry draws .upon ideas and techniques o f mathematics and s t a t i s t i c s . I t i s convenient t o d i s t i n g u i s h various classes o f problems a r i s i n g i n h i l l s l o p e

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morphometry ( i l l u s t r a t e d here by examples from the f i e l d of m o d e l l i n g ) .

( i ) Methodological problems (e.g. what degree of s i m p l i f i c a t i o n i s permissible?)

( i i ) T h e o r e t i c a l problems (e.g. what are the geomorphological grounds f o r t h i s model?)

( i i i ) Technical problems (e.g. what i s the mathematical basis o f t h i s model?)

( i v ) E m p i r i c a l problems (e.g. i s a p a r t i c u l a r model a r e a l i s t i c d e s c r i p t i o n of a given f i e l d p r o f i l e ? )

A l l these problems are important and need t o be considered c a r e f u l l y .

Any choice of research problem i s usually a r e f l e c t i o n of an i n v e s t i g a t o r ' s i n t e r e s t s and competence and h i s perception of the importance of problems w i t h i n a d i s c i p l i n e . The choice made here r e f l e c t s a stro n g i n t e r e s t i n methodological, t h e o r e t i c a l and t e c h n i c a l questions and a b e l i e f t h a t e m p i r i c a l i n v e s t i g a t i o n i s worthless without the backing o f sound ideas and methods. Contrary t o the opinions o f some f i e l d w o r k e r s i n geomorph-ology, such backing i s not a v a i l a b l e unless s p e c i a l i s t i n v e s t i g a t i o n i s undertaken i n t o methodological, t h e o r e t i c a l and t e c h n i c a l questions.

Discussion of t h i s issue o f t e n s l i d e s i n t o a p u e r i l e debate over the r e l a t i v e merits o f 'armchair' and 'muddy boots' approaches i n geomorphology. The simple answer.

_ II _

i s t h a t such approaches are complementary, not competing, and t h a t i t i s not a matter of one approach being superior and the other being i n f e r i o r . But t h i s s t i l l leaves room f o r argument over d e t a i l s ( e x a c t l y how do these approaches complement each o t h e r ? ) , and the d e t a i l s are not a matter f o r a b s t r a c t discussion: antagonists must s t a r t t o consider s p e c i f i c circumstances.

1.2 Aims and s t r u c t u r e

The aims of t h i s t h e s i s are as f o l l o w s . ( i ) To provide c r i t i c a l and comprehensive reviews of

work i n modelling and data analysis i n h i l l s l o p e morphometry, concentrating p a r t i c u l a r l y on continuous models (Chs. 2, 3) and p r o f i l e analysis (Ch. 8). I t i s f e l t t h a t the absence of such reviews i s a b a r r i e r t o progress, and t h a t as w e l l as h i n d e r i n g communication w i t h i n each f i e l d , i t impedes understanding o f models and methods by workers outside.

( i i ) To place procedures on a f i r m mathematical basis and t o evaluate t h e i r p r a c t i c a l u t i l i t y , c o n centrating p a r t i c u l a r l y on a u t o c o r r e l a t i o n analysis (Ch. 7 ) , p r o f i l e analysis (Ch. 8) and model curve f i t t i n g (Ch. 9 ) . I n each case i t i s shown t h a t e x i s t i n g geomorphological p r a c t i c e has considerable t e c h n i c a l shortcomings; a l t e r n a t i v e procedures are explained and evaluated e m p i r i c a l l y .

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( i i i ) To consider the e m p i r i c a l r o l e of h i l l s l o p e morphometry i n geomorphology. This e n t a i l s consideration of the hypotheses put forward i n the l i t e r a t u r e f o r the f i e l d area (Ch. 4) and analysis o f the i m p l i c a t i o n s o f morphometric r e s u l t s f o r these hypotheses (Chs. 6, 8, 9 ) . The aim i s not t o solve f i e l d problems or to provide new i n t e r p r e t a t i o n s of the geomorphology of the f i e l d area: t h a t would r e q u i r e much more work on forms, deposits and processes than has been p o s s i b l e .

( i v ) To set the f i e l d o f h i l l s l o p e morphometry w i t h i n a methodological and t h e o r e t i c a l context, p a r t i c u l a r l y by r e l a t i n g ideas on continuous models t o geomorphological th e o r y , the philosophy of science, and the methodology o f the n a t u r a l and environmental sciences (Ch. 2 ) . Here scholarship can provide a c l e a r f o r m u l a t i o n of genuine problems and a dismissal of pseudoproblems.

The chapter s t r u c t u r e i s as f o l l o w s .

Chapters 2 and 3 give a review of h i l l s l o p e p r o f i l e modelling. Chapter 2, on p r i n c i p l e s , i s almost e n t i r e l y non-mathematical: i t examines methodological and t h e o r e t i c a l issues at l e n g t h . This chapter should serve as a nontechnical i n t r o d u c t i o n f o r the general geomorphologist. Chapter 3 gives a d e t a i l e d and comprehensive review of the models which have been proposed; i t i s presented i n a u n i f i e d n o t a t i o n , and includes some new r e s u l t s .

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Chapter 4 suppliea background i n f o r m a t i o n on the 2

f i e l d area, a 100 km square centred on B i l s d a l e i n the North York Moors. I t discusses the hypotheses which have been put forward and i d e n t i f i e s the problems which may be attacked using morphometry. Chapter 5 b r i e f l y describes the sampling and measurement procedures used to c o l l e c t h i l l s l o p e p r o f i l e data f o r t h i s study. Chapter 6 describes the f i e l d p r o f i l e s , o u t l i n i n g the v a r i a t i o n s which e x i s t i n p r o f i l e form and i n angle and curvature frequency d i s t r i b u t i o n s . Some t e c h n i c a l innovations are employed.

Chapters 7, 8 and 9 place p a r t i c u l a r procedures on a f i r m mathematical basis and evaluate t h e i r p r a c t i c a l u t i l i t y . A u t o c o r r e l a t i o n analysis (Ch. 7) i s a r e l a t i v e l y new technique i n h i l l s l o p e geomorphology: i t s usefulness has not so f a r been examined s y s t e m a t i c a l l y . P r o f i l e analysis (Ch. 8) i s a f i e l d w i t h several competing methods: an attempt i s made t o separate the wheat from the cha f f . Model curve f i t t i n g (Ch. 9) has not received c a r e f u l a t t e n t i o n t o date: even the r e l a t i v e l y simple case discussed here, a one-parameter n o n l i n e a r model, presents some challengi n g problems.

Chapter 10 presents the conclusions of the t h e s i s .

1.3 Presentation

Figures and Tables are not d i s t i n g u i s h e d but are both regarded as E x h i b i t s ( c f . Tukey, 1977). This p r a c t i c e

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f l o u t s convention, but saves separate l a b e l l i n g systems and encourages h y b r i d d i s p l a y s . E x h i b i t s are l a b e l l e d a l p h a b e t i c a l l y w i t h i n chapters (e.g. 2A, 2B), whereas chapters, sections and subsections are l a b e l l e d numerically using a common h i e r a r c h i c a l n o t a t i o n .

Algebraic n o t a t i o n i s intended t o be consistent w i t h i n i n d i v i d u a l chapters and i s c o l l a t e d at the end of each chapter. I t has r e g r e t t a b l y not been found possible t o use a s i n g l e system o f n o t a t i o n throughout the t h e s i s .

Appendix I l i s t s p r o f i l e data and Appendix I I gives b r i e f d e s c r i p t i o n s o f computer programs.

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Chapter 2

PRINCIPLES OF MODELLING HILLSLOPE PROFILES

About t h i r t y years ago there was much t a l k t h a t g e ologists ought only t o observe and not t h e o r i s e ; and I w e l l remember someone saying t h a t a t t h i s r a t e a man might as w e l l go i n t o a g r a v e l - p i t and count the pebbles and describe the colours. How odd i t i s t h a t anyone should not see t h a t a l l observation must be f o r or against some view i f i t i s t o be of any s e r v i c e .

Charles Darwin i n a l e t t e r t o Henry Fawcett, 1861,quoted by P. B. Medawar, I n d u c t i o n and i n t u i t i o n i n s c i e n t i f i c thought, p. 11.

2.1 Geomorphology and h i l l s l o p e morphometry

2.2 P h i l o s o p h i c a l issues i n modelling

2.3 Approaches i n modelling

2.4 Major geomorphological problems i n modelling

2.5 Summary

2.6 Notation

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2 . 1 Geomorphology and h i l l s l o p e morphometry

The science o f geomorphology studies landforms and r e l a t e d processes, and aims t o describe and e x p l a i n the form of the land surface. I d e a l l y such d e s c r i p t i o n and explanation should be rooted i n a systematic theory which presents a coherent account o f form, process and development.

Three systematic approaches are e s p e c i a l l y note­worthy : ( i ) the land surface viewed as a continuous rough surface (Evans, 1972 ; Mark, 1975)

( i i ) the land surface viewed as a hierarchy of drainage basins (Leopold et a l , 1 9 6 4 ; Chorley, 1969; Gregory and W a l l i n g , 1973; Douglas, 1977)

( i i i ) the land surface viewed as a set of h i l l s l o p e s corresponding t o maximum gradient paths between drainage divides and drainage l i n e s (Carson and Kirkby, 1972; Young, 1972).

These approaches a l l have great value. The idea of a continuous rough surface i s the most general, whereas drainage basins and h i l l s l o p e s are most r e a d i l y i d e n t i f i e d where f l u v i a l (slope and stream) processes are dominant. On the other hand, they are f u n c t i o n i n g systems as w e l l as n a t u r a l geometric e n t i t i e s . H i l l s l o p e s are conveniently simpler than drainage basins; however, reduction t o one h o r i z o n t a l dimension loses the e f f e c t s o f plan curvature. Most of the landsurface i s composed of v a l l e y slopes (Young, 1972, 1) and so geomorphology ' i s by necessity mainly a study o f slopes' (Ahnert, 1 9 7 1 , 3 ) .

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The place o f h i l l s l o p e study w i t h i n geomorphology, and the place of morphometry i n h i l l s l o p e geomorphology, deserve more d e t a i l e d examination. Recent d i v e r s i f i c a t i o n of approaches w i t h i n geomorphology has o f t e n seemed tantamount to d i s i n t e g r a t i o n of the d i s c i p l i n e . Geomorphology i s i n motley d i s a r r a y , a 'bandwaggon parade 1, to use Jennings' (1973) picturesque expression. There has been much concern t h a t d i f f e r e n t 'schools* i n geomorphology f a i l t o under­stand one another. Indeed, Chorley (1967, 59) asked whether 'the study o f landforms s t i l l e x i s t s as a d i s c r e t e s c h o l a r l y e n t i t y ' and commented upon 'the i n a b i l i t y of workers t o i d e n t i f y broad common ob j e c t i v e s of even the most general character, or even t o communicate to one another t h e i r mutual o b j e c t i o n s ' .

Hence i t i s important t o o u t l i n e a view of h i l l -slope morphometry as p a r t of a ' p l u r a l i s t ' geomorphology i n which the existence of d i f f e r e n t approaches i s recognised and resolved (Butzer, 1973)• Butzer i d e n t i f i e d f o u r major d i r e c t i o n s o f primary research i n the d i s c i p l i n e :

(.i) Q u a n t i t a t i v e study o f geomorphological processes ( i i ) Q u a n t i t a t i v e analysis of landforms

( i i i ) Q u a n t i t a t i v e and q u a l i t a t i v e study of sediments ( i v ) Systematic, r e g i o n a l studies o f complex land-

form e v o l u t i o n through time and i n the wake of environmental change.

Research may also be c l a s s i f i e d according t o the geomorphological systems which are of primary i n t e r e s t

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(e.g. h i l l s l o p e s , drainage basins, topographic s u r f a c e s ) . A c r o s s - c l a s s i f i c a t i o n y i e l d s a simple m a t r i x r e p r e s e n t a t i o n of contemporary geomorphology which allows h i l l s l o p e morphometry t o be p r e c i s e l y located w i t h i n current research (2A). A glance at t h i s diagram shows t h a t h i l l s l o p e morphometry can be l i n k e d t o other approaches t o h i l l -slopes (row l i n k a g e s ) , and to other branches of morpho­metry (column l i n k a g e s ) .

This m a t r i x r e p r e s e n t a t i o n r e f l e c t s a s i m p l i f i e d yet s t r u c t u r e d view of the important d i r e c t i o n s of current research. I t may seem e n t i r e l y u n c o n t r o v e r s i a l . Nevertheless, i n s t r e s s i n g approaches based on r e p l i c a t e d systems (such as h i l l s l o p e s or drainage basins) t h i s view to some extent stands opposed t o a s t r o n g t r a d i t i o n i n geomorphology, which concentrates on i n t e r e s t i n g 'features' and by comparison neglects supposedly 'featureless* areas. This bias has had unfortunate consequences: '. . . the geomorphologist at the present r a t e o f knowledge can o f t e n say remarkably l i t t l e by way of d e s c r i p t i o n or explanation about an ordinary " f e a t u r e l e s s " r o l l i n g landscape' (Lewin, 1969, 84) . However, an a l l e g e d l y f e a t u r e l e s s f l u v i a l landscape can be discussed i n terms of i t s c o n s t i t u e n t h i l l s l o p e s and drainage basins. A concern f o r a t y p i c a l and s t r i k i n g forms should be supplemented by an analysis of o r dinary landscapes, which i s of equal i n t e r e s t and importance.

The analyses considered i n t h i s t h e s i s are morphometric.involving the use of mathematical models and

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of methods of sampling, measurement and data analysis ( c f . the d e f i n i t i o n i n Ch. 1.1 above). These analyses must not be u n r e f l e c t i v e , so t h i s chapter continues w i t h discussions of p h i l o s o p h i c a l issues, approaches and major geomorphological problems i n modelling. (Questions of sampling, measurement and data analysis receive a t t e n t i o n i n Chapters 5 to 9 below).

2.2 P h i l o s o p h i c a l issues i n modelling 2.2.1 C r i t i c a l r a t i o n a l i s m and the i n e v i t a b i l i t y of theory

Discussion of p h i l o s o p h i c a l issues i s rare i n geomorphology, and the d i s c i p l i n e i s dominated by what Reynaud (1971> 95-9) c a l l e d 'the spontaneous philosophy of geomorphologists', an a n t i t h e o r e t i c a l empiricism und e r l a i n by methodological p r i n c i p l e s which are f r e q u e n t l y i m p l i c i t r a t h e r than e x p l i c i t . I n contrast t h i s . t h e s i s adopts the viewpoint of c r i t i c a l r a t i o n a l i s m set out by Popper (Popper, 1972, 1976; Medawar, 1969; Magee, 1973). According t o t h i s view, i t i s best t o formulate ideas as c l e a r l y as p o s s i b l e ; and to subject them t o severe c r i t i c i s m . This a p p l i e s , f o r example, at a methodological l e v e l (such as when a methodological p r i n c i p l e i s i n question).and at an e m p i r i c a l l e v e l (such as when a hypothesis i s i n q u e s t i o n ) . I t i s i n t e r e s t i n g t o note the broad s i m i l a r i t y of G i l b e r t ' s methodological ideas ( c f . G i l b e r t , 1886; G i l l u l y , 1963; K i t t s , 1973).

Such a view i s i n e v i t a b l e once one r e a l i s e s t h a t i t i s impossible t o work without assumptions e i t h e r at a

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methodological l e v e l or a t an e m p i r i c a l l e v e l . There are o f t e n occasions when the assumptions are t r i v i a l or uncon-t r o v e r s i a l , but i n general i t i s valuable t o review the assumptions made i n a piece of work, and t o assess t h e i r v a l i d i t y .

Some theory about the world i s i n e v i t a b l e . Even i f we attempt merely t o observe, we use i m p l i c i t theory about what i s noteworthy, and are i n f l u e n c e d by preconceptions. '. . . The " f a c t s " t h a t enter our knowledge are already viewed i n a c e r t a i n way and are, t h e r e f o r e , e s s e n t i a l l y i d e a t i o n a l . . . Experience arises together w i t h t h e o r e t i c a l assumptions not before them, and an experience without theory i s j u s t as incomprehensible as i s ( a l l e g e d l y ) a theory without experience 1 (Peyerabend, 1975, 19» 168).

2 .2 .2 . Key terms discussed

I n t e r e s t i n p h i l o s o p h i c a l matters i s o f t e n associated w i t h a preoccupation w i t h questions o f d e f i n i t i o n , meaning and terminology. I t i s , however, generally possible t o avoid arguments over d e f i n i t i o n s . '. . . Since a l l d e f i n i t i o n s must use undefined terms, i t does n o t , as a r u l e , matter whether we use a term as a p r i m i t i v e term or as a defined term' (Popper, 1972, 58; c f . Popper, 1966, 9-21; Geach, 1976, Ch. 9 ) . Nevertheless, some key terms need t o be discussed i n d e t a i l , because they may be u n f a m i l i a r , or because they are i n p r a c t i c e used i n d i f f e r e n t senses.

One valuable d i s t i n c t i o n i s between e p i s t e m o l o g i c a l and o n t o l o g i c a l questions. O n t o l o g i c a l questions are about

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the w o r l d , i t s character and i t s h i s t o r y . Epistemological questions are about knowledge of the w o r l d , or more p r e c i s e l y about the v a l i d i t y of claims t o knowledge of the world. This simple d i s t i n c t i o n can help t o c l a r i f y controversies ( c f . Watson, 1969; Levins, 1970; Morales, 1975), as w i l l be seen below.

The term theory i s here used i n a general way t o denote a set of ideas used i n d e s c r i p t i o n and explanation. These ideas may be i m p l i c i t or e x p l i c i t ; and i f e x p l i c i t , expressed v e r b a l l y or mathematically.

The term system denotes 'a c o l l e c t i o n o f components which are e i t h e r a c t i n g upon other components, being acted upon, or mutually i n t e r a c t i n g ' ( B i r t w i s t l e e t a l , 1973, 13): t h a t i s , a c o l l e c t i o n of components which are i n t e r r e l a t e d i n some way ( c f . Margalef, 1968, 2; Chapman, 1977, 79-82). I n p a r t i c u l a r , 'system' i s usually t o be understood as ' r e a l - w o r l d system of i n t e r e s t ' (Wilson, 1972, 31): i n t h i s t h e s i s , the systems o f i n t e r e s t are h i l l s l o p e s or f l u v i a l systems at smaller or l a r g e r scales.

The term model denotes ' a s i m p l i f i e d p i c t u r e of the system' ( B i r t w i s t l e et_al , 1 9 7 3 , 1*0 • I n p a r t i c u l a r , 'model' i s usually t o be understood as 'formal r e p r e s e n t a t i o n of a theory' (Wilson, 1972, 32); i n f o r m a l schemes, whether v e r b a l , g r a p h i c a l o r i m p l i c i t , are not here considered as models.

These d e f i n i t i o n s c o n t r a s t w i t h usages: common i n geomorphology. Many workers use the term 'model' i n a

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c a t h o l i c sense ( e s p e c i a l l y Chorley, 1967; Thornes and Brunsden, 1977). Model and system concepts are o f t e n confused (tangled up) or c o n f l a t e d (taken as one) i n the geographical and geomorphological l i t e r a t u r e (e.g. Harvey, 1969, Ch. 23; Chorley and Kennedy, 1971; Carson and Ki r k b y , 1972, Chs. 1 and 2; Andrews, 1975, 8-10;

Sugden and John, 1976, 4 ) . I t i s t r u e t h a t according t o the d e f i n i t i o n s given above models themselves may be regarded as systems ( B i r t w i s t l e et a l , 1973, 15) but such c o n f l a t i o n i s not encouraged i n t h i s t h e s i s ; a sharp d i s t i n c t i o n i s made between the two concepts. A system i s e s s e n t i a l l y a p a r t of the p h y s i c a l w o r l d (an o n t o l o g i c a l e n t i t y , i n a sense) whereas a model i s e s s e n t i a l l y a human c r e a t i o n (an epistemological e n t i t y , i n a sense).

Both 'model' and 'system' are used i n a great v a r i e t y of senses both w i t h i n and outside geomorphology, and there seems t o be no very strong reason f o r regarding any p a r t i c u l a r senses as e s s e n t i a l l y c o r r e c t . Any choice i s admittedly a r b i t r a r y . That made here at le a s t seems simple and s t r a i g h t f o r w a r d , and i t does allow various methodological and e m p i r i c a l issues t o be formulated c l e a r l y . I n several ways, one may ask how 'good' a model i s as a re p r e s e n t a t i o n of a given system. Even f o r m u l a t i n g such questions i s g r e a t l y aided by a sharp d i s t i n c t i o n between 'model' and 'system'.

I f geomorphology aims t o e x p l a i n the form o f the land surface, then i t i s necessary t o consider the.

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c r i t e r i a f o r an i d e a l e x planation. This i s a large issue, much debated both i n philosophy and the sciences. Two c r i t e r i a deserve s p e c i a l emphasis (Popper, 1972, 191-3). F i r s t l y , an 'explanation' should be independently t e s t a b l e and not ad hoc or c i r c u l a r . Secondly, i t should r e l a t e a set of c o n d itions t o one or more u n i v e r s a l laws and i n i t i a l c o n d i t i o n s . These c r i t e r i a remain as l o g i c a l i d e a l s even though many supposed explanations i n geomorphology f a i l t o s a t i s f y them. (The character o f the ' u n i v e r s a l laws' invoked i n geomorphology w i l l be discussed below).

2.2.3. Why use mathematics?

I t i s reasonable t o ask why the use of mathematics, an a b s t r a c t and formal e x e r c i s e , should be o f value i n under­standing geomorphological systems. Why should a s t r i n g of symbols on a sheet o f paper have any relevance t o the behaviour of masses of s o i l , rock and water? This apparently naive question raises some deep p h i l o s o p h i c a l problems, yet there do not seem to be more than a few s c a t t e r e d comments upon the issue i n the geomorphological l i t e r a t u r e . I t i s t a c i t l y assumed e i t h e r t h a t the use o f mathematics i s f u t i l e , i r r e l e v a n t or p r e t e n t i o u s , or t h a t as a standard s c i e n t i f i c procedure i t needs l i t t l e or no j u s t i f i c a t i o n .

I t i s d i f f i c u l t t o f i n d a s a t i s f a c t o r y answer t o t h i s q u e s t i o n , and some alleged s o l u t i o n s are unconvincing. Atiyah (1976, 292) suggested t h a t mathematics may be viewed as the science of analogy, and t h a t 'the widespread a p p l i c a b i l i t y of mathematics i n the n a t u r a l sciences, which

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has i n t r i g u e d a l l mathematicians o f a p h i l o s o p h i c a l bent, arises from the fundamental r o l e which comparisons play i n the mental process we r e f e r to as "understanding"'. This suggestion appears to be l i t t l e more than a r e f o r m u l a t i o n of the problem, f o r equally s t r i k i n g questions immediately a r i s e : how and why are comparisons i n v o l v e d i n understanding?, are some kinds ' b e t t e r ' than o t h e r s , and i f so, i n what sense and why? And so on.

Hence a d e t a i l e d c o n s i d e r a t i o n of the a c t u a l and possible ro l e s o f mathematics i n geomorphology i s i n order. I n what f o l l o w s , mathematics i s taken as 'given', a set o f formal languages which may be of use i n d e s c r i p t i o n or explanation. But i t should be s t a t e d t h a t the character of mathematics i s i t s e l f a matter f o r considerable p h i l o s o p h i c a l dispute (e.g. Kfirner, 1960, 1971; Lakatos, 1962, 1976a, 1976b;

K r e i s e l , 1965; S t e i n e r , 1975).

One f a c t f r e q u e n t l y allowed t o confuse the issue i s t h a t geomorphologists o f t e n f i n d mathematical n o t a t i o n u n f a m i l i a r , p u z z l i n g or d i s t u r b i n g . Many would f e e l happy w i t h the s t r i n g of symbols 'the slope i s steep' but not w i t h the s t r i n g ' > y '. (The two could be construed as equivalent w i t h conventional i n t e r p r e t a t i o n s of 3 , j J and > and appropriate i n t e r p r e t a t i o n s o f z, x and Y ) . The u n f a m i l i a r i t y o f algeb r a i c n o t a t i o n i s an important educational and psychological i s s u e , and the mathematical weakness o f most geomorphologists has f a r from t r i v i a l consequences , but n e i t h e r has much bearing on the p h i l o s o p h i c a l issue of the value of mathematics as a mode of expression.

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N a t u r a l l y i t would be i n a p p r o p r i a t e t o use mathematics i n s i t u a t i o n s where ordinary language serves the purpose as w e l l or b e t t e r . The example j u s t given might seem an e x c e l l e n t case i n p o i n t . Notice, however, t h a t as soon as we wish t o say how steep a p a r t i c u l a r slope i s , or t o define 'steep', mathematics becomes the appropriate medium: numerical answers are req u i r e d f o r these questions. Moreover, the customary arguments from abuse or misuse are overplayed: arguing generally against mathematical a p p l i c a t i o n s i n geomorphology on the grounds t h a t some past a p p l i c a t i o n s have been mistaken or misleading (which i s c e r t a i n l y t r u e ) i s l o g i c a l l y on a par w i t h blaming a weapon f o r a crime. No approach bears a guarantee of success, and i t i s absurd to ask f o r one. There are many cases, a c t u a l or p o t e n t i a l , i n which 'the mathematical formalism may be h i d i n g as much as i t reveals' (Schwartz, 1962, 360). Hence i t i s not a matter of asking f o r j u s t i f i c a t i o n of a mathematical approach, but o f seeking a c l e a r view of how and why mathematics might be u s e f u l .

Another common misconception i s the view t h a t the usefulness o f l i n g u i s t i c symbolism i s obvious while the usefulness of mathematical symbolism requires explanation. Close examination shows t h a t both kinds of symbolism r a i s e deep problems ( c f . Craik, 1967, Ch. 5; Skellam, 1972, 15) .

I t i s thus i l l e g i t i m a t e t o object t o mathematical symbolism as symbolism unless we also ob j e c t t o l i n g u i s t i c symbolism.

I t seems l i k e l y t h a t mathematics might be u s e f u l t o geomorphology i n a v a r i e t y of ways, nor i s such m u l t i p l i c i t y p a r t i c u l a r l y s u r p r i s i n g : i t also applies t o ordinary language.

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( i ) The idea o f mathematics as a science o f analogy or p a t t e r n , noted by At i y a h (1976), i s c e r t a i n l y o f importance. 'Any p a t t e r n we see i n the universe w i l l be one f o r which a mathematical treatment i s p o s s i b l e ; conversely, whenever a new mathematical i n s i g h t occurs, we are able t o recognise new kinds of p a t t e r n s . I f any o f these occur i n nature, we have a t o t a l l y unexpected a p p l i c a t i o n o f the theory. And t h i s i s how mathematics gets i t s power; f o r a p a t t e r n which i s hard t o recognise i n one area may be obvious i n another. By t a k i n g i n s p i r a t i o n from the second we discover the existence of the f i r s t ' (Anon, 1973, 658). N a t u r a l l y the a p p l i c a t i o n of analogies must be c a r e f u l : the relevance of an a t t r a c t i v e analogy can e a s i l y be exaggerated and analogies r e q u i r e independent t e s t i n g ( c f . Wilson, 1969).

( i i ) Ordinary language i s e s s e n t i a l l y t o p o l o g i c a l and i t i s possible t o give a ve r b a l account of the important v a r i a b l e s i n a system and i n d i c a t e the connections and r e l a t i o n s h i p s between them. Conversely many o f the t o p o l o g i c a l 'box-and-arrow' diagrams i n the geomorphological l i t e r a t u r e could be t r a n s l a t e d i n t o words without appreciable loss of content. However, mathematical representations allow alge b r a i c and a r i t h m e t i c s p e c i f i c a t i o n o f r e l a t i o n s h i p s which i n d i c a t e the form and size of connections ( c f . Nelder, 1972, 368) and are thus more i n f o r m a t i v e . I t i s more i n f o r m a t i v e t o specify t h a t a h i l l s l o p e obeys a power f u n c t i o n than t o specify t h a t i t i s curved, and more i n f o r m a t i v e s t i l l t o spe c i f y parameter values.

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( i i i ) An algebraic argument has the merit t h a t i n p r i n c i p l e assumptions are made e x p l i c i t , d e r i v a t i o n s are shown, and the character and form of s o l u t i o n s are made p l a i n . The l o g i c a l s t r u c t u r e of the argument i s exposed t o p u b l i c s c r u t i n y , to c r i t i c i s m and t o t e s t i n g ( c f . Ziman, 1968, ^5). I n c o n t r a s t , a ve r b a l argument may be a s t o n i s h i n g l y d i f f i c u l t t o evaluate.

( i v ) An e n t i r e l y algebraic argument divorced from e m p i r i c a l evidence i s usu a l l y regarded w i t h suspicion. Nevertheless, i t i s important t o note the p o s s i b i l i t y t h a t mathematics may be used as 'a t o o l f o r gaining q u a l i t a t i v e i n s i g h t i n t o r e a l phenomena' (Smith and Br e t h e r t o n , 1972,

1507). I f assumptions made are demonstrably weak, or i f r e s u l t s are q u a l i t a t i v e l y s t a b l e under p e r t u r b a t i o n s of the axioms, then i t may be possible t o produce 'robust theorems' ( c f . Schwartz, 1962, 357; Levins, 1970, 76: Levin, 1975, 785).

A l t e r n a t i v e l y there are methods f o r i n v e s t i g a t i n g the q u a l i t a t i v e p r o p e r t i e s o f p a r t i a l l y s p e c i f i e d systems

(May, 1973; Levins, 197*1).

(v) One p o i n t of view goes beyond t h i s and stresses the l o g i c a l character of models, used as means of d e r i v i n g the consequences of i n i t i a l assumptions. Their e m p i r i c a l r e a l i s m ( t r u t h or f a l s i t y ) i s regarded as secondary. Lewontin (1963,

224) argued th a t models are not contingent, but a n a l y t i c ; models should never be s a i d 'to be t r u e or f a l s e i n an e m p i r i c a l sense'. 'A model i s e s s e n t i a l l y a c a l c u l a t i n g engine designed t o produce some output f o r a given i n p u t . . .

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Models cannot produce r e a l l y "new" knowledge, but can only demonstrate what i s e n t a i l e d by the theory from which the model i s built» (225). S i m i l a r l y Morales (1975, 335) deplored 'the mistake of using models as mi r r o r s of r e a l i t y r a t h e r than as h e u r i s t i c c o n s t r u c t s 1 . Models tend to be 'attacked, or defended, as o n t o l o g i c a l l y t r u e or f a l s e r a t h e r than e p i s t e m o l o g i c a l l y u s e f u l and i n s i g h t f u l or obfuscating and i n e f f e c t u a l ' (337). Neither Lewontin nor Morales advocated e m p i r i c a l i r r e l e v a n c e : they merely request a c l e a r view of

^ models i n which t h e i r l o g i c a l character i s appreciated.

( v i ) I t i s sad but t r u e t h a t q u a l i t a t i v e impressions obtained i n the f i e l d may be s e r i o u s l y misleading as w e l l as imprecise. '. . . A property which seems p e r f e c t l y apparent, or an "obvious" r e l a t i o n of cause and e f f e c t , may upon c a r e f u l measurement and analysis prove to be exa c t l y the reverse of the "apparent" or the "obvious"' (Leopold e t a l , 1964, 8 ) . Thus i n 1712 i t was generally b e l i e v e d t h a t Northamptonshire was the highest county i n England: r e l i a b l e methods of height measurement were only developed l a t e r ( P o r t e r , 1977, 223). Recent psych o l o g i c a l work on ge o l o g i c a l observation has undermined the widely-held b e l i e f t h a t the impressions of experienced i n v e s t i g a t o r s are o b j e c t i v e (Chadwick, 1975, 1976). Hence the value of q u a n t i t a t i v e measurement i s great.

( v i i ) Where some geomorphological response i s the r e s u l t of several f a c t o r s , i t i s important t o determine t h e i r r e l a t i v e importance. This can only be done by q u a n t i t a t i v e methods (van Hise, 1904, 605; J e f f r e y s , 1918, 179).

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( v i i i ) A precise statement can be more e a s i l y r e f u t e d than a vague one, and can t h e r e f o r e be b e t t e r t e s t e d (Popper, 1972, 356; c f . Ahnert, 1971, 11). I n c e r t a i n a p p l i c a t i o n s , a high degree of p r e c i s i o n may be spurious or unnecessary, as has o f t e n been poin t e d out, but such s u p e r f l u i t y needs t o be demonstrated r a t h e r than asserted.

2 .2 .4 . Modelling and s i m p l i f i c a t i o n

While the a c t u a l or p o t e n t i a l r o l e of mathematics i n geomorphology has received l i t t l e d e t a i l e d examination, issues of s i m p l i f i c a t i o n have been discussed more f r e q u e n t l y . I n p a r t i c u l a r , i t has o f t e n been al l e g e d t h a t models are unduly s i m p l i f i e d . Consider, f o r example, the f o l l o w i n g remarks. ( i ) 'Other scholars . . . have sought t o c o n t r i b u t e t o the r e s o l u t i o n of morphological problems by a mathematical treatment. Given the complexity of phenomena such a treatment can only be applied t o very simple forms' ( H o i , 1957, 198; t r a n s l a t e d from French). ( i i ) 'The d i r e c t a t t a c k by mathematical methods would seem t o o f f e r very l i m i t e d chances o f success. Dealing as we i n e v i t a b l y are w i t h i n f i n i t e l y v a r i a b l e mixtures o f s o l i d s , l i q u i d s and gases, i t i s manifest t h a t the parameters i n our imagined equations w i l l not be constant, but themselves unmanageably v a r i a b l e * (Wooldridge, 1958, 32).

( i i i ) *. . . Slope p r o f i l e s are usually too i r r e g u l a r t o be described i n t h e i r e n t i r e t y by formulae' ( P i t t y , 1970, 18) .

( i v ) ' Q u a n t i t a t i v e models almost i n v a r i a b l y lead t o s i m p l i f i c a t i o n and t o an i n c r e a s i n g distance t o r e a l i t y beyond permissible l i m i t s ' (Bttdel, 1975, 2 ) .

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Clearly such charges need t o be considered w i t h care. While much w i l l depend on the d e t a i l s o f geomorphologist's purpose and geomorphological system, some general remarks on s i m p l i f i c a t i o n are i n order.

I n the f i r s t p l ace, geomorphological systems are widely recognised to be complex. Thornbury (1954, 21) regarded the p r i n c i p l e t h a t 'complexity of geomorphic e v o l u t i o n i s more common than s i m p l i c i t y ' as a 'fundamental concept1- of geomorphology, w h i l e Schumm (1973, 1977; Schumm and Parker, 1973) has even proposed a p r i n c i p l e of complex response. Yet i t would be wrong t o overemphasise the degree of complexity found i n geomorphological systems. Milovidova (1970) showed how few of the l o g i c a l l y possible landform types a c t u a l l y occur i n a given r e g i o n ; Connelly (1972)

put forward a s i m i l a r view supported by r e s u l t s obtained w i t h entropy measures from a l t i t u d e data.

I n any case, i t " i s desirable t h a t models should be simple. I f a model i s t o be of any use, i t must be a s i m p l i f i e d r e p r e s e n t a t i o n of the system o f i n t e r e s t . An exact copy would be useless i n explanation (Hanson, 1971,

81) . The d e s i r a b i l i t y of s i m p l i c i t y i s encapsulated i n a celebrated l o g i c a l maxim known as Ockham's razor; i t s c o r r e c t v e r s i o n , i s n o t , however, as sharp as i s o f t e n supposed: L f r u s t r a f i t per p l u r a quod potest e q u a l i t e r f i e r i per pauciora' or *. . . i t i s vain t o do by more what can equally be done by fewer' ( L e f f , 1975, 35; c f . Anderson, 1963,

176; Skfellam, 1972, 27) .

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I n h i l l s l o p e geomorphology, s i m p l i f i c a t i o n i s j u s t i f i a b l e i n the many instances when the i n t e r e s t i s ' i n the average p r o p e r t i e s of a h i l l s i d e , not i n a l l of i t s i n t r i c a t e d e t a i l s ' (Scheidegger, 1970, 3; c f . K i r k b y , 1974, 2; M i z u t a n i , 1974, 4 ) . N a t u r a l l y , there are occasions on which the d e t a i l s are of i n t e r e s t , but the p o i n t i n t h i s context i s t h a t i n modelling p r o f i l e s the aim i s merely t o account f o r the o v e r a l l form of the p r o f i l e , and not f o r a l l the i n f i n i t e ( s i c ) v a r i a b i l i t y (pace Wooldridge), nor f o r the p r o f i l e i n i t s e n t i r e t y (pace P i t t y ) .

U n f o r t u n a t e l y , the t a i l can also wag the dog. S i m p l i c i t y may be forced upon the modeller by the need f o r a n a l y t i c a l t r a c t a b i l i t y (Schwartz, 1962; Klemes, 1974). 'For complicated phenomena l i k e surface weathering and erosion i t i s i m p r a c t i c a l t o include more than a few of the known p h y s i c a l e f f e c t s . An attempt t o do so would lead t o a cumbersome model; indeed, a d e t a i l e d study o f the f l u i d motion alone would be possi b l e only i n h i g h l y i d e a l i s e d s i t u a t i o n s . I t seems p r e f e r a b l e instead t o use as simple a model as p o s s i b l e , f o r i f we describe only the o v e r a l l macroscopic behaviour of the f l u i d we can r e l y l a r g e l y on conservation laws, and these r e t a i n t h e i r v a l i d i t y even though the d e t a i l s of the small-scale motion are unknown' (Luke, 1974, 4035).

I t i s d i s t u r b i n g t o f i n d t h a t opponents of modelling o f t e n regard i t as unnecessary t o su b s t a n t i a t e charges of o v e r s i m p l i f i c a t i o n . The p o s s i b i l i t y t h a t models might be us e f u l has been dismissed ex cathedra by eminent h i s t o r i c a l and c l i m a t i c geomorphologists l i k e Wooldridge and Budel.

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Yet i t i s not cl e a r t h a t denudation chronologies and c l i m a t i c -geomorphological r e g i o n a l i s a t i o n s are s e l f - e v i d e n t l y innocent of o v e r s i m p l i f i c a t i o n : they are ma n i f e s t l y s i m p l i f i e d schemes which d e l i b e r a t e l y omit a mu l t i t u d e of d e t a i l s . Furthermore, apparently simple models may have very complicated behaviour (May, 1976a): i t may be d i f f i c u l t t o assess the s i m p l i c i t y of a model i t s e l f .

Models should not be subjected to summary t r i a l and execution on a charge of o v e r s i m p l i f i c a t i o n . I t i s necessary t o hear the case at length.

2.2.5 Models, domains and t e s t i n g

The domain can be defined as the set of systems which s a t i s f y the assumptions underlying the model (cf..Cohen, 1966,

66 on 'domain o f a p p l i c a b i l i t y 1 ; Harvey, 1969, 89; Scheidegger, 1970, 150 on 'domain of a p p l i c a t i o n ' ) . S t r i c t l y speaking, a model should be te s t e d e m p i r i c a l l y on a system which belongs t o the domain. However, systems belonging t o the domain may w e l l be very rare (or nonexistent) i f the assumptions behind a model are at a l l i d e a l i s e d . I n p r a c t i c e , as the geographer Guelke (1971, 47) remarked i r o n i c a l l y , the discrepancies between a model and r e a l i t y have o f t e n been 'explained' by demonstrating t h a t the t e s t s i t u a t i o n was not a p p l i c a b l e t o i t . Guelke was r i g h t l y c r i t i c a l of such p r a c t i c e : 'Whatever the d i f f i c u l t i e s t h a t a s c i e n t i f i c i n v e s t i g a t o r might encounter i n attempting t o t e s t h i s models he may not regard the l o g i c a l p a r t of them as beyond c r i t i c i s m because the i d e a l conditions f o r which they were constructed do not e x i s t ' .

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There are f u r t h e r d i f f i c u l t i e s . I n the f i r s t place, i t may be d i f f i c u l t t o i d e n t i f y systems belonging t o the domain i f we have no independent evidence on the mode of h i l l s l o p e development. C l e a r l y i t would be c i r c u l a r t o i n f e r past processes from present forms and then use those inferences t o i d e n t i f y an appropriate model. Secondly, the idea of s i m p l i f i c a t i o n i s l i n k e d w i t h t h a t of approximation: we would r a r e l y expect an exact f i t to model p r e d i c t i o n s . This s t i l l leaves the important question of deciding how much discrepancy i s acceptable. I n p r a c t i c e t h i s may come down t o working w i t h a poor model on the grounds t h a t i t i s the best we have.

The p o s s i b i l i t y of e q u i f i n a l i t y i s important i n any discussion of t e s t i n g . E q u i f i n a l systems are such t h a t p a r t i c u l a r system states may be reached i n d i f f e r e n t ways (von B e r t a l a n f f y , 1950). I n geomorphological terms, a p a r t i c u l a r morphology may be produced by d i f f e r e n t processes or d i f f e r e n t combinations o f processes. E q u i f i n a l i t y i s now widely accepted i n geomorphology ( c f . Chorley, 1962,

1964; Cooke and Warren, 1973; Cooke and Reeves, 1976),

sometimes i n the guise of 'convergence' (Wilhelmy, 1958;

Twidale, 1971; Douglas, 1976; Gossmann, 1976) or 'homology' (King, 1953; P i t t y , 197D, although i t i s s t i l l o c casionally ignored ( f o r example see Scheidegger, 1970, 120, 143).

The main methodological i m p l i c a t i o n of e q u i f i n a l i t y i s t h a t i t s existence l i m i t s inference about formative processes from morphological evidence. This applies t o

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mathematical explanations as w e l l as t o v e r b a l explanations (Leopold e t a l , 1964, 500; among o t h e r s ) . For example, .a good f i t obtained w i t h a model based on a p a r t i c u l a r set of process assumptions does not neces s a r i l y imply t h a t the assumptions are r e a l i s t i c . However, i t does seem l i k e l y t h a t e q u i f i n a l i t y i s not an absolute c o n d i t i o n but r e l a t i v e both t o the degree of p r e c i s i o n and t o the set o f d e s c r i p t o r s which are used. Be t h a t as i t may, e q u i f i n a l i t y i s c e r t a i n l y a fliajor d i f f i c u l t y i n model t e s t i n g .

I d e n t i f y i n g the domain, deciding on the discrepancy acceptable and e q u i f i n a l i t y of form are a l l important problems i n t e s t i n g , which must be combined w i t h a r e a l i s a t i o n t h a t h i g h l y complex models tend to be both u n h e l p f u l and i n t r a c t a b l e . As has been poin t e d out, however, a d i f f e r e n t view i s p o s s i b l e , i n which the e m p i r i c a l r e a l i s m o f a model i s regarded as secondary, and i t s main r o l e i s t h a t o f a l o g i c a l v e h i c l e f o r d e r i v i n g the necessary consequences of i n i t i a l assumptions. There i s much t o be s a i d f o r such a p o i n t o f view. U l t i m a t e l y , however, the c h a r a c t e r i s t i c t h a t d i s t i n g u i s h e s a s c i e n t i f i c theory from a mathematical argument i s i t s a p p l i c a b i l i t y t o e m p i r i c a l r e a l i t y .

2.3 Approaches t o modelling

A v a r i e t y of approaches have been used i n modelling h i l l s l o p e p r o f i l e s . These can be c l a s s i f i e d i n a simple way using f i v e dichotomies (2B). ( i ) Static/dynamic: according t o time content (2 . 3 - 1 . )

( i i ) D e t e r m i n i s t i c / s t o c h a s t i c : according to p r o b a b i l i t y content (2 . 3-2.)

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( i i i ) Phenomenological/representational: according t o process content (2. 3. 3-)

( i v ) A n a l y t i c a l / s i m u l a t i o n : according t o t r a c t a b i l i t y (2.3.4. .)

(v) Discrete/continuous: models t r e a t i n g h i l l s l o p e s as combinations o f d i s c r e t e components and models t r e a t i n g h i l l s l o p e s as continuous curves (Cox, 1977c). References t o reviews o f d i s c r e t e models are given i n Ch. 3.1 and a discussion of d i v i d i n g p r o f i l e s i n t o d i s c r e t e components i s given i n Ch. 8.

2 .3 .1 . S t a t i c and dynamic models

The d i f f e r e n c e between s t a t i c and dynamic models i s simple: only dynamic models include elapsed time as a v a r i a b l e . S t a t i c models of h i l l s l o p e p r o f i l e s p r e d i c t form at one time, o f t e n w i thout any process i m p l i c a t i o n s . Dynamic models o f h i l l s l o p e p r o f i l e s p r e d i c t e i t h e r some k i n d of i n v a r i a n t or e q u i l i b r i u m form; or a series of successive forms, t h a t i s , the e v o l u t i o n of the h i l l s l o p e system. The idea of a s t a t i c model i s s t r a i g h t f o r w a r d , but ideas of e q u i l i b r i u m and e v o l u t i o n r e q u i r e d e t a i l e d examination.

The choice between e q u i l i b r i u m and ev o l u t i o n a r y models depends i d e a l l y on whether the time i t takes f o r a system of i n t e r e s t t o reach e q u i l i b r i u m i s short or long r e l a t i v e to the time span being considered (K i r k b y , 1974, 3; c f . Schumm and L i c h t y , 1965). However, the decision must gene r a l l y be taken i n ignorance, since knowledge of r e a c t i o n and r e l a x a t i o n times (Wolman and Gerson, 1978; Graf, 1977)

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and of rates of h i l l s l o p e r e t r e a t (Young, 197*0 i s s t i l l fragmentary.

Ideas of e q u i l i b r i u m have a long h i s t o r y i n geomorphology. Although most recent work f o l l o w s the pioneer studies of St r a h l e r (1950), Leopold and Maddock (1953), Hack (1960) and Chorley (1962), these authors acknowledge the i n f l u e n c e of G i l b e r t (1880, 1909) (on whom c f . Pyne, 1976). Moreover, ideas of steady s t a t e formed one strand i n c l a s s i c a l ' unifo r m i i t a r i a n i s m ' , e s p e c i a l l y t h a t of L y e l l ( c f . Hooykaas, 1970; Rudwick, 1970, 1971; Gould, 1975, 1977; Porter, 1977).

With such a background, i t i s not s u r p r i s i n g t h a t ' e q u i l i b r i u m ' has appeared i n geomorphology i n a v a r i e t y of d i f f e r e n t guises ( c f . Young, 1970a; 1972, 96-102 on concepts of e q u i l i b r i u m , grade and u n i f o r m i t y i n h i l l s l o p e geomorphology; Chorley and Kennedy, 1971, 201-3 on kinds of e q u i l i b r i i i m i n p h y s i c a l geography; T r i c a r t and C a i l l e u x , 1972 on morphoclimatic e q u i l i b r i u m ; Statham, 1977, Ch. 1 on mechanical and chemical e q u i l i b r i u m i n geomorphology). The issue i s f u r t h e r complicated by a c e r t a i n amount of confusion and disagreement over the meaning of some key terms: witness the treatment of 'dynamic e q u i l i b r i u m ' , ' e q u i l i b r i u m ' , ' q u a s i - e q u i l i b r i u m ' , 'steady-state' and 'time-independence' i n the t e x t s of Easterbrook (1969, 428), Small (1970, 189), P i t t y (1971, 70) ,

Gregory and Wa l l i n g (1973, 18-19), Garner (1974, 29) , Ruhe (1975a, 86) , Butzer (1976, 81-2) , Twidale (1976, 424),

Douglas (1977, 227-8) , Rice (1977, 222-4) and Schumm (1977,

4-5) . No c l e a r consensus emerges from these t e x t s about

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whether these terms denote d i s t i n c t c o n d i t i o n s , nor about how they may be d i s t i n g u i s h e d . Moreover, these t e x t s i n d i c a t e t h a t the m a j o r i t y of geomorphologists understand e q u i l i b r i u m t o be e s s e n t i a l l y a matter o f 'adjustment' or 'balance 1, and t h a t geomorphologists o f t e n use e q u i l i b r i u m terms i n an inexact and metaphorical fashion.

One way t o cut through such confusion, disagreement and in e x a c t i t u d e i s t o r e t u r n to primary sources i n a b i d t o i s o l a t e the fundamental ideas. The c l a s s i c paper by Hack (1960) i s the primary source on 'dynamic e q u i l i b r i u m ' theory. Two strands of thought are i n t e r t w i n e d i n t h i s paper. One i s c r i t i c a l , both of the Davisian theory i n p a r t i c u l a r , and more generally of h i s t o r i c i s m i n geomorphology, the idea t h a t explanations i n geomorphology must be h i s t o r i c a l explanations (an idea very much a l i v e : c f . Bttdel, 1975).

The other i s c o n s t r u c t i v e : the dynamic e q u i l i b r i u m theory i t s e l f . I t i s cu r i o u s , and u n f o r t u n a t e , t h a t while the f i r s t aspect has o f t e n been discussed, the second has l a r g e l y escaped c r i t i c a l e v a l u a t i o n .

The c e n t r a l idea i n the dynamic e q u i l i b r i u m theory i s t h a t of constant form. I t i s hypothesised (according t o Hack ' I t i s assumed') t h a t ' w i t h i n a s i n g l e e r o s i o n a l system a l l elements of the topography are mutually adjusted so t h a t they are downwasting at the same r a t e ' (Hack, 1960, 85) .

The hypothesis of constant form could be t e s t e d i n p r i n c i p l e by measuring rates of downwasting. Even i f the

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hypothesis were upheld, however, i t i s c l e a r t h a t constant form alone cannot e x p l a i n the form of (say) a p a r t i c u l a r h i l l s l o p e p r o f i l e . As Ahnert (1967, 24) point e d out, 'a pe r i o d of uniform downwearing must be preceded by one of d i f f e r e n t i a l downwearing' t o e s t a b l i s h the e x i s t i n g topography at a greater a l t i t u d e . Or, to put i t another way, i t needs to be shown, f i r s t l y , t h a t the system w i l l converge towards a constant form c o n d i t i o n , and, secondly, t h a t t h i s i s a sta b l e c o n d i t i o n . Constant form must be f i r s t a t t a i n e d and then maintained ( c f . Morse, 1949 and Lewontin, 1969 on stab l e and unstable e q u i l i b r i a ) . Hack (1960, 86) asserted t h a t 'as long as d i a s t r o p h i c Qsc. t e c t o n i c ] forces operate gradually enough so t h a t a balance can be maintained by erosive processes, then the topography w i l l remain i n a st a t e of balance even though i t may be ev o l v i n g from one form t o another'. This may w e l l be t r u e ; but i t takes more than asseveration t o e s t a b l i s h a case. Moreover, i t i s not at a l l c l e a r t h a t 'a s t a t e of balance' does not preclude e v o l u t i o n 'from one form t o another'. The dynamic e q u i l i b r i u m theory becomes t o t a l l y comprehensive, and hence t o t a l l y vacuous, w i t h the p a r e n t h e t i c a l admission (Hack, 196o, 94) t h a t 'erosional energy changes through time and hence forms must change'. No d e f i n i t e hypotheses of any ki n d accompany t h i s unexceptionable statement. A c r i t i c i s m made by the philosopher Gellner (1968, 165) i n another context applies t o Hack's theory w i t h p a r t i c u l a r f o r c e : ' I f a man says - " I have the idea X, which applies t o

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things except i n as f a r as i t does not" - we pay scant a t t e n t i o n t o him: a l l ideas have the property of a p p l y i n g , except i n as f a r as they do not. To p o s t u l a t e one o f them w i t h such a proviso i s not much of an achievement 1.

Hence Hack's (1960) dynamic e q u i l i b r i u m theory i s a combination of two components: an imprecise hypothesis o f 'adjustment' and 'balance', and a precise hypothesis of constant form, which i s not supported independently. The f i r s t , which receives most emphasis i n Hack's paper, i s an a l t e r n a t i v e t o hypotheses of ' r e l i c t ' forms or ' p o l y c y c l i c ' development made by c l i m a t i c or h i s t o r i c a l geomorphologists. The second, produced but not discussed i n d e t a i l , has been neglected u n t i l q u i t e r e c e n t l y . I t i s , however, s t r a i g h t ­forward t o formulate mathematically, and i s thus the component of dynamic e q u i l i b r i u m theory which f i n d s e x p l i c i t expression i n models of h i l l s l o p e development.

The idea of 'constant form' needs mathematical d e f i n i t i o n . Suppose t h a t a h i l l s l o p e p r o f i l e i s represented by a curve

z = z ( x , t ) Here z denotes height above the base of the slope, x h o r i z o n t a l distance from the d i v i d e , and t elapsed time. The f o l l o w i n g p o s s i b i l i t i e s must be d i s t i n g u i s h e d .

( i ) The slope p r o f i l e i t s e l f remains constant

| £ = 0 ; z ( x , t ) = z(x) ot

( i i ) The r a t e of downwearing i s constant over space and time; t h i s contains ( i ) as a s p e c i a l case

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7>t

( i i i ) The r a t e o f downwearing i s constant over space; t h i s contains ( i i ) as a s p e c i a l case

I t seems l i k e l y t h a t ( i ) - ( i i i ) would a l l be regarded as 'dynamic e q u i l i b r i u m 1 , 'steady-state' or 'time-independent' conditions by many geomorphologists. But since a l l these terms are f r e q u e n t l y used i n other senses, a d i f f e r e n t term i s p r e f e r a b l e : 'constant form' (Smith and Bretherton, 1972)

i s p e r f e c t l y adequate.

Note t h a t these ' e q u i l i b r i u m ' states are here defined i n terms o f p r o p e r t i e s which remain constant or i n v a r i a n t . This procedure would seem t o have wider a p p l i c a b i l i t y . Hypotheses o f constant s o i l depth or ' c h a r a c t e r i s t i c form' are f u r t h e r examples i n which i n v a r i a n t s can be s p e c i f i e d unequivocally ( c f . Carson and Kir k b y , 1972), and such a cl e a r s p e c i f i c a t i o n might reduce the vagueness and confusion f r e q u e n t l y c h a r a c t e r i s t i c of statements on e q u i l i b r i u m i n geomorphological l i t e r a t u r e . However, the procedure cannot be applied to a l l kinds of e q u i l i b r i u m c o n d i t i o n s ; a class of counterexamples are states defined by v a r i a t i o n a l p r i n c i p l e s (e.g. minimisation o f work: Carson and Kirk b y , 1972, 4; K i r k b y , 1977b). Furthermore, i n v a r i a n t forms may not be e q u i l i b r i u m forms, although i n large p a r t t h i s i s a t e r m i n o l o g i c a l issue. According t o Kirkby (197^, 9-10) c h a r a c t e r i s t i c forms are not e q u i l i b r i u m forms.

St = - ( t > , & ^ 0

- 36 -

The idea of constant form f i n d s i t s strongest q u a l i t a t i v e statement i n Hack (1960), although i t may be traced t o G i l b e r t (1880, 1909). Constant form hypotheses have been incorporated i n several models of h i l l s l o p e development, although not always w i t h reference t o dynamic e q u i l i b r i u m theory. I n reviewing t h i s work, i t i s important t o resolve a fundamental ambiguity by s p e c i f y i n g whether (a) constant form p r e v a i l s throughout geomorphological h i s t o r y , or (b) landforms converge on a constant form c o n d i t i o n .

I n e i t h e r case, ( i ) above i s an extremely implausible hypothesis. E i t h e r downwasting must be balanced exactly by u p l i f t , or i f u p l i f t i s not included i n the model, downwasting must be i d e n t i c a l l y zero: n e i t h e r a l t e r n a t i v e seems at a l l l i k e l y (Schumm, 1963; Ahnert, 1970a; Smith and B r e t h e r t o n , 1972, 1512).

I n case ( a ) , i f ( i i ) above holds, then there i s a s t r a i g h t f o r w a r d s o l u t i o n :

z ( x , t ) = z ( x , 0 ) - o f t The model was discussed b r i e f l y by Scheidegger (1961;

1970, 132-4) and Pollack (1968, 1969), although without reference t o any e q u i l i b r i u m ideas. I f ( i i i ) h o lds, then we have the g e n e r a l i s a t i o n :

z ( x , t ) = z(x , 0 ) - oct

t i s elapsed time.

Neither o f these models i s very i n t e r e s t i n g . I t seems u n l i k e l y t h a t e i t h e r ( i ) or ( i i ) would h o l d f o r long periods

where tx i s the time average, A \ « (V) it J o

- 37 -

of g e o l o g i c a l time. Moreover, i t i s necessary t o specify an i n i t i a l p r o f i l e t o o b t a i n any p a r t i c u l a r s o l u t i o n , and to e x p l a i n the i n i t i a l p r o f i l e i n order t o e x p l a i n the p r o f i l e at time t . Both of these requirements are d i f f i c u l t t o meet.

I n case (b) we seek a constant form s o l u t i o n to one or more equations s p e c i f y i n g h i l l s l o p e development. This was f i r s t done by J e f f r e y s (1918); i n recent years the approach has been used by Smith and Bretherton (1972), Luke (1974), Hirano (1975, 1976) and Kirkby (1976a, 1976b; Wilson and Kirkby, 1975). I t i s no longer necessary t o spe c i f y i n i t i a l p r o f i l e s and the s t a b i l i t y p r o p e r t i e s of the constant form s o l u t i o n can be i n v e s t i g a t e d a n a l y t i c a l l y .

E q u i l i b r i u m , i n the guise of constant form, plays a dual r o l e i n such models. I t i s a mathematical convenience, a s i m p l i f y i n g assumption which makes i t easier t o ob t a i n closed-form s o l u t i o n s . I t may serve as a f i r s t approximation to other s o l u t i o n s , as Kirkby has shown. I t also represents a p h y s i c a l hypothesis, and n a t u r a l l y r e quires e m p i r i c a l t e s t i n g . However, some t h e o r e t i c a l support f o r constant form s o l u t i o n s i s provided by the s t a b i l i t y r e s u l t s of Je f f r e y s (1918) and Smith and Bretherton (1972), which i n d i c a t e the conditions under which constant form w i l l be maintained through feedback.

I f there are no grounds f o r expecting any k i n d of

e q u i l i b r i u m , then e v o l u t i o n a r y models are necessary, which

p r e d i c t a series o f successive h i l l s l o p e p r o f i l e s .

- 38 -

Evolutionary models are very d i f f i c u l t t o t e s t s a t i s f a c t o r i l y . H i l l s l o p e development i s d i f f i c u l t t o observe except i n s p e c i a l circumstances, and, c o n t r a r i w i s e , the s p e c i a l circumstances are so s p e c i a l t h a t doubt must be cast on t h e i r t y p i c a l i t y (e.g. badlands s t u d i e d by Schumm, 1956a, 1956b). The major d i f f i c u l t y i s c l e a r l y the discrepancy which may a r i s e between human li f e s p a n s and time spans of geomorphological i n t e r e s t .

Given but one p r o f i l e f o r each h i l l s l o p e , i t i s possible t o use the series of p r o f i l e s p r e d i c t e d by the model as a set of templates, and choose the most r e a l i s t i c ( c f . P i t t y , 1972 on Davisian and Penckian p r e d i c t i o n s ) . This i s a procedure almost forced upon us unless we know the appropriate elapsed time from other evidence ( c f . Chappell, 1974). Such a procedure i s u n s a t i s f a c t o r y because i t i s more d i f f i c u l t t o r e f u t e a model i n these circumstances.

An a t t r a c t i v e s o l u t i o n t o the t e s t i n g problem i s t o seek a s p a t i a l series o f h i l l s l o p e p r o f i l e s which can be t r e a t e d as i f i t were a temporal s e r i e s . This k i n d of s o l u t i o n i s o f t e n described as invo k i n g e r g o d i c i t y or the ergodic hypothesis, but such use of terminology i s sometimes u n j u s t i f i e d ( f o r a f a i r l y rigorous statement of e r g o d i c i t y see Scheidegger, 1970, 267). A looser term such as 'space-time transformation* i s p r e f e r a b l e ( c f . Chorley and Kennedy, 1971, 277-80 f o r a p p l i c a t i o n s i n h i l l s l o p e geomorphology; Thornes and Brunsden, 1977, 23-5 more g e n e r a l l y ) . Despite widespread enthusiasm f o r the idea, i t seems t o be app l i c a b l e

- 39 -

only i n very s p e c i a l s i t u a t i o n s (e.g. Savigear, 1952). I n p a r t i c u l a r , the assumption t h a t p r o f i l e s w i t h i n a drainage basin f o l l o w the same sequence but at d i f f e r e n t rates appears to be f a l s e (Carson and Kirkby, 1972, 9, 405).

Evolutionary models f r e q u e n t l y r e q u i r e the s p e c i f i c a t i o n of an i n i t i a l p r o f i l e z ( x , 0 ) . This i s d i f f i c u l t t o supply i n most cases, but there are some exceptions, notably the work o f Mizutani (1974) on volcanoes and slag heaps.

2 .3 .2 . D e t e r m i n i s t i c and s t o c h a s t i c models

I f a model includes one or more random v a r i a b l e s each s p e c i f i e d by a p r o b a b i l i t y d i s t r i b u t i o n , i t i s s t o c h a s t i c . Otherwise i t i s d e t e r m i n i s t i c .

There i s a con t i n u i n g controversy i n geography (Harvey, 1969, 260-3) , i n geology (Watson, 1969, 491-2; Mann, 1970;

Raup, 1977; Whitten, 1977), i n geomorphology (Leopold and Langbein, 1963; Scheidegger and Langbein, 1966; Howard, 1972;

Shreve, 1975; Thornes and Brunsden, 1977), and indeed i n many other d i s c i p l i n e s , over the extent t o which explanations couched i n p r o b a b i l i t y terms are s a t i s f a c t o r y . Debate on t h i s issue can be traced t o preSocratic philosophy.

Two questions have o f t e n been c o n f l a t e d . F i r s t , there i s the o n t o l o g i c a l issue o f whether the world i s d e t e r m i n i s t i c or s t o c h a s t i c , e i t h e r as a whole or i n p a r t . This i s a d i f f i c u l t issue which seems e n t i r e l y open at present given the u n c e r t a i n status of s t o c h a s t i c models i n quantum mechanics. The question may even be undecidable i n p r i n c i p l e

- 40 -

since the f a i l u r e of a d e t e r m i n i s t i c model need not be ascribed to the s t o c h a s t i c character of nature.

Second, there i s the epistemological issue o f whether s t o c h a s t i c explanations i n v o k i n g random v a r i a b l e s are s a t i s f a c t o r y . Many scholars f e e l uneasy about hypothesising f a c t o r s which by d e f i n i t i o n are unpredictable except i n p r o b a b i l i t y terms, and i t has been suggested t h a t chance i s merely a l a b e l f o r our ignorance, used t o cover a residue of f a c t as yet unexplained. Hence only d e t e r m i n i s t i c explanations can be f u l l y s a t i s f a c t o r y . I t does seem, however, t h a t a p a r t i c u l a r stance on t h i s e p i s t e m o l o g i c a l issue need not e n t a i l a p a r t i c u l a r o n t o l o g i c a l view. I t would be e n t i r e l y p o s s i b l e t o use s t o c h a s t i c explanations while h o l d i n g t h a t the world i s e s s e n t i a l l y d e t e r m i n i s t i c , and vice versa; and, a f o r t i o r i , t o use one or other k i n d without committing oneself t o any ontology. I n Monod's (1974, 110-2) terminology, one can recognise ' o p e r a t i o n a l u n c e r t a i n t y ' .while not necess a r i l y a d m i t t i n g the existence of ' e s s e n t i a l u n c e r t a i n t y ' .

A f u r t h e r p o i n t which needs some c l a r i f i c a t i o n i s the meaning of the term 'random'. Three senses need t o be d i s t i n g u i s h e d :

( i ) Random i n the sense of apparently haphazard or ch a o t i c , a r e p o r t o f a s u b j e c t i v e impression: t h i s i s i n large p a r t a psy c h o l o g i c a l matter.

( i i ) Random i n the sense of equal and independent p r o b a b i l i t i e s .

- 41 -

( i i i ) Random i n the sense of a random v a r i a b l e characterised by a p r o b a b i l i t y d i s t r i b u t i o n : t h i s i s the standard mathematical sense and the sense adopted here. I n t h i s sense, random = s t o c h a s t i c .

Whatever the p h i l o s o p h i c a l issues, the a t t i t u d e of mathematicians t o the i n c l u s i o n of random va r i a b l e s i n models i s ge n e r a l l y pragmatic. W h i t t l e (1970, 19), i n a t e x t on p r o b a b i l i t y t h e o r y , took as a premise 'that there i s a c e r t a i n amount of v a r i a b i l i t y which we cannot e x p l a i n but must accept', while Bard (1974, 18) s i m i l a r l y wrote t h a t 'unpredictable disturbances are as much parts of p h y s i c a l r e a l i t y as are the underlying exact q u a n t i t i e s which appear i n the model'.

Random v a r i a b l e s may appear i n s t o c h a s t i c models i n many d i f f e r e n t ways; here we mention two ( c f . Watson, 1972, 39-40) (a) value at data p o i n t = value o f d e t e r m i n i s t i c f u n c t i o n

+ random e r r o r (b) value at data p o i n t = value at a p o i n t on a random

f u n c t i o n . Mathematically these model f a m i l i e s are not r e a l l y d i s t i n c t

the second can be regarded as a s p e c i a l case o f the f i r s t i n which the d e t e r m i n i s t i c f u n c t i o n i s i d e n t i c a l l y zero. However, i n p r a c t i c e , models of type ( a ) , which may be c a l l e d s t o c h a s t i c e r r o r models, are usu a l l y q u i t e d i s t i n c t from models o f type ( b ) , which may be c a l l e d s t o c h a s t i c process models. I n case ( a ) , v a r i a b i l i t y i s s p l i t i n t o a systematic p a r t , approximated by a d e t e r m i n i s t i c f u n c t i o n , and a r e s i d u a l or e r r o r p a r t , t r e a t e d as a random v a r i a b l e .

- 42 -

The need f o r a random v a r i a b l e a r i ses f rom the f a c t o f

m o d e l l i n g l i f e t h a t no n o n t r i v i a l data se r i es would ever

be f i t t e d exac t l y by a d e t e r m i n i s t i c model: sampling

v a r i a t i o n , measurement e r r o r , incor rec tness o f f u n c t i o n a l

f o r m , and u n c o n t r o l l e d v a r i a b l e s i n t e r v e n e . I n case ( b ) ,

data are regarded as a r e a l i s a t i o n o f a s t o c h a s t i c process ,

which i s a mathematical process ope ra t i ng i n t ime and/or

space according t o p r o b a b i l i t y laws.

The usual idea i s t h a t d e t e r m i n i s t i c f u n c t i o n s capture

'smooth ' behaviour w h i l e s t o c h a s t i c (random) v a r i a b l e s mop

up the remaining ' r o u g h ' behaviour . This idea i s sub jec t

t o two r e s e r v a t i o n s . F i r s t l y , ' r o u g h ' components need not

be t r e a t e d i n a p r o b a b i l i s t i c or s t o c h a s t i c manner, e s p e c i a l l y

i n e x p l o r a t o r y data ana lys i s ( c f . Tukey, 1977; M c N e i l , 1977).

Secondly, there are some d e t e r m i n i s t i c processes w i t h

extremely rough (apparent ly random) behaviour (May, 1976a;

Lorenz, 1976), and, converse ly , some s t o c h a s t i c processes

w i t h extremely smooth (apparen t ly d e t e r m i n i s t i c ) b e h a v i o u r

(Cohen, 1976; May, 1976b). This i s mentioned l a r g e l y

f o r completeness: these processes have not been a p p l i e d

as ye t i n geomorphology.

2 .3 .3 . Phenomenological and r e p r e s e n t a t i o n a l models

The d i s t i n c t i o n between phenomenological and represen t ­

a t i o n a l models i s based on a d i s t i n c t i o n between

phenomenological and r e p r e s e n t a t i o n a l t heo r i e s made by

the ph i losopher Bunge (1964). A phenomenological model

- H3 -

attempts only t o capture the phenomena (sc . su r face appearances);

a phenomenological model o f a h i l l s l o p e p r o f i l e at tempts on ly

an approximat ion o f h i l l s l o p e f o r m , and n e i t h e r i n assumptions

nor i n d e t a i l e d s t r u c t u r e does i t t r y t o r e f l e c t geo-

morpho log ica l processes or p h y s i c a l p r i n c i p l e s . A rep­

r e s e n t a t i o n a l model i s more amb i t i ous , a iming t o represent

the u n d e r l y i n g processes as w e l l as su r face appearances,

i d e a l l y r ep re sen t ing them i n terms o f p h y s i c a l p r i n c i p l e s .

Terminology here i s an awkward ma t t e r . These terms

are not e n t i r e l y s a t i s f a c t o r y , but more f a m i l i a r and less

cumbersome a l t e r n a t i v e s seem unsu i t ab le i n o ther ways.

P a r a l l e l d i s t i n c t i o n s have been drawn between • e m p i r i c a l '

and ' r a t i o n a l ' models i n geomorphology (Mackin , 1963;

Young, 1972, 18); between ' e m p i r i c a l ' and ' c o n c e p t u a l '

models ( C l a r k e , 1973) or between ' o p e r a t i o n a l ' and

' p h y s i c a l ' models (Klemes, 197*0 i n hyd ro logy ; between

' e m p i r i c a l ' and ' t h e o r e t i c a l ' models i n ecology (Wieger t ,

1975); and between 'homomorphic' and ' i s o m o r p h i c ' models

i n pedology (Hugget t , 1975). None o f these p a i r s i s very

s a t i s f a c t o r y i n c a p t u r i n g a con t ras t i n process con ten t .

The term 'process-response model ' ( W h i t t e n , 1964;

Carson and K i r k b y , 1972; Young, 1972, Ch. 10) , considered

here equ iva len t t o ' r e p r e s e n t a t i o n a l model*, i s not used,

p a r t l y because i t lacks an antonym, and p a r t l y t o avoid

con fus ion w i t h the r a t h e r d i f f e r e n t term 'process-response

system' (Chorley and Kennedy, 1971, Ch. 4 ) .

- u n ­

it i s w i d e l y accepted i n geomorphology t h a t , as f a r as

p o s s i b l e , landforms and r e l a t e d processes should be

exp la ined i n terms o f mechanical and chemical p r i n c i p l e s

( S t r a h l e r , 1952; Ya t su , 1966; Carson, 1971; Statham, 1977).

This i s merely an example o f a more general a t t i t u d e - t ha t

explana t ions should make re fe rence to a c t u a l mechanisms.

'To e x p l a i n a phenomenon, t o e x p l a i n some p a t t e r n o f

happenings, we must be able to descr ibe the causal mechanism

which i s respons ib le f o r i t ' (Har re , 1972, 178). Hence there

i s a des i re to r e l a t e g e o l o g i c a l knowledge to p h y s i c a l theory

( K i t t s , 197^), or to r e l a t e form t o process i n geography

(Harvey, 1969).

Although a quest f o r causal exp lana t ion leads to the

c o n s t r u c t i o n o f r e p r e s e n t a t i o n a l models o f h i l l s l o p e p r o f i l e s ,

phenomenological models may s t i l l be o f considerable

d e s c r i p t i v e value ( c f . Cur ry , 1967, 267). For example,

i f upslope c onve x i t i e s may be approximated by power

f u n c t i o n s , the parameter values p rov ide a simple and

e f f i c i e n t means o f comparing d i f f e r e n t convex i t i e s (Hack

and G o o d l e t t , 1960). Fur thermore, the importance o f

d e s c r i p t i o n should not be underplayed: d e t a i l e d and

systemat ic d e s c r i p t i o n has i t s place alongside explana tory

theo ry .

I f the aim i s t o b u i l d r e p r e s e n t a t i o n a l models, how

i s t h i s t o be done? Four l e v e l s may be d i s t i n g u i s h e d i n

process s tudy , not neces sa r i l y e x c l u s i v e , s equen t i a l or

exhaus t ive .

- 45 -

( i ) Recogni t ion o f processes f rom i n c i d e n t a l evidence

( e . g . the supposed i d e n t i f i c a t i o n o f creep f rom bending

t r e e s , b u l g i n g w a l l s , e t c . ) .

( i i ) Measurement o f ra tes o f o p e r a t i o n . This a l lows a

dec i s i on on the processes t o be modelled to be based on

q u a n t i t a t i v e evidence.

( i i i ) I d e n t i f i c a t i o n o f c o n t r o l l i n g v a r i a b l e s . This

a l lows a choice o f c o n t r o l l i n g f a c t o r s t o be i n c l u d e d i n

any model. Usual ly h i l l s l o p e p r o f i l e models have been

based upon the assumption tha t process ra tes are e s s e n t i a l l y

f u n c t i o n s o f p r o f i l e geometry ( e s p e c i a l l y g r a d i e n t , d is tance

f rom d i v i d e , cu rva tu re^ . Non-geometric c o n t r o l s such as

mantle s t r e n g t h , m o i s t u r e , t e x t u r e and v e g e t a t i o n have

rece ived less a t t e n t i o n f rom m o d e l l e r s , but have o f t e n

been i n c l u d e d i n f i e l d i n v e s t i g a t i o n s . •

( i v ) E l u c i d a t i o n o f p h y s i c a l mechanisms. Processes

are analysed i n terms o f mechanical and chemical p r i n c i p l e s :

many models f a l l sho r t o f such i n t e g r a t i o n . At some s tage ,

an e m p i r i c a l or phenomenological approach must be

employed. H i l l s l o p e p r o f i l e models cannot i n p r a c t i c e be

based on ' f undamen ta l ' p h y s i c a l theor i e s such as quantum

theory or r e l a t i v i t y theory ( c f . Schoener, 1972, 390 on

e c o l o g i c a l models) . The ' u n i v e r s a l laws ' invoked i n

i d e a l geomorphological explana t ions are u s u a l l y those of

mechanics or chemistry ( c f . Ch. 2 . 2 . 2 ) .

A p r i n c i p l e which i s w ide ly accepted i n mode l l i ng

h i l l s l o p e development i s t h a t a mass balance r a t h e r than an

- 46 -

energy balance provides an appropr ia te framework. Fluxes

o f energy pe r fo rming geomorphological work are a n e g l i g i b l e

component o f h i l l s l o p e energy budgets (Carson and K i r k b y ,

1972, 28; Young, 1972, 21; K i r k b y , 1974, 2-3; c f . Hare,

1973, 188). Hence f l u x e s o f m a t e r i a l are the c e n t r a l concern,

and c o n t i n u i t y equations a l low these to be handled system­

a t i c a l l y ( e . g . Carson and K i r k b y , 1972, 107-9; Wilson and

K i r k b y , 1975, 205-6; K i r k b y , 1976b, 9-10) .

While 'phenomenological 1 and • r ep re sen ta t iona l 1 are

presented here as p o l a r oppos i t e s , i t must be admit ted t h a t

i n p r a c t i c e models e x h i b i t continuous g r ada t i on i n process

con ten t .

2.3 .4 . A n a l y t i c a l and s i m u l a t i o n models

A n a l y t i c a l models i d e a l l y take the form of sets o f

equations possessing s o l u t i o n s i n c losed f o r m , whereas s i m u l a t i o n

models i nc lude those expressed i n the form o f computer programs

s p e c i f y i n g sequences o f ope ra t ions . (No o the r category o f

s i m u l a t i o n models w i l l be considered h e r e ) . I n p r a c t i c e ,

these classes o f model i n t e r g r a d e : s o l u t i o n s t o many

equations can only be obta ined us ing methods o f numer ica l

ana lys i s which must be implemented on a computer.

An i d e a l s i t u a t i o n may be sketched as f o l l o w s . The

model le r w r i t e s down a set o f equations ( u s u a l l y o rd ina ry

or p a r t i a l d i f f e r e n t i a l equa t ions , i f a dynamic model i s

be ing c o n s t r u c t e d ) , which represent e m p i r i c a l knowledge,

p h y s i c a l p r i n c i p l e s and any c o n s t r a i n t s which must be

s a t i s f i e d ( e . g . conserva t ion o f mass or ene rgy ) . These

- 47 -

equations are then so lved i n genera l us ing ' s t a n d a r d ' methods

and i n p a r t i c u l a r by i n s e r t i n g i n i t i a l and boundary c o n d i t i o n s .

I f t h i s i d e a l was always ob t a ined , there would be no

need f o r s i m u l a t i o n models. The s t o r y i s , however, something

o f a f a b l e , not l e a s t because d i f f e r e n t i a l equations are

s trange beasts . Many apparent ly simple equations possess

no simple c losed fo rm s o l u t i o n , and i t i s f r e q u e n t l y necessary

t o compromise, by seeking an approximate s o l u t i o n or a

p a r t i c u l a r k i n d o f s o l u t i o n , such as an e q u i l i b r i u m s o l u t i o n .

I n the l a t t e r case, the mathematical f a c t t h a t a p a r t i c u l a r

s o l u t i o n e x i s t s does not support the p h y s i c a l hypothesis

t h a t such a s o l u t i o n w i l l be a t t a i n e d . Such a hypothesis

i s a f u r t h e r statement r e q u i r i n g j u s t i f i c a t i o n .

I n c o n t r a s t , a s i m u l a t i o n model i s much eas ie r to

b u i l d , r e q u i r i n g only an elementary knowledge o f computer

programming (FORTRAN, r a t h e r than a s p e c i a l s i m u l a t i o n

language such as SIMULA ( B i r t w i s t l e e t a l , 1973), has

gene ra l l y been used i n geomorphology). I t i s u s u a l l y

poss ib l e t o b u i l d models more complex than ( t r a c t a b l e )

a n a l y t i c a l models, w h i l e some d i f f i c u l t i e s f a c i n g

a n a l y t i c a l models (such as the exis tence o f thresholds

and the need t o model magnitude and frequency d i s t r i b u t i o n s )

may be o f l i t t l e account: such f ea tu re s can be handled

e a s i l y . Correspondingly , the danger e x i s t s t h a t a h i g h l y

complex model w i l l be imposs ib le t o i n v e s t i g a t e s y s t e m a t i c a l l y ,

and i t w i l l never be c l ea r which r e s u l t s are genuine and

which a r t e f a c t u a l : t o use May's (1974, 682) d e l i g h t f u l

- 48 -

express ion , the model may be 1 a m u l t i - p a r a m e t e r , computerised

Goon show'. Howard (1972) has g iven a sober d i scuss ion o f

the problems o f computer s i m u l a t i o n i n geomorphology, w h i l e

Moon (1975) has r epor t ed a p ioneer s e n s i t i v i t y ana lys i s o f

Ahne r t ' s (1973) model. Such s e n s i t i v i t y analyses are v i t a l l y

necessary as complements t o development sequences produced

by s i m u l a t i o n runs ( f o r an e x c e l l e n t e c o l o g i c a l example, c f .

S t ee l e , 1974).

2.4 Major geomorphological problems i n mode l l i ng

Several k inds o f compl i ca t i ng f ea tu re s are c h a r a c t e r i s t i c

o f h i l l s l o p e systems, and hence should i d e a l l y be r e f l e c t e d

i n models o f h i l l s l o p e p r o f i l e s . These are examined i n

t u r n below.

2 .4 .1 . Polygenesis

'Po lygenes i s ' i s a term d e s c r i b i n g the common ( i f not

u n i v e r s a l ) s i t u a t i o n i n which a l andform has been produced

by a combinat ion o f d i f f e r e n t processes. Polygenesis i s a

major k i n d o f complexi ty f r e q u e n t l y found f o r h i l l s l o p e

systems. I n so f a r as h i l l s l o p e s are p o l y g e n e t i c , models

should r e f l e c t such an o r i g i n , a l though i n c o r p o r a t i o n o f

polygenesis should p r e f e r a b l y r e s t on q u a n t i t a t i v e evidence

about the r e l a t i v e importance o f d i f f e r e n t processes.

However, var ious k inds o f polygenesis need t o be

d i s t i n g u i s h e d . F i r s t l y , the re are s i t u a t i o n s i n which

d i f f e r e n t processes are a c t i n g more or less s imul taneous ly

( f o r example, creep and rainwash).. Secondly, there are

- 49 -

s i t u a t i o n s i n which processes have very d i f f e r e n t r e t u r n

f requenc ies ( f o r example, creep and l a rge - sca l e f a i l u r e s ) .

T h i r d l y , l a t e T e r t i a r y and Quaternary c l i m a t i c change may

have l e d t o the succession o f d i f f e r e n t su i t e s o f processes

(Young, 1972, 240-6); f o r example, a l t e r n a t i o n between

c r y o n i v a l and warmer cond i t i ons i n present day humid

temperate areas ( e . g . Rapp, 1967; B l a c k , 1969 and c f .

Ch. 4 .4.3 b e low) .

These s i t u a t i o n s may not be equa l ly p r o b l e m a t i c . I f

h i l l s l o p e s are e s p e c i a l l y sub jec t t o l a rge - sca l e f a i l u r e when

steep and to s lower processes when g e n t l e , t h e i r fo rm at

any t ime may r e f l e c t one or the o ther r a t h e r than a combinat ion

hence r e l a t i v e l y simple models may s t i l l be a p p l i c a b l e , at

l e a s t t o i n d i v i d u a l h i l l s l o p e components. C l i m a t i c change

i s not p rob lemat ic i f slopes are e s s e n t i a l l y r e l i c t , or i f

the con t ras t between d i f f e r e n t regimes has been unduly

exaggerated.

2 .4 .2 . Feedback

Feedback loops are o f grea t importance i n geomorpho-

l o g i c a l systems (Mel ton , 1958; K i n g , 1970; Twidale et a l ,

1974, 1977; C r o z i e r , 1977) and any r e a l i s t i c model must

thus mimic the major loops i n o p e r a t i o n . The most

genera l and most bas ic feedback r e l a t i o n s h i p i s between

process and f o r m . Not only do processes a f f e c t fo rms , but

forms a f f e c t processes, bo th i n genera l and on h i l l s l o p e s

(Chor ley , 1964, 71; Ahner t , 1971, 3-4; Young, 1972, 104-5).

- 50 -

As Smith and Bre the r ton (1972, 1506) put i t , the p h y s i c a l

landscape may be i d e a l i s e d as a t ime-dependent , s e l f -

fo rming s u r f a c e .

Even t h i s bas ic r e l a t i o n s h i p i s not always m i r r o r e d i n

r e p r e s e n t a t i o n a l models. Some s t o c h a s t i c process models

( c f . Ch. 2 . 3 . 2 , Ch. 3 ) , which are based on the idea t h a t

e m p i r i c a l data se r i e s = r e a l i s a t i o n o f s t o c h a s t i c process ,

do not a l low feedback, because the r e l a t i o n between

genera t ing process and generated se r ies i s asymmetric: the

c h a r a c t e r i s t i c s o f the se r ies do not a f f e c t those o f the

s t ochas t i c process . I n t h i s sense a t l e a s t the s t o c h a s t i c

process p o s t u l a t e d i s not analogous t o the geomorphological

processes i n o p e r a t i o n .

2 .4 .3 . Thresholds

Many geomorphological systems c o n t a i n th resholds or

d i s c o n t i n u i t i e s (Chorley and Kennedy, 1971, 236-40; Reynaud,

1971, 47-50; Schumm, 1973, 1977). I n the case o f h i l l s l o p e s ,

good examples are p rov ided by the th resho lds which must be

crossed be fo re slope f a i l u r e occurs ( e . g . Carson, 1976). The

exis tence o f d i s c o n t i n u i t i e s poses d i f f i c u l t i e s f o r the

usual approach t o mode l l i ng p h y s i c a l systems, cen t red around

o rd ina ry and p a r t i a l d i f f e r e n t i a l equa t ions , as Souchez

(1966a, 212) and Aronsson (1973, 2) have remarked i n a

h i l l s l o p e m o d e l l i n g con tex t . The use o f d i f f e r e n t i a l

equations i s g e n e r a l l y based on the assumption t h a t bo th

f u n c t i o n s and d e r i v a t i v e s vary smoothly and con t i nuous ly .

- 51 -

Hence there i s good mathematical reason f o r a sharp d i v i s i o n

between the models o f s o i l and rock mechanics, mainly

concerned w i t h the charac ter o f f a i l u r e c o n d i t i o n s , and the

models of p r o f i l e development considered he re , mainly

app l i cab le t o h i l l s l o p e s sub jec t t o s lower processes which

may be 'averaged' over whatever d i s c o n t i n u i t i e s are

present ( c f . Carson and K i r k b y , 1972, 110).

N a t u r a l l y t h i s d i v i s i o n i s u n f o r t u n a t e f rom a

geomorphological p o i n t o f view. I f h i l l s l o p e development

must be a t t r i b u t e d t o a combination o f threshold-dependent

and threshold- independent processes, then a model o f

h i l l s l o p e development should r e f l e c t such combinat ion.

This can be done, t o a c e r t a i n e x t e n t , i n a s i m u l a t i o n

model (see Ch. 3-3 below f o r examples).

2.4 .4 . Magnitude and frequency

Most geomorphological processes are i n t e r m i t t e n t i n

t h e i r a c t i o n : even apparent ly continuous processes such as

s o i l creep may take place as a se r ies o f ' m i c r o c a t a s t r o p h e s ' .

Hence the magnitude and frequency o f geomorphological events

need to be considered both i n general (Wolman and M i l l e r ,

1960; Leopold e t a l , 1964, 67-94; Wolman and Gerson, 1978) and

f o r h i l l s l o p e s (Carson and K i r k b y , 1972, 102-4; Young,

1972, 85-7; 1974, 74-5; S t a r k e l , 1976).

The 'Wolman-Mi l le r t h e s i s ' i s t h a t events o f

i n t e rmed ia t e magnitude and frequency have most geomorphological

impact : major events have l i t t l e e f f e c t i n t o t a l because they

- 52 -

are r a r e ; f r equen t events have l i t t l e e f f e c t i n t o t a l because

they are minor . This p r i n c i p l e c e r t a i n l y holds i f the

frequency d i s t r i b u t i o n o f magnitudes above some t h r e s h o l d i s

lognormal , and the r e l a t i o n s h i p between work achieved and

event magnitude ( ag a in , expressed above a t h r e s h o l d ) i s a

power f u n c t i o n . However, a l though the most va luable and

p rovoca t ive g e n e r a l i s a t i o n a v a i l a b l e on magnitude and

f requency , the Wolman-Mil ler t he s i s should not be considered

as e s t ab l i shed t r u t h . One impor tan t q u a l i f i c a t i o n (Wolman

and Gerson, 1978) i s t h a t extreme events o f p a r t i c u l a r

magnitude and frequency must be seen i n r e l a t i o n t o the

ra tes a t which recovery o f s p e c i f i c forms takes place

between recur rences . For example, the speed o f

r evege t a t i on o f bare ground i s a major c o n t r o l o f the

e f f e c t i v e n e s s o f extreme events .

Almost a l l continuous slope models handle the

magnitude-frequency issue by ( i m p l i c i t l y ) averaging over

frequency d i s t r i b u t i o n s and r ep re sen t ing processes as

continuous i n t ime (but see K i r k b y , 1976a, 1976b f o r an

excep t ion ; and a lso P r i c e , 1974, 1976 on a l l u v i a l f a n

d e p o s i t i o n ) . This i s almost c e r t a i n l y necessary i f a n a l y t i c a l

s o l u t i o n s are sought , w h i l e conversely i t i s not a necessary

assumption f o r s i m u l a t i o n models. Averaging over d i s t r i b u t i o n s

may be reasonable i f processes such as creep and rainwash

are i n q u e s t i o n , bu t probably not i n the case o f l a r g e -

scale f a i l u r e .

- 53 -

2 .4 .5- L a t e r a l i t y

A p r o f i l e i s only an i d e a l i s a t i o n , v a l i d t o the ex ten t

t ha t h o r i z o n t a l or p l an curva ture (and thus l a t e r a l sediment

f l u x ) can be neg lec ted . U l t i m a t e l y the aim must be the

m o d e l l i n g o f l andsur faces , and some o f the models

reviewed below attempt t o do t h i s (a l though Ch. 3 does

not cover a l l landsurface models) . Compl ica t ion comes

not on ly i n the form of another h o r i z o n t a l dimension ( e . g .

squar ing computer s torage requirements) bu t a lso i n

q u a l i t a t i v e l y new fea tu res which should be i n c o r p o r a t e d ,

e s p e c i a l l y the i n t e r a c t i o n o f a drainage network and the

i n t e r v e n i n g slopes (Sprun t , 1972; Armstrong, 1976).

The d i scuss ion i n Carson and Ki rkby (1972, 390-6)

leads t o an encouraging r e s u l t : p r o f i l e models are s t ab l e

i n the sense t h a t s l i g h t curvatures l ead t o only s l i g h t l y

d i f f e r e n t p r e d i c t i o n s . Contrary evidence would imply t h a t

only sur face models can be at a l l r e a l i s t i c .

2 .4 .6 . L i t h o l o g i c a l v a r i a t i o n and s o i l p r o p e r t i e s

The i n f l u e n c e o f l i t h o l o g i c a l v a r i a t i o n upon h i l l s l o p e

morphology has o f t e n been r epo r t ed ( c f . Young, 1972, Ch. 17

f o r a b r i e f rev iew) and the importance o f rock c h a r a c t e r i s t i c s

i n h i l l s l o p e development needs l i t t l e emphasis. I t i s ,

however, very d i f f i c u l t t o i nco rpo ra t e rock p r o p e r t i e s i n

models o f l andform development i n a s a t i s f a c t o r y manner.

Al though i t seems c l e a r t h a t the r e l a t i v e importance o f

l i t h o l o g i c a l v a r i a t i o n should be assessed q u a n t i t a t i v e l y ,

- 54 -

r e l a t i o n s h i p s between rocks and r e l i e f have o f t e n been

discussed w i t h o u t any re fe rence t o morphometric v a r i a b l e s

(Cox, 1973, 3 ) . I n many cases ' r e s i s t a n t ' beds s tand out

as steps i n slope p r o f i l e s : how impor tan t are they i n the

o v e r a l l landscape? are they merely m i c r o - or meso-features

on the scale o f the slope p r o f i l e , or can l i t h o l o g i c a l

v a r i a t i o n be h e l d t o dominate the landscape? These are

p r o p e r l y q u a n t i t a t i v e ques t ions .

I f the l i t h o l o g y u n d e r l y i n g a g iven p r o f i l e i s f a i r l y

homogeneous, then i t may be pe rmi s s ib l e t o omit rock

p r o p e r t i e s f rom a model f o r t h a t p r o f i l e a lone , a l though

c l e a r l y any comparison o f p r o f i l e s on d i f f e r e n t l i t h o l o g i e s

should not f o l l o w t h i s p r a c t i c e . Extreme he te rogene i ty

might a lso be taken as near-homogeneity i f much o f the

v a r i a b i l i t y i s on a mic ro-sca le and can be averaged o u t .

One poss ib le example would be a r a p i d l y a l t e r n a t i n g

sandstone-shale succession.

I n the more common in t e rmed ia t e s i t u a t i o n , a naive

t a c t i c i s t o ass ign d i f f e r i n g ' r e s i s t a n c e ' values t o

d i f f e r e n t s t r a t a . Several dynamic slope models a l l ow

such v a r i a t i o n s .

Three d i f f i c u l t i e s deserve no te . F i r s t l y , r e s i s t ance

values should be supp l i ed independent ly i n order t o avo id

the c i r c u l a r arguments u n f o r t u n a t e l y c h a r a c t e r i s t i c o f

r o c k - r e l i e f s tud ies (Ya t su , 1966, 9-10; Sparks, 1971, 370;

T r i c a r t and C a i l l e u x , 1972, 17). I f r e s i s t ance values are

- 55 -

given i n an a r b i t r a r y f a s h i o n , drawing upon i n t u i t i v e ideas

o n l y , then n o t h i n g i s done to break the c i r c l e . However,

the s i t u a t i o n cannot be improved s u b s t a n t i a l l y w i t h o u t

overcoming some fo rmidab le problems o f d e f i n i t i o n , sampling

and measurement. Secondly, i f a s o i l cover i s p re sen t , and

h i l l s l o p e development i s t r a n s p o r t - l i m i t e d , r e s i s t ance i s a

mat te r o f s o i l p r o p e r t i e s r a t h e r than rock p r o p e r t i e s : t o

use Chor ley ' s (1959, 503) metaphor, 'bedrock i s not t o be

considered a parent o f the r e l a t e d topography, but r a t h e r a

g randparen t ' . This s imple p o i n t i s not always observed i n

models. T h i r d l y , r e s i s t ance i s f r e q u e n t l y d e f i n e d i n

h i l l s l o p e models as a s i n g l e - v a l u e d p r o p e r t y , whereas i t i s

w e l l known t h a t seve ra l d i f f e r e n t p r o p e r t i e s o f rocks and

s o i l s a f f e c t ra tes o f m o b i l i s a t i o n and t r a n s p o r t ( e . g .

Bryan, 1968, 1977; Sparks, 1971, Ch. 2; Thornes, 1975;

Statham, 1977, Ch. 2 ) .

The l i t h o l o g y may a l l o w extens ive chemical removal

o f m a t e r i a l f rom the h i l l s l o p e , a l though most models focus

on mechanical removal . Carson and Ki rkby (1972, 257-71)

reviewed appropr ia te models f o r chemical removal , w h i l e i n

l a t e r work Ki rkby (1976 a ) suggested the simple approximat ion

t h a t chemical downwasting i s s p a t i a l l y cons tan t , which he

regarded as i n v a l i d only when chemical removal i s un impor tan t .

However, the context o f t h i s remark i s a model which assumes

homogeneous l i t h o l o g y .

S o i l depth has been i n c l u d e d i n seve ra l models, u s u a l l y

by p o s t u l a t i n g a r e l a t i o n s h i p o f some k i n d between s o i l

depth and weather ing r a t e . The main d i f f i c u l t y i s t h a t

- 56 -

v i r t u a l l y n o t h i n g i s known e m p i r i c a l l y about t h i s r e l a t i o n s h i p

(Young, 1972, 46) a l though there i s no shortage o f hypotheses

i n the l i t e r a t u r e ( e . g . C u l l i n g , 1965, 246; Souchez, 1966a,

190; Carson and K i r k b y , 1972, 104-6; and others be low) .

2.4 .7 . Boundary c o n d i t i o n s

What can be s a i d about the behaviour o f slope endpoints

(Carson, 1969, 77) and about t e c t o n i c , i s o s t a t i c and e u s t a t i c

rates? Assumed answers t o these quest ions appear as

boundary cond i t i ons i n some h i l l s l o p e models.

I n p a r t i c u l a r , ( i ) i s removal o f m a t e r i a l a t the

base impeded or unimpeded? ( S t r a h l e r , 1950; Savigear , 1952;

Mel ton , 1960; Carson and K i r k b y , 1972, 139-40).

( i i ) i s the stream downcut t ing and/or moving l a t e r a l l y ?

(Smith and B r e t h e r t o n , 1972).

( i i i ) are d i v i d e s m i g r a t i n g l a t e r a l l y ? (Carson and

K i r k b y , 1972, 396-7).

A l l these poss ib l e forms o f endpoint behaviour occur i n na ture

and deserve i n v e s t i g a t i o n .

There are some grounds f o r supposing tha t t e c t o n i c

and i s o s t a t i c movements need not be modelled e x p l i c i t l y .

Schumm (1963) argued t h a t a v a i l a b l e q u a n t i t a t i v e evidence

on ra tes o f downwearing and u p l i f t supports the c l a s s i c

Davis ian hypothesis o f r e l a t i v e l y r a p i d u p l i f t f o l l o w e d

by r e l a t i v e l y long s t i l l s t a n d s ( c f . a lso Carson and K i r k b y ,

1972, 21-5) . I f t h i s were c o r r e c t , u p l i f t could be

subsumed i n an i n i t i a l c o n d i t i o n , much as Davis (1909) d i d

- 57 -

i n h i s c y c l i c a l scheme. But i t seems dangerous t o r e l y on

t h i s i n t e r p r e t a t i o n , mainly because i n some regions epe i rogenic

movement seems t o have occurred over a long p e r i o d o f t i m e .

A more impor tan t reason f o r no t mode l l i ng t e c t o n i c

and i s o s t a t i c movements e x p l i c i t l y i s t h a t i f the area

a f f e c t e d by u p l i f t i s l a rge r e l a t i v e t o the l eng th o f the

p r o f i l e , i t may be poss ib l e t o t r e a t u p l i f t as a r e g i o n a l

r a t h e r than a l o c a l f a c t o r , or t o t r e a t i t i n d i r e c t l y ,

through r i v e r downcut t ing . This i s a less r e s t r i c t i v e

assumption than Schumm1s quas i -Dav i s i an hypo thes i s , and

thus renders the l a t t e r unnecessary.

The case o f e u s t a t i c f l u c t u a t i o n s i s more complex.

Since sea l e v e l has v a r i e d over the l a t t e r p a r t o f g e o l o g i c a l

t i m e , i t has been suggested t h a t pulses o f downcut t ing

( ' r e juvena t ion*)have t r a v e l l e d up v a l l e y s l e ad ing to the

f o r m a t i o n o f v a l l e y - i n - v a l l e y fo rms , marked by breaks o f

slope and even t e r races (Sparks, 1960, 220-4; Young,

1972, 239-40). However d e t a i l e d process s tud ies have

revealed t ha t such pulses may be q u i c k l y damped upstream;

t h a t the morpho log ica l response to ' r e j u v e n a t i o n * may be

complex; and t h a t forms a t t r i b u t a b l e t o ' r e j u v e n a t i o n '

are e q u i f i n a l (Leopold e t _ a l , 1964, 258-66, 442-5;

Chor ley , 1965a, 28; Schumm and Parker , 1973; Schumm,

1973, 1977). Hence i t i s d i f f i c u l t t o know whether

e u s t a t i c e f f e c t s should be modelled at a l l .

K i rkby (1971» 28) has suggested t h a t the theory o f

k inemat ic waves ( L i g h t h i l l and Whitham, 1955) could be used

- 58 -

to examine k n i c k p o i n t propagation, but t h i s idea has apparently not been fo l l o w e d up. Luke (1972, 1971*) b r i e f l y remarked on the r e l a t i o n s h i p of h i s models t o kinematic wave theory.

2.5 Summary

( i ) H i l l s l o p e geomorphology studies the forms, processes and development of h i l l s l o p e systems. H i l l s l o p e s are u s e f u l l y viewed as r e p l i c a t e d systems, r a t h e r than as unique or r e s t r i c t e d f e a t u r e s , and t h e i r study i s r e l a t e d to t h a t of other r e p l i c a t e d systems, notably drainage basins and topographic surfaces. W i t h i n h i l l s l o p e geomorphology, morphometry - the q u a n t i t a t i v e analysis of land form - i s

a major approach alongside h i s t o r i c a l , process and sediment studies. (2.1)

( i i ) Since theory i s i n e v i t a b l e , i t i s necessary t o examine ideas and assumptions c r i t i c a l l y . A few t r o u b l e ­some terms need t o be defined c a r e f u l l y : i n p a r t i c u l a r , a d i s t i n c t i o n i s drawn here between 'system' ( i . e . r e a l -w orld system of i n t e r e s t ) and 'model' ( i . e . s i m p l i f i e d formal r e p r e s e n t a t i o n ) . The r o l e o f mathematics i n geomorphology, and the r e l a t i o n s h i p between modelling and s i m p l i f i c a t i o n , are f r e q u e n t l y misunderstood: attempts are made t o c l a r i f y the underlying issues. The t e s t i n g of a model i n i t s domain, i n c l u d i n g the problem o f e q u i f i n a l i t y , needs close a t t e n t i o n . (2.2)

( i i i ) Approaches t o modelling h i l l s l o p e p r o f i l e s are considered using a simple c l a s s i f i c a t i o n based on f i v e dichotomies: static/dynamic ( i n c l u d i n g an extended examination of the important idea of e q u i l i b r i u m ) ; d e t e r m i n i s t i c / s t o c h a s t i c

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( i n c l u d i n g a discussion o f the r o l e of p r o b a b i l i t y i n models); phenomenological/representafcional ( i n c l u d i n g a discussion of the r e p r e s e n t a t i o n of geomorphological processes); a n a l y t i c a l / s i m u l a t i o n ( e s p e c i a l l y the p a r t i c u l a r advantages and l i m i t a t i o n s of each k i n d ) ; and discrete/continuous ( f o r f u r t h e r discussion see Chs. 3.1 and 8 below).(2.3)

( i v ) Major geomorphological problems i n modelling h i l l s l o p e p r o f i l e s are assembled and evaluated (polygenesis, feedback, t h r e s h o l d s , magnitude and frequency, l a t e r a l i t y , l i t h o l o g i c a l v a r i a t i o n and s o i l p r o p e r t i e s , boundary c o n d i t i o n s ) . (2.4)

2.6 Notation

d i n ordinary d e r i v a t i v e or i n i n t e g r a l t , t ' elapsed time x h o r i z o n t a l coordinate z v e r t i c a l coordinate oi r a t e of downwearing X average ra t e of downwearing

T constant i n p a r t i a l d e r i v a t i v e i n i n t e g r a l modulus

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Chapter 3

A REVIEW OF CONTINUOUS MODELS OF HILLSLOPE PROFILES

... Prince Papadiamantopoulos turned o u t , i n s p i t e of hi s wonderfully promising t i t l e and name, to be a p e r f e c t l y serious i n t e l l e c t u a l l i k e the r e s t o f us. More serious indeed; f o r I discovered, t o my h o r r o r , t h a t he was a f i r s t - c l a s s g e o l o g i s t and could under­stand the d i f f e r e n t i a l c a l c u l u s .

Aldous Huxley, Those barren leaves, Pt. Ch. 1.

3.1 I n t roduct i o n

3.2 S t a t i c models

3.3 Dynamic models

3.4 Notation

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3.1 I n t r o d u c t i o n

This chapter gives a f a i r l y complete review of models of h i l l s l o p e p r o f i l e s which t r e a t a h i l l s l o p e as a continuous curve. Work on modelling topographical p r o f i l e s and land-surfaces i s discussed when i t i s o f d i r e c t i n t e r e s t , but not studies o f stream and r i v e r p r o f i l e s (e.g. Tanner, 1971).

Related work on h i l l s l o p e s t r e a t e d as combinations of d i s c r e t e components ( f o r example, f r e e face and debris slope) i s not reviewed here, p a r t l y because the a p p l i c a b i l i t y of such models w i l l not be f u r t h e r considered i n t h i s t h e s i s , and p a r t l y because there are good b r i e f reviews by Carson and Kirkby (1972, l 4 0 - 7 ) , Scheidegger (1970, 120-32) and Young (1972, 105-9), which remain e s s e n t i a l l y up-to-date. By c o n t r a s t , the f i e l d of continuous modelling lacks a comprehensive and up-to-date review. Models o f s p e c i a l features such as t a l u s slopes or mass f a i l u r e s are not covered.

There i s l i t t l e consensus i n the f i e l d about n o t a t i o n , and there are even some workers who demonstrably use i n c o n s i s t e n t or i n a p p r o p r i a t e n o t a t i o n . A u n i f i e d n o t a t i o n has been adopted here which should a i d comparison, and which may even encourage s t a n d a r d i s a t i o n .

While an attempt has been made at completeness, there i s c l e a r l y i n s u f f i c i e n t space t o l i s t every major equation i n the l i t e r a t u r e , l e t alone t o provide f u l l proofs o f every r e s u l t , or t o cast the discussion i n the rigorous s t y l e of p r o f e s s i o n a l mathematicians. The surest guide t o the character

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of p a r t i c u l a r models i s the o r i g i n a l l i t e r a t u r e : here only an overview i s provided.

The sharpest dichotomy i n p r a c t i c e between sets o f models c l a s s i f i e d according t o the scheme o u t l i n e d above (Ch. 2.3) i s between s t a t i c and dynamic models, which are reviewed below i n Chs. 3.2 and 3.3 r e s p e c t i v e l y . A thematic review seems most n a t u r a l f o r s t a t i c models, w h i l e a h i s t o r i c a l review w i t h connective summary appears best f o r dynamic models.

The p r o f i l e n o t a t i o n adopted here as standard i s given i n 3A.

3.2 S t a t i c models

A l i s t of s t a t i c models proposed f o r h i l l s l o p e p r o f i l e s i s given i n 3B. This l i s t includes some models used f o r h i l l s l o p e components, glaci a l l y - m o u l d e d p r o f i l e s and topographical p r o f i l e s . Over the l a s t century, and e s p e c i a l l y over the l a s t twenty years, many f u n c t i o n a l forms have been discussed i n the l i t e r a t u r e , u s u a l l y as phenomenological r a t h e r than r e p r e s e n t a t i o n a l models. I f a h i l l s l o p e resembles an arc of a c i r c l e or a Gaussian curve, then t h a t i s t h a t . I f n o t h i n g more i s claimed, then n o t h i n g more need be discussed. Here three o f the more popular classes o f model are s i n g l e d out f o r a t t e n t i o n , together w i t h some s t o c h a s t i c models which are r e l a t i v e l y novel.

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3B

S t a t i c models o f h i l l s l o p e p r o f i l e s

Model BINOMIAL or NORMAL (GAUSSIAN)

Re ference Tylo r 1875

Thompson 1942, 121 Ashton 1976

Comments Cf. o u t l i n e o f Mt. Tabor (now I s r a e l ) H i l l o u t l i n e s Mt. Piper, V i c t o r i a , A u s t r a l i a

POWER SERIES POLYNOMIAL

Tyl o r 1875 Lake 1928

Savigear 1956, 1962 Troeh 1964, 1965

Ruhe & Walker 1968 Lewin 1969

Clark 1970 Klei s s 1970 Ongley 1970

Young 1970b

Doornkamp & King 1971

Parabola. Hirwaun, Wales. Parabola. Components. Gwynedd, Wales. Devon and Cornwall. 'Paraboloids of r e v o l u t i o n ' f i t t e d t o three-dimensional forms - quadratic i n two dimensions Iowa I n c l . q u i n t i c . Yorkshire Wolds. Parabola. Western U.S.A. Quadratic. Components. Iowa. Linear. Components. N.S. Wales Components. Mato Grosso, B r a z i l . L inear, q u a d r a t i c , cubic discussed

Woods 1972, 1974, 1975 Colorado and Kentucky. Cf. Cox 1975

Parsons 1973, 1976b

Blong 1975 Cox 1975 Toy 1977

Linear. Components. I t a l y , Morocco. Linear. N. I s l a n d , New Zealand Cf. Woods Linear. U.S.A.Cubic f o r 'average' p r o f i l e

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3B (continued)

Model

CATENARY

Reference

Davis 1916

Comments

G l a c i a l trough cross p r o f i l e s , Montana. Idea mentioned by s e v e r a l l a t e r workers.

ARC OP CIRCLE Lake 1928

Juvigne 1973

Components. Gwynedd, Wales

Components. Belgium.

POWER FUNCTION Hack & Goodlett 196O

White 1966

Drury & Nisbet 1971

B u l l 1975, 1977 > Components

Cox 1977a Toy 1977 -J

Svensson 1959

Graf 1970

Doornkamp & King 1971^

Drewry 1972

T i l l 1973

King 1974 J

Aniya 197^

Graf 1976a

r V i r g i n i a . Cf. Drury & Nisbet

Ohio

Cf. Hack & Goodletl

C a l i f o r n i a , Arizona, Utah. Cf. Cox

Cf. B u l l

^U.S.A.

G l a c i a l trough c r o s s -p r o f i l e s

G l a c i a l cirque long-p r o f i l e s

EXPONENTIAL Dury 1966, 1970

Dury e t a l 1967

Dury 1972

Tangent-distance r e g r e s s i o n s , Pediments

N.S. Wales

Semi-logarithmic r e g r e s s i o n s Pediments.

S. England

- 66 -

3B (continued)

Model Reference Comments

Rune 1975b, 1977 P r o f i l e s . Iowa, N. Mexico, Hawaii, Puerto Rico. Cf. Cox

Cox 1977b Cf. Ruhe. Bridge & Beckraan 1977 S. E. Queensland

ASYMPTOTIC GOMPERTZ CURVE

Lewin 1969 Yorkshire Wolds

COSINE CURVE

Clark 1970 Western U.S.A.

RECTANGULAR HYPERBOLA

Clark 1970 Doornkamp & King 1971

Western U.S.A.

LOGARITHMIC Milne 1878, 1879 Volcano slopes Kl e i s s 1970 Components, Iowa Doornkamp & King 1971 Toy 1977 Attempted & dismissed f o r

concavity. U.S.A.

ANGLE LINEAR FUNCTION OF INDEX

P i t t y 1970 Abrahams & Parsons 1977

Components Components.N. S. Wales

ARMA Thornes 1972, 1973 Ice l a n d and unnamed area Haining 1977a Cf. Thornes

- 67 -

3B (continued)

Model Reference Comments

FRACTIONAL BROWNIAN FUNCTIONS

Mandelbrot 1975a, 1975b, 1975c, 1977

S t o c h a s t i c models

CYCLOID Bridge & Beckman 1977 S. E. Queensland

OTHERS Rune 1967 D e t e r m i n i s t i c ( o r i g i n a l not seen)

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3.2.1 Power series polynomial

This class of model has the general form

P a r t i c u l a r cases include p = 0 ( 1 ) 1 (or l i n e a r ) z = t> 0 + b,x

p = 0(1)2 (or q u a d r a t i c ; i n c l u d i n g parabolas) 2

z = b Q + b-j x + b^x

p = 0(1)3 ( o r cubic) z = b Q + b- x + b 2 x ^ + b^x^

A s i m i l a r f a m i l y of models has the general form

Most authors use low-order polynomials: the highest order employed i n h i l l s l o p e p r o f i l e models appears t o be 7 (Toy, 1977). A l i n e a r model w i l l n a t u r a l l y only be a good approximation of a h i l l s l o p e which has a nearly constant gradient. A quadratic model i s appropriate f o r a smooth convex or concave slope; a cubic allows an i n f l e x i o n ; and higher orders allow i n c r e a s i n g l y complicated forms.

Polynomial models are phenomenological: the idea t h a t they can 'explain' h i l l s l o p e p r o f i l e s (Woods, 1974, 4l6) i s absurd, unless an extremely weak n o t i o n of explanation i s adopted ( c f . Ch. 2.2.2 above). Since the form o f the f u n c t i o n i s not derived from geomorphological theory, the best t h a t can be hoped f o r i s a parsimonious summary o f the data (Cox, 1975, 489; c f . Lewin, 1969, 72).

The l i n e a r model has been used (Doornkamp and King, 1971; Blong, 1975; Toy, 1977) t o estimate average angle 9 from the r e l a t i o n

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dz = t a n 8 dx

This seems p o i n t l e s s , i f only because 9 can be c a l c u l a t e d d i r e c t l y from

tan 9 = z d

3.2.2 Power functions

I n i t s simplest form, a power f u n c t i o n i s represented by b z, - z = ax d

although other versions are also found. Power fu n c t i o n s have been f i t t e d separately t o i n d i v i d u a l components ( c o n v e x i t i e s , s t r a i g h t slopes, c o n c a v i t i e s ) . They do not allow i n f l e x i o n s . The model i s again e s s e n t i a l l y phenomenological, although one of Kirkby's models has a power f u n c t i o n s o l u t i o n (see below). The reader i s r e f e r r e d t o a debate on the ' a l l o m e t r i c 1

i n t e r p r e t a t i o n o f power f u n c t i o n s ( B u l l , 1975, 1976; Cox, 1977a; B u l l , 1977) which w i l l not be prolonged here. I t i s also of i n t e r e s t t o note the use of power f u n c t i o n s t o model g l a c i a l trough cross p r o f i l e s and g l a c i a l cirque long p r o f i l e s .

3.2.3 Exponential f u n c t i o n

I n i t s simplest form, the exponential model has the form

z = ae -bx , b>0 where e i s the transcendental number 2.71828 ... , the base of n a t u r a l logarithms. The model i s such t h a t z tends t o

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0 a s y m p t o t i c a l l y as x tends tb i n f i n i t y : i t i s . an ever-more-

gentle concavity. Since the model w i l l not allow i n f l e x i o n s ,

i t can hardly be appropriate f o r e n t i r e p r o f i l e s , unless an

upper convexity i s absent or n e g l i g i b l e . Nevertheless i t

has been a p p l i e d to e n t i r e p r o f i l e s (Ruhe, 1975b; c f . Cox,

1977b; Ruhe, 1977) .

A more complicated r e l a t i v e i s i m p l i c i t i n the tangent-

distance r e g r e s s i o n s of Dury (1966, 1970; et a l , 1967) on

pediment p r o f i l e s , and another r e l a t i v e i s i m p l i c i t i n the

'semi-logarithmic' r e g r e s s i o n used, but not explained i n

d e t a i l , by Dury ( 1972) .

3 . 2 . 4 S t o c h a s t i c process models

I n the l a s t few y e a r s , a new kind of s t a t i c model has

been introduced, f o l l o w i n g the i d e a that h i l l s l o p e p r o f i l e s

may be regarded as r e a l i s a t i o n s of s p a t i a l s t o c h a s t i c p r o c e s s e s .

Thornes (1972, 1973) has ap p l i e d a u t o r e g r e s s i v e models,

moving average models and mixed autoregressive-moving average

models borrowed from time s e r i e s a n a l y s i s to s e r i e s of

measured slope angles. These models are a l l s p e c i a l cases

of the general ARMA model (Box & J e n k i n s , 1976; C h a t f i e l d ,

1975, 4 1 - 5 1 ) . The a p p l i c a t i o n of these models to h i l l s l o p e

p r o f i l e s faces s e v e r a l problems, which deserve d i s c u s s i o n .

F i r s t l y , the models are e s s e n t i a l l y phenomenological.

Secondly, i n time s e r i e s a n a l y s i s i t i s u s u a l l y n a t u r a l to

suppose that the present i s i n f l u e n c e d by the p a s t , but not

v i c e v e r s a : the arrow of time l i m i t s p l a u s i b l e s p e c i f i c a t i o n s .

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On the other hand, a point on a slope i s r e l a t e d not only to

points upslope (downslope sediment f l u x ) but a l s o to points

downslope (e.g. b a s a l u n d e r c u t t i n g ) . Thus the u n i l a t e r a l

ARMA models need to be re p l a c e d by b i l a t e r a l models which

allow i n f l u e n c e s to operate i n both d i r e c t i o n s . This problem

was recognised by Thornes (1976, 5 9 ) , although Church (1972,

80) took a d i f f e r e n t view i n a study of f l u v i o g l a c i a l stream

p r o f i l e s : he argued that i t was u n l i k e l y t h at c o n t r o l from

downstream would be e f f e c t i v e f o r long d i s t a n c e s . However,

those b i l a t e r a l models of s p a t i a l s t o c h a s t i c processes

which have been d i s c u s s e d i n the l i t e r a t u r e ( W h i t t l e , 195^;

Haining, 1977a, 1977b) seem inappropriate f o r the geomorph­

o l o g i c a l case i n which upslope and downslope i n f l u e n c e s

d i f f e r i n kind. T h i r d l y , feedback between form and process

i s not captured by these models ( c f . Ch. 2 . 4 . 3 ) . F o u r t h l y ,

i t seems p o s s i b l e that i n c e r t a i n r e s p e c t s r e s u l t s may be

a r t e f a c t s of the measured lengths used i n p r o f i l e survey

(Thornes, 1973) .

Mandelbrot (1975a, 1975b, 1975c, 1977) considered a

v a r i e t y of s t o c h a s t i c process models f o r l a n d s u r f a c e s . The

most i n t e r e s t i n g are based on the c l a s s of f r a c t i o n a l

Brownian f u n c t i o n s . Given points P 1, P" which l i e i n a

r e a l E u c l i d e a n space, a f r a c t i o n a l Brownian func t i o n B^ i s

defined by a property of l o c a l d i f f e r e n c e s

B H (P») - B R (P") = AB r, say AB„ i s drawn from a normal (Gaussian) d i s t r i b u t i o n with mean

n zero and varia n c e jp ,P l l|2H j o r

AG ~ N ( 0 , (P'P 'I 2 ")

- 72 -

H, the parameter of the p r o c e s s , l i e s i n the i n t e r v a l 0 < H < 1 .

Mandelbrot's model encompasses both p r o f i l e s (space has

dimension 1) and s u r f a c e s (dimension 2 ) . H « O J turns out

to generate quite r e a l i s t i c l a n d s u r f a c e s . Although a

Poisson approximation to a Brownian s u r f a c e was motivated by

an i d e a of random f a u l t s , t h i s i s g e o l o g i c a l l y u n r e a l i s t i c

and the model i s best regarded as phenomenological. I t i s

here c l a s s e d as s t a t i c : Mandelbrot, however, was not c l e a r

about whether the r e a l E u c l i d e a n space i n which points P l i e

could be i n t e r p r e t e d as p h y s i c a l space-time, which would

make h i s model a dynamic model.

Parsons (1973, 1976b) and Graf (1976b) have explored

the use of Markov-type ideas and t r a n s i t i o n p r o b a b i l i t y

schemes f o r h i l l s l o p e modelling. Such an approach t r e a t s

h i l l s l o p e s as combinations of d i s c r e t e components, and

w i l l not be d i s c u s s e d here i n f u r t h e r d e t a i l .

3. 3 Dynamic models

The review of dynamic h i l l s l o p e p r o f i l e models which

follows i s h i s t o r i c a l , by dates of authors' f i r s t key

p u b l i c a t i o n s i n the f i e l d . This i s the l i n e of l e a s t

r e s i s t a n c e : no other sequence appears at a l l s a t i s f a c t o r y ,

however. Some readers may p r e f e r to read the connective

summary (3.3.22) f i r s t , and then to r e f e r to modellers of

p a r t i c u l a r i n t e r e s t .

3.3.1 J e f f r e y s

J e f f r e y s ( 1 9 l 8 ) 4 b e s t known for h i s d i s t i n g u i s h e d work

i n geophysics, published the f i r s t continuous model of

- 73 -

h i l l s l o p e development, although i t was not presented as such. This paper i s of more than h i s t o r i c a l i n t e r e s t : i t s ideas and r e s u l t s remain of s i g n a l importance, and i t has been sadly neglected by recent workers ( c f . Cox, 1977c, f o r an a p p r e c i a t i o n ) .

J e f f r e y s gave a dynamical treatment o f the flow o f surface water d u r i n g r a i n and then considered denudation by viscous flow. The r a t e of denudation w i t h uniform s o i l was shown t o be a f u n c t i o n o f d tan 0 , where d i s the depth o f water and 0 slope angle. I f t h i s product i s constant, there i s an i n t e r e s t i n g case: 'The surface sinks at a uniform r a t e a l l over r e t a i n i n g i t s size and shape, but p r o g r e s s i v e l y s i n k i n g . This represents one case of the "peneplain"' ( J e f f r e y s , 1918, 184).

I f a surface z = z ( x , y , t ) i s such t h a t contours are p a r a l l e l t o the y-axis then the form of the 'peneplain' can be derived. This s p e c i a l case i s c l e a r l y t h a t of a h i l l s l o p e p r o f i l e z = z ( x , t ) . I t i s given parametric a l l y , by

z = a - b (2 cosec 9 - s i n 9 ) x = c - b (cosec 0 cot 9 - cos 9 )

where a i s a f u n c t i o n of time and b and c are constants. This constant form p r o f i l e i s concave upwards and almost p a r a b o l i c except near the d i v i d e where i t i s nearly v e r t i c a l . This l a s t p r e d i c t i o n i s not very r e a l i s t i c , but was discussed at some length by J e f f r e y s .

- 74 -

The i d e a o f c o n s t a n t f o r m , a l t h o u g h p r e s e n t i n t h e

' d y n a m i c e q u i l i b r i u m t h e o r y 1 o f G i l b e r t (1880, 1909),

a p p e a r s t o h a v e b e e n i n t r o d u c e d i n d e p e n d e n t l y b y J e f f r e y s .

He p r o v i d e d some m o t i v a t i o n f o r t h i s i d e a by c o n s i d e r i n g

t h e s t a b i l i t y o f t h e p e n e p l a i n u n d e r s m a l l d i s t u r b a n c e s .

He a r g u e d t h a t s u c h a p e n e p l a i n was s t a b l e f o r c o r r u g a t i o n s

r u n n i n g a c r o s s t h e s l o p e b u t n o t f o r t h o s e r u n n i n g down

t h e s l o p e . T h i s k i n d o f i n s t a b i l i t y was t h o u g h t t o be

c o u n t e r a c t e d much o f t h e t i m e b y s o i l f r i a b i l i t y a n d

v e g e t a t i o n .

T h i s p a p e r i s r e m a r k a b l e f o r i t s e l e g a n t a n d r i g o r o u s

a p p r o a c h a n d i t s c o n c e r n t o e l u c i d a t e t h e m e c h a n i c s

o f s u r f a c e w a t e r f l o w a n d e r o s i o n . I t o r i g i n a t e d t w o

o f t h e m o s t v a l u a b l e i d e a s o f h i l l s l o p e m o d e l l i n g :

c o n s t a n t f o r m a n d s t a b i l i t y u n d e r p e r t u r b a t i o n s . C o n s t a n t

f o r m i s i n t r o d u c e d as a h y p o t h e s i s , n o t as a t h e o r e m :

J e f f r e y s made i t c l e a r t h a t i t was ' a n i n t e r e s t i n g c a s e ' ,

b u t d i d n o t c l a i m w i d e r v a l i d i t y f o r t h e i d e a . H o w e v e r ,

t h e r e s u l t s o f s t a b i l i t y a n a l y s e s p r o v i d e some m o t i v a t i o n

f o r s u c h a h y p o t h e s i s . H e r e t h e r e i s a s t r i k i n g p a r a l l e l

w i t h t h e l a t e r w o r k o f S m i t h ( s e e b e l o w ) .

3.3.2 de M a r t o n n e a n d B i r o t

de M a r t o n n e a n d B i r o t (19^*0, i n a p a p e r o n s l o p e

d e v e l o p m e n t i n h u m i d t r o p i c a l c l i m a t e s , s u g g e s t e d t h a t

r e g o l i t h d e p t h w ( m e a s u r e d n o r m a l t o t h e s u r f a c e i n t h i s

c a s e ) f o l l o w e d

- 75 -

w h e n c e , u s i n g g e o m e t r i c a l a n d t r i g o n o m e t r i c a l i d e n t i t i e s , t h e

h i l l s l o p e f o l l o w s

a l t h o u g h de M a r t o n n e a n d B i r o t s h o w e d a c a v a l i e r d i s r e g a r d

f o r t h e d i s t i n c t i o n s b e t w e e n 6 ( s m a l l d i f f e r e n c e ) , d

( o r d i n a r y d i f f e r e n t i a l ) a n d ) . The movemen t o f t h e

s t r e a m a t t h e s l o p e b a s e was g i v e n b y a q u a d r a t i c

2: = b , t + b 0 t 2

b 1 ^

The s o l u t i o n i s cumbersome a n d g i v e n p a r a m e t r i c a l l y .

C h o o s i n g p h y s i c a l l y a d m i s s i b l e v a l u e s f o r t h e b a s a l s l o p e ,

de M a r t o n n e a n d B i r o t c o n s i d e r e d t h e e f f e c t s o f d i f f e r e n t

d o w n c u t t i n g r e g i m e s .

T h i s m o d e l i s n o t v e r y w e l l p r e s e n t e d . The c e n t r a l /-v

p r e m i s e ~ = a s i n 0 l e a d s t o t h e p r e d i c t i o n t h a t w

i n c r e a s e s f a s t e s t on v e r t i c a l s l o p e s , w h i c h i s a b s u r d i f

o n l y b e c a u s e v e r t i c a l s l o p e s do n o t c a r r y r e g o l i t h s : h e n c e

t h e n e e d t o c h o o s e ' p h y s i c a l l y a d m i s s i b l e ' v a l u e s . The

m a i n f e a t u r e o f i n t e r e s t i s t h e t r e a t m e n t o f d i f f e r e n t

d o w n c u t t i n g r e g i m e s p e r m i t t e d b y t h e q u a d r a t i c .

3.3.3 C u l l i n g

C u l l i n g (1960) s u g g e s t e d t h a t t h e d i f f u s i o n e q u a t i o n

m i g h t s e r v e as a p h e n o m e n o l o g i c a l m o d e l f o r p r o f i l e s a n d

l a n d s u r f a c e s . I n one h o r i z o n t a l d i m e n s i o n

I + / 2 M 7)7. 3t

( H e r e t h e p a r t i a l d i f f e r e n t i a t i o n s y m b o l ^ has b e e n u s e d ,

~ a — 3x

- 76 -

i . e . , r a t e o f d o w n c u t t i n g i s p r o p o r t i o n a l t o h o r i z o n t a l r a t e

o f change o f g r a d i e n t . One s a l i e n t a d v a n t a g e o f t h i s

e q u a t i o n , w h i c h a l s o a p p l i e s t o h e a t c o n d u c t i o n , i s t h a t

i t s p r o p e r t i e s a r e w e l l u n d e r s t o o d ( e . g . C a r s l a w a n d J a e g e r ,

I t i s e a s y t o show t h a t t h e d i f f u s i o n e q u a t i o n f o l l o w s

f r o m an a s s u m p t i o n t h a t s e d i m e n t f l u x S i s p r o p o r t i o n a l t o

g r a d i e n t , i . e .

C u l l i n g (1963, 1965) a r g u e d t h a t t h e d i f f u s i o n e q u a t i o n

was e s p e c i a l l y a p p l i c a b l e t o s o i l c r e e p . The h y p o t h e s i s

t h a t c r e e p i s t h e r e s u l t o f n u m e r o u s r a n d o m l y d i r e c t e d

movemen t s o f m i n u t e e x t e n t made b y i n d i v i d u a l p a r t i c l e s

was shown t o l e a d t o a d i f f u s i o n e q u a t i o n . V a r i o u s

m o d i f i c a t i o n s a n d g e n e r a l i s a t i o n s o f t h e m o d e l w e r e a l s o

d i s c u s s e d , w h i c h a l l o w f o r n o n - r a n d o m i n f l u e n c e s a n d mass

t r a n s p o r t . A l l t h r e e p a p e r s i n c l u d e d s e v e r a l w o r k e d e x a m p l e s

f o r p a r t i c u l a r i n i t i a l a n d b o u n d a r y c o n d i t i o n s .

W h i l e t h e p h y s i c a l v a l i d i t y o f C u l l i n g ' s s t o c h a s t i c

h y p o t h e s i s i s d o u b t f u l , t h e d i f f u s i o n m o d e l r e m a i n s

a v a i l a b l e f o r p r o c e s s e s w i t h s e d i m e n t f l u x p r o p o r t i o n a l t o

g r a d i e n t . I n d e e d , i f c r e e p i s r e g a r d e d as p r o p o r t i o n a l t o

s i n 0 ( « t a n 9 f o r Q<20° ) t h i s i s an a p p r o p r i a t e m o d e l

i r r e s p e c t i v e o f t h e m i c r o s c a l e m e c h a n i c s w h i c h u n d e r l i e t h e

p r o c e s s .

1959).

S = a t a n Q 3z a 3 *

Prom c o n t i n u i t y

1 U (-^)

- 77 -

3.3•^ S c h e i d e g g e r

S c h e i d e g g e r p r o p o s e d s e v e r a l m o d e l s o f h i l l s l o p e

d e v e l o p m e n t i n a s e r i e s o f p a p e r s p u b l i s h e d i n t h e 1960s: a

c o n v e n i e n t summary was p r o v i d e d b y S c h e i d e g g e r (1970,

C h s . 3 .5 , 3 .6 , 5 . 8 ) .

A f a m i l y o f l i n e a r m o d e l s h a v e t h e g e n e r a l f o r m

| 5 = - o ^ , a. > 0

I n p a r t i c u l a r c a s e s ( i ) f = 1 ( c o n s t a n t f o r m ) ( i i ) f = z

( i i i ) f = —•—- . I n e a c h case s t r a i g h t f o r w a r d a n a l y t i c a l

s o l u t i o n s e x i s t , b u t t h e m o d e l s a r e a d m i t t e d t o be

s i m p l i s t i c a t b e s t .

A f a m i l y o f n o n l i n e a r m o d e l s s u g g e s t e d as an

i m p r o v e m e n t h a s t h e g e n e r a l f o r m

I - JO(1)1 lit The m u l t i p l y i n g f a c t o r

r e f l e c t s t h e a s s u m p t i o n t h a t ' w e a t h e r i n g ' a c t s n o r m a l l y t o

t h e s l o p e . S i n c e s e c 9 i n c r e a s e s as 0 i n c r e a s e s b e t w e e n

0° a n d 9 0 ° , t h e e f f e c t o f t h i s f a c t o r i n t h e n o n l i n e a r

m o d e l s i s t h a t s t e e p e r s l o p e s d o w n w a s t e r e l a t i v e l y f a s t e r

as c o m p a r e d w i t h t h e l i n e a r m o d e l s . S i n c e a n a l y t i c a l

s o l u t i o n s a r e o n l y a v a i l a b l e i n s p e c i a l c i r c u m s t a n c e s ,

- 78 -

n u m e r i c a l s o l u t i o n s a r e g e n e r a l l y n e c e s s a r y . The s p e c i a l

c a se s s u g g e s t e d a r e ( i ) t o ( i i i ) a b o v e , t h e l a s t b e i n g

r e g a r d e d as m o s t r e a l i s t i c .

M o d i f i c a t i o n s a n d g e n e r a l i s a t i o n s t o t h e s e n o n l i n e a r

m o d e l s i n c l u d e g e n e r a l i s a t i o n s t o a l l o w f o r l i t h o l o g i c a l

v a r i a t i o n a n d f o r e n d o g e n e t i c e f f e c t s ( u p l i f t o r s u b s i d e n c e ) .

A l t h o u g h t h e s e m o d e l s a r e m o t i v a t e d b y v a r i o u s r e m a r k s

a b o u t s u p p o s e d p r o c e s s e s , t h e y a r e b e s t r e g a r d e d as

p h e n o m e n o l o g i c a l . The a s s u m p t i o n t h a t d e n u d a t i o n a c t s

n o r m a l l y t o t h e s l o p e seems more a p p r o p r i a t e f o r w e a t h e r i n g -

l i m i t e d d e v e l o p m e n t t h a n f o r t r a n s p o r t - l i m i t e d d e v e l o p m e n t .

W r i t i n g i n a more g e n e r a l c o n t e x t a b o u t l a r g e s c a l e

l a n d s c a p e d e v e l o p m e n t , S c h e i d e g g e r (1970, C h . 5.8) m o t i v a t e d

a d i f f u s i o n e q u a t i o n

b y a n e x t e n d e d a n a l o g y b e t w e e n l a n d s c a p e s a n d t h e r m o d y n a m i c

s y s t e m s . I l l u s t r a t i v e s o l u t i o n s show t h e d e c a y o f an

i d e a l i s e d r a n g e a n d o f an i d e a l i s e d s l o p e b a n k . M o d i f i c a t i o n s

a n d g e n e r a l i s a t i o n s o f t h i s m o d e l w e r e d i s c u s s e d b y

S c h e i d e g g e r a n d L a n g b e i n (1966, 3 -5) . The a n a l o g y r e m a i n s

an a n a l o g y ; t h e s e m o d e l s a r e a l s o p h e n o m e n o l o g i c a l .

3.3.5 T a k e s h i t a

T a k e s h i t a (1963) d i s c u s s e d v a r i o u s m o d e l s f o r d i f f e r e n t

k i n d s o f p r o c e s s e s a n d p r e s e n t e d r e s u l t s o f s i m u l a t i o n s

o b t a i n e d w i t h f i n i t e d i f f e r e n c e s c h e m e s . The m o d e l s

3z 3*

- 79 -

i n c l u d e d e r o s i o n r a t e s n o r m a l t o t h e s l o p e a n d d e r i v a t i v e s

w i t h r e s p e c t t o a r c l e n g t h : t h e y h a v e b e e n t r a n s l a t e d h e r e

i n t o p a r t i a l d i f f e r e n t i a l e q u a t i o n s o f f u n c t i o n s a n d d e r i v a t i v e s

e x p r e s s e d v e r t i c a l l y a n d h o r i z o n t a l l y .

^ . I f * , el S u b d u i n g r e c e s s i o n f o l l o w s

i n t h e case o f l i f t - a n d - d r o p p r o c e s s e s o r t h e s i m i l a r

r e l a t i o n

1>Z ~b 2? . J. U e 7 3 t 3 x

i n t h e case o f s o i l c r e e p , u n c o n c e n t r a t e d w a s h a n d

• aqueous s o l i f l u c t i o n 1 . P a r a l l e l r e c e s s i o n f o l l o w s

1* = b -ue = b

i n t h e case o f r i l l w a s h a n d l a r g e - s c a l e l a n d s l i d e s w h i l e

s t e e p e n i n g r e c e s s i o n f o l l o w s

i

i n t h e ca se o f h e a d - d e p e n d e n t p r o c e s s e s s u c h as f a l l ,

g u l l y i n g a n d m u d f l o w . I n p r a c t i c e t h e s e modes o f r e c e s s i o n

a r e f r e q u e n t l y c o m b i n e d .

T a k e s h i t a a p p e a r s t o h a v e b e e n u n a w a r e t h a t a n a l y t i c a l

s o l u t i o n s a r e a v a i l a b l e f o r some o f these e q u a t i o n s . T h e i r

m a j o r i n t e r e s t l i e s i n t h e p r o c e s s m o t i v a t i o n s u p p l i e d

b y T a k e s h i t a .

- 80 -

3.3.6 Y o u n g

Y o u n g (1963) b u i l t a s e t o f w h a t w o u l d now be c a l l e d

s i m u l a t i o n m o d e l s , a l t h o u g h h i s s i m u l a t i o n s w e r e n o t a s s i s t e d

b y any c o m p u t e r - e v i d e n t l y a H e r c u l e a n l a b o u r . The m o d e l s

a r e c e n t r e d a r o u n d a c o n t i n u i t y p r i n c i p l e : e i g h t e q u a t i o n s

p r e s e n t e d i n an e l a b o r a t e a r g u m e n t ( Y o u n g , 1963, ^9-50)

a r e i n e s s e n c e n o t h i n g more t h a n a s i n g l e c o n t i n u i t y e q u a t i o n

^1 + h . 0

3x -at Much o f t h e i n t e r e s t o f t h e s e l i e s i n t h e i r r e l a t i o n

t o Y o u n g ' s f i e l d e x p e r i e n c e , t o g e t h e r w i t h t h e v a r i e t y

o f c o n d i t i o n s i n v e s t i g a t e d . I n i t i a l s l o p e s i n c l u d e d a

s t r a i g h t 35° s l o p e w i t h l e v e l s u r f a c e a b o v e , a 35° s l o p e

w i t h l e v e l s u r f a c e s a b o v e a n d b e l o w , a n d a l e v e l s u r f a c e .

S e d i m e n t f l u x e s w e r e g i v e n b y t h e f o l l o w i n g e q u a t i o n s

S = C-l s i n 9 ( i )

S = C 2 Q 2 ( i i )

S = C 3 x s i n G ( i i i )

S = C j x s i n 9 , d i s t a n c e s o n c o n c a v i t y d o u b l e d ( i v )

S = C/, w s i n 0 ( v )

( i ) a n d ( v ) may r e p r e s e n t c r e e p a n d ( i i i ) w a s h . ( i v ) may

y i e l d a n a p p r o x i m a t i o n t o t h e e f f e c t s o f c a t e n a r y v a r i a t i o n

i n t e x t u r e . ( i i ) i s an e x a m p l e o f c o n d i t i o n s i n w h i c h

s e d i m e n t t r a n s p o r t i s v e r y much f a s t e r on s t e e p s l o p e s t h a n

on g e n t l e s l o p e s .

- 81 -

I n a d d i t i o n , ' d i r e c t r e m o v a l ' was i n c l u d e d i n some

m o d e l s . T h i s t e r m c o v e r s p r o c e s s e s s u c h as s o l u t i o n w h i c h

r e m o v e m a t e r i a l f r o m t h e s l o p e p r o f i l e i n a t i m e s h o r t e r

t h a n t h e t i m e s t e p u s e d i n s i m u l a t i o n . A s s u m p t i o n s

e m p l o y e d w e r e

S = c5 , 0X> S = Cg s i n 9

The r a t e o f w e a t h e r i n g was i n c l u d e d i n t w o m o d e l s t o p r o d u c e

r e l a t i v e e s t i m a t e s o f s o i l d e p t h

W = C 7 w s i n 8

W = C g / w

A v a r i e t y o f s l o p e b a s e c o n d i t i o n s w e r e c o n s i d e r e d :

n o v e r t i c a l e r o s i o n , w i t h b a s a l r e m o v a l i m p e d e d o r u n i m p e d e d

v a r y i n g r a t e s o f u n i f o r m v e r t i c a l e r o s i o n ; a n d a l t e r n a t i n g

p e r i o d s o f v e r t i c a l e r o s i o n a n d n o e r o s i o n w i t h u n i m p e d e d

b a s a l r e m o v a l . One f u r t h e r f e a t u r e o f i n t e r e s t was t h a t

r a p i d mass movement was a s sumed a t a n g l e s a b o v e 3 5 ° ,

r e d u c i n g t h e a n g l e t o 35° .

These s i m u l a t i o n m o d e l s , w h i c h h a v e i n f l u e n c e d many

s u b s e q u e n t w o r k e r s , c o n t a i n many i n t e r e s t i n g i d e a s , w h i c h

i n g e n e r a l a r e b e t t e r i n v e s t i g a t e d e i t h e r a n a l y t i c a l l y o r

b y c o m p u t e r s i m u l a t i o n . P a r s o n s (1976a) ha s r e c e n t l y

i n v e s t i g a t e d s i m i l a r m o d e l s .

3.3.7 T r o f i m o v

T r o f i m o v a n d h i s c o w o r k e r s a t K a z a n h a v e p u b l i s h e d

s e v e r a l p a p e r s s i n c e 1964 on c o n t i n u o u s m o d e l s o f h i l l -

s l o p e d e v e l o p m e n t , m o s t l y i n R u s s i a n . I t i s n o t p o s s i b l e

- 82 -

t o g i v e h e r e a n a d e q u a t e summary a n d a s s e s s m e n t o f e a r l i e r

w o r k .

T r o f i m o v a n d M o s k o v k i n (1976a) d e r i v e d an e x p r e s s i o n

d e f i n i n g t h e s t a b l e e q u i l i b r i u m p r o f i l e o f a s l o p e

d e v e l o p i n g u n d e r s h e e t f l o o d e r o s i o n , w h i c h t u r n s o u t t o

be c o n c a v e . They (1976b) u s e d a c o n t i n u i t y e q u a t i o n

f o r

S = q p w h e r e q i s w a t e r d i s c h a r g e a n d p s e d i m e n t c o n c e n t r a t i o n

2 7" p = ( a x + b x + c )

The s o l u t i o n was o b t a i n e d u s i n g L e g e n d r e p o l y n o m i a l s : i t

i s a sum o f an i n f i n i t e s e r i e s . T r o f i m o v a n d M o s k o v k i n

n o t e d an i m p o r t a n t p r o p e r t y o f t h i s k i n d o f s o l u t i o n :

i n i t i a l d e t a i l s become i n c r e a s i n g l y i r r e l e v a n t .

T h e y a l s o d e r i v e d a s o l u t i o n o f t h e d i f f u s i o n e q u a t i o n

f o r t h e b o u n d a r y c o n d i t i o n o f c o n s t a n t b a s a l r e c e s s i o n , i . e .

z b = z ( b t , t ) = 0

3.3.8 Souchez

S o u c h e z (1966a; c f . 1961, 1963, 1964, 1966b) c o n s i d e r e d

t h e d e v e l o p m e n t o f a h i l l s l o p e p r o f i l e u n d e r ' v i s c o u s ' a n d

' p l a s t i c ' mass m o v e m e n t s . I t i s , u n f o r t u n a t e l y , d i f f i c u l t

t o f o l l o w h i s a r g u m e n t : i n a d d i t i o n t o s e v e r a l m i n o r e r r o r s ,

t h e p r e s e n t a t i o n i s m a r r e d b y a d e e p - s e a t e d c o n f u s i o n o v e r

c o o r d i n a t e s y s t e m s . The f o l l o w i n g i s a r e c o n s t r u c t i o n .

- 83 -

B o t h v i s c o u s a n d p l a s t i c f l o w f o l l o w

dW _ T — T c , X > T c

w h e r e v i s s p e e d o f mass movement

u d e p t h b e l o w s u r f a c e , m e a s u r e d n o r m a l l y

T = < r a s i n G t a n g e n t i a l s t r e s s

C s p e c i f i c w e i g h t

T t c r i t i c a l s t r e s s

? J c o e f f i c i e n t o f v i s c o s i t y

( c f . S t r a h l e r , 1952, 926).

F o r v i s c o u s f l o w T = 0. I f s p e e d v = 0 a t

u = u Q t h e n

b y i n t e g r a t i o n , a n d a v e r a g e s p e e d

_ — I p"0 cr SCKV 0 a V = H 0 V 4 = &n

0

The s e d i m e n t f l u x

3YJ

S - u. V = SUA 9 0

3 \ ") e < 2.0°

F r o m c o n t i n u i t y

- 84 -

T h i s i s t h e d i f f u s i o n e q u a t i o n y e t a g a i n .

F o r p l a s t i c f l o w T t > 0 : t h e s y s t e m e x h i b i t s a

t h r e s h o l d . I t i s s t r a i g h t f o r w a r d t o o b t a i n e x p r e s s i o n s

f o r v , v a n d S b u t t h e c o n t i n u i t y e q u a t i o n i s v a l i d o n l y

i f t h e c o n d i t i o n s f o r mass movement a r e met e v e r y w h e r e .

A p a r t f r o m t h e d i f f i c u l t i e s s u r r o u n d i n g h i s

p r e s e n t a t i o n , t h e p r o b l e m w i t h S o u c h e z ' s m o d e l i s t h a t t h e r e

seems t o be l i t t l e p h y s i c a l j u s t i f i c a t i o n f o r p o s i t i n g e i t h e r

v i s c o u s o r p l a s t i c f l o w ( S c h e i d e g g e r , 1970, 9 7 - 8 ; Y o u n g , 1 9 7 2 ,

1 1 2 - 3 ) . I t i s i n t e r e s t i n g , h o w e v e r , t o see a n o t h e r m o t i v a t i o n

f o r t h e d i f f u s i o n e q u a t i o n , as a l o w - a n g l e a p p r o x i m a t i o n t o

h i l l s l o p e d e v e l o p m e n t u n d e r v i s c o u s f l o w .

3.3-9 A h n e r t

A h n e r t (1966, 1970b, 1971, 1972, 1972b, 1973, 1976a,

1976b, 1977; see a l s o M o s l e y , 1973, M o o n , 1975, 1977)

has b e e n d e v e l o p i n g s i m u l a t i o n m o d e l s o f h i l l s l o p e a n d

l a n d s u r f a c e d e v e l o p m e n t o v e r t h e l a s t d e c a d e . The e m p h a s i s

h e r e i s o n t h e l a t e s t v e r s i o n s (FORTRAN p r o g r a m s ; COSLOP

f o r h i l l s l o p e s , SL0P3D f o r l a n d s u r f a c e s ) . The b a s i c

s t r u c t u r e o f b o t h p r o g r a m s i s s i m i l a r , a n d e a c h i s w r i t t e n

i n a m o d u l a r f a s h i o n .

W e a t h e r i n g r a t e d e p e n d s o n s o i l t h i c k n e s s . I t may be

m e c h a n i c a l ( d e c r e a s i n g e x p o n e n t i a l l y w i t h t h i c k n e s s ) ,

c h e m i c a l ( i n c r e a s i n g a n d t h e n d e c r e a s i n g ) o r a c o m b i n a t i o n

o f t h e t w o . More ' r e s i s t a n t ' s t r a t a , e i t h e r h o r i z o n t a l ,

- 85 -

S t a r t

R e p e a t

m a i n

l o o p

READ i n i t i a l p r o f i l e / s u r f a c e a n d p r o g r a m p a r a m e t e r s

W e a t h e r i n g

D o w n c u t t i n g ( i f a n y )

Was te t r a n s p o r t

L a n d s l i d i n g ( i f a n y )

WRITE r e s u l t s

E n d

v e r t i c a l o r d i p p i n g , may be i n c o r p o r a t e d . R e s i s t a n c e

r e s u l t s i n r e d u c e d w e a t h e r i n g r a t e s . I n f i l t r a t i o n

p r o p e r t i e s may a l s o v a r y w i t h l i t h o l o g y .

D o w n c u t t i n g t a k e s p l a c e a t one o r t w o p o i n t s . A

v a r i e t y o f modes o f b a s e l e v e l l o w e r i n g a r e p o s s i b l e .

Was te t r a n s p o r t may be b y s p l a s h , w a s h , p l a s t i c

f l o w , v i s c o u s f l o w o r s l i d i n g . S e d i m e n t f l u x f o r s p l a s h

f o l l o w s c .

a n d f o r w a s h

S = r d ° 2 s i n c 3 0

- 86 -

w h e r e r i s a r e s i s t a n c e v a r i a b l e ( m o d e l l e d as a p o w e r

f u n c t i o n o f s o i l t h i c k n e s s ) a n d d i s d e p t h o f f l o w .

P l a s t i c f l o w d e p e n d s o n a t h r e s h o l d w 1

S cC (" H ^ ^ - U ' ) H S iA 0 £ '

w h i l e v i s c o u s f l o w o c c u r s a t a l l g r a d i e n t s

D e b r i s s l i d i n g o c c u r s s o t h a t no s l o p e a b o v e 35° c a r r i e s

a r e g o l i t h . ( N o t e t h a t r o c k f a c e s a b o v e 45° a r e a l l o w e d ) .

These a s s u m p t i o n s o n w a s t e t r a n s p o r t ( A h n e r t , 1976b,

1977) a r e b a s i c a l l y s e m i - e m p i r i c a l . They a r e a g r e a t

i m p r o v e m e n t on p r e v i o u s s e t s o f a s s u m p t i o n s i n e a r l i e r

v e r s i o n s o f A h n e r t ' s m o d e l s . The m o s t o b v i o u s a b s e n t e e s

a r e o p t i o n s f o r r o t a t i o n a l f a i l u r e as o p p o s e d t o

t r a n s l a t i o n a l f a i l u r e ( c f . H u t c h i n s o n , 1968) a n d f o r

g r a d i e n t - d e p e n d e n t c r e e p

S cC ta*\ 0

The o p t i o n f o r v i s c o u s f l o w i s c l e a r l y m e a n t t o be t h e

a l t e r n a t i v e t o s u c h c r e e p .

A s s u m p t i o n s a b o u t w e a t h e r i n g a n d d o w n c u t t i n g seem

r e l a t i v e l y p l a u s i b l e , b u t t h i s i s p r o b a b l y a r e f l e c t i o n

o f p r e s e n t i g n o r a n c e a b o u t t h e s e m a t t e r s .

The g r e a t s t r e n g t h o f t h e s e m o d e l s i s a l s o t h e i r

g r e a t w e a k n e s s . M o d u l a r s t r u c t u r e a n d v e r s a t i l i t y c o m b i n e

t o make t h e m a t t r a c t i v e t o o l s , a l t h o u g h n a t u r a l l y t h e y

m u s t be u s e d w i t h c a r e . More s e n s i t i v i t y a n a l y s e s ,

f o l l o w i n g Moon (1975) , w o u l d be v a l u a b l e .

- 87 -

3. 3.10 D e v d a r i a n i

D e v d a r i a n i ( 1 9 6 7 a ) d r e w an a n a l o g y b e t w e e n s t r e a m l o n g -

p r o f i l e d e v e l o p m e n t a n d t h e c o n d u c t i o n o f h e a t i n a

t h e r m o d y n a m i c s y s t e m , t h u s m o t i v a t i n g a d i f f u s i o n e q u a t i o n

= <x?3 •at

w i t h g e n e r a l s o l u t i o n

w h e r e t h e a ' ' s , t h e b ' s a n d t h e c ' s a r e c o n s t a n t s , o r d e r e d

so t h a t b 2 < b ^ < . . . < b & o . As t i m e t i n c r e a s e s , t h e

f i r s t t e r m comes t o d o m i n a t e t h e s e r i e s so t h a t

z = a ' e s ^ n c x

d r o p p i n g t h e s u b s c r i p t i = 1. As t^>c<5, z - » 0 ,

The m o d e l was g e n e r a l i s e d t o a l l o w f o r s p a t i a l l y

v a r i a b l e r o c k p r o p e r t i e s , s t r e a m d i s c h a r g e , e t c . , a n d

f o r t h e e x i s t e n c e o f a l i m i t i n g p r o f i l e z | ( ^ ( x ) f o r

w h i c h i ' * = 0 . The s o l u t i o n now t a k e s t h e f o r m

D e v d a r i a i i s u g g e s t e d t h a t m o d e l s o f t h i s k i n d w e r e

a p p l i c a b l e t o h i l l s l o p e s d e v e l o p i n g u n d e r mass movement

o r s u r f a c e w a s h .

A s s u m i n g a g a i n t h a t as t i n c r e a s e s , t e r m s i n d e x e d

b y i > 2 become n e g l i g i b l e , t h e r i g h t - h a n d t e r m becomes

- 88 -

t h e p r o d u c t o f a ' e " b t , a f u n c t i o n o f t i m e a l o n e , a n d

c ( x ) , a f u n c t i o n o f d i s t a n c e a l o n e . This i s r e m a r k a b l y

s i m i l a r t o c h a r a c t e r i s t i c f o r m s o l u t i o n s s o u g h t b y

K i r k b y ( s e e b e l o w ) , a l t h o u g h i n t h e l a t t e r c a s e , t h e

t e r m c o r r e s p o n d i n g t o z ( ( x ) i s o f t e n i d e n t i c a l l y z e r o .

The m a i n d i f f i c u l t y w i t h t h i s m o d e l i s t h e f o r m o f

t h e s o l u t i o n w h i c h i n c l u d e s t h e sum o f an i n f i n i t e

s e r i e s . I n g e n e r a l , t h e s e a r e n o g r o u n d s f o r a s s u m i n g

t h a t t h e b ' s a n d t a r e s u c h t h a t t e r m s o t h e r t h a n t h e

f i r s t may be n e g l e c t e d . The f o r m o f t h e l i m i t i n g p r o f i l e

z j ( . ( x ) i s n o t s p e c i f i e d t h e o r e t i c a l l y , n o r i s i t

s u g g e s t e d how i t m i g h t be d e t e r m i n e d f r o m f i e l d d a t a .

T h i s a l s o a p p l i e s t o t h e f u n c t i o n s c ^ ( x ) . M o r e o v e r ,

t h e i d e a o f a l i m i t i n g p r o f i l e s t a n d s i n n e e d o f d e t a i l e d

g e o m o r p h o l o g i c a l j u s t i f i c a t i o n .

I n a l a t e r p a p e r , D e v d a r i a n i (1967b) s u g g e s t e d an

e q u a t i o n f o r s e d i m e n t f l u x

S = - o , ^ ^X

w h e r e z i s h e i g h t a b o v e t h e l i m i t i n g p r o f i l e , so t h a t r e l

z ( x , t ) = z | ( x ) + z r e l ( x , t )

Thus

The e q u a t i o n f o r S was g e n e r a l i s e d t o a = a ( x ) .

U s i n g a c o n t i n u i t y e q u a t i o n w i t h a n e n d o g e n e t i c t e r m

2>X 2>t

- 89 -

i t may be s e e n t h a t

w i t h s o l u t i o n r e q u i r e d f o r z r e i ( x , t ) . The m o d e l was a g a i n

r e g a r d e d as a p p l i c a b l e t o b o t h h i l l s l o p e a n d s t r e a m p r o f i l e s .

A s e r i e s s o l u t i o n was o b t a i n e d f o r g e n e r a l f ( x , t ) , a n d

some s p e c i a l c a s e s w e r e i n v e s t i g a t e d . T h i s m o d e l i s s u b j e c t

t o t h e k i n d o f c r i t i c i s m s made a b o v e , b u t r e m a i n s o f

i n t e r e s t as a p h e n o m e n o l o g i c a l m o d e l t a k i n g u p l i f t i n t o

a c c o u n t .

3.3.11 H i r a n o

I n a d d i t i o n t o t h e p a p e r s d i s c u s s e d b e l o w ( H i r a n o ,

1968, 1972, 1975, 1976), H i r a n o h a s p u b l i s h e d t e n o t h e r

p a p e r s i n J a p a n e s e .

H i r a n o (1968) p r o p o s e d a c o m p o s i t e l i n e a r m o d e l

•at a * 1 3x S o l u t i o n s w e r e p r e s e n t e d f o r t h e c a s e o f a ' f i n i t e m o u n t a i n ' ,

s y m m e t r i c a l a b o u t a d i v i d e a t x = 0 , f o r t h e s p e c i a l c a s e

c = 0, a n d f o r a v a r i e t y o f b o u n d a r y c o n d i t i o n s . L i t h o l o g i c a l

v a r i a t i o n s w e r e t r e a t e d b y l e t t i n g a , b a n d c become

f u n c t i o n s o f x , z a n d t a n d e n d o g e n e t i c e f f e c t s b y

a d d i n g a f u n c t i o n f t o t h e m o d e l

2rfc ^x1 ^ I n p a r t i c u l a r , t h e e n d o g e n e t i c f u n c t i o n was assumed

s e p a r a b l e

f ( x , t ) = X ( x ) T ( t )

- 90 -

Examples of T were instantaneous impulse, uniform e f f e c t and exponential decay. An example of X was a step f u n c t i o n

X = 1 x < x^

representing a f a u l t scarp.

The basic equation, a second order l i n e a r p a r t i a l d i f f e r e n t i a l equation, permits a wide v a r i e t y of s o l u t i o n s depending on parameter values, i n i t i a l c o nditions and boundary c o n d i t i o n s .

tz

Hirano suggested t h a t the term u. r--^ represented creep (which i s p l a u s i b l e ) ; t h a t the term (> = represented wash (thereby assumed distance-independent); and was a t a loss t o provide a process i n t e r p r e t a t i o n f o r the term cz. I n f a c t , the model i s e s s e n t i a l l y phenomenological, and Hirano's attempt t o provide a process m o t i v a t i o n i s unconvincing. The r a t i o n a l e f o r the model i s , i n large p a r t , i t s t r a c t a b i l i t y which stems from i t s l i n e a r i t y i n the d e r i v a t i v e s .

Hirano (1972) r e p o r t e d some t e s t i n g o f t h i s model on a f a u l t scarp and a v a l l e y w a l l i n the H i r a Mountains, Japan.

Hirano (1975) motivated the l i n e a r model

b = 0 x > x^ b

^ 2 3z. + b a. •ax as an approximation to

2 1)2-

2>x 7fc

- 91 -

i t s e l f regarded as a model of the combined e f f e c t s of creep and wash. He presented several constant form s o l u t i o n s (although w ithout reference t o e a r l i e r work d e r i v i n g such s o l u t i o n s ) and some more general s o l u t i o n s , both a n a l y t i c a l and numerical.

Most r e c e n t l y Hirano (1976) discussed a g e n e r a l i s a t i o n of h i s l i n e a r model t o two h o r i z o n t a l dimensions

7t = a ( + Y ) + ^ + ^ J + f 6 t ' ^ ( I n some a p p l i c a t i o n s the f u n c t i o n f was taken t o be i d e n t i c a l l y zero). This equation can be transformed i n t o a d i f f u s i o n equation by s u b s t i t u t i o n . Hirano p a i d e s p e c i a l a t t e n t i o n to constant form s o l u t i o n s

0 + U 3x

and t o the r e l a t i o n s h i p between channel and v a l l e y slopes.

3.3.12 Pollack

Pollack (1968) discussed various simple models b r i e f l y w i t h s p e c i a l emphasis on conversion o f p a r t i a l d i f f e r e n t i a l equations t o d i f f e r e n c e equations. The only o r i g i n a l model i s the u n i n t e r e s t i n g equation

= £ < 0 a t

where ^ i s a random v a r i a b l e .

I n a l a t e r and somewhat ina c c e s s i b l e paper Pollack (1969) proposed a modified d i f f u s i o n model

- 92 -

where f\ and f describe the c h a r a c t e r i s t i c s of d i f f e r e n t s t r a t a i n f l u e n c i n g the rates and p o s i t i o n of stream down-c u t t i n g and v a l l e y widening. Numerical experiments w i t h the model attempted t o simulate' the development of the Grand Canyon, w i t h some success. For an accessible summary of t h i s model, see Harbaugh and Bonham-Carter ( 1 9 7 0 , 5 3 1 - 5 ) .

3 . 3 . 1 3 Aronsson

Aronsson (1973) discussed a model i n which denudation i n t e n s i t y f i s a f u n c t i o n o f p o s i t i o n f (x, z ) . Denudation was assumed to act normally t o the slope, w i t h t r a n s p o r t o f m a t e r i a l so r a p i d t h a t loose s u r f i c i a l debris does not a f f e c t the process (although i n some a p p l i c a t i o n s a p r o t e c t i v e r e g o l i t h was assumed to cover the h i l l s lope s u r f a c e ) . Hence the model i s ap p l i c a b l e t o w e a t h e r i n g - l i m i t e d development, r a t h e r than t r a n s p o r t - l i m i t e d development.

I f P ( x , z) i s a p o i n t i n i t i a l l y i n s i d e the slope and t ( x , z) the time elapsed before P appears at the surface, then t i s given by

where the minimum i s taken over a l l curves which connect P and the i n i t i a l p r o f i l e z = z ( x , 0 ) , and where s i s the arc length o f the curve. This procedure resembles the use o f Fermat's p r i n c i p l e i n op t i c s ( c f . Gelfand and Fomin, 1963, Appendix I ) . Knowledge of the f u n c t i o n t ( x , z) i s i n p r i n c i p l e s u f f i c i e n t t o describe the development of a h i l l s l o p e p r o f i l e . Curves o f the form t ( x , z) = constant describe p r o f i l e s at p a r t i c u l a r times.

2

t ( x , z) ds

min Us)

- 93 -

Unf o r t u n a t e l y , no general method i s known f o r c a l c u l a t i n g t h i s f u n c t i o n . However, a numerical procedure developed by Rydman i s ap p l i c a b l e t o some s p e c i a l cases i n which the i n i t i a l p r o f i l e i s l i n e a r and denudational i n t e n s i t y ( i n v e r s e l y p r o p o r t i o n a l t o res i s t a n c e ) i s constant f o r each o f a set of p a r a l l e l h o r i z o n t a l layers which make up the h i l l s i d e . Aronsson presented the r e s u l t s o f a series o f s i m u l a t i o n experiments showing diagrammatically the sequences of h i l l s l o p e development under various h y p o t h e t i c a l c o n d i t i o n s .

These r e s u l t s are i m p l a u s i b l e : many p r o f i l e s i n clude v e r t i c a l or overhanging components, generally only possible i n stron g bedrock. Moreover, the procedure demands a s p e c i f i c a t i o n o f an i n i t i a l p r o f i l e z ( x , 0) which i s r a r e l y p r a c t i c a b l e .

The c e n t r a l question i s , however, whether the Fermat-type p r i n c i p l e i s a n a t u r a l model f o r h i l l s l o p e development. I n an e a r l i e r , more d i f f i c u l t , paper Aronsson (1970) proved t h a t the p r i n c i p l e was a consequence of c e r t a i n axioms. Hence the question becomes whether these axioms are appropriate. This i s i n doubt, p a r t i c u l a r l y because of the assumption t h a t h i l l s l o p e development a f t e r some time t depends only on the p r o f i l e a t t (Aronsson, 1970; independence assumptions, 675; Condition D, 676).

3.3.1*1 Gossmann

Gossmann (1970, 1976) used a basic equation f o r sediment f l u x

_ 9 l l -

rh^9 6 > 9C

S = O^ Z Sin 9 Cos 9 -\- o '

0 e 9C

Here the f i r s t term represents wash: the m u l t i p l y i n g f a c t o r s i n 9 cos 9" takes a maximum at 9 = ^5° and i s intended t o mimic the f a c t t h a t wash erosion reaches a maximum a t intermediate angles (although the peak appears t o be much below 45°: Horton, 19^5). The second term represents processes which are independent o f distance x yet dependent on a t h r e s h o l d angle 0C . This f l u x equation was combined w i t h a c o n t i n u i t y equation ( u n f o r t u n a t e l y misquoted)

35 + ^ = c Results from these equations f o r d i f f e r e n t parameter

values and boundary conditions were presented g r a p h i c a l l y by Gossmam(1976). These r e s u l t s stem from a s i m u l a t i o n version o f the model. They were compared w i t h f i r s t l y , semi-arid and a r i d conditions ('pedimentation' and surface wash assumed dominant)] secondly, ' p e r i g l a c i a l ' c o n d i t i o n s ( s o l i f l u c t i o n and surface wash); and t h i r d l y , t r o p i c a l wet and dry c o n d i t i o n s . Special equations were also employed; f o r example, i n the p e r i g l a c i a l case the equation

sm 39 OS 38 O £ 0 s 30s

0 8 >30c

i s meant t o mimic the decrease of s o l i f l u c t i o n caused by removal of f i n e s by surface and subsurface wash. The term s i n 39 cos 30 reaches a maximum value at 0 = 15° (although

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t h i s maximum i s 2, not 1 as s t a t e d by Gossmann).

The most valuable f e a t u r e of Gossmann's paper i s the l i n k forged between h i l l s l o p e modelling and c l i m a t i c geomorphology, e s p e c i a l l y as p r a c t i s e d by Btidel and h i s d i s c i p l e s , although many of the un d e r l y i n g theses are accepted u n c r i t i c a l l y ; ( c f . Stoddart, 1969). On the other hand, while h i l l s l o p e development i s modelled i n a r a t i o n a l way, the form o f sediment f l u x equations i s not always motivated c o n v i n c i n g l y .

Subsequent s i m u l a t i o n work i n a s i m i l a r s t y l e (Rohdenburg e t a l , 1976) used the basic sediment f l u x equations

S = a tan 0 S = a x m s i n 1 1 9

and explored the extent t o which p a r t i c u l a r forms are pr o c e s s - s p e c i f i c , paying s p e c i a l a t t e n t i o n t o basal c o n d i t i o n s .

3.3.15 Kirkby

Kirkby (1971, 1976a, 1976b, 1977a; Carson and Kirkby, 1972; Wilson and Kir k b y , 1975) has produced some of the most valuable models of h i l l s l o p e development at present a v a i l a b l e .

The f i r s t f a m i l y o f models ( K i r k b y , 1971; Carson and Kirkby, 1972, 107-9, ^33-6) are best described through an example.

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Sediment f l u x i s given by

1 dX I S — CK.X

and must satisfy a c o n t i n u i t y equation

— + — - o whence

1 3* L '5* / = 0

Kirkby s o u g h t ' c h a r a c t e r i s t i c form' s o l u t i o n s o f the k i n d z ( x , t ) = X ( x ) . T ( t )

i.e . , the v a r i a b l e s were assumed separable. This k i n d of s o l u t i o n i s o f t e n associated w i t h exponential decay

T ( t ) = c 1 e " c 2 t

(Wilson & Kir k b y , 1975 , 2 1 5 - 7 ; but beware typ o g r a p h i c a l e r r o r s i n eqns. 6 . 2 1 4 , 6.218, 6 . 2 2 4 ) . With the boundary conditions o f h o r i z o n t a l l y f i x e d d i v i d e and h o r i z o n t a l l y and v e r t i c a l l y f i x e d base w i t h unimpeded removal, an approximation t o X(x) i s given by

*6<) ' M D or

Z X u n J

]

Since both z/z d and x/x^ are dimensionless t h i s f u n c t i o n gives the shape of the p r o f i l e . The scale of the p r o f i l e s

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varies as z^ since i s assumed constant. I n t h i s case z^ must decline e x p o n e n t i a l l y f o r the s o l u t i o n t o be v a l i d : and, indeed, t h i s i s consistent w i t h the e m p i r i c a l f i n d i n g (but on a d i f f e r e n t scale, and over space r a t h e r than time) t h a t denudation r a t e i s p r o p o r t i o n a l t o r e l i e f (Ahnert, 1970a). Furthermore, some s i m u l a t i o n r e s u l t s suggest t h a t h i l l s l o p e s may converge towards a s t a t e i n which

dz = -cz , c > 0 dt

(Kirkby, 1976a, 259-60).

The s p e c i a l case m = 0, n = 1 y i e l d s the f a m i l i a r d i f f u s i o n equation, discussed by C u l l i n g and others. Conversely, i t may be seen t h a t Kirkby has generalised the d i f f u s i o n equation, which now appears i n i t s proper perspective as merely one possible t r a n s p o r t law. Exact r e s u l t s a v a i l a b l e f o r t h i s case encourage the contention t h a t e m p i r i c a l convergence towards c h a r a c t e r i s t i c form s o l u t i o n s may be q u i t e r a p i d .

I n i t i a l c o n d itions do not need to be s p e c i f i e d t o obta i n c h a r a c t e r i s t i c form s o l u t i o n s . A f u r t h e r p o i n t of i n t e r e s t i s t h a t the approximation t o X (x) quoted above i s a power f u n c t i o n , discussed above as a s t a t i c model.

Signal advantages o f the fa m i l y o f models of which t h i s i s an example i n c l u d e , f i r s t l y , the e m p i r i c a l r e a l i s m o f the sediment t r a n s p o r t laws f o r many i n d i v i d u a l processes such as creep and wash; secondly, the i n t e r e s t i n g and

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important idea of c h a r a c t e r i s t i c form; and t h i r d l y , the way i n which the model generalises the d i f f u s i o n model. The main disadvantage of the s o l u t i o n obtained f o r X (x) i s i t s dependence on a r e s t r i c t i v e set o f boundary c o n d i t i o n s .

(Note t h a t i n the more general case

I • m (STw(-%r& The n i n the right-hand term i s erroneously omitted from Carson and K i r k b y , 1972, 433, eqn. Bl)

(For a p p l i c a t i o n s o f these models, see Kirkby and Ki r k b y , 1974, 1976; Richards, 1977).

The second fa m i l y of models (Wilson & Kirkby, 1975, 186-8, 199-200, 216-7; K i r k b y , 1976a) a l l centre on the sediment t r a n s p o r t law

For example, w i t h a c o n t i n u i t y equation

and constant form

the s o l u t i o n i s

( (Wilson & K i r k b y , 1975, 199-200).

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The sediment t r a n s p o r t law used f o r t h i s f a m i l y seems t o be q u i t e good at representing the combined e f f e c t s of creep and wash, and leads f a i r l y r e a d i l y t o a v a r i e t y o f s o l u t i o n s (other examples are given i n the references c i t e d ) .

The t h i r d f a m i l y o f models ( K i r k b y , 1976a, 1976b, 1977a) are e x e m p l i f i e d by a s i m u l a t i o n model which incorporates c l i m a t i c and h y d r o l o g i c a l v a r i a b l e s ( K i r k b y , 1976a). For other work i n c l u d i n g h y d r o l o g i c a l and p e d o l o g i c a l processes, see Kirkby (1976b, 1977a).

Daily r a i n f a l l f a l l i n g on a h i l l s l o p e p r o f i l e i s p a r t i t i o n e d i n t o overland flo w , subsurface f l o w , e v a p o t r a n s p i r a t i o n and change i n s o i l water storage. Annual volumes are computed using an assumed frequency d i s t r i b u t i o n of d a i l y r a i n f a l l . Sediment fluxes f o r creep, wash and s o l u t i o n are computed as func t i o n s o f h y d r o l o g i c a l and morphometric v a r i a b l e s , and the r a t e o f downwearing c a l c u l a t e d from a c o n t i n u i t y equation.

Creep (or splash and unconcentrated wash i n a r i d areas) f o l l o w s

S = 10 tan 9 cm2 y r - 1

and s o i l wash

S = 170 q 2 tan ® cm2 y r " 1

where q i s annual overland flow f l u x i n m2 y r - 1 . S o l u t i o n i s c a l c u l a t e d separately f o r each oxide using an approach developed from t h a t of Carson and Kirkby (1972).

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Simulation experiments reported were f o r a f i x e d slope base. Input was an i n i t i a l slope p r o f i l e , c l i m a t i c parameters, and rock and s o i l parameters. Output included slope p r o f i l e s , flow volumes, rates of lowering and s o i l thickness. The r e s u l t s included simulations o f the v a r i a t i o n s i n slope p r o f i l e development and sediment y i e l d w i t h c l i m a t i c parameters.

These s i m u l a t i o n models, by e x p l i c i t l y i n c o r p o r a t i n g c l i m a t e , hydrology and pedology, represent a great leap forward i n h i l l s l o p e modelling.

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3.3-16 Smith and Bretherton

Smith and Bretherton (1972; c f . also Smith, 1971, 197*0 presented two very i n t e r e s t i n g models i n an extremely important t h e o r e t i c a l paper. Both were motivated by the example o f sand-clay badlands, which have l i t t l e v e getation and a f a i r l y impermeable s u b s t r a t e , yet are asserted t o embody the fundamental aspects o f drainage basin e v o l u t i o n while r e t a i n i n g r e l a t i v e s i m p l i c i t y .

The 'smooth su r f a c e 1 model i s aimed at the basic problem: how does an i n i t i a l surface z = z ( x , y, t ) which slopes only i n the x - d i r e c t i o n , f a l l i n g monotonically away from a divide at x = 0, evolve under r a i n f a l l - p r o d u c e d denudation, and how does i t come to assume a form s i m i l a r t o t h a t o f known drainage basins?

The assumptions made i n t h i s model were set out very c l e a r l y . The substrate i s homogeneous, r a i n f a l l i s uniform and steady,and there are no losses o f water through evaporation or i n f i l t r a t i o n . The surface z i s smooth i n the sense t h a t i t i s (at l e a s t ) twice continuously d i f f e r e n t i a b l e i n the space domain, and ( a t l e a s t ) once continuously d i f f e r e n t i a b l e i n the time domain. Both water and sediment are assumed t o flow d o w n h i l l ( r a t h e r than, f o r example, i n the d i r e c t i o n o f the fr e e water s u r f a c e ) , and are constrained by c o n t i n u i t y equations.

The key assumption, however, i s t h a t o f a sediment t r a n s p o r t law

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S = f ( g , q) where g = j V z j = gradient and S i s constrained by the i n e q u a l i t i e s

The operator 7 i s the d i r e c t i o n a l d i f f e r e n t i a l operator which y i e l d s the vector f i e l d of maximum gradient when a p p l i e d to landsurface a l t i t u d e z. Taking the modulus gives the s c a l a r f i e l d of maximum gr a d i e n t ,

i s , there i s no plan curvature. V z J = g would equal i f and only i f — =0, t h a t

This model was s p e c i a l i s e d f i r s t t o one h o r i z o n t a l dimension t o solve f o r topographic p r o f i l e s z = z ( x , t ) w i t h the boundary conditions t h a t no water or sediment crosses the d i v i d e . Solutions obtained were, f i r s t l y , •constant form' s o l u t i o n s f o r which j£ — —(X ; the conditions under which these are convex or concave were i n v e s t i g a t e d ; secondly, time-dependent s o l u t i o n s f o r the s p e c i a l case

S = c q m g n ; m, n > 0 provided t h a t there i s no sediment sink

z dx = constant o

and t h a t 2n^m, n + l^p m . The s o l u t i o n s have the form

= o ; * > where C2 = l/(2m - n)

x b = c 3 t c 2

and the constants c-j^, c^ r e f l e c t the no sediment sink c o n s t r a i n t

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Roughly speaking, i t seems t h a t i f 0 < m < l there w i l l be an i n f l e x i o n o f the slope p r o f i l e , but i f m>l, the p r o f i l e w i l l be everywhere concave. Smith and B r e t h e r t o n i n t e r p r e t e d these s o l u t i o n s as models o f stream channels and noted t h a t they imply continuous d e c l i n i n g development: steady s t a t e i s not achieved because the time-dependent term never vanishes.

The problem of i n i t i a t i o n of c h a n n e l - l i k e features was then t r e a t e d using the method of s t a b i l i t y analysis which i s standard i n hydrodynamics (e.g. Chandrasekhar, 196l; see also A l l e n , 1970, 6 l - 5 ) . The c e n t r a l question i s whether constant form s o l u t i o n s are stable under p e r t u r b a t i o n s o f i n f i n i t e s i m a l amplitude. I n one h o r i z o n t a l dimension i t turns out t h a t f o r any t r a n s p o r t law S = f ( g , q) the constant form s o l u t i o n i s s t a b l e : f o r example, k n i c k p o i n t s ( o f i n f i n i t e s i m a l amplitude) are always removed. I n two h o r i z o n t a l dimensions, the r e s u l t i s t h a t channel-like forms must grow on c o n c a v i t i e s , but w i l l disappear on c o n v e x i t i e s . Smith and Bretherton suggested t h a t these s t a b i l i t y r e s u l t s may be extended from the constant form s o l u t i o n s t o the more general time-dependent s o l u t i o n s .

According t o Smith and B r e t h e r t o n , a t r a n s p o r t law, t o be r e a l i s t i c , should have an associated constant form surface which i s convex i n the upper p o r t i o n and concave i n the lower p o r t i o n : i t i s then termed a 'landscape-forming t r a n s p o r t law'. Such a law should be able t o account i n broad terms f o r the existence of a

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sta b l e channel network, which r e f l e c t s an approximate balance between p o s i t i v e feedback processes which cause the necessary i n s t a b i l i t y f o r channel growth and negative feedback processes which give the necessary s t a b i l i t y t o check u n b r i d l e d channel growth.

A 'di s c r e t e channel model' i s necessary, according t o Smith and Br e t h e r t o n , because n e i t h e r a n a l y t i c a l nor computer methods draw r e s u l t s on the l a t e r stages of

, channel development from the smooth surface model. The aim i s t o model a continuous surface c o n s i s t i n g o f d i s c r e t e streams and v a l l e y s , but t h i s involves a major d i f f i c u l t y : the need t o spe c i f y a law o f l a t e r a l channel m i g r a t i o n .

Given t h a t f l u x e s of water and sediment enter a channel from both side slopes, how does the stream move sideways? Or does i t remain approximately f i x e d h o r i z o n t a l l y , downcutting i n the same place? Smith and Bretherton hypothesised t h a t a stream moves away from the side slope c o n t r i b u t i n g the l a r g e r sediment f l u x , and so there i s a tendency t o equalise f l u x e s .

A model o f a two-valley system, w i t h side slopes at an angle o f repose, and r e l a t i v e l y s t r a i g h t streams w i t h channel slopes much lower than side slopes, was subjected t o a s t a b i l i t y a n a l y s i s . The constant form s o l u t i o n turned out t o be unstable t o p e r t u r b a t i o n s of i n f i n i t e s i m a l amplitude. On the other hand, a model w i t h h o r i z o n t a l l y f i x e d streams produced embarrassing r e s u l t s :

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small streams become perched high up on the v a l l e y side slopes of other streams. Since n e i t h e r model seems very r e a l i s t i c e m p i r i c a l l y or very w e l l developed, the problem of l a t e r a l channel m i g r a t i o n remains unsolved.

Both the approaches and the r e s u l t s set out i n t h i s important paper merit long discussion. The use of a general model i s l i n k e d w i t h the aim of e x p l a i n i n g t y p i c a l or q u i n t e s s e n t i a l f l u v i a l topography i n q u a l i t a t i v e terms. I t f o l l o w s t h a t the Smith and Bretherton models cannot be employed to e x p l a i n d i f f e r e n c e s i n h i l l s l o p e form over space or time. However, i t seems p o i n t l e s s t o condemn the models f o r f a i l i n g t o f u l f i l an aim f o r which they were not designed.

The use of the t r a n s p o r t law S = f ( g , q ) , together w i t h other features o f the model, was motivated by badland s i t u a t i o n s i n which surface wash i s the dominant geomorpho-l o g i c a l process. Despite t h i s , the r e s u l t s o f the models seem t o carry over t o a l a r g e extent t o s i t u a t i o n s i n which other t r a n s p o r t processes are important.

The use of constant form s o l u t i o n s i s probably the greatest weakness o f the model, and needs g r e a t e r j u s t i f i c a t i o n . Moreover, i t i s unclear how f a r s t a b i l i t y under p e r t u r b a t i o n s of i n f i n i t e s i m a l amplitude i m p l i e s s t a b i l i t y under p e r t u r b a t i o n s o f f i n i t e amplitude. The only c e r t a i n t y i s t h a t the mathematics are f a r more d i f f i c u l t (Lewontin, 1969).

A l l i n a l l , however, these models are among the best now a v a i l a b l e , not l e a s t i n the l i n k s they forge between p r o f i l e modelling and surface modelling.

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3.3.17 Luke

Luke (1972, 1971*, 1976) i s a mathematician i n t e r e s t e d i n geomorphology. I n a recent s e r i e s o f papers he has shown some r e s u l t s o f a q u a l i t a t i v e , geometric approach.

Luke (1972) considered both the general case o f a landsurface z = z ( x , y, t ) developing according t o

and the s p e c i a l case o f a h i l l s l o p e p r o f i l e developing according t o

Here f i s an ' e m p i r i c a l l y determined f u n c t i o n ' , although i n Luke's examples i t i s always a phenomenological f u n c t i o n of u n c e r t a i n o r i g i n .

Luke showed t h a t , given an i n i t i a l p r o f i l e or surface, these equations may be solved f o r any time t using the method of c h a r a c t e r i s t i c s . This i s something o f a Py r r h i c v i c t o r y , since the determination of f and the i n i t i a l p r o f i l e are gene r a l l y n o n t r i v i a l . Luke suggested a method f o r determining f from a c t u a l p r o f i l e d , but t h i s depends on knowing the time t . Moreover, s o l v i n g by c h a r a c t e r i s t i c s may y i e l d m u l t i v a l u e d s o l u t i o n s ( v e r t i c a l or overhanging s l o p e s ) : t h i s d i f f i c u l t y i s met by choosing the lowest z value, a procedure which requires more j u s t i f i c a t i o n (but c f . Luke, 1976, f o r a note on m u l t i v a l u e d p r o f i l e s ) .

3 t 3

2>X •at or

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The landsurface model i s i n t e r e s t i n g but extremely crude. Stream downcutting takes place on predetermined paths i n the (x, y) plane and i s combined w i t h h i l l s l o p e e v o l u t i o n .

I n one example, Luke p o i n t e d out t h a t a f t e r a c e r t a i n time the p r o f i l e depended only on the f u n c t i o n f : a r e s u l t which i s re l e v a n t i n p a r t i c u l a r t o Aronsson's axioms (see above).

Luke (197*0 considered a surface developing according to

where f , again, i s an 'e m p i r i c a l f u n c t i o n ' and g i s gradient

This model i s a g e n e r a l i s a t i o n o f one developed by Smith and Bretherton (1972) and applies mainly t o surfaces subject t o surface wash. U n f o r t u n a t e l y , Luke's treatment of i t f a i l s between two s t o o l s . The general discussion i s vague about geomorphological i m p l i c a t i o n s , while s p e c i a l s o l u t i o n s are given only f o r i m p l a u s i b l e s i t u a t i o n s which lack i n t e r e s t . F i n a l l y , not a word i s s a i d t o motivate the constant form s o l u t i o n s which are derived.

3.3.18 Huggett

Huggett (1973a, 21-8; see also 1973b, 1975) put forward a model f o r landsurface development i n the wider context of work on 'soil-landscape systems'. The key equation i s

f (3,1 - S ) •at

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where Dx, Dy are d i f f u s i o n c o e f f i c i e n t s , and v x , v are speeds (Huggett wrote ' v e l o c i t i e s ' ) o f bulk flow i n x,y d i r e c t i o n s . Huggett suggested p u t t i n g v x = — ~ , v v = — — (which would be i n c o r r e c t dimensionally) and

y Tfj

he modified the model t o allow D x and Dy t o be funct i o n s o f x, y and z.

This model i s a nonlinear d i f f u s i o n equation w i t h mass t r a n s p o r t terms. I t must be solved n u m e r i c a l l y , e s p e c i a l l y i n i t s more complex form. I t i s best classed as phenomenological, since although Huggett i n t e r p r e t e d various terms o f the complex model i n geomorphological terms not a l l the e f f e c t s are very p l a u s i b l e . Apart from some s i m u l a t i o n r e s u l t s ( i n c l u d i n g the case o f a heterogeneous bedrock) the model has been l i t t l e developed or applied.

3.3.19 Mizutani

Using p h y s i c a l p r i n c i p l e s , Mizutani (197*0 proposed two equations f o r the depth of erosion or d e p o s i t i o n measured normal t o the surface i n conditions where surface wash and mass wasting are the dominant processes.

where 1 i s slope length measured along the surface

A = a 9 - sifrv 6 )* (ii)

where Q i s a c r i t i c a l angle.

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These equations were manipulated t o y i e l d p a r t i a l d i f f e r e n t i a l equations f o r h i l l s l o p e development,(i) becomes

w i t h the s i m p l i f y i n g assumptions (a) 1 = x; (b) ( ) 2

= t a n 2 ^ may be neglected; (c) n = 1. Assumptions (a) and (b) both imply gentle slopes.

I n the case m = 2 the equation may be solved a n a l y t i c a l l y using some cunning s u b s t i t u t i o n s . I n f a c t , a d i f f e r e n t manipulation using only assumptions (a) and (b) would y i e l d

which resembles a model proposed by Kirkby (1971).

Using (a) (b) (c) equation ( i i ) reduces t o

Mizutani suggested a g e n e r a l i s a t i o n

which may be solved using the method o f c h a r a c t e r i s t i c s .

These models were modified t o produce upslope c o n v e x i t i e s , d i v i d e recession, p l a t e a u d i s s e c t i o n and r a d i a l v a l l e y development, and t o allow f o r l i t h o l o g i c a l v a r i a t i o n : i n s h o r t , a v a r i e t y of i n t e r e s t i n g a p p l i c a t i o n s ,

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A f u r t h e r p o i n t t o commend i s the r e p r e s e n t a t i o n a l character of the models, which are q u i t e c l o s e l y r e l a t e d t o p h y s i c a l p r i n c i p l e s .

E a r l i e r work by Mizutani i s quoted and summarised i n h i s 1974 paper. For l a t e r work i n Japanese, see Mizutani (1976).

3.3.20 Grenander

Grenander (1975; 1976, 402-11).considered the height

of a landsurface z as a f u n c t i o n o f l o c a t i o n on a c i r c l e x

of u n i t circumference, and o f time t . The landsurface i s

considered t o be the r e s u l t o f i n t e r a c t i o n between an

erosion mechanism given by a d i f f u s i o n equation

and a t e c t o n i c mechanism which operates instantaneously at d i s c r e t e time p o i n t s . The t e c t o n i c mechanism i s a Poisson p o i n t process i n time , and a set of independent s t a t i o n a r y p e r i o d i c processes i n space. The t o t a l mass of the landform

Z dbc = constant o

which i m p l i e s t h a t 'average u p l i f t ' I J -F (x) A* = 0

0

Grenander showed t h a t the s t o c h a s t i c p a r t i a l d i f f e r e n t i a l equation y i e l d e d by combining erosion and t e c t o n i c mechanisms defines a height f i e l d i n s t a t i s t i c a l e q u i l i b r i u m . He derived expressions f o r time and space autocovariance f u n c t i o n s ,

- I l l -

c a r r i e d out some s i m u l a t i o n experiments, and discussed the est i m a t i o n o f p a t t e r n parameters from s p a t i a l data and optimal r e t r o s p e c t i o n o f past topographical p r o f i l e s .

Although e l e g a n t l y and r i g o r o u s l y developed, Grenander's model i s not p a r t i c u l a r l y i n s t r u c t i v e . The assumptions are phenomenological and o f dubious realism. The treatment o f a c i r c u l a r p r o f i l e i s d i f f i c u l t t o r e l a t e t o other work. On a naive view, the f i n d i n g o f s t a t i s t i c a l e q u i l i b r i u m i s not s u r p r i s i n g given t h a t t o t a l mass i s conserved. F i n a l l y , a l l the important closed form r e s u l t s c o n t a i n sums o f doubly i n f i n i t e s e r i e s which would be d i f f i c u l t t o use i n p r a c t i c e .

Freiberger and Grenander (1977) considered the case of a surface developing according t o a s t o c h a s t i c p a r t i a l d i f f e r e n t i a l equation under the c o n s t r a i n t o f constant t o t a l mass. The basic process i s a two-dimensional d i f f u s i o n mechanism w i t h two f o r c i n g terms, one a d d i t i v e random noise, the other a t a n g e n t i a l force f i e l d r e p r e s e n t i n g drag forces supposed t o act on the surface from the i n t e r i o r o f the e a r t h . According t o the l a t t e r the occurrence, amplitude and d i r e c t i o n o f p o i n t disturbances are a l l random v a r i a b l e s , and these disturbances are propagated over the surface according t o a s p e c i f i e d i n f l u e n c e f u n c t i o n . A n a l y t i c a l r e s u l t s were given f o r a square i n the ( x , y) plane, which was mapped on t o a torus t o avoid boundary e f f e c t s . These r e s u l t s cover s t o c h a s t i c c h a r a c t e r i s t i c s of various f i e l d s , optimal data

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compression and noise suppression, and the s t a t i s t i c a l geometry o f random surfaces. The case of a height f i e l d on a sphere was also examined.

The planar case was simulated w i t h a d i s c r e t e space, d i s c r e t e time FORTRAN program. The i n i t i a l surface was f l a t .

The authors d i d not supply any d e t a i l e d p h y s i c a l j u s t i f i c a t i o n o f t h e i r assumptions, but i n v i t e d comments from g e o l o g i s t s and geographers on the v a l i d i t y o f t h e i r model. I t does seem t o bear very l i t t l e r e l a t i o n to c u r r e n t ideas on geomorphological processes, whether exogenetic or endogenetic.

3.3.21 New r e s u l t s

I n t h i s s e c t i o n some new r e s u l t s are presented which extend Kirkby's models.

Constant form s o l u t i o n s of the form

can be derived f o r the equations

1 1 = 0 2>x 3t

Combining equations

^ l > x ( - * ) J - u

I n t e g r a t i n g , and using a boundary c o n d i t i o n S = 0, x = 0,

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Rearranging j ^_

2»c V o . /

I n t e g r a t i n g again, and using the d e f i n i t i o n z = z^, x = 0

I—y*\4-v\ where k = — 7\

Using the d e f i n i t i o n z = 0, x = xfe

0 - *A " ^ " ( i ) ^

D i v i d i n g through

| - l - ( M ZA V

This constant form s o l u t i o n , which i s exact, i s equal to the approximation derived by Kirkby (1971) t o the space-dependent f u n c t i o n which appears i n the variables-separable s o l u t i o n

z ( x , t ) = X (x) T ( t ) 3.3.22 Connective summary

A h i s t o r i c a l review o f ideas and r e s u l t s i n dynamic h i l l s l o p e modelling over the l a s t s i x t y years must c l e a r l y be supplemented by a connective summary. This i s attempted here i n a c r o s s - c l a s s i f i c a t i o n o f dynamic models (3C) and

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i n an index o f p a r t i c u l a r t o p i c s (3D). Note t h a t i t i s not always easy t o decide which category should receive each model i n 3C. I n p a r t i c u l a r , the categories 'phenomenological 1

and ' r e p r e s e n t a t i o n a l ' i n t e r g r a d e continuously. The d i c t i o n a r y i n 3D i s not exhaustive o f the t o p i c s covered by p a r t i c u l a r workers, but i t does serve as a systematic guide t o ideas and r e s u l t s o f value.

Some general remarks on dominant themes are i n order by way of conclusion.

One r e c u r r e n t assumption i s t h a t h i l l s l o p e s may be t r e a t e d as s e l f - m o d i f y i n g geometric systems. While such an assumption i s a mathematical version of the basic idea t h a t process and form are r e l a t e d by feedback ( c f . Ch. 2.4.3 above), i t has the important consequence t h a t distance, height and e s p e c i a l l y gradient are regarded i n most models as the major c o n t r o l s of processes. By c o n t r a s t , independent v a r i a t i o n s i n c l i m a t e , hydrology and s o i l p r o p e r t i e s have received l i t t l e a t t e n t i o n . L i t h o l o g i c a l p r o p e r t i e s have been incorpo r a t e d i n several models, but generally i n an extremely crude manner ( c f . Ch. 2.4.7 above).

F a i l u r e and s o l u t i o n are the most f r e q u e n t l y neglected processes. I n the case o f f a i l u r e , neglect can be a t t r i b u t e d t o i n t r a c t a b i l i t y : threshold-dependent processes cannot be handled e a s i l y i n models based on d i f f e r e n t i a l equations ( c f . Ch. 2.4.4 above). I n the case of s o l u t i o n , ignorance i s probably the major cause of neglect.

- 117 -

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- 118 -

Continuous h i l l s l o p e development models have usually been devised f o r t r a n s p o r t - l i m i t e d s i t u a t i o n s : weathering-l i m i t e d s i t u a t i o n s have received l i t t l e a t t e n t i o n .

C o n t i n u i t y equations are a f i r m and indispensable basis f o r h i l l s l o p e m o d e l l i n g , even though the value and importance of c o n t i n u i t y p r i n c i p l e s has only become f u l l y apparent q u i t e r e c e n t l y .

Two key ideas, constant form and s t a b i l i t y p r o p e r t i e s , although introduced i n the e a r l i e s t work i n t h i s f i e l d , have been rediscovered since 1972 ( c f . Ch. 2.3.1 above). The stra t e g y of seeking robust q u a l i t a t i v e r e s u l t s i s a s t r i k i n g i n n o v a t i o n i n a f i e l d which has u n t i l r e c e n t l y been characterised by a marked lack o f mathematical s o p h i s t i c a t i o n .

While many modellers o f h i l l s l o p e development have also addressed the more challenging and more general problem of surface development, r e l a t i v e l y l i t t l e progress has been made on t h i s f r o n t , l a r g e l y because plan curvature e f f e c t s , s l o p e endpoint behaviour and network development remain poorly understood ( c f . Ch. 2.4.6, 2.4.8 above).

There i s l i t t l e r e l a t i o n s h i p between s t a t i c and dynamic models o f h i l l s l o p e p r o f i l e s : the major exception i s t h a t power f u n c t i o n s a r i s e as e q u i l i b r i u m ( i n v a r i a n t ) s o l u t i o n s t o some dynamic models.

Most dynamic models have been d e t e r m i n i s t i c . Ideas of s p a t i a l s t o c h a s t i c processes introduced q u i t e r e c e n t l y have f a i l e d to y i e l d important new i n s i g h t s , although the

- 119 -

use of s t o c h a s t i c e r r o r s i n model f i t t i n g remains an i n t e r e s t i n g p o s s i b i l i t y ( c f . Ch. 2 .3.2 above, Ch. 9 below).

Both a n a l y t i c a l and s i m u l a t i o n approaches are o f value. Simulation i s necessary t o handle thresholds and frequency d i s t r i b u t i o n s , although s i m u l a t i o n should be accompanied by s e n s i t i v i t y analyses ( c f . Ch. 2 .3.4 above).

Despite i n c r e a s i n g knowledge about process r a t e s , c o n t r o l s and mechanisms, phenomenological models remain popular. I f models are t o be o f explanatory value, t h e i r process content must be exposed t o searching c r i t i c a l examination ( c f . Ch. 2 .3 .3 above).

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3.4 Notation

a, b, c constants or f u n c t i o n s w i t h l o c a l meaning BJJ f r a c t i o n a l Brownian f u n c t i o n d, d e depth o f water, depth of erosion d i n ordinary d e r i v a t i v e or i n i n t e g r a l D d i f f u s i o n c o e f f i c i e n t e base of n a t u r a l l o g a r i t h m s , 2 .71828. . .

f , f f u n c t i o n , i t s d e r i v a t i v e g gradient H parameter of B^ i s u b s c r i p t k Kirkby parameter 1 length m, n exponents N ( ,) Normal (Gaussian) d i s t r i b u t i o n p i n t e g e r power i n polynomial P, P', P" poi n t s i n space q discharge r resistance s arc l e n g t h S sediment f l u x t elapsed time T f u n c t i o n o f t only u, u Q depths normal t o surface v, v speed, average speed w, w* r e g o l i t h depth, t h r e s h o l d W r a t e of weathering x, x^ h o r i z o n t a l coordinate, o f slope base

- 121 -

X f u n c t i o n of x only y h o r i z o n t a l coordinate z, z b, z d v e r t i c a l coordinate, of slope base, o f d i v i d e z l i m > z

r e l height of l i m i t i n g p r o f i l e , height above i t -r% r a t e of downwearing £ d i f f e r e n c e operator £ random v a r i a b l e Yj v i s c o s i t y fi 0 fl angle, average angle, c r i t i c a l angle p sediment concentration 0- s p e c i f i c weight ^ summation operator r t c t a n g e n t i a l s t r e s s , c r i t i c a l value °C i s p r o p o r t i o n a l t o ^ i n p a r t i a l d e r i v a t i v e V d i r e c t i o n a l d i f f e r e n t i a l operator co i n f i n i t y

, ~ i s drawn from, approximately equals | i n i n t e g r a l | \ modulus

tends t o

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Chapter 4

THE FIELD AREA

The nobly s i l e n t h i l l s loom up on high In peace that s t i l l s my question whence or why.

Goethe, Faust, Pt. I I , Act IV.

4.1 Introduction

4.2 Geological background

4.3 Geomorphological interpretations

4.4 Quaternary events

4.5 Summary

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4.1. Introduction

The f i e l d area i n which slope p r o f i l e s were measured for t h i s thesis i s the 10km x 10km g r i d square SE 59, which l i e s i n the western part of the North York Moors. I t i s approximately bisected by the valley of Bilsdale. The area was chosen mainly f o r i t s r e l a t i v e l y simple geological and geomorphological hi s t o r y .

The general character of the Jurassic upland was w e l l described by Fox-Strangways (1.892, 408). 'The Jurassic rocks of Yorkshire form an isolated range of h i l l s cut o f f from the rest of the county, and from the elevated ground composed of other geological formations, by a series of large valleys, which form the great lines of drainage of t h i s part of England. From t h e i r peculiar geological construction these h i l l s present a bold front to the north and weat, overlooking the great plains of the Tees and the Ouse, while to the south and east they gradually f a l l away to low ground beneath the escarpment of the Wolds, or are cut o f f by the sea'.

In t h i s chapter, a review of knowledge of geological and geomorphological history i s presented f o r the f i e l d area, drawing where appropriate on studies made i n other parts of the Moors and on f i e l d observations. The c r i t e r i o n of relevance i s that geological and geomorphological findings should have direct or i n d i r e c t implications f o r h i l l s l o p e p r o f i l e morphometry.

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4.2 Geological background

4 . 2 . 1 Jurassic s t r a t a

A l l the s o l i d rocks of the f i e l d area date, from the Jurassic Period (190-140 m i l l i o n y r ) . They were mapped for the Geological Survey by Pox-Strangways et a l (1885,

1886, 1892) : the r e s u l t i n g Memoirs remain the most valuable accounts f o r geomorphologists, despite a large volume of subsequent work ( c f . Hemingway, 1974 f o r a comprehensive review).

Salient characteristics of the various Jurassic formations are given i n 4A, which i s based on Geological Survey sources, except that the Estuarine Beds have been termed Deltaic (Hemingway, 19*19) . The Middle Jurassic nomenclature proposed by Hemingway and Knox (1973) i s not employed because i t cannot be correlated exactly with Geological Survey mapping units. I t i s , however, clearly preferable to the older terminology.

One important characteristic of the geological succession i s the marked v e r t i c a l and l a t e r a l v a r i a b i l i t y found w i t h i n formations. The l i t h o l o g i c a l descriptions given i n 4A are generalised: the examples of v e r t i c a l v a r i a b i l i t y given i n 4B provide a counterweight to such generalisation.

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4B Examples of s e c t i o n s i n B i l s d a l e showing l i t h o l o g i c a l

v a r i a b i l i t y

1. Novey House, L a d h i l l Beck, i n Grey Limestone S e r i e s

i n . cm. (rounded) Shales with f o s s i l s G r i t Hard, sandy s i l i c e o u s beds 36 90 Sandy s h a l e s 60 150 S i l i c e o u s and calcareous beds 48 120 Shales

2. Blow G i l l Farm, i n E l l e r Beck Bed

Sandstone Shale 60 150 Thin ironstone 4 10 Shale 36 90 Ironstone 6 15 Shale

3. Tarn Hole Beck, i n Ironstone S e r i e s

Ferruginous shale 60 150 Ironstone - 5 13 Shale 24 60 Ironstone 15 38 Shale 42 105 Sandy band 4 10 Shale 48 120 Ironstone 4 10 Micaceous sandstone 24 60 Shale 144 365 Ironstone 21 53 Shale 60 150

Sources: G e o l o g i c a l Survey Memoirs

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4.2.2. Structural history

The s t r u c t u r a l history of the North York Moors has been r e l a t i v e l y simple ( c f . Kent, 1974).

The dominant post-Cretaceous movement has been easterly t i l t i n g ( c f . Smalley, 1967). As the top of the Chalk subsided beneath the Tertiary i n the North Sea basin, the Pennine area correspondingly rose by 500 to ^00 m. This movement has been regarded as contemporary with Alpine orogenesis, and thus essentially Miocene, but the evidence from the North Sea supports a picture of continuing subsidence and i t thus seems l i k e l y that the r i s e of the Pennine area was a long-continued epeirogenic movement

i n i t i a t e d i n the early Tertiary.

In addition, the area of the Moors has suffered a n t i c l i n a l warping with an amplitude on the Dogger of at least 500 m. r e l a t i v e to the Derwent Valley syncline to the south. The crest of the f o l d i s divided i n t o separate culminations by crossing meridional trends. A series of domes may be distinguished from west to east along the Cleveland a n t i c l i n e : the Chop Gate dome, the Danby Head dome, the Egton dome and the Robin Hood's Bay dome. The f i r s t of these takes i t s name from Chop Gate (559996) i n SE 59.

Taking account of uncertainty about magnitude and s p a t i a l v a r i a t i o n , i t can be seen that post-Cretaceous u p l i f t i n the f i e l d area has been of the order of 100 to 1000 m, an average of some 1.5 to 15 mm per 1000 yr.

Known fa u l t s are few i n number and r e l a t i v e l y minor.

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4. 3 Gee-morphological interpretations 4.3.1 The influence of l i t h o l o g y

Archibald Geikie was i n l i t t l e doubt about the influence of structure and l i t h o l o g y when he wrote his prefatory Notice to the Eskdale and Rosedale Memoir (Pox-Strangways e t _ a l , 1885).

'. . . This region may be regarded as a vast model exemplifying, i n a s t r i k i n g manner, the relations of topographical feature to the nature and disposition of the rocks underneath. The s t r a t a being nearly horizontal and l i t t l e disturbed by dislocations, the valleys r a d i a t i n g from the tableland can be traced out as the results of erosion, with a precision and completeness unattainable i n other d i s t r i c t s of the country where the geological structure i s less simple'.

Pox-Strangways (1892, Ch. 17; 1894) argued the case i n greater d e t a i l . He regarded the forms of h i l l s and valleys i n the Jurassic upland as 'entirely due to sub-a e r i a l agents' (1892, 411) and related large-scale forms, such as the drainage pattern, to structure, and small-scale forms to l i t h o l o g i c a l v a r i a t i o n .

1. . . The main features of the d i s t r i c t . . . occur where there i s the greatest geological difference between succeeding s t r a t a , f o r instance where a t h i c k bed of porous sandstone succeeds to a considerable thickness of shale; by the weathering away of which the rock stands out i n a bold feature overlooking the beds below. The thickest beds

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of shale are the Lias and Oxford Clay, and therefore i t i s j u s t above t h e i r outcrop that we get the main features of the d i s t r i c t ' ( 1 8 9 2 , 417).

Broadly similar views have been put forward by Elgee (1912, Ch. 12), Palmer (1973, Ch. U) and de Boer (1974, 281). Palmer (1956) examined the relationship between d i f f e r e n t i a l weathering and t o r formation at the Bridestones (SE 8791) i n the eastern Tabular H i l l s . The only quantitative study of the relationship between lit h o l o g y and r e l i e f , however, i s the work of Gregory and Brown (1966) i n Eskdale. They discussed areal frequency d i s t r i b u t i o n s of slope angle f o r d i f f e r e n t geological formations, derived from a comparison of a morphological map with Geological SurYey sheets. Weighted mean angles and resistance values from t h e i r paper are reproduced i n 4C.

Although these results are extremely i n t e r e s t i n g , the methods and interpretations of Gregory and Brown require c r i t i c i s m on several grounds.

( i ) Morphological mapping i s an unsatisfactory method of data c o l l e c t i o n . I t i s predicated on the assumption of geomorphological atomism (see Ch. 8.2 below), i t lacks r e p l i c a b i l i t y and i t ignores i n t e r n a l v a r i a b i l i t y .

( i i ) As Doornkamp and King (1971, 126) pointed out more generally, comparison of a morphological map and a geological map 'is only a h e l p f u l exercise i f i t i s known that the geology was not mapped from surface form i n the f i r s t instance. "Feature mapping" of geology i s bound to

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4c

Weighted mean angle and resistance for various geological formations i n parts of Eskdale according to Gregory and

Brown (1966)

Formation Weighted mean angle (deg) Resistance Kellaways 2.26 High Cornbrash 4.92 Very high Upper Deltaic 3.36 Moderate Moor Grit 3.00 Very high Grey Limestone 3.92 Very high Middle Deltaic 5.03 High E l l e r Beck 12.95 Very high Lower Deltaic 7.11 High Dogger 22.24 Very high Alum Shale Low Jet Rock r 9-32 Moderate Grey Shale J Low Ironstone Series 4.59 High Sandy Series 6.49 High Lower Lias 8.72 Moderate

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lead to the conclusion that there i s a clear relationship between geology and slope form. Where there has been s u f f i c i e n t confidence i n t h i s relationship f o r feature mapping to have taken place, the correlation between form and geology may indeed be close. Feature mapping of the geology tends, however, to overstate a relationship which may not always be so precise i n r e a l i t y * .

( i i i ) The resistance values given by Gregory and Brown, without any explanation of derivation, are open to objection. Presumably they are based on impressions obtained i n the f i e l d of physical characteristics of the various rocks. Some shales, f o r example, may l i t e r a l l y f a l l to pieces i n the hand, while even w e l l weathered g r i t s can be broken only with d i f f i c u l t y . While such impressions are unlikely to be misleading, they may be i r r e l e v a n t . Where a s o i l cover i s present, 'resistance' properties are pedological, not l i t h o l o g i c a l , and the very idea of a single measure of resistance i s dubious since several d i f f e r e n t processes are operating ( c f . Ch. 2.4.6 above).

( i v ) Neither the strength of the relationship between lithology and slope form, nor the possible v a r i a t i o n i n the relationship with scale, i s explored i n any d e t a i l .

(v) The existence of p a r t i c u l a r modal or characteristic angles i s not demonstrated convincingly ( c f . Speight, 1971, fo r a f u l l e r discussion of th i s issue).

( v i ) Gregory and Brown tend to play down l i t h o l o g i c a l v a r i a t i o n w i t h i n formations ( f o r example, t h e i r Fig. 7

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misleadingly describes each mapping unit as either shale, or sandstone, or limestone, or g r i t : q u a l i f y i n g symbols denoting variable succession or l a t e r a l v a r i a t i o n do not s u f f i c i e n t l y offset the impression of un i f o r m i t y ) .

( v i i ) Palmer (1973, 45) c r i t i c i s e d Gregory and Brown for neglecting the position of the rock i n r e l a t i o n to the streams, a major control i n 'an area that i s s t i l l i n a r e l a t i v e l y youthful stage of landform evolution'. For example, moderately steep slopes occurring i n the r e l a t i v e l y weak Lower Lias can be explained by t h e i r proximity to recent r i v e r incisions i n some valley f l o o r s . While Palmer's c r i t i c i s m may be couched i n questionable term­inology, t h i s kind of relationship must be remembered. However, the position of s t r a t a i n i t s e l f explains nothing: what are important are the processes associated with that position.

Some f i e l d observations i n SE 59 throw l i g h t on the question of l i t h o l o g i c a l influence and can best be considered i n terms of scale. Much of the l i t h o l o g i c a l v a r i a t i o n i s on a microscale, and i t finds morphological expression at that scale. This i s most clearly seen i n stream long p r o f i l e s . Hard sandstone or gritstone bands frequently outcrop as small wate r f a l l s or rapids, commonly 1 m - 4 m i n height (e.g. Blow G i l l , 52 8932; Tarn Hole, around 593981; Arns G i l l , 533965; Proddale confluence, 518970; L a d h i l l Beck, 548925). Valleyside crags and tors

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1ft "rJ

MD V a l l e y s i d e t o r above Tarn Hole

Escarpment of the Hambleton and Tabular H i l l s

135

!

4F Streamside b l u f f i n T r i p s d a l e

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(e.g. above Tarn Hole, 4D) are frequently larger features, but the essential point remains v a l i d : 'features' a t t r i b u t a b l e to l i t h o l o g i c a l v a r i a t i o n are often microforms when compared with complete p r o f i l e s .

On a larger scale, the r e l a t i o n between r e l i e f and mapping units i s clear i n the broad contrast between escarpments emphasised especially by Fox-Strangways (1892, 4 l 4 ) . In p a r t i c u l a r , the Middle Oolite corresponds to the escarpment of the Hambleton and Tabular H i l l s ( i l l u s t r a t e d i n 4E). What remains least clear, i r o n i c a l l y enough, i s the importance of l i t h o l o g i c a l v a r i a t i o n at the scale of the h i l l s l o p e p r o f i l e : a major task of a morphometric approach to h i l l s l o p e p r o f i l e s i s to throw some l i g h t on t h i s issue.

F i n a l l y , two b l u f f s cut i n the Upper Liassic shales i n Tripsdale (584995) provide small i l l u s t r a t i o n s of Palmer's c r i t i c i s m s (one i s shown i n 4F). Although apparently very unresistant, these b l u f f s are steep solely because of t h e i r p o s i t i o n , being undercut by the stream.

4.3.2. Supposed planation surfaces (see 4G and 4H)

Davis (1895), w r i t i n g on 'The development of certain English r i v e r s ' , regarded the topography as the result of subaerial rather than marine denudation, and suggested that i t was i n the mature stage of a second cycle of denudation. However, the North York Moors, as only a

- 137 r

B i l s d a l e from the e a s t , showing the s u b h o r i z o n t a l i n t e r f l u v e

• i

*»

KM 4H I n c i s e d v a i l ey i n T n p s d a l e , of the kind a t t r i b u t e d t

r e j u v e n a t i o n

- 138 -

small p a r t o f the area discussed by Davis, received but passing reference. A more d e t a i l e d i n t e r p r e t a t i o n of 'The ge o l o g i c a l h i s t o r y of the r i v e r s o f east Yorkshire' was given by Reed (19Q1), who p o s t u l a t e d s i x cycles o f denudation, of which not a l l were cycles i n the Davisian sense. Superimposition of the drainage from a Cretaceous cover was f o l l o w e d by e a r l y T e r t i a r y p l a n a t i o n . Versey (1939) regarded much o f the upland surface of the Moors as p a r t of a 'Wolds Peneplane [ s i c ] ', the very highest p a r t s standing above the Peneplane as monadnocks. He also placed p l a n a t i o n i n the e a r l y T e r t i a r y , f o l l o w e d by u p l i f t , warping and f a u l t i n g . Hemingway (1958, 24; 1966, 16), on the other hand, considered p l a n a t i o n t o have been marine, although he d i d not argue t h i s case i n any d e t a i l .

Peel and Palmer (1955) introduced a d i s s e n t i n g note. Since the 'peneplain' truncates the Cleveland Dyke, dated by Dubey and Holmes (1929) at 26 m i l l i o n years (but see below), they i n f e r r e d a l a t e T e r t i a r y age, thus producing an i n t e r p r e t a t i o n s i m i l a r t o t h a t suggested f o r the 'peneplain' of southern England. They also suggested t h a t u p l i f t of the peneplain was discontinuous.

Palmer (1967) l a t e r showed w i l l i n g n e s s t o e n t e r t a i n a hypothesis of marine p l a n a t i o n . 'There i s no s t r o n g argument against most of the surfaces having an i n i t i a l l y marine o r i g i n . . . the absence of contemporary marine o r , f o r t h a t matter, r e s i d u a l land deposits i s no embarrassment, except t o those who h o l d the conservative view t h a t the

- 139 -

upland surfaces coincide e x a c t l y w i t h reconstructed T e r t i a r y surfaces' (1967, 17).

Palmer (1973) has r e c e n t l y extended t h i s view, producing an imperfect echo of Davis nearly eighty years e a r l i e r : 1. . . The main features of the r e l i e f may be s a i d t o r e s u l t from the i n c i s i o n of streams duri n g the e a r l y p a r t of the present uncompleted cycle o f erosion i n t o a peneplain produced towards the end o f a previous c y c l e 1

(1973, 22). The peneplain on the Moors i s ' c e r t a i n l y of l a t e - T e r t i a r y age' (32) since i t truncates m i d - T e r t i a r y f o l d s and fault-produced i r r e g u l a r i t i e s .

However, '. . . what has been taken f o r a peneplain i s r e a l l y a stepped surface, each step represented by a series of h i l l t o p s and b e v e l l e d spurs t h a t l i e w i t h i n a r e s t r i c t e d height range . . . Consequently a more appropriate model than a simple t i l t e d peneplain i s one where the peneplain was covered by the sea and then rose i n t e r m i t t e n t l y around the Cleveland axis t o allow the sea t o t r i m benches i n i t ' (33).

A ' t e n t a t i v e model* f o r denudation chronology i n n o r t h ­east Yorkshire was given by Palmer (1973, Ch. 5) and some key f i g u r e s are reproduced i n 4 l , together w i t h metric e q u i v a l e n t s , r e v i s e d Quaternary stage names ( M i t c h e l l e t a l , 1973) and Pliocene dates (Berggren ?1973)•

I n the Pliocene (according t o Palmer) there was discontinuous u p l i f t of the peneplain from the sea, while

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4i Heights of marine benches and v a l l e y widening stages i n north-east Yorkshire according t o Palmer (1973) Figures i n f e e t (m)

Age Height

RNARY

Flandrian Devensian Ipswichian Wolstonian Hoxnian Anglian Cromerian Beestonian Pastonian Baventian Antian Thurnian Ludhamian Waltonian

?80 (24)

225-175 (69-53)

280-250 (85-76)

350-300 (107-91)

550-H75 (167-145)

1.8 m i l l i o n years 750-625 (229-191)

1100-850 (335-259) 1500-1200 (457-366)

5 m i l l i o n years

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during the Quaternary g l a c i a l l y c o n t r o l l e d s e a - l e v e l f l u c t u a t i o n s were superimposed upon the p a t t e r n . The assignment o f benches higher than 550 f t . t o the Pliocene, and those lower t o the Quaternary, i s based on an analogy w i t h southern England. The absence of diagnostic deposits once again i s 'no embarrassment':

'There are no known beach or s e a - f l o o r deposits on the benches,which i s hardly s u r p r i s i n g since there are hardly any g l a c i a l deposits e i t h e r t h a t are older than

The wasting of the i n t e r f l u v e s by weathering and s o i l movement took care of such d e p o s i t s , as i t has taken care of any Pliocene s o i l s ' (Palmer, 1973, 51).

The i n t e r p r e t a t i o n given by Palmer t o h i l l s l o p e development i s r e l a t e d to t h i s ' t e n t a t i v e model' of denudation chronology. The h i l l s l o p e s around the Bridestones were described as rejuvenated (Palmer, 1956), and the h i l l s l o p e s of the Moors as a whole were described i n e s s e n t i a l l y Davisian terms, emphasising stage of development and a t t r i b u t i n g v a l l e y - i n - v a l l e y forms t o r e j u v e n a t i o n (Palmer, 1973, Chs. 4 and 5 ) .

de Boer (1974), reviewing studies of the physiographic e v o l u t i o n of Yorkshire, s t r u c k a more s c e p t i c a l note, w r i t i n g of a growing a p p r e c i a t i o n of 'the u n c e r t a i n t i e s of e x p l a n a t i o n ' (271). He p o i n t e d out t h a t the most recent d a t i n g of the Cleveland Dyke ( a t l e a s t 58 m i l l i o n years, Evans e t a l , 1973) i m p l i e s t h a t e a r l y T e r t i a r y

the Weichselian I sc. Devensian , or l a s t g l a c i a l age.

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p l a n a t i o n i s 'once more a tenable hypothesis' ( 2 7 2 ) . de Boer favoured the idea of a mainly s u b a e r i a l , p o l y c y c l i c denudation h i s t o r y , p o s t u l a t e d f o r the Moors by Davis, Reed and Versey, i n c l u d i n g superimposition of drainage from a Cretaceous cover.

Such ideas on p l a n a t i o n surfaces are open t o o b j e c t i o n on a v a r i e t y o f methodological grounds. An extended discussion of these grounds isunnecessarybecause there are e x c e l l e n t accounts elsewhere: i t i s s u f f i c i e n t t o rehearse the major p o i n t s b r i e f l y , and t o c i t e appropriate references.

The various i n t e r p r e t a t i o n s put forward over a p e r i o d of e i g h t y years d i f f e r i n several respects, p a r t i c u l a r l y on the number, date and o r i g i n o f p l a n a t i o n surfaces, but they a l l share the major assumption t h a t accordance of f l a t s i m p l i e s former b a s e - l e v e l l i n g ( c f . L i n t o n , 1964, 1 1 8 ) .

However, the existence of a p l a n a t i o n surface cannot be i n f e r r e d simply from accordance; i t must be demonstrated independently, and the a l t e r n a t i v e hypothesis must be considered t h a t the present landforms can be explained without r e s o r t t o several cycles ( c f . Leopold e t a l , 1964 ,

4 9 9 - 5 0 0 ) . Such hypotheses were put forward by contemporaries of Davis, such as Tarr ( 1 8 9 8 ) , Shaler (1899) and Smith ( 1 8 9 9 ) ; by workers on G i p f e l f l u r e n , or summit surfaces i n mountain ranges ( c f . Daly, 1905; B a u l i g , 1952, 1 7 3 - 7 ;

H e w i t t , 1972 , 2 7 - 3 0 ) ; and by proponents of the 'dynamic e q u i l i b r i u m ' approach t o landscape e v o l u t i o n (Hack, 196O;

Chorley, 1962 , 1965a, 1 9 6 5 b ) .

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Hypotheses of p l a n a t i o n surfaces are very d i f f i c u l t t o t e s t s a t i s f a c t o r i l y , f o r two basic reasons. F i r s t l y , land­scapes i n c l u d i n g p l a n a t i o n surfaces can assume a very large v a r i e t y of forms, given t h a t a peneplain may be u n d u l a t i n g , u p l i f t e d , t i l t e d , warped, f a u l t e d , dissected, and even destroyed almost completely, y e t s t i l l be detectable by convinced en t h u s i a s t s . (The grossest anomalies can also be explained as former monadnocks.) Secondly, there i s a lack of independent evidence on r e s i d u a l deposits and t e c t o n i c h i s t o r y .

A f i n a l p o i n t i s t h a t any analogies w i t h the supposedly b e t t e r understood geomorphological e v o l u t i o n of southern England, and e s p e c i a l l y w i t h the celebrated scheme of Wooldridge and L i n t o n ( 1 9 5 5 ) , should take account of the f a c t t h a t d e t a i l e d g e o l o g i c a l and s o i l mapping i n t h a t area has overturned accepted hypotheses (Worssam, 1973; Hodgson e t _ a l , 197^ ; Shepard-Thorn, 1975; Catt and Hodgson, 1 9 7 6 ) .

These objections go f a r beyond the s c e p t i c a l remarks of de Boer ( 1 9 7 4 ) . Any a l t e r n a t i v e scheme of denudation h i s t o r y would have t o meet these o b j e c t i o n s , which would be u n l i k e l y i n the present s t a t e of knowledge. A more r e a l i s t i c task f o r a morphometric approach i s t o provide q u a n t i t a t i v e evidence on the h i l l s l o p e forms which e x i s t .

4 .4 Quaternary events 4 . 4 . 1 . Glaciations

The Quaternary, the l a s t p e r i o d of g e o l o g i c a l time,

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included s i x c o l d phases i n B r i t a i n . G l a c i a t i o n s are known to have occurred i n the l a s t t h r e e , Anglian, Wolstonian and Devensian (Sparks and West, 1972, 4-5). I n Yorkshire, very l i t t l e i s known about g l a c i a t i o n s before the Devensian (Penny, 197^)• The Devensian g l a c i a l l i m i t , however, i s c l e a r l y defined i n the f i e l d area. During the l a s t g l a c i a t i o n much of the North York Moors stood above the i c e . I n g r i d

2 square SE 59 only about 0.1 km i s covered by g l a c i a l d e posits, i n Scugdale i n the extreme northwest. The escarpment of the Moors here acted as a rampart against the i c e . The date of the Late Devensian g l a c i a l maximum i n England was about 18000 B.P. (Penny, 1974, 254; Jones, 1977).

The 'Older D r i f t ' ( i . e . pre-Devensian) on the Moors i s n e i t h e r abundant nor r e v e a l i n g . None has been reported from SE 59. I t may represent Anglian or Wolstonian g l a c i a t i o n s or both ( B i s a t , 1940; Penny, 1974; c f . Elgee, 1912 and Versey, 1939 f o r the o l d e r idea of T e r t i a r y marine d e p o s i t i o n ) . SE 59 may have been g l a c i a t e d during one or both o f these stages, although no features have apparently been a t t r i b u t e d to g l a c i a l e r o s i o n , and nothing i s known f o r c e r t a i n about t h i s p o s s i b i l i t y . As Palmer (1973» 51) remarked, the Moors are u n l i k e l y t o have supported t h e i r own g l a c i e r s , but would r a t h e r have been overridden by an icesheet from outside. 4.4.2 Supposed overflow channels

I n h i s paper hypothesising 'A system o f g l a c i e r - l a k e s i n the Cleveland H i l l s ' Kendall (1902) b r i e f l y discussed some

- 145 -

features t o the n o r t h and west of the f i e l d area, which he i d e n t i f i e d as former overflow channels t h a t drained p r o g l a c i a l lakes impounded between the icesheet and the northern escarpment. Two 'channels' i n p a r t i c u l a r might have allowed water t o escape southwards i n t o Raisdale and B i l s d a l e , and hence i n t o SE 59.

( i ) Between Carlton Bank and Cringle Moor at 528032, regarded by Kendall (1902, 514) as of minor importance, and not mentioned by Elgee (1908), Best (1956) or A r n e t t (1971a) i n subsequent geomorphological accounts. I n the f i e l d t h i s f e a t u r e i s not very convincing as an overflow channel: a broad f l a t t i s h c o l , i t could have allowed overflow i f a lake e x i s t e d , but there i s no sign of strong l i n e a r erosion (see 4 J ) .

( i i ) I n the Ingleby Greenhow Corner: 'The r e - e n t r a n t angle of the Cleveland escarpment at the eastern end i s breached by a splendid overflow-channel' (Kendall, 1902, 514). This f e a t u r e i s d i f f i c u l t t o place from i t s d e s c r i p t i o n , and reference t o 'the eastern end' of the r e - e n t r a n t angle i s p u z z l i n g . A l l l a t e r workers have construed the d e s c r i p t i o n as a reference t o the c o l at Hasty Bank (573032), or have independently regarded t h i s f eature as an overflow channel. But i f Kendall meant Hasty Bank, why d i d he not say so d i r e c t l y ? I f he meant somewhere e l s e , where could i t be?

Best (1956) included Hasty Bank i n a group of h i g h -l e v e l channels which he a t t r i b u t e d t o an e a r l i e r g l a c i a t i o n ,

- 146 -

4J Supposed overflow channel, C a r l t o n Bank

• K Mi

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i 1

V»V ' .n n

v a w n ,11

4K Supposed overflow channel, Hasty Bank

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on the grounds of t h e i r weathered appearance and t h e i r lack of a s s o c i a t i o n w i t h l o w - l e v e l channels i n the Vales of York and Stokesley. A r n e t t (1971a, 14-15) suggested t h a t meltwater pouring through the Hasty Bank channel l e d t o considerable enlargement of B i l s d a l e , although no other geomorphologist appears t o have suggested meltwater erosion beyond the channels. The Hasty Bank c o l i s more convincing i n the f i e l d as a candidate overflow channel: i t f a l l s away r a p i d l y i n a steep-sided v a l l e y towards B i l s d a l e (see 4K).

The presence of cols at these s i t e s can c r e d i b l y be a t t r i b u t e d t o the beheading o f Raisdale and B i l s d a l e by scarp r e t r e a t (Fox-Strangways e t a l , 1885, 56; Elgee, 1912, 236; Palmer, 1973, 41), but there appears t o be no stro n g independent evidence f o r the hypothesis of overflow channels. Moreover, p r o g l a c i a l drainage of the k i n d p o s t u l a t e d i s now be l i e v e d t o be rare ( c f . Bonney, 1915, who made t h i s o b j e c t i o n Kendall, 1916, who ignored i t ; Sissons, 1960, i 9 6 l and Gregory, 1962a, 1962b, 1965, who r e i n t e r p r e t e d Kendall's work i n Eskdale).

4.4.3 Cryonival conditions

During the Quaternary the North York Moors repeatedly experienced cold c l i m a t e s , i n c l u d i n g c r y o n i v a l conditions i n which f r o s t and snow a c t i v i t y was s i g n i f i c a n t . (The term ' c r y o n i v a l ' i s used as a more precise a l t e r n a t i v e t o the debased term ' p e r i g l a c i a l * : c f . L i n t o n , 1969).

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S e v e r a l geomorphologists have described c r y o n i v a l

f e a t u r e s and deposits from the North York Moors. Palmer

(1973, 5 8) voiced a general i n t e r p r e t a t i o n : 'Although information

i s not as good as i t should be, one gains the impression t h a t

the h i l l s l o p e s and t h e i r a s s o c i a t e d deposits owe much of t h e i r

present appearance to the e f f e c t s wrought by a rigorous

p e r i g l a c i a l c l i m a t e ' .

Dimbleby (1952) i d e n t i f i e d a system of f o s s i l i c e wedges

i n the e a s t e r n Tabular H i l l s , but f a i l e d to f i n d any e l s e ­

where i n the Moors. These wedges were a s s o c i a t e d with t i l l ,

e r r a t i c pebbles and s o l i f l u c t i o n m a t e r i a l and were assigned

to a pre-Devensian period.

Palmer (1956) regarded v a l l e y i n f i l l i n Dovedale as a

s o l i f l u c t i o n d e p o s i t , although he was i n c l i n e d to play down

the r o l e of c r y o n i v a l conditions i n t o r formation and

h i l l s l o p e development. S o l i f l u c t i o n deposits were a l s o

i d e n t i f i e d by Gregory (1965) i n E s k d a l e ; by T u f n e l l (1969)

on Murk Mire Moor, Levisham Moor, and Lockton Low Moor, and

around Robin Hood's Bay; by Imeson (1970, 1974) i n the

upper p a r t of Bransdale; by Bendelow and C a r r o l l (1976) around

P i c k e r i n g Moor and Troutsdale; and by Jones (1977) i n K i l d a l e .

According to Arnett Q-971a, 18) i n c r y o n i v a l conditions

' c o l l u v i a l a c t i v i t y on the slopes was f a r g r e a t e r than at

p r e s e n t ' : he c i t e d 'angular rubble' found beneath s o i l p r o f i l e s

i n Caydale.

Gregory (1966) described s m a l l ' n i v a t i o n benches' i n

E s k d a l e , a t t r i b u t i n g them to snow patch e r o s i o n , and

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assigned i n a c t i v e mass movement phenomena ( r o t a t i o n a l slumps, earth flow tongues) to c r y o n i v a l c o n d i t i o n s f a l t h o u g h w i t h o u t arguing the case i n d e t a i l . T u f n e l l (1969) a l l u d e d t o f r o s t s h a t t e r e d sandstone on the North York Moors and t o erected stones on Murk Mire Moor. Palmer (1973> 57) c i t e d angular boulders exposed by peat f i r e s as evidence f o r f r o s t r i v i n g . Catt e t a l (1974) reported loess i n the Hambleton H i l l s .

Simmons and C u n d i l l (1974) quoted evidence t h a t peat growth i n two l a n d s l i p bogs (NZ 683028, SE 674996) s t a r t e d i n the e a r l y Plandrian; t h i s gives minimum ages f o r the land­s l i p s , which may have been associated w i t h c r y o n i v a l c o n d i t i o n s .

Many sections i n SE 59 show m a t e r i a l which i s c l e a r l y c o l l u v i a l r a t h e r than i n s i t u (e.g. stream side sections at 592979 and 508977; t r a c k s i d e sections at 538954 and 549907). The c o l l u v i a l o r i g i n i s c l e a r l y demonstrated when sandstone cobbles and boulders are i n c o r p o r a t e d i n a mantle r e s t i n g on shales, as at 592979 and 508977. The thickness of colluvium i s not g r e a t , being generally of the order of 1 m. Many other s e c t i o n s , however, are small and not obviously c o l l u v i a l . Whether the colluvium i s a s o l i f l u c t i o n deposit i s d i f f i c u l t t o say, although i t i s p l a u s i b l e when the colluvium incorporates very coarse m a t e r i a l . Surface boulders are also common ( f o r example, a conspicuous spread around 549918), but i t i s not c l e a r t h a t these must a l l be a t t r i b u t e d t o c r y o n i v a l f r o s t r i v i n g . An apparently i n a c t i v e and w e l l vegetated l a n d s l i p scar such as Kay Nest (583985 ; see 4L) may also be p l a u s i b l y a t t r i b u t e d to c r y o n i v a l

- 1^0 -

• l . :.' i

L a n d s l i p sea j f n v N©st

4M G u l l y i n g i n Black Intake

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conditions of the e a r l y P l a n drian or e a r l i e r .

Assessment of c r y o n i v a l i n f l u e n c e on h i l l s l o p e

development i s r e l a t i v e l y easy i n p r i n c i p l e . I t i s a

matter of e s t a b l i s h i n g how much development must be

a t t r i b u t e d to ( i ) pre Quaternary time; ( i i ) r e l a t i v e l y

warm conditions during the Quaternary; ( i i i ) c r y o n i v a l

conditions during the Quaternary; ( i v ) p o s t g l a c i a l time;

and a l s o to g l a c i a l e r o s i o n i f t h i s operated. This i s

b a s i c a l l y a question of the r a t e s of slope r e t r e a t during

these periods and t h e i r durations (Young, 1974, 44).

Unfortunately, t h e i r durations are s t i l l very much a

matter f o r controversy, while r a t e s of r e t r e a t must n a t u r a l l y

be i n f e r r e d from contemporary observations of forms,

processes and d e p o s i t s .

Two theses of c r y o n i v a l i n f l u e n c e may be d i s t i n g u i s h e d

(Carson and Kirkby, 1972, 322).

( i ) (Weak t h e s i s ) Many h i l l s l o p e s d i s p l a y c r y o n i v a l

f e a t u r e s and deposits apparently l i t t l e modified. Therefore,

p o s t g l a c i a l slope r e t r e a t has been n e g l i g i b l e and h i l l s l o p e s

may be regarded as r e l i c t .

( i i ) (Strong t h e s i s ) I n c r y o n i v a l periods a marked

a c c e l e r a t i o n of slope processes took p l a c e , leading to

profound a l t e r a t i o n s of p r o f i l e geometry and b a s a l accumu­

l a t i o n s of c r y o n i v a l deposits.

Such t h e s e s , both i n general and as f a r as the North

York Moors are concerned, must be considered i n the l i g h t of

the following p o i n t s .

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F i r s t l y , there i s a l a c k of q u a n t i t a t i v e evidence on

past slope r e t r e a t under c r y o n i v a l p r o c e s s e s . Prom the

a v a i l a b l e evidence, Young (1972, 244) has suggested that

t o t a l r e t r e a t was of the order of 1 - 10 m. I f t h i s i s

c o r r e c t , c r y o n i v a l i n f l u e n c e may w e l l have been exaggerated.

Secondly, there are problems of r e p o r t i n g b i a s and

sampling b a s i s . Reports of c r y o n i v a l f e a t u r e s and deposits

are frequent i n the l i t e r a t u r e , but workers f a i l i n g to f i n d

any i n an area w i l l w r i t e about something e l s e i n s t e a d

(Young, 1972, 241). Large and s t r i k i n g f e a t u r e s and s e c t i o n s

are most . l i k e l y to be reported, leading to a b i a s e d

estimate of the importance of c r y o n i v a l a c t i v i t y .

T h i r d l y , as Young (1972, 244 j 1974, 74) argued, the

unmodified c o n d i t i o n of c r y o n i v a l f e a t u r e s and deposits

does not imply that p o s t g l a c i a l a c t i v i t y i s l e s s intense

than c r y o n i v a l . I t may be accounted f o r by the shortness of

p o s t g l a c i a l time.

F o u r t h l y , i t i s d i f f i c u l t to say how many t r a c e s of

c r y o n i v a l a c t i o n have been destroyed or trun c a t e d , or to

ass e s s the o v e r a l l importance of p r e g l a c i a l or i n t e r g l a c i a l

r e t r e a t ( W i l l i a m s , 1968, 3 i i ; Sparks and West, 1972, 116).

F i f t h l y , many of these f e a t u r e s reported (e.g. wedges,

ere c t e d stones, angular boulders and even n i v a t i o n benches)

appear to be only micro- or meso- forms compared with the

s c a l e of the h i l l s l o p e p r o f i l e s .

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S i x t h l y , the c r y o n i v a l o r i g i n of many features i s not always c l e a r cut. The c r i t e r i a used to i d e n t i f y c r y o n i v a l deposits are 'often very vague' (Carson and Ki r k b y , 1972, 322) while there i s growing a p p r e c i a t i o n of the d i f f i c u l t i e s o f d i s t i n g u i s h i n g s o l i f l u c t i o n deposits from other c o l l u v i a l deposits (Benedict, 1976).

Seventhly, i t i s d i f f i c u l t t o understand how any appreciable m o d i f i c a t i o n of p r o f i l e geometry could take place w i t h o u t a great deal of rock weathering t o supplement downslope movement of the mantle (Carson and Ki r k b y , 1972, 323).

E i g h t h l y , i t i s not c l e a r t h a t c r y o n i v a l processes would even tend t o produce a r a d i c a l l y d i f f e r e n t p r o f i l e geometry, although e x i s t i n g theory on c r y o n i v a l slope development i s very weak ( c f . Carson and K i r k b y , 1972, Ch. 12; Jahn, 1975, Ch. 17; French, 1976, Ch. 7). For example, while lobes and terraces are d i s t i n c t i v e products o f g e l i f l u c t i o n (Washburn, 1973, 189; Embleton and King, 1975, 112-9; Benedict, 1976; French, 1976, 139-41) these are commonly micro- or meso- forms: formation of such features i s not i n e v i t a b l y associated w i t h fundamentally d i f f e r e n t p r o f i l e shapes.

Hence the many report s of apparently c r y o n i v a l features and deposits must be i n t e r p r e t e d c a u t i o u s l y . There are several grounds f o r doubting a t h e s i s of profound c r y o n i v a l i n f l u e n c e .

4.4.4 Contemporary erosion

Imeson (1970, 1971a, 1971b, 1973, 1974) has s t u d i e d contemporary erosion i n upper Bransdale. He c l e a r l y

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documented the r e l a t i o n s h i p between heather burning and s o i l erosion (Imeson, 1971a) . The broader p i c t u r e i s one i n which unvegetated areas and channel sides c o n t r i b u t e much

p more sediment than vegetated areas, although i n a 19 km basin most i s deposited as colluvium or al l u v i u m and does not pass the o u t l e t (Imeson, 1974).

I n c i d e n t a l observations by Palmer (1973, 59) and i n SE 59 underscore the importance of vegetation e s t a b l i s h e d

A s y s t e m a t i c a l l y by Imeson. R i l l i n g , sheet wash and d i s s e c t i o n of peat can be observed, e.g. above Crookleth Crags, around 555968. These are o f t e n associated w i t h overgrazing and burning. There i s occasional s l i g h t g u l l y i n g , notably along footpaths. The f o o t p a t h up from Beacon Guest i s deeply i n s e t around 56^965, suggesting former g u l l y i n g . I t i s q u i t e w e l l vegetated now, which may r e f l e c t the great decrease i n i t s use. S i m i l a r l y the o l d fo o t p a t h through Black Intake (around 575992) seems to have been g u l l i e d and then s t a b i l i s e d , although there are f u r t h e r signs of erosion at the present (see 4M). The most s t r i k i n g example, at 586916, i s a case i n which a g u l l y has exposed bedrock over a reach of 7m, removing 90 cm of s o i l (see 4N). Occasional l a n d s l i p s , as at 543905 ( a l l dimensions ~ 20 m; see 40), have been a c t i v e r e c e n t l y . A f u r t h e r i n t e r e s t i n g feature i s the 'bunker' or 'sheep scar' (McVean and Lockie, 1969* 29; Evans, 1974),a crescent-shaped scar enlarged by sheep, as at 543907, 532927 or 532935.

Apart from areas of peat e r o s i o n , however, a l l these bare areas s u f f e r i n g erosion are minor and l o c a l i s e d .

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.

AM

G u l l y i n g near Roppa Wood

• ' I

40 L a n d s l i p at Hawnby H i l

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4.5 Summary

( i ) Geological Survey accounts o f the Jurassic formations i n the f i e l d area remain the most valuable f o r the geomorph-o l o g i s t , even though the usefulness of t h e i r s t r a t i g r a p h i c a l c l a s s i f i c a t i o n i s l i m i t e d by v e r t i c a l and l a t e r a l v a r i a t i o n s i n l i t h o l o g y ( 4 . 2 . 1 ) . Post-Cretaceous s t r u c t u r a l h i s t o r y can be summarised very simply as a combination of epeirogenic t i l t i n g and a n t i c l i n a l warping ( 4 . 2 . 2 ) .

( i i ) The f o l l o w i n g theses have been advanced about the geomorphological development o f the f i e l d area, although none has received very much c r i t i c a l examination.

(a) The the s i s o f profound l i t h o l o g i c a l i n f l u e n c e (4 .3 .1 )

(b) The th e s i s o f p o l y c y c l i c denudation h i s t o r y (4 .3 .2 )

(c) The the s i s of p r o g l a c i a l lake overflow channels (4 .4 .2 )

(d) The the s i s o f profound c r y o n i v a l i n f l u e n c e (4 .4 .3 )

( i i i ) There i s complete ignorance about the e f f e c t s o f pre-Devensian g l a c i a t i o n s on the f i e l d area (4 .4 .1 )

( i v ) The question of scale of feature i s c r u c i a l , p a r t i c u l a r l y as f a r as features a t t r i b u t a b l e t o l i t h o l o g i c a l or c r y o n i v a l i n f l u e n c e are concerned ( 4 . 3 . 1 , 4 . 4 . 3 ) .

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Chapter 5

SAMPLING AND MEASUREMENT PROCEDURES

' H i l l . Yes, t h a t was i t . But i t i s a hasty word f o r a t h i n g t h a t has stood here ever since t h i s p a r t of the world was shaped.'

.R.R. To l k i e n , The Lord of the Rings, Bk. I l l , Ch.

5.1 I n t r o d u c t i o n 5.2 Sampling 5.3 Measurement 5.4 Notation

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5.1 I n t r o d u c t i o n

I n t h i s chapter sampling and measurement procedures used to c o l l e c t h i l l s l o p e p r o f i l e data i n the f i e l d area are discussed i n the l i g h t o f general p r i n c i p l e s . Although t h i s study does not o f f e r innovations i n procedures, the need t o give a b r i e f r e p o r t allows the i n t r o d u c t i o n of some method­o l o g i c a l ideas p r e v i o u s l y neglected i n h i l l s l o p e p r o f i l e s t u d i e s .

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5.2 Sampling

The sampling problem i s simple i n essence. Given a set of objects (a p o p u l a t i o n ) , choose a smaller subset (a sample) f o r d e t a i l e d study. There are two fundamentally d i f f e r e n t approaches: ( i ) Choose objects q u i t e a r b i t r a r i l y , f o r example, those which are i n t e r e s t i n g or accessible. ( i i ) Choose objects according to some d e f i n i t e r u l e designed to ensure t h a t the sample i s repr e s e n t a t i v e of the pop u l a t i o n .

This d i s t i n c t i o n corresponds f a i r l y c l o s e l y t o t h a t o f t e n made between purposive and p r o b a b i l i t y sampling (e.g. Harvey, 1969, 356-69), but such terms are not very appropriate: procedures covered by ( i ) may lack a d e f i n i t e purpose, i n the sense t h a t choice may be e s s e n t i a l l y haphazard; wh i l e procedures covered by ( i i ) may not be p r o b a b i l i s t i c , i n the sense t h a t the ' d e f i n i t e r u l e ' i s d e t e r m i n i s t i c .

I t i s c l e a r t h a t unless some d e f i n i t e r u l e i s followed t o ensure a repr e s e n t a t i v e sample, there can be no stron g grounds f o r g e n e r a l i s i n g from the sample t o the p o p u l a t i o n , and any statements made about the sample should not be a t t r i b u t e d wider a p p l i c a b i l i t y . This should always be recognised e x p l i c i t l y : whether i t i s important depends on the purpose o f the exercise.

Sampling theory i s , i n large p a r t , a study o f the d e f i n i t e r u l e s devised to generate representative samples i n various circumstances. I t w i l l be worthwhile considering

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d i f f e r e n t kinds of sampling problems t o see how f a r they a r i s e i n h i l l s l o p e geomorphology, not le a s t because there i s some controversy and confusion i n t h i s f i e l d over sampling p r i n c i p l e s and procedures. (The c l a s s i f i c a t i o n used here i s not exhaustive. The terminology employed i s o r i g i n a l ) .

The c l a s s i c a l sampling problem arises when the pop u l a t i o n consists of d i s c r e t e i n d i v i d u a l s which are a set i n the s t r i c t sense; t h a t i s , they possess no n a t u r a l o r d e r i n g and are not indexed by time or space coordinates. Random s e l e c t i o n i s the fundamental s o l u t i o n t o the c l a s s i c a l sampling problem: objects are l a b e l l e d numerically and a sample chosen using a l i s t of pseudorandom numbers or some equivalent procedure. Although a v a r i e t y of other procedures e x i s t s , i n essence they are m o d i f i c a t i o n s o f simple random s e l e c t i o n (e.g. s t r a t i f i c a t i o n , m u l t i s t a g e sampling). See Stua r t (1976) f o r an i n t r o d u c t i o n to c l a s s i c a l sampling.

C l a s s i c a l sampling procedures might be appropriate i n h i l l s l o p e geomorphology i f the landsurface could be regarded as a combination of d i s c r e t e u n i t s , a view here termed geomorphological atomism ( f o r f u l l e r d i s cussion, see Ch. 8.2 below). This view i s open t o o b j e c t i o n as a p a r t i a l t h e o r y , but here i t i s s u f f i c i e n t t o note two l i m i t a t i o n s t o an a t o m i s t i c approach t o h i l l s l o p e p r o f i l e sampling. F i r s t l y , an i n i t i a l survey stage i s necessary t o define the u n i t s o f a landscape before such u n i t s can be used as a framework f o r choosing a sample of p r o f i l e s . Secondly, a t t r i b u t e s of neighbouring u n i t s would tend t o be a u t o c o r r e l a t e d , i m p l y i n g

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t h a t random s e l e c t i o n i s not a p p r o p r i a t e .

The s e r i a l sampling problem arises when the p o p u l a t i o n i s a single-valued continuous series indexed by time or space coordinates. Systematic s e l e c t i o n i s the fundamental s o l u t i o n t o the s e r i a l sampling problem: thus time series are generally recorded at r e g u l a r i n t e r v a l s , while various g r i d s ( w i t h square or t r i a n g u l a r meshes) are i n c r e a s i n g l y being recognised as the p r e f e r r e d class of sampling schemes f o r s p a t i a l series ( c f . Holmes, 1970; Evans, 1969, 1972; McCammon, 1975; but see Hammersley, 1975* f o r a d i s s e n t i n g note on abstruse t e c h n i c a l grounds). Many continuous series a r i s e i n h i l l s l o p e geomorph-ology, and p o i n t sampling schemes have been used t o measure slope over a length centred at a p o i n t (e.g. S t r a h l e r , 1956; Juvigne, 1973) or t o measure s o i l p r o p e r t i e s or process rates i n p l o t s centred on a p o i n t (e.g. Reynolds, 1975a, 1975b; Anderson, 1977). Systematic schemes have n o t , however, been u n i v e r s a l l y used. I n process studies p a r t i c u l a r l y , some k i n d of s t r a t i f i c a t i o n i s common, where sampling i s l i n k e d t o an experimental design which aims t o assess the e f f e c t s of c o n t r o l l i n g v a r i a b l e s .

The path sampling problem arises when the population i s a set of paths on a surface. I t does not seem t o a r i s e outside geomorphology, nor i s there a theory of paths on surfaces which would lead t o recommendations about sampling procedures ( c f . Longuet-Higgins, 1962; Switzer, 1976 f o r theory on random s u r f a c e s ) . Since h i l l s l o p e p r o f i l e s are

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maximum-gradient paths, t h i s means t h a t there i s no f i r m t h e o r e t i c a l basis f o r choosing h i l l s l o p e p r o f i l e samples. By d e f a u l t , h i l l s l o p e p r o f i l e sampling has been t r e a t e d as a s e r i a l sampling problem, a problem o f s e l e c t i n g p o i n t s , whether these are t o be endpoints or intermediate p o i n t s on p r o f i l e s .

I t i s f i r s t necessary t o decide whether the t a r g e t p o p u l a t i o n o f h i l l s l o p e p r o f i l e s includes a l l p r o f i l e s or merely some subset. The most important c o n s i d e r a t i o n here i s plan curvature, and,as a f i r s t approximation, p r o f i l e s may be d i v i d e d i n t o three classes.

(a) P r o f i l e s w i t h n e g l i g i b l e plan curvature ( s t r a i g h t c o n t ours), e.g. on v a l l e y sides. ' N e g l i g i b l e 1 t o be defined o p e r a t i o n a l l y (see, f o r example, Abrahams and Parsons, 1977).

(b) P r o f i l e s convex i n p l a n , e.g. on spurs (c) P r o f i l e s concave i n p l a n , e.g. i n v a l l e y heads

Usually e i t h e r (a) alone i s regarded as the t a r g e t p o p u l a t i o n , or ( a ) , (b) and (c) combined.

Given the t a r g e t p o p u l a t i o n , i t i s possible t o choose a set of po i n t s on a map and i d e n t i f y p r o f i l e s which extend upslope and downslope from those p o i n t s . (Any p r o f i l e s not belonging to the t a r g e t p o p u l a t i o n w i l l n a t u r a l l y have t o be r e j e c t e d ) . Whatever the d e t a i l s of p o i n t s e l e c t i o n , t h i s approach i s at best only an approximate s o l u t i o n t o the h i l l s l o p e p r o f i l e sampling problem.

- 163 -

From a geomorphological p o i n t of view, a more d i r e c t and n a t u r a l approach t o p r o f i l e sampling i s t o use the stream network as a basis f o r s e l e c t i o n . Broadly s i m i l a r schemes of t h i s k i n d have been used or suggested by Arne t t (1971b), Chorley and Kennedy (1971, 50-55), Young e t a l (1974, 17 -19) , Summerfield (1976) and Abrahams and Parsons (1977). I n the simplest s i t u a t i o n , slopes w i t h s t r a i g h t contours have bases at or near the midpoints of l i n k s i n the stream network, w h i l e slopes w i t h curved contours have bases at or near source or j u n c t i o n nodes. This approach presupposes w e l l -i n t e g r a t e d f l u v i a l topography, i n which h i l l s i d e s are i n cle a r and unambiguous r e l a t i o n s h i p s w i t h streams.

While ideas from sampling th e o r y , such as random and systematic s e l e c t i o n and s t r a t i f i c a t i o n , appear i n such schemes, the choice of the stream network as sampling framework i s d i f f i c u l t t o evaluate t h e o r e t i c a l l y , however a t t r a c t i v e i t may be f o r the geomorphologist. Moreover, the i d e n t i f i c a t i o n of the stream network i t s e l f may not be s t r a i g h t f o r w a r d . Stream sources may be d i f f i c u l t t o locate and some permanent watercourses may appear to lack geomorph­o l o g i c a l s i g n i f i c a n c e . There i s much t o be s a i d f o r the view t h a t the v a l l e y network r a t h e r than the stream network i s appropriate f o r a p r o f i l e sampling framework. Indeed, one i d e a l would be f o r a sample of p r o f i l e s t o be chosen by an i t e r a t i v e s e l e c t i o n procedure which i d e n t i f i e d a repre s e n t a t i v e set from a s u f f i c i e n t l y d e t a i l e d a l t i t u d e m a t r i x .

- 164 -

However a sample of p r o f i l e s i s chosen, i t i s q u i t e l i k e l y t h a t some p r o f i l e s w i l l prove unsurveyable (e.g. land use may not permit survey; access may be forbidden or dangerous). Hence i n p r a c t i c e i t may be necessary t o modify an i n i t i a l l y chosen sample. There i s , n a t u r a l l y , no guarantee t h a t such forced omissions w i l l not induce biases i n rep r e s e n t a t i o n .

The t a r g e t p o p u l a t i o n f o r t h i s study was the set of h i l l s l o p e p r o f i l e s w i t h s t r a i g h t contours. P r o f i l e s w i t h n e g l i g i b l e plan curvature are simplest t o i n t e r p r e t , and they are p a r t i c u l a r l y appropriate f o r t e s t i n g h i l l s l o p e models which neglect l a t e r a l i t y .

As a f i r s t step the stream network i n the f i e l d area was d e l i m i t e d from Ordnance Survey 1:63,360 and 1:25,000 sheets and a l i s t drawn up of stream 'systems' and 'major subsystems', the terms being used i n one-off senses (5A). A l i s t was then prepared of 'possible p r o f i l e s ' using s p e c i f i c c r i t e r i a (5B).

This l i s t includes 19 p r o f i l e s . I n f a c t only 11 of these were surveyed, f o r two reasons. As research progressed, i t became cl e a r t h a t an adequate treatment of methodological, t h e o r e t i c a l and t e c h n i c a l questions - the major f o c i of the t h e s i s , i n short - would imply t h a t the treatment of e m p i r i c a l questions i n the p r o j e c t would be less extensive than was o r i g i n a l l y envisaged. I n a d d i t i o n , t h i s number of p r o f i l e s seemed s u f f i c i e n t t o allow i l l u s t r a t i o n and examination of the methods developed i n t h i s study.

- 165 -5A

Stream systems and ma.ior subsystems i n SE 59

1. Stream systems i . e . d i s c r e t e networks w i t h i n f i e l d area ( i ) Scugdale Beck system - flows i n t o Leven, and thence

i n t o Tees ( i i ) Rye system - flows i n t o Dement

( i i i ) L a d h i l l Beck system - flows i n t o Rye ( i v ) Seph system - flows i n t o Rye (v) R i c c a l system - flows i n t o Rye

2. Major subsystems ( i i ) Rye system Stonymoor Sike, Proddale Beck, Arns G i l l , Wheat Beck, Locker Beck, B l o w g i l l , Eskerdale Beck, Thorodale Beck* ( i v ) Seph system Raisdale Beck, Hollow Bottom Beck, Ledge Beck (Tripsdale Beck, Tarnhole Beck), Pangdale Beck, T o d h i l l Beck (v) R i c c a l system Bogmire G i l l , B o n f i e l d G i l l , P o t t e r House Beck*

Notes * ad hoc names devised by author (a) Some short streams i n upper B i l s d a l e are not marked beyond the B1257 road. Carlton Watercourse (marked on 1:25,000) i s c l e a r l y a r t i f i c i a l . The upper course o f Kyloe Cow Beck i s also problematic. (b) The 'major subsystems'are the l a r g e r t r i b u t a r i e s , l a r g e r i n terms o f mainstream length and/or v a l l e y depth: however, the decisions on these were f a i r l y s u b j e c t i v e , and no precise c r i t e r i a used.

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5B

Possible p r o f i l e s and a c t u a l p r o f i l e s

Possible p r o f i l e Crest GR Base GR Scugdale 522995 520989 Stonymoor Sike 508978 515980 X Proddale Sike 522970 524965 X Arns G i l l 528962 524965 X Wheat Beck 506947 505960 Locker Beck 510937 510927 Blow G i l l 534946 531953 X Eskerdale Beck 519921 510927 Thorodale Beck* 516907 515915 L a d h i l l Beck 549924 555926 X Raisdale Beck 546996 540987 Hollow Bottom Beck 554982 551987 X Tripsdale Beck 581984 575987 X Tarn Hole Beck 590976 598974 X Fangdale Beck 560948 553948 X T o d h i l l Beck 571912 575905 X Bogmire G i l l 595929 585930 B o n f i e l d G i l l 600962 594964 X Potter House Beck* 594912 590605 * ad hoc names devised by author X a c t u a l l y surveyed ( c f . 6L)

S e l e c t i o n procedure Take 'systems' and 'major subsystems' as given. Choose one

p r o f i l e f o r each subsystem and one f o r each system w i t h o u t sub­systems. Take a base approximately midway along r e l e v a n t stream l i n k , or along p a r t o f relevant l i n k w i t h i n f i e l d area. Val l e y s i d e should have approximately s t r a i g h t contours and be r e l a t i v e l y f r e e of obstructions such as roads. I t would be convenient t o have p r o f i l e s on opposite sides o f a r i d g e .

A r b i t r a r y w i t h i n these broad s p e c i f i c a t i o n s .

- 167 -

I t i s necessary, t h e r e f o r e , t o stress the l i m i t a t i o n s of t h i s sample of p r o f i l e s . F i r s t l y , the number o f p r o f i l e s i s r a t h e r small t o allow c l e a r c u t g e n e r a l i s a t i o n s about the f i e l d area. Secondly, a haphazard element entered i n t o the choice of p r o f i l e s , given t h a t research plans narrowed i n focus while data c o l l e c t i o n proceeded i n t e r m i t t e n t l y , and t h a t the o r i g i n a l sample was not completed. T h i r d l y , i t i s not demonstrated convincingly t h a t the p r o f i l e s e l e c t i o n procedure used produces a rep r e s e n t a t i v e sample, although i t i s not c l e a r p r e c i s e l y what other procedure would achieve t h i s aim.

When slope endpoints were located i n the f i e l d , i t was sometimes found necessary t o move s l i g h t l y , t o avoid l o c a l o b s t r u ctions or l o c a l plan curvature ( c f . 6L below f o r a c t u a l endpoint p o s i t i o n s ) .

These p r o f i l e s are shown on a map i n 5C, together w i t h major streams and major watersheds.

- 168 -

TR

\ HO

ST

TA ***

1 1

(A

• PR

%AR

B O . ;

I

f

9<! FA

LA

f

fo**>

TO

s N

I •+4 1 p r o f i t watershed

5C Measured p r o f i l e s , major streams and major watersheds i n SE 59

- 169 -

5.3 Measurement

A p r o f i l e may be recorded i n two ways, which are numerically equivalent:

( i ) As a series of measured lengths, 1^; i = l , . . . , n , together w i t h a corresponding series o f measured angles ^ ; i = 1,...,n.

( i i ) As a series o f coordinates, x^, z^; i = l , . . , n . N a t u r a l l y i f ( i ) i s chosen, and constant 1 i s used, then the h o r i z o n t a l i n t e r v a l , Ax , say, w i l l not be constant i n general; and conversely, i f ( i i ) i s chosen, and constant Ax i s used, then 1 w i l l not be constant i n general.

The method of measured lengths and angles was used i n the present study and a t t e n t i o n here i s focused on t h i s case.

I t seems cle a r t h a t measured lengths should be constant and r e l a t i v e l y s hort ( P i t t y , 1967; c f . Young et a l , 197^, 31-3). Many h i l l s l o p e p r o f i l e s have been surveyed w i t h v a r i a b l e measured lengths. I n extreme cases, breaks and changes o f slope have been i d e n t i f i e d v i s u a l l y and used t o bound measured lengths. N a t u r a l l y such a survey procedure prejudices any i n t e r p r e t a t i o n of p r o f i l e data, and (what i s worse) prejudices i t i n an unknown and complicated manner. The case f o r constant measured lengths i s thus very strong.

Gerrard and Robinson (197D discussed the choice o f measured length i n some d e t a i l . They warned t h a t the r e s u l t s of slope p r o f i l e survey may vary considerably w i t h the measured length adopted. This warning was i l l u s t r a t e d by angle

- 170 -

frequency d i s t r i b u t i o n s obtained from repeated surveys using d i f f e r e n t measured lengths. They also warned t h a t m i c r o r e l i e f would i n f l u e n c e readings made at very short measured lengths. I t does seem, however, t h a t such warnings may be s l i g h t l y exaggerated. The e f f e c t of repeated survey i s t o mix scale v a r i a t i o n w i t h measurement e r r o r ( G i l g , 1973), while t h e i r homily against m i c r o r e l i e f r e s t s on a dichotomy between m i c r o r e l i e f and 'the true nature of the slope' (Gerrard and Robinson, 1971, 50) which seems dubious. There do not appear t o be any f i r m p h y s i c a l grounds f o r d i s t i n g u i s h i n g sharply between two components of topography, w h i l e r e s u l t s from s p e c t r a l analysis (e.g. Sayles and Thomas, 1978) support the contrary idea t h a t v a r i a t i o n s can be observed at a l l s p a t i a l frequencies of geomorphological i n t e r e s t .

A sh o r t measured length (5 m. or les s ) seems p r e f e r a b l e t o allow d e t a i l e d recording of h i l l s l o p e p r o f i l e s . N a t u r a l l y superfluous d e t a i l can always be ignored or removed, wh i l e d e t a i l f i n e r than t h a t recorded cannot be added w i t h o u t resurvey.

A pantometer ( P i t t y , 1968) was used t o survey h i l l s l o p e p r o f i l e s i n t h i s study. I t was made by Dr. E. W. Anderson and produces angle measurements f o r a constant measured length of 1.52 m. (5 f t . ) (Anderson, 1977, 100-102). Angles were measured to the nearest 0.5° ( c f . Young e t a l , 1974, 13). The procedure near the cr e s t o f each p r o f i l e was t o continue measurement along an orthogonal u n t i l the crest was c l e a r l y passed: the exact p o s i t i o n of the cr e s t was i d e n t i f i e d l a t e r from the data.

- 171 -

Measured angle s e r i e s , together w i t h notes on ve g e t a t i o n , n a t u r a l and a r t i f i c i a l f e a t u r e s , and basal stream c h a r a c t e r i s t i c s , are given i n Appendix I .

5•4 Notation

i s u b s c r i p t 1 measured length n number of observations x h o r i z o n t a l coordinate z v e r t i c a l coordinate A d i f f e r e n c e operator 0 angle

- 172 -

Chapter 6

DESCRIPTION OF MEASURED PROFILES

... Without words, there i s no p o s s i b i l i t y of

reckoning of Numbers; much l e s s e of Magnitudes,

of Swiftnesse,of Force, and other t h i n g s , the

reckonings whereof are necessary to the being,

or w e l l - b e i n g of man-kind.

Thomas Hobbes, Lev i a t h a n , Ch. 4.

6.1 P r o f i l e form

6.2 Angle and curvature frequency d i s t r i b u t i o n s

6.3 Frequency d i s t r i b u t i o n s at d i f f e r e n t s c a l e s

6.4 Summary

6.5 Notation

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6.1 P r o f i l e form

I f l j _ ; i = l , . . . , n and 9^; i = 1,... ,n denote measured

lengths and measured angles i n c r e s t to base sequence, then

p r o f i l e coordinates may be c a l c u l a t e d from

j X: = E I; Cos B-

J i=l L L

2: = E l t sen GL

P r o f i l e height z d and p r o f i l e length may be derived from

r\

\ Z Lj Cos 0 t

and average angle 0 i s given by a r c t a n ( z ^ / x ^ ) ( c f . 3A).

These r e l a t i o n s were used to produce p r o f i l e coordinates

f o r the eleven p r o f i l e s measured i n the f i e l d a r e a .

Dimensionless p l o t s , i n which z / z d i s shown ag a i n s t x/x^,

are given i n 6A to 6K. Number of observations n, estimated

p r o f i l e dimensions z^ and x b , average angle 9 , endpoint

l o c a t i o n s and bedrock geology are shown f o r each p r o f i l e

i n 6L. I d e n t i f i e r s are here introduced f o r each p r o f i l e

and f o r each g e o l o g i c a l formation (see 6M f o r k e y ) .

The p r o f i l e s f a l l r e a d i l y i n t o four c l a s s e s :

( i ) BO, ST, PR, PA: the four g e n t l e s t p r o f i l e s , a l l

on the formation d e l .

- 174 -

o o

CO

G o

LU

UJ

o CC zr

I LU CC

0.00 20 40 60 80 R E L A T I V E D I S T A N C E

6A P r o f i l e BO

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oo

\ — LD o CO I — I

UJ

o CE=r _ J

CO

rvi

80 60 20 140 0.00 R E L A T I V E D I S T A N C E

6B P r o f i l e ST

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o o

r—H

CO

\ I—

\ CD LU

LJJ

\ — i

— _ J

nr

o

O

20 80 0.00 yo 60

R E L A T I V E D I S T A N C E 6C P r o f i l e PR

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o o

v.

o CO

\

— CD o

to

\ LU

LU

LU 0C

O 0.00 20 yo 60 80 1.00 R E L A T I V E D I S T A N C E

6D P r o f i l e PA

- 178 -

o o

"v. o CO

— \

CO U J

U J

_ J LU CO

CM

c 0.00 2 0 40 60 80 1.00

R E L A T I V E D I S T A N C E 6E P r o f i l e FA

- 179 -

3

CO

s \ \ \

O

1.00 80 60 40 20 0.00 R E L A T I V E D I S T A N C E 6 F P r o f i l e LA

i

- 180 -

o o

V CO

(Do CO I — LxJ

— o CE _ J U J CO

CM

1.00 80 60 40 20 0.00 R E L A T I V E . D I S T A N C E 6G P r o f i l e TR

- 181 -

o o

CO \

G o CO

IT LU

h-_ J LU CO

CM

80 1 60 20 40 0.00 R E L A T I V E D I S T A N C E

6H P r o f i l e AR

- 182 -

o o

CD

G o CO I — I

LxJ

LU

CE =r _ J LU c r

0.00 20 40 60 80 R E L A T I V E D I S T A N C E

61 P r o f i l e TA

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o o

CO

— to i —

LJJ 31

U J \

— CT=r \

\ LU r r

o

0.00 20 40 80 60 R E L A T I V E D I S T A N C E 6J P r o f i l e HO

- 184 -

o o

\ CO

G o CO

LU >

o CT rr _ l

CO

o

o 80 1 60 40 0.00 20 R E L A T I V E D I S T A N C E

6K P r o f i l e TO

- 185 -

( i i ) FA, LA: the next g e n t l e s t p r o f i l e s , both g l s / d e l .

( i i i ) TR, AR, TA: the next g e n t l e s t , a l l with bases

on u l i .

( i v ) HO, TO: the s t e e p e s t , c r o s s i n g four and f i v e

formations r e s p e c t i v e l y .

The dimensionless p l o t s f a c i l i t a t e comparison of

p r o f i l e shape. They give an immediate impression of

o v e r a l l form. The v e r t i c a l exaggeration of each p l o t i s

cot G (= x b / z d ) .

( i ) The four g e n t l e s t p r o f i l e s , a l l on d e l , are

s l i g h t l y convex i n o v e r a l l form, with steepening towards

the base (6A to 6D).

( i i ) The next two, both on g l s / d e l , are a l s o s l i g h t l y

convex. Indeed PA and FA are almost i d e n t i c a l i n shape

( i . e . dimensionless form) (6E and 6F).

( i i i ) TR, AR and TA, a l l with bases on u l i , are more

stron g l y convex, with suggestions of d i s t i n c t breaks of

slope bounding separate components. The most s t r i k i n g

feature on these p r o f i l e s i s Tarn Hole Crag (estimated

at 50°) on TA (6G to 6 l ) . ( i v ) HO and TO are the s t e e p e s t p r o f i l e s . The most

dramatic feature on HO i s the f a i l u r e s c a r below the crag.

TO i s a remarkably steep p r o f i l e on the scarp of the

Tabular H i l l s , being steepest near the c r e s t (6J and 6K).

Dimensionless p l o t s are used here p a r t l y as one

method of s t a n d a r d i s e d comparison, and p a r t l y because

dimensionless curves o r i g i n a l l y obtained by Kirkby (197D

- 186 -

CO <D

<u bO <D ! C D

•d

oo O N O N H c— V O CO V O K N

i n vo V O oo O N O N O rH

rH rH rH

O N rH

rvi O o i n O N CO rH K N i n i n VO -=T vo

-p CD

•a o K N o K N V O V D oo i n co rH

rH

•=r K N O J rH V O O N i n c— OJ O

c K N •=r CNJ i n

rH c— O c— O N O O N K N K N O N K N t— m m CO m •=r

KN. CNJ O vo K N rH O o i n i n rH rH rH rH H rH

O N K N K N t— -=r O N V O O VO V O rH O

•=r K N i n •=r K N

vo

o

• H H

CD •d —

o o

co • H • H rH CD rH rH CD

rH • H \ <M >3 H rH • H • H rH bO o bO CD CD rH rH CD O X

O •d - d •d •d o rH \ \ \ \ o rH H rH rH CO CO rH rH CO rH bO

CD CD CD a> CD CD rH rH CD CD rH CD o Q. O XS •a •d -d hO bO •d XJ 50 •d rH KJ

rH (0 K CNJ t~~ o CNJ CO i n o - CNJ [— CM o

rH V O c— t— i n CNJ OO V O t— OO rH rH O N O N O N O N O N O N O N O N O N O N O N • H CD O C O •=r vo O N CO CNJ O N O K N OO X ! O o CNJ K N i n •=r CO CNJ O N i n r—

cd V O m m i n i n i n i n i n i n i n i n •d

re

« CO O m rH CO CO O N l>- -=r vo i n cd vo C O vo i n -=T CNJ CO vo t— CO O

,a> •P O N O N O N O N O N O N O N O N O N O N O N S CO i n i n t— .=3- CNJ i n t— O N o O N

CD O N rH CNJ K N i n i n t— CNJ O N m r>-FH m m i n i n i n i n i n i n m i n m

B O O

B rH CD CD CD CD cd rH X O o o CQ CQ O 9> • H • H • H CD CD CD CD U CO CO * ra CQ CQ • * • CQ •p o CQ co

CQ

CQ CO PH OH EH < EH EH

u CD

• H <+n • H -P C CD

•d

o CQ

EH CO

CcJ EH <

EH

O o EH

a>

a o -p •p

•d rH CD CD O •d O CD • H CD rH rH rH CQ rH O rH CJ rH rH cd rH o

CQ

CD B cd cd rH -d •rH % • H >> •d • H •d • H CO bO O "in c •d O bO PH CQ rH

o o FH q !d • H a u rH O -p FH cd cd FH FH cd O

CQ co CM PH EH < E H K o

EH

- 187 -

6M. Key t o g e o l o g i c a l f o r m a t i o n i d e n t i f i e r s

F o r m a t i o n I d e n t i f i e r

Lower Ca lca reous G r i t l e g

O x f o r d Clay o x f

K e l l a w a y s Rock k e l

Comb r a s h c o r

D e l t a i c Beds d e l

Grey L imes tone S e r i e s g l s

Dogger dog

Upper L i a s u l i

M i d d l e L i a s m l i

- 188 -

w i l l be f i t t e d t o these p r o f i l e s . A g a i n s t t h i s must be s e t an i m p o r t a n t r e s e r v a t i o n , t h a t shape ( d i m e n s i o n l e s s f o r m ) must n o t be c o n s i d e r e d i n d e p e n d e n t l y o f s i z e : h i l l s l o p e p r o f i l e s must n o t be assumed i s o m e t r i c .

6.2 Angle and c u r v a t u r e f r e q u e n c y d i s t r i b u t i o n s

The s t u d y o f ang l e f r e q u e n c y d i s t r i b u t i o n s i s an

acc e p t e d p a r t o f h i l l s l o p e geomorphology ( c f . S t r a h l e r ,

1950, 1956; Young, 196l , 1972; Gregory and Brown , 1966;

P i t t y , 1969, 1970; S p e i g h t , 1971; G e r r a r d and R o b i n s o n ,

1971; J u v i g n e , 1973; S t a t h a m , 1975; Ca r son , 1976, 1977;

Evans , 1977 among s e v e r a l o t h e r s ) . Ang le f r e q u e n c y

d i s t r i b u t i o n s have been c o m p i l e d v a r i o u s l y f o r a l l k i n d s

o f s l o p e s and f o r p a r t i c u l a r k i n d s , such as s t r a i g h t

components , w h i l e t h e r e has been i n t e r e s t b o t h i n t h e

g e n e r a l f o r m o f f r e q u e n c y d i s t r i b u t i o n s , and i n t he

i d e n t i f i c a t i o n and i n t e r p r e t a t i o n o f modal o r c h a r a c t e r i s t i c

a n g l e s .

C u r v a t u r e i s a p r o p e r t y w h i c h has r e c e i v e d l e s s

a t t e n t i o n . I t i s p r o p e r l y d e f i n e d as t h e r a t e o f change

o f ang l e 8 w i t h a rc l e n g t h s , t h a t i s — ( c f . F e r g u s o n ,

1973 on r i v e r meande r s ) . I n h i l l s l o p e geomorpho logy , a

v a r i e t y o f r e l a t e d c u r v a t u r e measures have been p r o p o s e d

( e . g . A h n e r t , 1970c, 78-9; Young , 1972, 137~ I»3;

Demirmen, 1975, 258-60). I n t h i s s t u d y , c o n s t a n t

measured l e n g t h s l ( = A s ) a l l o w t h e use o f t he s i m p l e s t

measure , t h e f i r s t f o r w a r d d i f f e r e n c e

= e c t l - e c

- 189 -

( c f . W i l k e s , 1966, 18; F e r g u s o n , 1975 on r i v e r m e a n d e r s ) .

Frequency d i s t r i b u t i o n s are most commonly summarised

by moment-based measures , such as mean, s t a n d a r d d e v i a t i o n ,

skewness and k u r t o s i s (see Evans 1972, 1977 f o r m a j o r examples

o f such an a p p r o a c h ) . The ma in d i s a d v a n t a g e o f moment-based

measures i s t h e i r l a c k o f r e s i s t a n c e ( r o b u s t n e s s ) t o w i l d

o b s e r v a t i o n s o r o u t l i e r s (see M o s t e l l e r and Tukey , 1977,

Ch. 10 f o r a con tempora ry i n t r o d u c t i o n t o r e s i s t a n c e and

r o b u s t n e s s ) . The a l t e r n a t i v e approach t a k e n he re i s t o

use r e s i s t a n t measures w h i c h are q u a n t i l e s ( o r d e r s t a t i s t i c s )

o r f u n c t i o n s o f q u a n t i l e s . As P i t t y (1970) p o i n t e d o u t ,

t h i s approach i s more a p p r o p r i a t e f o r the k i n d s o f f r e q u e n c y

d i s t r i b u t i o n s c h a r a c t e r i s t i c o f h i l l s l o p e p r o f i l e d a t a ,

w h i c h o f t e n i n c l u d e o u t l i e r s .

Given a sample o f d a t a , t h e o r d e r s t a t i s t i c s a re

t he v a l u e s r a n k e d i n n u m e r i c a l o r d e r , say

s m a l l e s t has r a n k 1 , . . . , l a r g e s t has r a n k n

f o r t h e sake o f a rgument . The minimum ( m i n ) and the maximum

(max) a re e a s i l y d e f i n e d as t h e ex t reme o r d e r s t a t i s t i c s .

The median (med) i s t h e m i d d l e r a n k e d v a l u e , d e f i n e d by t h e

f o l l o w i n g r u l e :

( i ) i f n i s odd

med = v a l u e w i t h r a n k ( n + l ) / 2

( i i ) i f n i s even

med = average o f v a l u e w i t h r a n k (n+2)/2 and v a l u e w i t h

r a n k n /2

- 190 -

F o r d e f i n i n g o t h e r q u a n t i l e s , i t i s h e l p f u l t o

i n t r o d u c e t h e average f u n c t i o n , ave ( ) , and the f l o o r

f u n c t i o n ( I v e r s o n , 1962, 12) , w h i c h g i v e s t he l a r g e s t

i n t e g e r l e s s t h a n o r e q u a l t o i t s a rgument : f o r example ,

f l o o r (4 .7) = 4 and f l o o r (4 .0) = 4.

The q u a r t i l e s ( l o q , upq) may be e s t i m a t e d q u i c k l y

by t h e f o l l o w i n g r u l e s :

( i ) i f n i s n o t d i v i s i b l e by 4 •4" 3

l o q = v a l u e w i t h r a n k o f f l o o r (~ ) * n + 3

upq = v a l u e w i t h r a n k o f n + 1 - f l o o r ( ) 4

( i i ) i f n i s d i v i s i b l e by 4 n + 3

l o q = ave ( v a l u e w i t h r a n k o f f l o o r ( ) , n e x t l a r g e r v a l u e )

n •+ 3 upq = ave ( v a l u e w i t h r a n k o f n + 1 - f l o o r ( ) ,

4-n e x t s m a l l e r v a l u e )

( F o r these e s t i m a t o r s o f q u a j f > t i l e s , see Andrews e t a l ,

1972; f o r t h e l a b e l s m i n , max, med, l o q , u p q , see M c N e i l , 1977;

f o r t h e l a b e l ave , see Blackman and Tukey , 1959).

Other q u a n t i l e s may be e s t i m a t e d by ana logous r u l e s :

the 5% and 95% p o i n t s (p5, p95) a re used h e r e .

The s p r e a d ( d i s p e r s i o n o r s c a l e ) o f a d i s t r i b u t i o n

may be measured by d i f f e r e n c e s o f t h e f o r m

an upper q u a n t i l e - i t s c o r r e s p o n d i n g l o w e r q u a n t i l e

Range (max - m i n ) , 90% s p r e a d (p95 - ?5) and m i d s p r e a d o r

i n t e r q u a r t i l e range ( u p q - l o q ) show i n c r e a s i n g r e s i s t a n c e t o

o u t l i e r s .

- 191 -

F i n a l l y q u a n t i l e - b a s e d measures o f ' a s y m m e t r y 1 and

' t a i l e d n e s s ' may be d e v i s e d as a l t e r n a t i v e s t o moment-based

skewness and k u r t o s i s . V a r i o u s such measures have o f t e n

been used i n s e d i m e n t o l o g y ( G r i f f i t h s , 1967, 107-8) , b u t

those used he re appear t o be new.

The r a t i o

upq - med

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measures asymmetry . I t has a l o w e r l i m i t o f 0, a v a l u e o f

1 f o r any s y m m e t r i c d i s t r i b u t i o n ( e . g . Gaussian o r n o r m a l ) ,

and i n d e f i n i t e l y l a r g e v a l u e s f o r i n c r e a s i n g l y r i g h t - s k e w e d

d i s t r i b u t i o n s .

An ' o u t e r ' q u a n t i l e may be d e f i n e d as one n e a r e r an

ext reme t h a n an ' i n n e r q u a n t i l e * . R a t i o s o f t h e f o r m

sp read between o u t e r q u a n t i l e s sp r ead between i n n e r q u a n t i l e s

can be used i n

v a l u e o f r a t i o f o r a d i s t r i b u t i o n - 1 v a l u e o f r a t i o f o r Gauss ian - 1

w h i c h i s a measure o f t a i l e d n e s s . I t has a l o w e r l i m i t o f

0, a v a l u e o f 1 f o r a Gaussian d i s t r i b u t i o n , and i n d e f i n i t e l y

l a r g e v a l u e s f o r i n c r e a s i n g l y l o n g - t a i l e d d i s t r i b u t i o n s . I f

o u t e r q u a n t i l e s are p5 and p95, and i n n e r q u a n t i l e s are l o q

and u p q , t h e n t h e Gauss ian has a s p r e a d r a t i o . o f T h i s

p a r t i c u l a r t a i l e d n e s s measure i s used h e r e .

Summary measures f o r ang le f r e q u e n c y d i s t r i b u t i o n s

o b t a i n e d f r o m t h e measured p r o f i l e s a re g i v e n i n 6N, w h i c h

- 192 -

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- 193 -

l i s t s m i n , p5, l o q , med, u p q , p95, max, r a n g e , 90% s p r e a d ,

m i d s p r e a d , asymmetry and t a i l e d n e s s . These measures are

based on a b s o l u t e measured angles ( t h a t i s , i g n o r i n g t h e

n e g a t i v e s i g n o f ang les on r e v e r s e d s l o p e s ) . S u p p o r t i n g

diagrams are g i v e n i n 60 and 6P,

Medians and q u a r t i l e s (60) a re e a s i l y r e l a t e d t o t h e

f o u r c l a s se s i d e n t i f i e d above ( 6 . 1 ) . BO, ST, PR, and PA

are c l e a r l y s i m i l a r w i t h low medians and m i d s p r e a d s ; FA and

LA have h i g h e r v a l u e s o f b o t h medians and m i d s p r e a d s . The

marked c u r v a t u r e o f TR i s r e f l e c t e d by l a r g e s p r e a d and

h i g h asymmetry . The r e m a i n i n g p r o f i l e s , AR, TA, HO and

TO have i n c r e a s i n g l y l a r g e medians and m i d s p r e a d s .

Medians and midspreads (6P) supplement t h i s p i c t u r e .

There i s a g e n e r a l t endency f o r midspreads t o i n c r e a s e

w i t h m e d i a n , a l t h o u g h TO ( r e l a t i v e l y low m i d s p r e a d ) and

p a r t i c u l a r l y TR ( v e r y low median) are d i s c r e p a n t .

Angle d i s t r i b u t i o n s m o s t l y v a r y f r o m a p p r o x i m a t e

symmetry t o modera te r i g h t - s k e w e d n e s s ; TR i s n o t a b l y

most e x t r e m e . They range f r o m b e i n g m o d e r a t e l y s h o r t -

t a i l e d t o b e i n g m o d e r a t e l y l o n g - t a i l e d .

. These r e s u l t s on asymmetry and t a i l e d n e s s r a i s e t he

p o s s i b i l i t y o f t r a n s f o r m i n g the d a t a t o more n e a r l y

Gauss ian v a l u e s ( c f . Evans , 1977 f o r a r e c e n t r e v i e w o f

t h i s i s s u e ) . A t r a n s f o r m a t i o n i s m o n o t o n i c i f , f o r a l l

v a l u e s , i t s a t i s f i e s the i d e n t i t y

r a n k o f t r a n s f o r m e d v a l u e = r a n k o f o r i g i n a l v a l u e

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- 196 -

The t r a n s f o r m a t i o n s employed f o r ang le d a t a are m o n o t o n i c o v e r t h e u s u a l i n t e r v a l and so t h e commuta t ive

p r o p e r t y

q u a n t i l e f r o m t r a n s f o r m e d d a t a = t r a n s f o r m o f o r i g i n a l

q u a n t i l e

can be used t o reduce t h e work i n c h o o s i n g a p p r o p r i a t e

t r a n s f o r m a t i o n s . As an example , t h e t r a n s f o r m I n t a n g e n t

has been used (6Q t o 6 S ) .

I n t h i s p a r t i c u l a r case t h e I n t a n g e n t t r a n s f o r m a t i o n

i s o n l y a m i x e d success . I t removes most o f t h e r e l a t i o n s h i p

between m i d s p r e a d and m e d i a n , e x c e p t t h a t TR remains

d i s c r e p a n t ( 6 R ) . Asymmetry and t a i l e d n e s s are sometimes

i m p r o v e d and sometimes worsened ( 6 S ) , t he p a t t e r n b e i n g

as f o l l o w s :

asymmetry & t a i l e d n e s s b e t t e r : PA, FA, AR, HO, TO

asymmetry b e t t e r , t a i l e d n e s s w o r s e : TR, TA

asymmetry w o r s e , t a i l e d n e s s b e t t e r : BO, LA

asymmetry & t a i l e d n e s s w o r s e : ST, PR.

C u r v a t u r e f r e q u e n c y d i s t r i b u t i o n s (6T) a re a l l

c e n t r e d n e a r z e r o , a p p r o x i m a t e l y s y m m e t r i c a l , and r a t h e r

l o n g e r - t a i l e d t h a n Gauss ian . They are d i s t i n g u i s h e d

m a i n l y by t h e i r s p r e a d s .

- 197 -

6Q. Some measures for In tangent transformed data

Profile med midspread asymmetry tailedness

BO -2.34 1.05 0.72 1.02

ST -2.10 0.5^ 0.58 1.85

PR -2.17 0.66 0.80 O.85

PA -2.17 0.70 0.89 1.49

FA -2.00 0.80 O.83 1.77

LA - 1 . 8 4 0.93 0.57 O.76

TR -2.25 1.81 1.07 0.93

AR -1.90 0.97 0.81 0. 88

TA -1.74 1.00 0.94 1.79

HO r l . 5 1 0.90 0.97 1.03

TO -1.09 0.66 1.00 -O.83

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- 199 -

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6 .3 Frequency d i s t r i b u t i o n s a t d i f f e r e n t s c a l e s

I t i s i m p o r t a n t t o c o n s i d e r w h e t h e r r e s u l t s depend

undu ly on measured l e n g t h , as a rgued by G e r r a r d and

Robinson (197D i n p a r t i c u l a r . One way o f i n v e s t i g a t i n g

t h i s q u e s t i o n i s t o average a d j a c e n t a n g l e v a l u e s , compute

c u r v a t u r e s as d i f f e r e n c e s be tween a d j a c e n t a v e r a g e s , and

r e c o m p i l e ang le and c u r v a t u r e f r e q u e n c y d i s t r i b u t i o n s .

(On a v e r a g i n g and d i f f e r e n c i n g me thods , see Blackman and

T u k e y , 1959; C u r r y , 1972; Cox, 1 9 7 3 ) .

I n g e n e r a l , g i v e n a s e r i e s u ^ ; i = l , . . . , n , an

a p p r o p r i a t e method w o u l d be t o c a l c u l a t e averages f o r

s u b s e r i e s o f l e n g t h n '

u j = ave ( v a l u e s i n s u b s e r i e s j ) j = l , . . . , p

u - = ave ( v a l u e s i n s u b s e r i e s j ) j = p + 1 ; n modulo n / + 0

where p i s g i v e n by

n - p n ' = n modulo n x

( i . e . , t he r e m a i n d e r on d i v i d i n g n by n ' )

The c o r r e s p o n d i n g d i f f e r e n c e s a re g i v e n by

A u j = u j + 1 - U j

V a r y i n g s u b s e r i e s l e n g t h n t h e n amounts t o a v e r a g i n g

and d i f f e r e n c i n g a t d i f f e r e n t s c a l e s .

I n t h e case o f h i l l s l o p e s , s u b s e r i e s o f ang les o f

l e n g t h n ' do n o t i n g e n e r a l c o r r e s p o n d t o measured l e n g t h s

o f n ' t i m e s t h e o r i g i n a l c o n s t a n t measured l e n g t h , because

s l o p e s may be l o c a l l y convex o r concave . I n a d d i t i o n , t h e

a p p r o p r i a t e a v e r a g i n g f u n c t i o n appears t o be

9 / £ cos- B )

- 202 -

A v e r a g i n g a n d d i f f e r e n c i n g w e r e c a r r i e d o u t f o r

s u b s e r i e s l e n g t h s n / o f 1 , 2 , 5 a n d 10 ( c o r r e s p o n d i n g t o

t o t a l m e a s u r e d l e n g t h s o f 1 . 5 2 m , 3 - 0 4 , 7 - 6 0 , 1 5 . 2 0 m ) .

I n i n t e r p r e t i n g t h e r e s u l t s t w o p o i n t s s h o u l d be b o r n e

i n m i n d . F i r s t l y , as n ' i n c r e a s e s , t h e n u m b e r o f v a l u e s

a v a i l a b l e t o c o m p i l e e a c h f r e q u e n c y d i s t r i b u t i o n d e c r e a s e s

( i t i s a p p r o x i m a t e l y n / n / ) , a n d r e s u l t s i n e v i t a b l y r e f l e c t

a c o m b i n a t i o n o f s c a l e v a r i a t i o n a n d s a m p l i n g v a r i a t i o n ,

t h e l a t t e r i n c r e a s i n g w i t h s u b s e r i e s l e n g t h . S e c o n d l y ,

s i n c e o r i g i n a l d a t a w e r e m e a s u r e d t o 0 . 5 ° , d i f f e r e n c e s o f

t h i s o r d e r a r e i n n o s e n s e s u r p r i s i n g .

R e s u l t s f o r a n g l e a r e g i v e n i n 6U f o r s e l e c t e d

m e a s u r e s . The p a t t e r n s shown a r e b r o a d l y as w o u l d be

e x p e c t e d . M i n i m a a n d m a x i m a a p p r o a c h e a c h o t h e r as

s u b s e r i e s l e n g t h i n c r e a s e s a n d bumps a n d r u t s a r e a v e r a g e d

o u t . (The m a x i m a f o r TA a r e an a p p a r e n t e x c e p t i o n , b u t

t h e s t a b i l i t y i n t h i s case r e f l e c t s a s t r i n g o f 12 m e a s u r e d

l e n g t h s o n T a r n H o l e C r a g w h i c h c o u l d n o t be t r a v e r s e d w i t h

a p a n t o m e t e r : an Abney l e v e l was u s e d t o e s t i m a t e t h e

o v e r a l l a n g l e , a n d t h e d i s t a n c e o v e r t h e s t r i n g was

m e a s u r e d b y t a p e ) . These f i g u r e s show q u i t e c l e a r l y

t h a t maximum a n g l e , f a v o u r e d b y S t r a h l e r ( 1 9 5 0 ) a n d o t h e r s ,

d e p e n d s s t r o n g l y o n m e a s u r e d l e n g t h . On a l l p r o f i l e s

e x c e p t T A , maximum s l o p e s f o r xif = 10 a r e 7 t o 25 d e g r e e s

g e n t l e r t h a n f o r n ' = 1 . More t h o u g h t m u s t be g i v e n t o

t h i s p r o b l e m o f s c a l e v a r i a t i o n b y t h o s e s e e k i n g s p e c i a l

p r o c e s s i n t e r p r e t a t i o n s f o r maximum a n g l e s . I t i s a l s o

- 203 -A n g l e f r e q u e n c y d i s t r i b u t i o n s : v a r i a t i o n o f summary m e a s u r e s w i t h s c a l e

P r o f i l e n ' m i n l o q med u p q max m i d s p r e a d

BO 1 - 1 0 . 0 3 . 0 5 . 0 8 . 0 4 1 . 0 5 . 0 2 - 3 . 0 3 . 5 5 . 5 7 - 7 3 6 . 0 4 . 2 5 0 . 3 3 .8 5 . 4 7 . 9 23 .2 4 . 1

10 0 . 8 3 . 7 5 . 7 8 . 1 16 .0 4 . 4

ST l - 1 4 . 5 5 . 0 7 . 0 8 .5 2 3 . 5 3 . 5 2 - 3 . 7 5 . 2 6 . 7 8 .2 2 0 . 7 3 . 0 5 1.3 5 . 6 6 . 6 8 . 1 18 .5 2 . 5

10 2 . 9 5 . 9 6 . 7 7 . 8 1 3 . 5 1.9

PR 1 - 1 . 5 4 . 5 6 . 5 8 .7 32 .5 4 . 2 2 1 .0 4 . 7 6 . 2 8 .9 2 0 . 0 4 . 2 5 2 . 5 4 .8 7 . 0 8 .7 1 4 . 4 3 . 9

10 3 . 4 4 . 7 7 . 1 8 .5 1 3 . 4 3 .8

PA 1 - 1 7 . 5 4 . 5 6 . 5 9 - 0 4 5 . 0 4 . 5 2 - 8 . 2 4 . 7 6 . 5 8 .5 4 4 . 8 3 . 8 5 - 1 . 7 4 . 8 6 . 5 7 . 8 3 6 . 1 3 . 0

10 0 . 2 4 . 9 6 . 5 8 . 0 3 5 . 7 3 . 1

PA 1 - 2 1 . 0 5 . 0 7 . 5 10.5 5 0 . 0 5 . 5 2 - 9 . 2 5 . 5 7 . 5 10 .0 4 4 . 2 4 . 5 5 - 1 . 0 5 .6 8 .0 9 . 5 3 6 . 6 3 . 9

10 1.7 5 . 5 8 . 1 9 . 4 3 0 . 9 3-9

LA 1 - 1 1 . 0 5 . 0 9 . 0 1 2 . 5 2 5 . 5 7 . 5 2 - 2 . 7 5 . 2 9 . 2 1 2 . 5 2 2 . 0 7 . 3 5 2 . 2 5 . 2 9 . 7 1 2 . 9 18 .0 7 . 7

10 3 . 2 4 . 8 9 . 8 1 2 . 9 16 .2 8 . 1

TR 1 - 5 . 0 2 . 5 6 . 0 1 5 . 0 4 6 . 0 1 2 . 5 2 - 2 . 0 2 . 5 6 . 0 1 4 . 5 4 3 . 2 1 2 . 0 5 - 0 . 3 2 . 5 6 . 1 1 5 - 3 3 6 . 0 1 2 . 8

10 0 . 1 2 . 7 6 . 2 18 .0 3 3 . 9 1 5 . 3

AR 1 - 2 . 0 5 . 0 8 .5 1 3 . 0 4 7 . 5 8 . 0 2 1 .2 5 -5 8 .9 1 3 . 5 4 3 . 2 8 . 0 5 2 . 3 5 . 7 7 . 9 1 3 - 7 4 1 . 6 8 . 0

10 3 . 4 6 . 4 8 . 1 1 4 . 6 3 7 . 7 8 .2

TA 1 - 1 1 . 5 5 . 5 10 .0 16 .0 5 0 . 0 10 .5 2 - 3 . 5 5 . 4 9 . 7 16 .0 5 0 . 0 10 .6 5 - 2 . 2 5 . 8 9 . 3 16 .2 5 0 . 0 10 .4

10 0 . 1 6 . 9 10 .0 16.8 5 0 . 0 9 . 9

HO 1 - 2 1 . 0 8 . 0 1 2 . 5 1 9 . 0 9 0 . 0 1 1 . 0 2 - 1 5 . 2 8 . 0 1 2 . 7 18.8 6 5 . 0 10.8 5 - 7 . 1 8 . 0 1 3 - 7 18 .2 5 2 . 7 10 .2

10 1.2 7 . 8 1 3 . 8 1 7 . 1 4 6 . 5 9 . 3

TO 1 -10 .5 1 3 . 5 18 .5 2 5 . 0 3 9 . 0 1 1 . 5 2 - 3 . 0 1 3 . 7 18 .7 2 4 . 7 3 5 . 0 1 1 . 0 5 3 . 2 1 3 . 8 18 .6 2 5 . 0 3 4 . 0 1 1 . 2

10 7 . 9 1 3 . 7 1 9 . 1 2 4 . 5 3 1 . 7 10.8

- 204 -

i m p o r t a n t t o n o t e f r o m t h e s e r e s u l t s t h a t o n l y on HO a n d

TA a r e a n g l e s o f 4 0 ° o r more o t h e r t h a n l o c a l i s e d . The

m i n i m a a r e o f l e s s i n t e r e s t : t h e y g e n e r a l l y a r i s e f r o m

l o c a l s l o p e r e v e r s a l s , o f t e n o f human o r i g i n ( c f . A p p e n d i x I ) .

By c o n t r a s t m e d i a n a n d m i d s p r e a d a r e r e l a t i v e l y s t a b l e

m e a s u r e s , as w o u l d b e e x p e c t e d f r o m t h e i r r e s i s t a n t

c h a r a c t e r i s t i c s .

R e s u l t s a r e d i s p l a y e d g r a p h i c a l l y f o r f o u r s e l e c t e d

p r o f i l e s , one f r o m e a c h o f f o u r g r o u p s d i s t i n g u i s h e d e a r l i e r

( B O , L A , AR, TO) i n 6V a n d 6W. The m a r k e d c o n t r a s t b e t w e e n

u n s t a b l e m i n a n d max ( 6 V ) a n d s t a b l e med a n d m i d s p r e a d (6W)

i s v e r y c l e a r .

R e s u l t s f o r c u r v a t u r e ( 6 X ) show a g a i n t h a t e x t r e m e s

a r e u n s t a b l e a n d t h e m e d i a n s t a b l e . T h e r e i s a l s o a

s y s t e m a t i c t e n d e n c y f o r q u a r t i l e s t o a p p r o a c h t h e m e d i a n ,

w h i c h i s e x p e c t a b l e o n g e o m e t r i c g r o u n d s ( n o t e a g a i n t h a t

t h e c u r v a t u r e m e a s u r e u s e d h e r e i s A 0 ; n o a c c o u n t has b e e n

t a k e n o f v a r y i n g Z \ s ) .

A v e r a g i n g a n d d i f f e r e n c i n g t h u s a p p e a r s t o be a

u s e f u l m e t h o d o f e x p l o r i n g s c a l e v a r i a t i o n , a n d o f

i d e n t i f y i n g s c a l e - v a r i a b l e a n d s c a l e - c o n s t a n t m o r p h o m e t r i c

m e a s u r e s . The q u e s t i o n o f s c a l e v a r i a t i o n w i l l a r i s e

a g a i n i n o t h e r ways i n t h e f o l l o w i n g t w o c h a p t e r s .

- 205 -

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- 206 -

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- 207 -

u r v a t u r e f r e q u e n c y d i s t r i b u t i o n s ; v a r i a t i o n o f summary m e a s u r e s w i t h s c a l e

P r o f i l e n ' m i n l o q med u p q max m i d s p r e a d

BO 1 -39-5 -2 .5 0.0 3.0 20.0 5.5 2 -26.2 -2 .0 0.2 2.1 20.5 4.1 5 -17.2 -0 .8 0.1 1.1 1 4 . 3 1.9

10 -10.0 -0 .7 0.2 1.2 6.5 1.9

ST 1 -37.0 -2 .0 0.0 2.5 28.5 4.5 2 -18.0 -1 .5 0.0 1.7 9.0 3.2 5 -6.4 -1 .2 0.1 1.1 10.0 3.3

10 -4 .9 -0 .8 0.2 1.0 7.5 1.8

PR 1 -25.0 -2 .0 0.0 1.5 19.0 3.5 2 -10.7 -1 .0 0.5 1.2 7.5 2.2 5 -6.4 -0 .7 0.3 1.1 3.7 1.8

10 -5 .4 -0 .5 0.5 1.2 2.8 1.7

PA 1 - 3 1 . 5 -2 .0 0.0 2.0 19.5 4.0 2 - 2 4 . 5 -1 .5 0.0 1.5 17.5 3.0 5 -27.3 -0 .8 0.0 1.5 10.0 2.3

10 -27.7 -0 .7 0.4 1.3 12.5 2.0

PA 1 -45.5 -2 .5 0.0 2.5 22.0 5.0 2 - 1 4 . 7 -1 .7 0.0 1.7 21.7 3.4 5 -11.4 -0 .8 0.2 1.1 5.9 1.9

10 -7 .5 -1 .0 0.7 1.5 8.3 2.5

LA 1 -23.0 -1 .5 0.0 1.5 20.0 3.0 2 -11.2 -1 .0 0.0 1.2 13.2 2.2 5 -3.6 -0.9 0.2 0.9 6.6 1.8

10 -3 .3 -0.6 0.3 0.9 7.3 1.5

TR 1 - 2 4 . 0 -2 .5 0.0 2.5 25.5 5.0 2 -16.7 -1 .7 0.0 2.0 19.2 3.7 5 -11.0 -1.2 0.2 1.5 19.4 2.7

10 -7-9 -1 .3 0.5 1.9 13.7 3.2

AR 1 -32.0 -2 .5 0.0 2.0 16.5 4.5 2 -19.5 -1 .5 0.3 1.5 10.7 3.0 5 -21.2 -1 .2 0.3 2.0 12.0 3.2

10 -17.3 -0 .9 0.3 2.1 10.1 3,0

TA 1 -30.5 -3 .0 0.0 3.0 20.5 6.0 2 -15.0 -2 .0 0.0 2.3 20.0 4.3 5 - 1 4 . 6 -1.6 0.0 1.7 18.1 3.3

10 -20.0 -1 .4 0.2 1.5 26.4 2.9

HO 1 -60.0 -3 .5 -0 .5 3.5 71.0 7.0 2 -29.0 -3 .2 -0 .3 3.0 35.5 6.2 5 -16.4 -2 .8 -0.6 3.4 20.4 6.2

10 -29.9 -2.6 -0 .5 2.2 21.2 4.8

TO 1 -27-5 -4 .0 0.5 3-7 27.5 7.7 2 -20.5 -2 .5 0.0 3.2 10.7 5.7 5 - 1 3 . 6 -1 .8 0.2 1.8 9.4 3.6

10 -9 .0 -2 .5 0.1 2.9 12.8 5.4

- 208 -

6.4. Summary

( i ) P r o f i l e f o r m i s d i s c u s s e d f o r t h e p r o f i l e s m e a s u r e d

i n t h e f i e l d i n t e r m s o f p r o f i l e d i m e n s i o n s a n d p r o f i l e

s h a p e . A f o u r - f o l d g r o u p i n g i s o u t l i n e d , w h i c h i s c l o s e l y

r e l a t e d t o v a r i a t i o n s i n b e d r o c k g e o l o g y ( 6 . 1 ) .

( i i ) A n g l e a n d c u r v a t u r e f r e q u e n c y d i s t r i b u t i o n s a r e

s u m m a r i s e d u s i n g q u a n t i l e - b a s e d m e a s u r e s w h i c h a r e c o n s i d e r e d

more a p p r o p r i a t e f o r d a t a c o n t a i n i n g w i l d o b s e r v a t i o n s .

M e d i a n a n d m i d s p r e a d o f a n g l e c a n be r e a d i l y i n t e r p r e t e d

i n t e r m s o f t h e f o u r - f o l d g r o u p i n g p r o p o s e d e a r l i e r ( 6 . 2 ) .

( i i i ) S p a t i a l a v e r a g i n g a n d d i f f e r e n c i n g o f a n g l e

s e r i e s t h r o w s l i g h t on t h e s c a l e v a r i a t i o n o f d i s t r i b u t i o n

summary m e a s u r e s . M i n i m u m a n d maximum a n g l e s a r e e x t r e m e l y

u n s t a b l e , b u t m e d i a n a n d m i d s p r e a d o f a n g l e s a r e

s a t i s f a c t o r i l y s t a b l e ( 6 . 3 ) .

- 209 -

6 • 5 . n o t a t i o n

d i n o r d i n a r y d e r i v a t i v e

i s u b s c r i p t

J s u b s c r i p t

1 m e a s u r e d l e n g t h

n n u m b e r o f o b s e r v a t i o n s

n ' n u m b e r i n s u b s e r i e s

P r e l a t e d t o n u m b e r o f s u b s e r i e s

s a r c l e n g t h

u v a l u e o f s e r i e s

u a v e r a g e v a l u e

X h o r i z o n t a l c o o r d i n a t e

*b s l o p e l e n g t h

z v e r t i c a l c o o r d i n a t e

z d s l o p e h e i g h t

A d i f f e r e n c e o p e r a t o r

9 a n g l e

9 a v e r a g e a n g l e

£ s u m m a t i o n o p e r a t o r

Mnemon ic s

ave ave r a g e

l o q l o w e r q u a r t i l e

max maximum

med m e d i a n

m i n m i n i m u m

P5 5% p o i n t

P95 95% p o i n t

upq u p p e r q u a r t i l e

- 2 1 0 -

C h a p t e r 7

AUTOCORRELATION ANALYSIS OF HILLSLOPE SERIES

S c h o o l m a s t e r : ' S u p p o s e x i s t h e n u m b e r o f s h e e p i n

t h e p r o b l e m ' . P u p i l : ' B u t , S i r , s u p p o s e x i s n o t

t h e n u m b e r o f s h e e p ' . [ I a s k e d P r o f . W i t t g e n s t e i n

was t h i s n o t a p r o f o u n d p h i l o s o p h i c a l j o k e , a n d he

s a i d i t w a s . ]

J . E . L i t t l e w o o d , A m a t h e m a t i c i a n ' s m i s c e l l a n y ,

p . 4 l .

7 . 1 The i d e a o f a u t o c o r r e l a t i o n

7 . 2 A u t o c o r r e l a t i o n a n d h i l l s l o p e p r o f i l e s

7 . 3 S t a t i o n a r i t y , n o n s t a t i o n a r i t y a n d a u t o c o r r e l a t i o n

7 . 4 C h o i c e o f e s t i m a t o r

7 . 5 E m p i r i c a l r e s u l t s

7 . 6 Summary

7 . 7 N o t a t i o n

- 2 1 1 -

7 . 1 The i d e a o f a u t o c o r r e l a t i o n

The i d e a o f a u t o c o r r e l a t i o n i s p r o b a b l y b e s t a p p r o a c h e d

f r o m t h e more w i d e l y - k n o w n i d e a o f c o r r e l a t i o n . The c o r r e l a t i o n

r b e t w e e n t w o v a r i a b l e s , u a n d v s a y , i s a m e a s u r e w i t h t h e

f o l l o w i n g g e n e r a l p r o p e r t i e s :

( i i ) r i s p o s i t i v e i f u a n d v a r e a s s o c i a t e d

d i r e c t l y , a n d n e g a t i v e i f t h e y a r e a s s o c i a t e d

i n v e r s e l y .

( i i i ) The a b s o l u t e v a l u e o f r i s 0 i f u a n d v a r e

n o t a s s o c i a t e d ( u n c o r r e l a t e d ) , 1 i f t h e y a r e

p e r f e c t l y a s s o c i a t e d , a n d b e t w e e n 0 a n d 1 f o r

i n t e r m e d i a t e c a s e s .

P a r t i c u l a r m e a s u r e s o f c o r r e l a t i o n may be d i s t i n g u i s h e d

a c c o r d i n g t o t h e p r e c i s e c r i t e r i o n o f ' a s s o c i a t i o n ' w h i c h i s

e m p l o y e d . I f l i n e a r a s s o c i a t i o n i s b e i n g c o n s i d e r e d , t h e n

i t i s n a t u r a l t o t a k e t h e P e a r s o n p r o d u c t - m o m e n t c o r r e l a t i o n

c o e f f i c i e n t

w h e r e c o v , v a r a n d s t d d e n o t e c o v a r i a n c e , v a r i a n c e a n d

s t a n d a r d d e v i a t i o n .

F o r a s e t o f o b s e r v a t i o n s u ^ , v ^ ; i = l , . . . , n we h a v e ,

c a n c e l l i n g a d i v i s o r common t o n u m e r a t o r a n d d e n o m i n a t o r ,

( i ) - I ^ ^ ^ I

r CoV (ui,v) cov ( a . v )

sU ( a.) SU 0) J[varCa) V2ir(v)]

7[(E(- -ar) (E(v l -vr) ] r

- 212 -

w h e r e u , v a r e means

P u t t i n g - u = a i , v-j_ - v = b-^ t h e n

r =

F o r any r e a l n u m b e r s a i , b ^ ; i = l , • • • » n t h e C a u c h y - S c h w a r z

i n e q u a l i t y ( e . g . S t e p h e n s o n , 1 9 7 1 , 1 4 ) g i v e s

whence

a n d

E l : k . t > t -

S * i *>; t3J r

The P e a r s o n c o r r e l a t i o n c o e f f i c i e n t i s n o t t h e o n l y

p o s s i b l e m e a s u r e o f c o r r e l a t i o n ] o t h e r c r i t e r i a o f

a s s o c i a t i o n l e a d t o o t h e r m e a s u r e s . I f m o n o t o n i c a s s o c i a t i o n

i s b e i n g c o n s i d e r e d , t h e n i t i s n a t u r a l t o c o m p u t e m e a s u r e s

- 213 -

b a s e d on r a n k s o n l y , s u c h as S p e a r m a n ' s r a n k c o r r e l a t i o n

c o e f f i c i e n t .

A u t o c o r r e l a t i o n i n v o l v e s c o r r e l a t i n g one v a r i a b l e

( s a y u ) w i t h i t s e l f , r a t h e r t h a n t w o v a r i a b l e s ( s a y u a n d

v ) w i t h e a c h o t h e r . I n p a r t i c u l a r , t a k e t h e ca se o f a o n e -

d i m e n s i o n a l s e r i e s u-^ ; i = l , . . . , n w h e r e t h e s u b s c r i p t s i

r e f e r t o p o s i t i o n i n a s e q u e n c e , w h e t h e r t e m p o r a l o r s p a t i a l .

( T h e case o f o b s e r v a t i o n s i n t w o o r more d i m e n s i o n s w i l l

n o t be c o n s i d e r e d h e r e . I n t h e t w o - d i m e n s i o n a l c a s e ,

a u t o c o r r e l a t i o n can be d e f i n e d b y a s t r a i g h t f o r w a r d a n a l o g y

w i t h t h e o n e - d i m e n s i o n a l ca se i f o b s e r v a t i o n s r e f e r t o a

r e g u l a r l a t t i c e : a d i f f e r e n t a p p r o a c h i s n e c e s s a r y f o r a n

i r r e g u l a r l a t t i c e , f o r w h i c h see C l i f f a n d O r d , 1 9 7 3 . )

A u t o c o r r e l a t i o n i s t h e c o r r e l a t i o n b e t w e e n a s e r i e s

a n d i t s e l f d i s p l a c e d . Thus by a n a l o g y w i t h t h e P e a r s o n

c o e f f i c i e n t

r = CQV(U. L . V l ^ i c ) k J [ w O t ) varOL + (J]

H e r e k i s t h e l a g o r s p a c i n g b e t w e e n o b s e r v a t i o n s u ^ ; i = l ,

. . . , n - k , w h i c h a r e b e i n g c o r r e l a t e d w i t h o b s e r v a t i o n s

u i + k i i + k = l + k , . . . , n . The c o v a r i a n c e i s k n o w n i n t h i s

ca se as t h e a u t o c o v a r i a n c e . F o r k = 0 , r ^ = r Q , t h e

c o r r e l a t i o n b e t w e e n a s e r i e s a n d i t s e l f , w h i c h i s e x a c t l y 1 .

F o r k = 0 , 1 , 2 , 3 , . . . » we h a v e t h e a u t o c o r r e l a t i o n f u n c t i o n ,

- 214 -

T Q , r - p v^t r - j * . . . . A p l o t o f t h e a u t o c o r r e l a t i o n f u n c t i o n

i s a c o r r e l o g r a m .

Some b a s i c t e x t s on a u t o c o r r e l a t i o n a n d r e l a t e d s u b j e c t s

a r e as f o l l o w s :

( i ) A g t e r b e r g ( 1 9 7 4 ) a n d S c h w a r z a c h e r ( 1 9 7 5 ) , r e l i a b l e

t e x t s i n t e n d e d f o r e a r t h s c i e n t i s t s ;

( i i ) K e n d a l l ( 1 9 7 3 ) a n d C h a t f i e l d ( 1 9 7 5 ) , g e n e r a l t e x t s

on t i m e s e r i e s a n a l y s i s .

7 . 2 A u t o c o r r e l a t i o n a n d h i l l s l o p e p r o f i l e s

A u t o c o r r e l a t i o n a n a l y s i s o f s p a t i a l s e r i e s i s now an

a c c e p t e d t e c h n i q u e i n g e o m o r p h o l o g y , e i t h e r i n i s o l a t i o n , o r

more u s u a l l y i n c o n j u n c t i o n w i t h s p e c t r a l a n a l y s i s o r m o d e l

f i t t i n g . I t h a s b e e n u s e d , f o r e x a m p l e , i n t h e s t u d y o f

s t r e a m a n d g u l l y p r o f i l e s ( M e l t o n , 1 9 6 2 ; B e n n e t t , 1 9 7 6 ;

R i c h a r d s , 1 9 7 6 ; A l e x a n d e r , 1 9 7 7 ) , o f s t r e a m p l a n s ( S p e i g h t ,

1 9 6 5 , 1 9 6 7 ; F e r g u s o n , 1 9 7 5 , 1 9 7 6 ) , a n d o f t o p o g r a p h i c

p r o f i l e s ( E v a n s , 1 9 7 2 , 3 3 - 6 ; D r e w r y , 1 9 7 5 ; P i k e a n d Rozema,

1 9 7 5 ; W e b s t e r , 1 9 7 7 ) . E m p i r i c a l a u t o c o r r e l a t i o n f u n c t i o n s

w e r e c o m p u t e d f r o m h i l l s l o p e p r o f i l e d a t a f i r s t b y

N i e u w e n h u i s a n d v a n den B e r g ( 1 9 7 1 ) , a n d s u b s e q u e n t l y b y

T h o r n e s ( 1 9 7 2 , 1 9 7 3 ) .

Why s t u d y a u t o c o r r e l a t i o n p r o p e r t i e s o f h i l l s l o p e

p r o f i l e d a t a ? T h e r e seem t o be f o u r m a i n r e a s o n s .

F i r s t l y , many s t a n d a r d s t a t i s t i c a l m e t h o d s assume

i n d e p e n d e n c e o f o b s e r v a t i o n s . I f a v a r i a b l e v i s i n d e p e n d e n t

- 215 -

o f u , t h e n t h e c o n d i t i o n a l p r o b a b i l i t y d i s t r i b u t i o n p r ( v j u )

i s i d e n t i c a l t o t h e m a r g i n a l d i s t r i b u t i o n p r ( v ) . L o o s e l y ,

k n o w i n g u p r o v i d e s n o e x t r a i n f o r m a t i o n w h i c h w o u l d h e l p i n

p r e d i c t i n g v . I f v i s i n d e p e n d e n t o f u , t h e n u a n d v a r e

u n c o r r e l a t e d , i . e . r ( u , v ) = 0. (The c o n v e r s e i s n o t

n e c e s s a r i l y t r u e , as may be s e e n f r o m a c o u n t e r e x a m p l e

( P l a c k e t t , 1971, 65): i f u i s s y m m e t r i c a l l y d i s t r i b u t e d

a b o u t 0, t h e n v = | u | a n d u a r e u n c o r r e l a t e d , e v e n t h o u g h

v may be p r e d i c t e d e x a c t l y f r o m u . )

H o w e v e r , s p a t i a l d a t a t e n d t o be s t r o n g l y a u t o c o r r e l a t e d ,

w h i c h i m p l i e s t h a t t a b u l a t e d s a m p l i n g d i s t r i b u t i o n s o f

v a r i o u s s t a n d a r d t e s t s t a t i s t i c s s u c h as S t u d e n t ' s t a n d

F i s h e r ' s F a r e d e f i n i t e l y i n v a l i d ( c f . M o r a n , 1973; H e p p l e ,

1974; B o x , 1976; H a g g e t t e t _ a l . , 1977, C h s . 10-11).

N i e u w e n h u i s a n d v a n den B e r g (1971) w e r e t h e f i r s t t o

d i s c u s s t h e p r o b l e m o f i n v a l i d i t y o f s t a n d a r d m e t h o d s i n a

h i l l s l o p e c o n t e x t . T h e i r r e s u l t s s u g g e s t e d t h a t a 10%

s a m p l e o f t a n g e n t s m e a s u r e d o v e r a d j a c e n t 10 m m e a s u r e d

l e n g t h s w o u l d c o n s i s t o f a p p r o x i m a t e l y i n d e p e n d e n t o b s e r v a t i o n s .

Hence t h e s t r e n g t h o f a u t o c o r r e l a t i o n p r e s e n t i n

h i l l s l o p e d a t a a f f e c t s t h e v a l i d i t y o f some s t a n d a r d

s t a t i s t i c a l t e s t s .

S e c o n d l y , i t " i s p o s s i b l e t h a t a u t o c o r r e l a t i o n f u n c t i o n s

a n d / o r a u t o c o v a r i a n c e f u n c t i o n s m i g h t s e r v e as d e s c r i p t o r s

o f s u r f a c e r o u g h n e s s a t d i f f e r e n t s c a l e s . I t i s a m a t t e r

o f common o b s e r v a t i o n t h a t s l o p e s a r e r o u g h , d i s p l a y i n g

- 216 -

' b u m p s ' , ' r u t s ' , ' m i c r o r e l i e f ' a n d e v e n ' n a n o r e l i e f ( Y o u n g ,

1972, 201). A u t o c o r r e l a t i o n w o u l d seem a p o s s i b l e i n v e r s e

m e a s u r e o f r o u g h n e s s , f o r t h e e x t e n t . to w h i c h v a l u e s o f

s u r f a c e g e o m e t r y c a n be p r e d i c t e d f r o m n e i g h b o u r i n g v a l u e s

i s i n v e r s e l y r e l a t e d t o s u r f a c e r o u g h n e s s .

T h i r d l y , some s t o c h a s t i c m o d e l s ( r e v i e w e d i n C h . 3

a b o v e ) y i e l d p r e d i c t i o n s o f a u t o c o r r e l a t i o n f u n c t i o n s w h i c h

c a n be t e s t e d e m p i r i c a l l y .

F o u r t h l y , m o d e l s w h i c h a r e c o m b i n a t i o n s o f d e t e r m i n i s t i c

a n d s t o c h a s t i c c o m p o n e n t s c a n be f i t t e d b y a s s u m i n g i n d e p e n d e n c e

o f s t o c h a s t i c e r r o r s . The r e s i d u a l s , o r d i f f e r e n c e s b e t w e e n

a c t u a l a n d p r e d i c t e d v a l u e s , s h o u l d t h e n be e x a m i n e d f o r

a u t o c o r r e l a t i o n ( P i t t y , 1970; c f . C h . 9 b e l o w ) .

The a u t o c o r r e l a t i o n p r o p e r t i e s o f h i l l s l o p e p r o f i l e

d a t a a r e t h u s o f i n t e r e s t f o r s e v e r a l r e a s o n s , a n d c l e a r l y

m e r i t f u r t h e r e x p l o r a t i o n .

7.3. S t a t i o n a r i t y , n o n s t a t i o n a r i t y a n d a u t o c o r r e l a t i o n

C a l c u l a t e d a u t o c o r r e l a t i o n s can be t a k e n s i m p l y as

d e s c r i p t i v e m e a s u r e s . F o r e x a m p l e , any m e a s u r e o f t h e f o r m

w h i c h s a t i s f i e s J r ^ I has a n a t u r a l g e o m e t r i c i n t e r p r e t a t i o n

n

T mm •"1=1 1=1

as t h e c o s i n e o f t h e a n g l e b e t w e e n t h e v e c t o r s ( a 8 L . ) n

a n d (b , . . . , b ) . I t c o u l d be a r g u e d t h a t a c t u a l v a l u e s

- 217 -

o f a u t o c o r r e l a t i o n s a r e t h e o n l y s u b j e c t o f i n t e r e s t : w h a t

v a l u e s m i g h t h a v e b e e n i s a h y p o t h e t i c a l i s s u e .

N e v e r t h e l e s s , one s t a n d a r d a p p r o a c h t o a u t o c o r r e l a t i o n

i s c e n t r e d on s u c h i n f e r e n t i a l q u e s t i o n s . The a s s u m p t i o n

w h i c h i s made i s t h a t t h e . e m p i r i c a l s e r i e s u ^ ; i = l , . . . , n

i s one p o s s i b l e r e a l i s a t i o n o f a c h a n c e o r s t o c h a s t i c

p r o c e s s U . The i n f e r e n t i a l p r o b l e m i s t o e s t i m a t e t h e

p a r a m e t e r s o f t h i s g e n e r a t i n g p r o c e s s .

A l m o s t a l l t h e t h e o r y b e h i n d s u c h an a p p r o a c h

c o n s i d e r s t h e s p e c i a l case o f s t a t i o n a r y s t o c h a s t i c p r o c e s s e s

w h i c h a r e d e f i n e d b y t h e c h a r a c t e r i s t i c t h a t e x p e c t a t i o n s

o f s t a t i s t i c s a r e e v e r y w h e r e t h e same: t h a t i s t o s a y , t h e

a v e r a g e s o v e r a l l p o s s i b l e r e a l i s a t i o n s a r e c o n s t a n t i n t i m e

a n d / o r s p a c e . I n p a r t i c u l a r , a p r o c e s s i s s e c o n d - o r d e r

s t a t i o n a r y i f e x p e c t a t i o n s o f f i r s t - o r d e r a n d s e c o n d - o r d e r

s t a t i s t i c s a r e i n v a r i a n t u n d e r t r a n s l a t i o n o f t h e o r i g i n .

( H i g h e r o r d e r s t a t i s t i c s m i g h t v a r y . ) Such a p r o c e s s i s

d e f i n e d b y c o n s t a n t mean

Here E i s t h e e x p e c t a t i o n o p e r a t o r ( s e e , f o r e x a m p l e , W h i t t l e

1 9 7 0 , C h . 2 ) .

E [ u ] = f t

a n d a u t o c o v a r i a n c e f u n c t i o n

r-n - \ u

- 218 -

P u t t i n g k = 0 we deduce constant variance

To

and s c a l i n g we deduce constant a u t o c o r r e l a t i o n f u n c t i o n

Synonyms f o r second-order s t a t i o n a r i t y are covariance, mean square, weak, wide-sense s t a t i o n a r i t y .

I t i s usually argued t h a t i t only makes sense t o compute the a u t o c o r r e l a t i o n f u n c t i o n i f the generating process i s second-order s t a t i o n a r y ( i n a g e o l o g i c a l context see Agterberg, 1 9 7 ^ , 3 1 ^ - 5 ; Schwarzacher, 1 9 7 5 , 1 6 6 ) .

Otherwise the a u t o c o r r e l a t i o n f u n c t i o n i s e f f e c t i v e l y a set of v a r i a b l e s , not a set of parameters, and i t i s indeed extremely d o u b t f u l whether there i s an e s t i m a t i o n problem i n the c l a s s i c a l sense. I f t h i s i s t r u e , i t i s very r e s t r i c t i v e : f o r i f s t a t i o n a r i t y i s a requirement, only s t r a i g h t slopes can be s t u d i e d by a u t o c o r r e l a t i o n analysis of angle data, and only slopes o f constant curvature ( s i n g l e c o n v exities or s i n g l e c o n c a v i t i e s ) by a u t o c o r r e l a t i o n analys of curvature data. An approach so r e s t r i c t e d would be geomorphologically useless.

However, attempts t o step outside the framework provided by s t a t i o n a r i t y face a general problem explained c l e a r l y by W h i t t l e ( 1 9 6 3 j 8 3 ) : ' I n dropping the assumption of s t a t i o n a r i t y , one i s l e f t w i t h scarcely any r e s t r i c t i o n

T-

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upon one's model. For t h i s reason, i t i s a l l the more d i f f i c u l t t o specify a model, or even t o specify some of the s t a t i s t i c a l p r o p e r t i e s o f the v a r i a t e s (such as f i r s t and second moments)'.

This view i s perhaps a l i t t l e conservative. Independence or dependence of observations i s an important i s s u e , whether or not there i s any question of a s t a t i o n a r y generating process. E m p i r i c a l a u t o c o r r e l a t i o n f u n c t i o n s might be o f some use as averaged d e s c r i p t o r s even i n nonstationary s i t u a t i o n s . And i n c r e a s i n g i n t e r e s t i s being shown by s t a t i s t i c i a n s i n nonstationary processes, notably i n the class of ARIMA processes discussed by Box and Jenkins ( 1 9 7 6 ) .

Nevertheless the standard response t o n o n s t a t i o n a r i t y i s t o operate on the e m p i r i c a l series t o produce a new, approximately s t a t i o n a r y , s e r i e s , usually by d i v i d i n g i t i n t o nonstationary and s t a t i o n a r y components. Detrending by polynomial approximation, v a r i a t e d i f f e r e n c i n g , the a p p l i c a t i o n of moving averages, demodulation and remodulation, and v a r i a n c e - s t a b i l i s i n g transformations have been the main methods used. None o f these are u n i v e r s a l l y e f f e c t i v e or free from secondary complications but i n processing geomorphological series the most important s i n g l e issue must be the p h y s i c a l r a t i o n a l e o f any method f o r producing approximate s t a t i o n a r i t y . For t h i s reason a u t o c o r r e l a t i o n analysis f i n d s i t s major a p p l i c a t i o n i n analysing r e s i d u a l s from process-based or process re l e v a n t models: any a r b i t r a r y

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s p e c i f i c a t i o n of t r e n d i s l i k e l y t o produce geomorphologically i r r e l e v a n t r e s u l t s . (However, f i r s t d i f f e r e n c i n g i s of i n t e r e s t , because curvature deserves a t t e n t i o n i n i t s own r i g h t . )

The sampling d i s t r i b u t i o n theory o f a u t o c o r r e l a t i o n functions of s t a t i o n a r y processes has been e x t e n s i v e l y i n v e s t i g a t e d . One approximate but f a i r l y robust r e s u l t f o r the n u l l case Pk. = 0 (k f a i r l y small) i s

Both bias and variance are 0 ("j^"). Bias may thus be neglected i n p r a c t i c e f o r n 2 0 0 . This a n a l y t i c a l r e s u l t i s broadly supported by Monte Carlo sampling experiments (Cox, 1 9 6 6 ; W a l l i s and O'Connell, 1 9 7 2 ) .

(The term s t a t i o n a r i t y i s used here t o denote invariance under t r a n s l a t i o n of the o r i g i n , r e f e r r i n g u n iformly t o temporal, s p a t i a l and t e m p o r a l - s p a t i a l s t o c h a s t i c processes. Note, however, t h a t homogeneity i s used by some authors f o r the s p a t i a l case. A f u r t h e r p o i n t of importance i s t h a t processes i n two or more dimensions may also be chara c t e r i s e d by the d i f f e r e n t p roperty of i s o t r o p y , invariance under r o t a t i o n of coordinate axes.)

7 . 4 Choice of estimator

Pour d i f f e r e n t estimators o f a u t o c o r r e l a t i o n are widely used. D i f f e r e n t means are d i s t i n g u i s h e d below by

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l £ ^ " • r i ^; , the mean of the whole series

U,' = — 52 , the mean of leading values

77* = — L V u.- , * the mean of lagging values

n - k £ '-He ( 1 ) (e.g. Kendall, 1 9 7 3 , 40; Schwarzacher, 1 9 7 5 , 164)

r. = — ^

This i s the exact analogue of the Pearson c o e f f i c i e n t , and, from the Cauchy-Schwarz i n e q u a l i t y , s a t i s f i e s — I T ^ |

(2) (e.g. Granger and Hatanaka, 1 9 6 4 , 8 ; Quimpo, 1 9 6 8 , 367)

1= 1

I n t h i s case the denominator only contains one variance term, the variance of the whole s e r i e s .

(3) (e.g.Kendall, 1973, 40; Agterberg, 1974, 315; Bath, 1974,

179; Drewry, 1975, 194; Richards, 1976, 77; Webster, 1977, 199)

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i-Hc r,

i-=l

I n t h i s case there i s a f u r t h e r s i m p l i f i c a t i o n : both terms

i n the covariance are r e f e r r e d to the mean of the whole

s e r i e s , r a t h e r than to separate means of l e a d i n g and lagging

v a l u e s .

(4) (e.g. Jenkins and Watts, 1968, 180-2; Pishman, 1 9 6 9 ,

82-3; C h a t f i e l d , 1975,24; Box and J e n k i n s , 1976, 32)

The f i n a l s i m p l i f i c a t i o n here i s t h a t the f a c t o r — has

been dropped.

What j u s t i f i c a t i o n could there be f o r using ( 2 ) , (3) or

(4) i n place of ( 1 ) ?

F i r s t l y , use of the g r o s s l y b i a s e d (M) i n p a r t i c u l a r

ensures p o s i t i v e estimates of variance spectrum o r d i n a t e s ;

otherwise meaningless negative estimates might be produced^;

This i s r e l e v a n t only i f s p e c t r a l a n a l y s i s i s being undertaken.

Secondly, the gross b i a s of (4) may be p r a c t i c a l l y

n eglected f o r k « n .

T h i r d l y , ( 2 ) , (3) and (4) may be c a l c u l a t e d more

q u i c k l y than ( 1 ) . This i s r e l e v a n t only i f computing

2 . ( i+fc

&) 1

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f a c i l i t i e s are not a v a i l a b l e .

F o u r t h l y , i t i s o f t e n asserted t h a t approximate estimators are more e f f i c i e n t ( i . e . have smaller variance) than the Pearson analogue estimator ( 1 ) , b a s i c a l l y because the denominator i s a constant, not a f u n c t i o n of l a g , and because the use of one mean f o r the whole serie s r a t h e r than separate means f o r leading and lagging values would reduce v a r i a b i l i t y . However, sampling experiments f o r s t a t i o n a r y processes suggest t h a t the gain may not be appreciable ( c f . Cox, 1966; W a l l i s and O'Connell, 1972; Lenton and Schaake, 1973 i n e s t i m a t i n g D i n n u l l and

Markov cases).

F i f t h l y , expectations o f ( 1 ) , (2) and (3) coincide under second-order s t a t i o n a r i t y . This i s the argument of computational s i m p l i c i t y i n another guise, and may be r e j e c t e d on the grounds t h a t e s t i m a t i o n should not assume second-order s t a t i o n a r i t y as a matter of course.

Note again t h a t only (1) s a t i s f i e s j v ^ J < i : the others may lead t o 'improper' estimates ( | > | ) , although i n p r a c t i c e t h i s i s l i k e l y only at long lags.

Examples of a u t o c o r r e l a t i o n analysis i n geomorphology (7A) include cases i n which estimators have not been s p e c i f i e d , cases i n which estimators have not been made completely c l e a r , and cases of i n c o r r e c t formulae. There does seem t o be a general preference f o r estimators (3) and CO, p a r t l y because several authors used s p e c t r a l a n a l y s i s .

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I t i s concluded t h a t the Pearson analogue est i m a t o r should be used, and a f o r t i o r i t h a t the chosen estimator should be s t a t e d and j u s t i f i e d . A safe procedure i s t o use both ( 1 ) and ( 3 ) : the d i f f e r e n c e between estimates i s a measure of second-order n o n s t a t i o n a r i t y , which i s o f i n t e r e s t i n i t s own r i g h t . Such a procedure i s also p r e f e r a b l e t o the elaborate and r a t h e r i m p r a c t i c a b l e t e s t s f o r n o n s t a t i o n a r i t y which have been proposed (e.g. P r i e s t l e y and Subba Rao, 1 9 6 9 ) .

7 . 5 E m p i r i c a l r e s u l t s

I n t h i s s e c t i o n e m p i r i c a l r e s u l t s are presented f o r a u t o c o r r e l a t i o n functions c a l c u l a t e d f o r each p r o f i l e , f o r angle and curvature s e r i e s , using two separate estimators f o r lags k = 0 ( 1 ) 5 0 . Various r u l e s o f thumb e x i s t f o r the minimum number of lags t o be computed, p a r t l y because a large number of a u t o c o r r e l a t i o n c o e f f i c i e n t s are u s u a l l y redundant, and p a r t l y because c o e f f i c i e n t s become i n c r e a s i n g l y unstable as k increases, since they are based on fewer p a i r s of values. Davis ( 1 9 7 3 , 2 3 6 ) , C h a t f i e l d ( 1 9 7 5 , 2 5 )

and Box and Jenkins ( 1 9 7 6 , 33) a l l recommended not going beyond n/4. The r u l e of thumb suggested here a f t e r some experimentation (which included computing a l l possible l a g s l ) i s t o compute up t o the smaller of 50 and f l o o r (n|4-). (For the f l o o r f u n c t i o n , see Ch. 6 . 2 above, or Iverson, 1 9 6 2 , 1 2 ) .

A u t o c o r r e l a t i o n s f o r lags k = 1 ( 1 ) 5 ( 5 ) 5 0 , f o r angle and c u r v a t u r e , f o r a l l p r o f i l e s and f o r 'Pearson' and 'abbreviated' e s t i m a t o r s , i . e . ( 1 ) and ( 3 ) of Ch. 7 . ^ , are

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displayed i n 7B, 7C, 7E and 7F. Differences between estimates are displayed i n 7D and 7G. These tables summarise 44 correlograms which cannot a l l be shown here. However, angle and curvature series and the r e s u l t i n g correlograms f o r the Pearson estimator are shown i n 7H t o 70 f o r two c o n t r a s t i n g p r o f i l e s , ST and AR.

I n i n t e r p r e t i n g these a u t o c o r r e l a t i o n s , two sets o f questions, geomorphological and s t a t i s t i c a l , must be considered. The most basic feature of 7B (Pearson f o r angle) i s the f a c t , expected on geomorphological grounds, t h a t angle a u t o c o r r e l a t i o n s do not i n general dampen t o zero. 143 out of 154 a u t o c o r r e l a t i o n s i n the t a b l e l i e outside the 0.99 confidence i n t e r v a l f o r P(t = 0, which was c a l c u l a t e d on the assumption th a t e s t i m a t o r bias could be

n o t i o n t h a t these angle observations can be taken as s t a t i s t i c a l l y independent. Nor would a sampling procedure akin t o t h a t suggested by Nieuwenhuis and van den Berg (1971) work f o r these data.

The next idea t o go i s the f a i r l y naive n o t i o n t h a t a u t o c o r r e l a t i o n f u n c t i o n s might capture the l o c a l property of surface roughness. The correlograms i n f a c t t e l l a st o r y o f o v e r a l l p r o f i l e shape, as i s c l e a r from the ranking on r (, p u t t i n g the p r o f i l e w i t h smallest value f i r s t : ST, BO, PR, TO, HO, LA, FA, PA, TR, TA, AR. This ranking can be compared w i t h the groups d i s t i n g u i s h e d i n Ch. 6.1, repeated here f o r convenience:

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( i ) BO, ST, PR, PA: the f o u r g e n t l e s t , s l i g h t l y convex ( i i ) FA, LA: the next g e n t l e s t , more convex ( i i i ) TR, AR, TA: the next g e n t l e s t , yet more convex ( i v ) HO, TO: the steepest, more complex i n form

Group ( i i i ) w i t h the strongest c o n v e x i t i e s (the strongest trends i n angle s e r i e s ) produces the three highest values of r ^ . Group ( i i ) produces r a t h e r lower values, while the multicomponent form o f group ( i v ) r e s u l t s i n s t i l l lower values. With one p u z z l i n g exception (PA), the nearly s t r a i g h t slopes o f group ( i ) have the lowest values: they almost lack trends i n angle s e r i e s .

The reason f o r t h i s r e s u l t , t h a t p o s i t i v e a u t o c o r r e l a t i o n s at s h o r t lags r e f l e c t p r o f i l e convexity ( o r c o n c a v i t y ) , can be seen by considering 7P. A s t r a i g h t slope would correspond t o a small region of the s c a t t e r diagram of leading against lagging values, w i t h i n which values would be f a i r l y evenly spread. By c o n t r a s t a convex slope would correspond t o a l a r g e r region of such a s c a t t e r diagram, o r i e n t e d at an angle approaching 45°• Hence the auto­c o r r e l a t i o n ( c o r r e l a t i o n between l e a d i n g and lagg i n g values) would tend t o be higher f o r the convex slope than f o r the s t r a i g h t slope. The g l o b a l property of convexity has an i n f l u e n c e on the l o c a l property of a u t o c o r r e l a t i o n , because a u t o c o r r e l a t i o n i s based on a combination of p a i r e d values from a l l p a r t s of the slope.

Marked convexity ( o r indeed concavity) of slope i s , i n s t a t i s t i c a l terms, marked n o n s t a t i o n a r i t y of angle s e r i e s .

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PROFILES

N

SCATTER DIAGRAMS

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CD

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convex straight

7P Schematic r e l a t i o n s h i p s between leading and lagging values

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I t has been considered whether a u t o c o r r e l a t i o n f u n c t i o n s c a l c u l a t e d from nonstationary series might be of use as sets of averaged d e s c r i p t o r s , contrary t o the orthodox view (Ch. 7.3 above). Moreover, the Pearson estimator of auto­c o r r e l a t i o n does not depend on the s t a t i o n a r i t y o f the series t o s a t i s f y -| < ^ \ , and i t allows leading and lagging values t o have d i f f e r e n t means and variances. I n these senses the Pearson e s t i m a t o r adjusts f o r nonstationary behaviour. Nevertheless, the r e s u l t s i n 7A show t h a t low-lag a u t o c o r r e l a t i o n s t e l l a s t o r y o f p r o f i l e shape, or of n o n s t a t i o n a r i t y . The naive idea t h a t low-lag auto­c o r r e l a t i o n s would r e f l e c t l o c a l roughness collapses and i t can be seen t h a t combination of r e s u l t s from d i f f e r e n t p a r t s o f the slope allows a u t o c o r r e l a t i o n s t o r e f l e c t o v e r a l l p r o f i l e shape.

Hence the i n t e r e s t of 7B appears t o be s t a t i s t i c a l r a t h e r than geomorphological, because p r o f i l e shape can be s t u d i e d more d i r e c t l y and more e f f i c i e n t l y by other means.

7C (abbreviated f o r angle) i s best viewed through 7D (Pearson - abbreviated f o r angle). Differences tend t o increase w i t h l a g . Apart from t h a t , the d i f f e r e n c e s serve as measures o f n o n s t a t i o n a r i t y . The r e s u l t s presented f o r each p r o f i l e are summarised by the median absolute valuer the median being taken over the lags l i s t e d , k = 1(1)5(5)50. The ranking (smallest f i r s t ) i s HO, ST, TO, LA, TA, PR, BO, TR, AR, FA and PA. There i s no simple s t o r y here, but i t

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i s c l e a r t h a t d i f f e r e n c e s between estimators can be appreciable even f o r low lags and series which v i s u a l l y appear t o be almost t r e n d - f r e e .

Pearson estimates tend t o exceed abbreviated estimates. The 154 values i n JO range from 0.243 t o -0.040. 129 are p o s i t i v e and 25 negative. This i m p l i e s t h a t the abbreviated estimator imparts a downward b i a s , and reduces the number of a u t o c o r r e l a t i o n s which would be c o r r e c t l y recognised as s i g n i f i c a n t l y d i f f e r e n t from the n u l l case.

Results f o r curvature (7E t o 7G) d i f f e r s t r i k i n g l y from those f o r angle. A l l values i n 7E are s t r o n g l y negative and f a l l outside the 0 .99 confidence i n t e r v a l f o r P^= 0. However, curvature a u t o c o r r e l a t i o n f u n c t i o n s dampen much more r e a d i l y than angle a u t o c o r r e l a t i o n f u n c t i o n s . Only 27 out o f 154 values i n 7E f a l l outside the 0.99 confidence i n t e r v a l f o r = 0, and only 16 out of 143 at lags of 2 or more.

7F (abbreviated f o r curvature) i s best viewed through 7G (Pearson - abbreviated f o r curvature) which shows a s t r a i g h t f o r w a r d p i c t u r e . A l l t a b u l a t e d d i f f e r e n c e s are very s m a l l , showing t h a t the f i r s t d i f f e r e n c i n g process which produces curvature i s s u f f i c i e n t t o y i e l d second-order s t a t i o n a r y s e r i e s . I t may be t h a t r ^ o f curvature i s a f a i r measure of roughness. Whether t h i s i s the case or n o t , i t i s f a i r l y conservative, ranging from -0.349 (PA) to -0.584 (ST).

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7.6 Summary

( i ) A u t o c o r r e l a t i o n p r o p e r t i e s o f h i l l s l o p e s e r i e s are of i n t e r e s t i f only because i t i s important t o know whether values are s t a t i s t i c a l l y dependent. Angle series are s t r o n g l y a u t o c o r r e l a t e d , whereas curvature series dampen much more r e a d i l y a f t e r strong negative values at l a g one. Otherwise, a u t o c o r r e l a t i o n of angle i s of f a i r l y l i m i t e d geomorphological i n t e r e s t . I t does not measure surface roughness, but r a t h e r tends t o r e f l e c t o v e r a l l p r o f i l e shape, which can be defined and measured more d i r e c t l y i n other ways.(7.2 and 7.5)

( i i ) The r e l a t e d problems of n o n s t a t i o n a r i t y and e s t i m a t o r choice deserve more a t t e n t i o n than i s customary i n geomorphology. I t i s important t o s t a t e and j u s t i f y the e s t i m a t o r used. The Pearson analogue esti m a t o r i s here recommended s t r o n g l y . (7.3 and 7.4)

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7.7 Notation

a,b E F i k n

N( , ) 0 r t u U j u', u" U V

V

T

f

2

I I I Mnemonics

cov med p r s t d var

r e a l numbers expectation operator Fisher's s t a t i s t i c s u b s c r i p t l a g number o f observations Normal (Gaussian) d i s t r i b u t i o n o f the order of a u t o c o r r e l a t i o n , c o r r e l a t i o n Student's s t a t i s t i c r e a l v a r i a b l e

means of whole s e r i e s , leading values, lagging values generating process r e a l v a r i a b l e mean of v autocovariance of process mean o f process a u t o c o r r e l a t i o n of process standard d e v i a t i o n of process summation operator i s drawn from given modulus

covariance median p r o b a b i l i t y standard d e v i a t i o n variance

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Chapter 8

PROFILE ANALYSIS

... I f a man's wit be wandering, l e t him study

the mathematics; f o r i n demonstrations, i f h i s

wit be c a l l e d away never so l i t t l e , he must

begin again ...

F r a n c i s Bacon, Essays L: Of s t u d i e s

8.1 The problem of p r o f i l e a n a l y s i s

8.2 Geomorphological c o n s i d e r a t i o n s

8.3 B a s i c p r i n c i p l e s

8.4 E x i s t i n g methods

8.5 A d d i t i v e e r r o r p a r t i t i o n s i n p r i n c i p l e

8.6 Additive e r r o r p a r t i t i o n s i n p r a c t i c e

8.7 Summary and d i s c u s s i o n

8.8 Notation

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8.1 The problem of p r o f i l e a n alysis

P r o f i l e analysis may be defined as the p a r t i t i o n o f a h i l l s l o p e p r o f i l e i n t o components which are r e l a t i v e l y homogeneous i n some e x p l i c i t sense. I n other words, w i t h i n a component the value of some v a r i a b l e (say u) i s approximately constant. (The term ' p a r t i t i o n ' i s used here both as a verb, to denote the process, and as a noun, t o denote the r e s u l t ) .

I f u i s angle 0 , then the components have approximately constant angle, and may be termed segments. I f u i s curvature — , then the components have approximately constant curvature, and may be termed elements. Since constant angle i m p l i e s constant (zero) curvature, segments are a subset o f elements (but c f . Young, 1971, 1972 and Parsons, 1977> who regarded segments and elements as d i s j o i n t s e t s ) . This terminology i s compared w i t h t h a t of other workers i n 8A.

For various reasons no need may be perceived f o r any s p e c i a l method of p r o f i l e a n a l y s i s . F i r s t l y , the existence and bounds o f d i s t i n c t components such as 'free face' or 'debris slope 1 may be judged e n t i r e l y obvious. Secondly, the number o f observations may not be s u f f i c i e n t l y l a r g e t o re q u i r e ( o r t o j u s t i f y ) s o p h i s t i c a t e d a n a l y s i s . T h i r d l y , basic features of the data may emerge q u i t e c l e a r l y from g r a p h i c a l analysis (e.g. P i t t y , 1969, 43-53), or from i n s p e c t i o n of averages and d i f f e r e n c e s ( c f . K u l i n k o v i c h e t al,1966; Hawkins and

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8A P r o f i l e analysis terminology

Reference (s) Leopold e t a l .

1964

S t r a i g h t

facet f a c e t Savigear 19 6 5

Savigear 1967 P i t t y 1969, 1970 Ahnert 1970c Caine 1971

Young 1971, 1972 segment Demirmen 1975 Jahn 1975 Graf 1976b Toy 1977

Curved

element segment

element

E i t h e r

segment

u n i t , component component segment segment, component u n i t segment sector component segment

This t h e s i s segment element component

Note: The term 'component slope' i s used by Young (1972,4) f o r a d i f f e r e n t purpose.

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Merriam, 1975). I n c o n t r a s t , a t t e n t i o n i s d i r e c t e d here towards s i t u a t i o n s i n which the existence o f components i s considered problematic, the number o f data i s reasonably large (number of values >100, say) and the need i s f e l t f o r a numerical method of p r o f i l e a n a l y s i s .

The p a r t i t i o n of a series of values u^; i = l , . . . , n i n t o k subseries may be regarded as the p l a c i n g of k-1 markers t o i n d i c a t e the bounds o f the chosen subseries. k may take any value between 1 and n.

P r o f i l e a n alysis may be considered as a co m b i n a t o r i a l problem ( c f . C l i f f e t _ a l , 1975, Ch. 2; t h e i r maximally constrained case i s equivalent t o p r o f i l e a n a l y s i s , but i s denoted by t h e i r Pig. 2.5b, not by F i g . 2.5a). The series u could be represented by a 'chain' w i t h n-1 ' l i n k s ' . Neighbouring values of u could be placed e i t h e r i n the same component or i n neighbouring components: each l i n k could be e i t h e r ' i n t a c t ' or 'broken 1. There are thus 2 possible p a r t i t i o n s of a p r o f i l e of n observations. This number increases e x p l o s i v e l y w i t h n, w i t h the simple but important i m p l i c a t i o n t h a t as n increases i t soon becomes imp r a c t i c a b l e to inspect a l l p ossible p a r t i t i o n s , and i t i s advisable, i f at a l l p o s s i b l e , t o use a method other than i n s p e c t i o n t o f i n d the best p a r t i t i o n , given some o p t i m a l i t y c r i t e r i o n .

The t o t a l number o f possible p a r t i t i o n s 2 n ~ 1 i s f o r a l l the p o s s i b i l i t i e s f o r k = l , . . . , n ; f o r example, the 1 p o s s i b i l i t y f o r k = l , the n-1 p o s s i b i l i t i e s f o r k=2, and so on. I n general

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^k-1^ m a y b e m u c h l e s s t n a n 2 n _ 1 , depending on k, but i t grows r a p i d l y w i t h n, and w i t h k < — . For example when n=100, 200 and k = 2 ( l ) 5

,100-1, ,200-1, k C k - l ) ( k-l> 2 99 199 3 4 851 19 701 i» 156 849 1 293 699 5 3 764 376 63 391 251

The problem o f p r o f i l e analysis i s f o r m a l l y equivalent t o some other problems i n data a n a l y s i s , such as the problem o f d i v i d i n g a time series i n t o epochs or periods (Fisher, 1958; Guthery, 197^; Harti g a n , 1975) and the problem o f s t r a t i g r a p h i c a l zonation met i n geology and palaeoecology (Gordon, 1973] Gordon and B i r k s , 1972; Hawkins and Merriam, 1973, 1975). I n each case, the fundamental issue i s the p a r t i t i o n of a one-dimensional series i n t o subseries which are homogeneous i n some sense: t h i s i s a s p e c i a l problem i n numerical c l a s s i f i c a t i o n . Such p a r t i t i o n problems, which are sometimes described as piecewise approximation or segmentation problems, a r i s e i n many d i s c i p l i n e s : see, f o r example, the papers which f r e q u e n t l y appear upon the subject i n the IEEE Transactions on Computers, such as t h a t by Blumenthal e t a l . (1977).

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The two-dimensional case i s also o f great i n t e r e s t : consider the r o l e of r e g i o n a l i s a t i o n i n , f o r example, geography (Grigg, 1967; Spence and Taylor, 1970; C l i f f et a l . , 1975; Haggett e t a l . , 1977), geomorphology ( G e l l e r t , 1972; Mather, 1972), geology (Henley, 1976) and t e r r a i n e v a l u a t i o n ( M i t c h e l l , 1973; O i l i e r , 1977).

The basic s i m i l a r i t y of p a r t i t i o n problems emerges most c l e a r l y from a formal statement. This allows ideas and techniques t o be borrowed where a p p r o p r i a t e , and leads to the embedding of p r o f i l e analysis w i t h i n numerical c l a s s i f i c a t i o n . I t i s both remarkable and unfortunate t h a t p r o f i l e analysis has not been recognised widely w i t h i n geomorphology as a numerical c l a s s i f i c a t i o n problem (although note some passing discussion i n Parsons, 1973).

The data u could i n general be vector-valued, leading t o a m u l t i v a r i a t e c l a s s i f i c a t i o n problem (cf« Webster, 1973; Hawkins and Merriam, 197*1; Hawkins, 1976), but the m u l t i v a r i a t e case of p r o f i l e analysis w i l l not be f u r t h e r examined here.

8.2 Geomorphological considerations

The primary purpose of p r o f i l e a n a l y s i s , as w i t h any c l a s s i f i c a t i o n , must be parsimony i n d e s c r i p t i o n . I f n data values may be summarised e f f i c i e n t l y by the a t t r i b u t e s of k components, where k « n , then p r o f i l e analysis y i e l d s a convenient s i m p l i f i c a t i o n . P r o f i l e analysis as discussed here i s a morphometric technique, although the method could be a p p l i e d t o any one-dimensional data s e r i e s . While i t may be hoped t h a t a p a r t i t i o n of a p r o f i l e might be of

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i n t e r e s t t o students of process or development, or to p r a c t i t i o n e r s of a p p l i e d d i s c i p l i n e s , such purposes are here regarded as secondary t o the aim o f morphometric d e s c r i p t i o n .

Nevertheless, i t i s important t o discuss the geomorph-o l o g i c a l grounds f o r p r o f i l e a n a l y s i s . C l e a r l y p r o f i l e analysis i s a v a l i d approach i f a h i l l s l o p e may be l e g i t i m a t e l y regarded as a combination o f d i s c r e t e components ( o r , more g e n e r a l l y , i f a landscape may be considered as a mosaic of d i s c r e t e u n i t s ) . Two views may be i d e n t i f i e d on t h i s issue: one emphasising the a t o m i s t i c character of the landscape, the other emphasising i t s continuous character. ( I t i s commonplace i n the p h y s i c a l sciences t o contrast a t o m i s t i c and continuum views, or p a r t i c l e and f i e l d t h e o r i e s : see, f o r example, the remarks o f Holton (1973, 1978) on these views as 'themata' i n the h i s t o r y o f s c i e n t i f i c thought.)

The a t o m i s t i c view f i n d s i t s strongest expression i n the morphological mapping procedures o f the ' S h e f f i e l d school', which are centred on 'the concept t h a t there i s a small basic i n d i v i s i b l e u n i t of t e r r a i n ' ( M i t c h e l l , 1973, 1»9). Waters (1958) and Savigear (1965) were the main proponents of morphological mapping, while Gregory and Brown (1966) and Doornkamp and King (1971, Ch. 6) provided f u r t h e r discussion and worked examples. Morphological mapping i n p a r t i c u l a r and geomorphological atomism i n general seem t o have three t h e o r e t i c a l bases, as f o l l o w s :

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( i ) The idea t h a t a p o l y c y c l i c denudation h i s t o r y associated w i t h a f l u c t u a t i n g base l e v e l would produce a landsurface which was a mosaic of slopes and f l a t s (Wooldridge, 1932; L i n t o n , 1951).

( i i ) The idea t h a t process d i s c o n t i n u i t i e s would be associated w i t h morphological d i s c o n t i n u i t i e s , found f o r example i n the work o f King (1967) and i n the so-c a l l e d 'Nine Unit Landsurface Model' (Dalrymple e t a l . , 1968; Conacher and Dalrymple, 1977).

( i i i ) The idea t h a t morphological d i s c o n t i n u i t i e s may be associated w i t h l i t h o l o g i c a l d i s c o n t i n u i t i e s .

While i n i n d i v i d u a l cases these ideas may be very p l a u s i b l e , i t s t i l l remains necessary t o t e s t such a view as p a r t of p r o f i l e a n a l y s i s , and thus t o consider the a l t e r n a t i v e p o s s i b i l i t y t h a t a h i l l s l o p e p r o f i l e i s e s s e n t i a l l y a smoothly-changing continuous curve. An emphasis on c o n t i n u i t y does not seem t o possess any t h e o r e t i c a l j u s t i f i c a t i o n . I t i s r a t h e r t h a t ideas of continuous curves and surfaces are both n a t u r a l and convenient f o r any modelling approach centred on d i f f e r e n t i a l equations, or f o r any morphometric approach centred on the s t a t i s t i c a l analysis o f s p a t i a l s e r i e s .

The atomism-continuity issue i s perhaps best approached by comparing v a r i a b i l i t y between and w i t h i n components. One possible p i t f a l l here i s t h a t v a r i a b i l i t y measures may be unduly s e n s i t i v e t o the measured l e n g t h ( s )

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used i n p r o f i l e survey ( c f . Gerrard and Robinson, 1971; Gerrard, 1978). This could be i n v e s t i g a t e d t o some extent by the aggregation approach used i n Ch. 6.3 above f o r analysing angle and curvature frequency d i s t r i b u t i o n s .

8.3 Basic p r i n c i p l e s

The f o l l o w i n g p r i n c i p l e s are suggested t o u n d e r l i e p r o f i l e a n a l y s i s .

( i ) O b j e c t i v i t y . A c a l l f o r o b j e c t i v i t y i m p l i e s t h a t methods should be e x p l i c i t and r e p l i c a b l e . I t does not amount t o a d e n i a l of the need or value o f i n d i v i d u a l i n t e r p r e t a t i o n or experience. I n p r a c t i c e a f u l l s p e c i f i c a t i o n of the a l g o r i t h m used i s r e q u i r e d ; f o r example, i n the form of a computer program. Making methods e x p l i c i t and r e p l i c a b l e means t h a t they can be discussed and evaluated, and, i f need be, modified or r e j e c t e d .

( i i ) Against adhoekery. Ad hoc procedures, such as a r b i t r a r y c u t - o f f or a l l o c a t i o n r u l e s , should be avoided as f a r as p o s s i b l e . Since p r o f i l e analysis i s a numerical c l a s s i f i c a t i o n problem i t seems eminently sensible t o embed the problem w i t h i n the f i e l d s of c l u s t e r a nalysis and numerical taxonomy ( c f . Jardine and Sibson, 1971; Sneath and Sokal, 1973; E v e r i t t , 197*1; Sokal, 197^; Hartigan, 1975). I f p r o f i l e analysis has s p e c i a l features which need t o be handled by s p e c i a l procedures, then t h i s case must be argued e x p l i c i t l y . As a d i s t i n g u i s h e d s t a t i s t i c i a n once wrote, 'we make no mockery o f honest adhockery' (Good, 1965, 56): but i f a systematic procedure i s a v a i l a b l e , adhockery deserves a l l the mockery i t may receive.

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( i i i ) Data a n a l y s i s . A c e r t a i n open-mindedness should be e n t e r t a i n e d about the basic features of the data. I n p a r t i c u l a r

(a) While many geomorphologists use the term ' m i c r o r e l i e f ' i t i s by no means c l e a r whether m i c r o r e l i e f i s r e a l l y a d i s t i n c t i v e source o f v a r i a t i o n i n p r i n c i p l e , l e t alone how i t may be d i s t i n g u i s h e d i n p r a c t i c e ( c f . Ch. 5-3 above).

(b) I t w i l l r a r e l y be c l e a r prima f a c i e how many components e x i s t , or indeed whether they r e a l l y e x i s t qua components.

(c) I t should be easy to vary the number of components k. While f o r a v a r i e t y o f geomorphological and psychological reasons the value o f k chosen w i l l o f t e n be between 1 and 9, i t i s nevertheless v i t a l t h a t i t should be s t r a i g h t f o r w a r d to consider d i f f e r e n t values o f k. A s a t i s f a c t o r y method f o r p r o f i l e a n alysis w i l l not prejudge the amount of d e t a i l r e q u i r e d by the user, while k should be changed i f only t o determine the s e n s i t i v i t y o f the r e s u l t i n g p a r t i t i o n s .

(d) I t i s important t h a t there should be some check of the v a l i d i t y o f supposed components i n the form o f a comparison of v a r i a b i l i t y between and w i t h i n components.

( i v ) Decency assumptions made e x p l i c i t . I f a method of data analysis i s regarded as a tr a n s f o r m a t i o n o f one set of numbers i n t o another s e t , i t can o f t e n be shown t h a t there are conditions i n which the transformations work best according t o given o p t i m a l i t y c r i t e r i a . The corresponding

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'decency assumptions' about the i d e a l character o f data should be made e x p l i c i t , and the consequences o f suboptimal conditions should be known as f a r as po s s i b l e . (The e x c e l l e n t term 'decency assumption' i s taken from Levins, 1970, 74).

(v) P i r e c t i o n - i n v a r i a n c e . Results from p r o f i l e analysis should not vary w i t h d i r e c t i o n of data processing, base t o crest or crest t o base ( c f . Gerrard, 197*0. Formally, r e s u l t s from the s e r i e s u i , . . . , u n and the reversed s e r i e s u n,...,U] L should be e q u i v a l e n t . A component i s a component whether one i s c l i m b i n g up or s l i d i n g down.

( v i ) P r i n c i p l e s and p r a c t i c e . A f u r t h e r requirement i s t h a t any method of p r o f i l e analysis must also be u s e f u l geomorphologically, which i s i n large p a r t an issue f o r the f i e l d w o r k e r . However, i t seems v i t a l t h a t t h i s p r i n c i p l e should not be allowed t o override other p r i n c i p l e s , so t h a t a s t a t i s t i c a l l y dubious method i s regarded as acceptable geomorphologically, merely because i t produces apparently sensible r e s u l t s .

8.4 E x i s t i n g methods

8.4.1 Ahnert's method

Ahnert (1970c) proposed t h a t segments ( i . e . components i n the terminology proposed here) should be regarded as s t r a i g h t (sc. segments) i f angle does not change twice i n the same d i r e c t i o n i n two successive measured lengths and t o t a l v a r i a t i o n i n angle does not exceed 3°. Concave and convex segments (sc. components) are then d i s t i n g u i s h e d by

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the d i r e c t i o n of angle increase.

The choice of 3° f o r angle range i s c l e a r l y a r b i t r a r y and i m p l i e s t h a t the number of segments (sc. components) i s s t r o n g l y i n f l u e n c e d by measured l e n g t h .

Juvigne (1973) independently proposed a broadly s i m i l a r method, which i s open to s i m i l a r o b j e c t i o n s .

8.4.2 Ongley's method

The method devised by Ongley (1970) tack l e s the problem of i d e n t i f y i n g ( r e c t i l i n e a r ) segments r a t h e r than t h a t of p r o f i l e analysis sensu s t r i c t o . Subseries o f p r o f i l e coordinates x and z are entered i n l o c a l regressions w i t h the i m p l i c i t model

Z = <x + + e

where oL ,(> are parameters and € s t o c h a s t i c e r r o r . Choice of subseries f o r regression i s determined by a complicated a l g o r i t h m : the program works i t s way up a p r o f i l e . A subseries i s accepted as a segment i f no r e s i d u a l exceeds a p r e s p e c i f i e d tolerance i n absolute s i z e . Note t h a t i n general segments may overlap.

Apart from the f a c t t h a t Ongley's method i s of no use f o r i d e n t i f y i n g elements, i t i s u n s a t i s f a c t o r y f o r several reasons ( c f . also Gerrard, 197*1, 1978). I n p r a c t i c e i t i s d i f f i c u l t t o choose an appropriate tolerance value. At high l e v e l s considerable overlaps occur; at low l e v e l s there may be a m u l t i t u d e of very short segments. I n e i t h e r case, i t i s not easy t o decide on a value f o r

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tolerance which would lead t o much b e t t e r r e s u l t s . Consequently i t may be necessary t o rerun the program several times before o b t a i n i n g output s a t i s f a c t o r y f o r the purpose i n mind.

The a l g o r i t h m f o r 'walking' the program upslope i s extremely clumsy. I t i s not s u r p r i s i n g t h a t i t may lead t o d i r e c t i o n - v a r i a n t r e s u l t s : i f data are reversed, and the program i n s t r u c t e d t o walk downslope, completely d i f f e r e n t r e s u l t s have been found t o occur (contrary t o the suggestion by Gerrard, 197*0. This seems to be a f a t a l d e fect.

Ongley's method was a p p l i e d t o survey data f o r p r o f i l e s TO and TR t o i l l u s t r a t e these remarks. Tolerance was set at 5m somewhat a r b i t r a r i l y . (For h i s slopes on the .Cobar p e d i p l a i n , New South Wales, Ongley used 0.3 f t . (90mm), but these slopes were e v i d e n t l y both gentle and smooth.) The data f o r TO and TR were read i n both base-crest and crest-base d i r e c t i o n s . Results f o r TO are given i n 8B, which l i s t s t e r m i n a l index numbers f o r each segment i d e n t i f i e d . The indexes run i n crest-base sequence i n both cases f o r c o m p a r a b i l i t y .

The most s t r i k i n g f eature of the TO r e s u l t s i s t h a t 5^ segments were i d e n t i f i e d i n one case, and 107 i n the other. More d e t a i l e d i n s p e c t i o n shows the lack of correspondence between the two sets of r e s u l t s : i n f a c t , no segment occurs i n both l i s t s . An a l t e r n a t i v e method of comparing r e s u l t s i s through frequency d i s t r i b u t i o n s o f segment lengths (8C): the d i f f e r e n c e s are q u i t e c l e a r .

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8B Results f o r Ongley's method on TO Tolerance = 5m.

Base-crest 1,44 6,45 9,46 11,47 13,48 14,49 16,50 18,51 20,52 21,54 22,55 25,56 26,57 28,58 30,59 32,60 33,61 34,62 36,63 39,64 40,65 42,66 41,67 43,68 43,69 45,71 46,72 48,73 49,74 50,77 52,78 54,79 55,80 56,82 57,83 58,85 59,87 60,89 61,90 62,91 63,94 64,97 65,100 66,102 67,103 68,106 69,107 70,112 71,113 72,116 73,177 74,204 75,287 76,309 # segments = 54 Crest-base

1,118 2,119 3,120 11,121 79,122 86,123 90,124 95,126 96,127 97,128 100,129 102,130 104,131 105,132 106,133 107,134

108,135 110,137 111,138 112,139 113,140 114,140 1 1 5 , l 4 l 116,142 117,142 118,142 119,143 120,142 121,147 122,150 123,153 124,155 125,158 126,157 127,158 128,162 129,168 130,174 131,180 132,189 133,195 134,200 135,206 136,220 137,219 138,224 139,225 142,226 152,227 154,228 155,231 156,232 157,233 161,234 164,235 170,236 171,237 172,238 173,239 174,240 176,241 182,242 183,243 184,244 186,245 188,246 190,253 192,254 196,255 199,256 201,257 203,258 206,259 207,260 209,261 211,262 214,265 216,266 217,267 219,268 222,269 224,270 225,271 226,274 233,275 235,276 236,277 237,278 238,279 239,280 240,284 241,285 242,289 243,291 244,293 245,294 246,294 248,292 249,291 250,290 251,292 252,293 253,293 254,296 255,300 256,308 257,309

# segments = 107

(# measured angles = 309)

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A user faced w i t h the r e s u l t s i n 8B and 8C and wishing t o continue would presumably rerun w i t h lower t o l e r a n c e , but what value would he choose? I n t e l l i g e n t guesswork would be the only guide.

The r e s u l t s f o r TR (8D and 8E) t e l l a s i m i l a r s t o r y . 237 segments are i d e n t i f i e d by processing i n one d i r e c t i o n , while only 93 are i d e n t i f i e d by processing i n the other d i r e c t i o n . Once again no segment occurs i n both l i s t s . There i s a s t r o n g tendency t o i d e n t i f y a m u l t i p l i c i t y o f segments of approximately s i m i l a r length as the program works i t s way round a concavity or convexity. This i s hardly s u r p r i s i n g since Ongley's method i s , a f t e r a l l , attempting t o i d e n t i f y segments.

Parsons (1973, 1976b) independently produced a s u p e r f i c i a l l y s i m i l a r method also based on l o c a l l i n e a r regressions. Various ad hoc devices reduce the i n f l u e n c e of o u t l i e r s and lead t o an analysis o f the p r o f i l e i n t o d i s j o i n t segments. This method i s , however, also d i r e c t i o n -v a r i a n t (Parsons, personal communication, 1977).

8.4.3 P i t t y ' s method

P i t t y (1970, 30-44) proposed regressions o f angle against index number w i t h the i m p l i c i t model

e = * + pi + e

where (X, are parameters and 6 s t o c h a s t i c e r r o r . A cusum t e s t was suggested f o r changes i n slope and a t e s t using

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8D R e s u l t s f o r O n g l e y ' s method on TR

T o l e r a n c e = 5m

B a s e - c r e s t

1 ,42 2,43 3,44 4 ,45 5 ,46 6 ,47 7 ,48 8 ,49 9 , 50 1 0 , 5 1 11,52 12,53 13 ,54 15,55 16 ,56 17 ,57

19 ,58 20 ,59 22,60 23,61 24,62 25,63 27 ,64 28 ,65 29,66 31 ,67 33 ,68 34 ,69 35 ,70 37 , 7 1 37,72 39,73 40,74 41 ,75 42 ,76 43,77 44 ,78 45,79 46,80 4 6 , 8 1 47,82 49,83 50 ,84 51,85 52 ,86 53 ,87 54,88 55 ,89 5 6 , 9 0 57,91 58,93 59 ,94 60 ,95 61 ,96 62 , 97 63 , 98 6 4 , 1 0 0 6 5 , 1 0 1 66 ,102 67 ,103 68 ,104 69 ,105 70,106 71,107 72,108 73,109 74,110 75,111 76,112 77,113 78,114 79,115 80,116 81,118 82,119 83,120 84,121 85,122 86,123 87,124 88,125 89 ,126 90,127 91,129 92 , 130 93,131 94,132 95,133 96,135 97,137 98,138 99,139 100 ,140 101 ,141 102,142 104 ,143

105,144 106,145 107,146 108 ,147 110 ,148 111 ,149 112 ,149 113 , 151 115,152 117,153 118,154 119,155 120,156 121 ,157 122 ,158 123 ,159 124 ,160 1 2 6 , 1 6 1 127,162 129,163 130,164 131 ,165 132 ,166 134 ,167 135,168 136 ,170 1 3 8 , 1 7 1 139,172 140,173 141,174 142,176 144 ,177 145,178 146,179 147 ,180 1 4 8 , 1 8 1 149,182 150,183 151,184 152,185 153,186 154,187 155,188 156,189 157,190 158,191 159,192 160,193 161,195 162,196 163,197 164,198 165,199 166,200 167 ,201 168,202 169 ,204 170,205 171,206 172,207 173,208 174,209 175 ,210 176,211 177,213 178 ,214 179 ,215 180,216 181,217 182,218 183,219 185,221 186,222 187,223 188 ,224 189,225 190,226 191,227 192,228 193,229 195,230 198,231 198,232 200,233 202,234 204,235 205,236 207,237 209,238 210,239 211 ,240 212 , 241 214 ,242 216 ,243 217 ,244 218 ,245 219 ,246 222 ,247 223 ,248 224,249 225,251 226,252 227,253 228,255 229,256 230 ,258 231,260 232,261 233,268 234 ,284 235,296 236,302 237,309 238,311 2 39,312 249,313 267,314 270,315 276,316 281,317 284,318 289,319 288 , 320 290,321 293,322 294,323 295,324 297,325 300,326 300,327 301,328 303,329 305,330 305 ,331 306,332 308,333 309,335 310,336 312,337 313 ,402 314 ,403

# segments = 237

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Crest-base 1,256 170,257

238,264 243,265 252,272 253,273 260,281 261,282 268,298 269,300 276,314 277,316 296,342 301,343 334,350 335,351 343,359 344,360 351,376 352,380 369,391 370,392 385,399 386,400 # segments = 9 3

178,258 245,266 254,272 262,285 270,303 278,323 310,344 337,352 345,361 353,381 374,393 387,401

186,259 247,267 255,272 263,287 271,306 279,331 316,345 338,353 346,362 354,384 376,394 388,402

198,260 248,268 256,274 264,290 272,307 280,333 318,346 339,354 347,365 355,387 379,398 389,403

205,261 249,269 257,277 265,293 273,309 281,339 326,347 340,355 348,368 357,388 381,396

216,262 250,270 258,278 266,295 274,309 283,240 330,348 341,356 349,370 361,389 382,397

231,263 251,271 259,279 267,297 275,312 288,341 331,349 342,358 350,374 365,390 384,398

(# measured angles = 403)

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regression output f o r breaks of slope.

P i t t y d i d not discuss the choice of angle subseries f o r regression. There i s considerable confusion i n h i s account between three l o g i c a l l y d i s t i n c t issues: the

the model (assessed perhaps by a g l o b a l l a c k - o f - f i t s t a t i s t i c ) and the independence o f r e s i d u a l s . I t i s unfortunate t h a t two t e s t s r a t h e r than one are proposed f o r t h i s problem: there seems t o be no j u s t i f i c a t i o n f o r t h i s c o mplication. The cusum t e s t has r e c e n t l y been shown t o be t o t a l l y unsuitable f o r a u t o c o r r e l a t e d data (Johnson and Bagshaw, 197**; Bagshaw and Johnson, 1975), w h i l e the t e s t f o r breaks of slope i m p l i c i t l y assumes the adequacy of the regression model which i s one o f the issues at stake. F i n a l l y , P i t t y ' s use of a l g e b r a i c n o t a t i o n i s i n c o n s i s t e n t , which makes understanding of h i s proposals d i f f i c u l t .

8.h.k Young's methods

Young (1971) proposed three techniques o f p r o f i l e a n a l y s i s : best segments a n a l y s i s , best elements analysis and best u n i t s analysis ( h e r e a f t e r BSA, BEA, BUA). They are a development of previous s u b j e c t i v e methods developed by Savigear and Young.

The v a r i a b i l i t y o f a subseries i n e i t h e r angle or curvature ( e i t h e r case denoted here by u) i s measured by i t s c o e f f i c i e n t of v a r i a t i o n i . e . standard d e v i a t i o n . 100

v a l i d i t y of the n u l l hypothesis H o:£=0, the adequacy o f

) J 2. mean

• lOO

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where u = > i t ^ - i and 1^ denotes measured length.

Curvature i s i n f a c t defined r a t h e r u n s a t i s f a c t o r i l y by Young as

100 (A ~Bt)

L, + La.

= 200 (e n - 6n-i)

1 = 2 , . . ^ - !

I n BSA a subseries i s accepted as a segment i f the c o e f f i c i e n t of v a r i a t i o n of angle does not exceed a p r e s p e c i f i e d maximum. I n BEA a subseries i s accepted as an e l e m e n t . i f the c o e f f i c i e n t of v a r i a t i o n of curvature does not exceed a second p r e s p e c i f i e d maximum. BUA allows f o r both segments and elements, depending on two p r e s p e c i f i e d maxima.

There i s an immediate d i f f i c u l t y here. Since segments are a subset of elements, BSA and BEA are not independent. Furthermore, the existence of BUA i s both unnecessary and confusing.

I n these techniques, a measured le n g t h may f a l l i n t o two. or more components. This d i f f i c u l t y has t o be resolved i f components are not t o overlap. For example, i n BUA, 'a measured length which f a l l s i n t o two or more slope u n i t s ,

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each w i t h i n the s p e c i f i e d maximum v a r i a b i l i t y , i s a l l o c a t e d t o the longest u n i t ; i f two u n i t s are of equal l e n g t h , i t i s a l l o c a t e d t o t h a t w i t h the lowest c o e f f i c i e n t o f v a r i a t i o n i f the c o e f f i c i e n t s are also equal, a l l o c a t i o n i s t o a segment i n preference t o an element' (Young, 1971, 5 ) . Hence since s i m i l a r i t y and c o n t i g u i t y c r i t e r i a do not always produce a s a t i s f a c t o r y p a r t i t i o n i n t o components, Young's method assigns measured lengths whose status i s i n doubt t o 'the longest acceptable u n i t ' , defined i n an ad hoc manner.

Another fundamental d i f f i c u l t y i s t h a t l i k e a l l r a t i o s , the c o e f f i c i e n t of v a r i a t i o n i s not a s t a b l e measure ( c f . Kendall and S t u a r t , 1969 , *»7-8). As denominators become smaller, values o f the c o e f f i c i e n t tend to become extremely l a r g e . This produces a bias towards sh o r t components f o r low mean angles or curvatures ( ' f l a t s ' become short segments, 'segments' become shor t elements). Young t a c k l e d t h i s d i f f i c u l t y by r e p l a c i n g means below 2 by a value of 2, but t h i s i s c l e a r l y not a very s a t i s f a c t o r y s o l u t i o n .

The c o e f f i c i e n t of v a r i a t i o n has the f u r t h e r d i s ­advantage (Lewontin, 1966; G i l b e r t , 1973, 5*0, t h a t i t r e s t s on the i m p l i c i t assumption t h a t standard d e v i a t i o n i s p r o p o r t i o n a l t o the mean.

Young (1971) recommended c o e f f i c i e n t s of v a r i a t i o n of 10.0 (angle) and 25.0 ( c u r v a t u r e ) , but remarked (personal communication, 1975) t h a t higher values might

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be necessary. 8F gives the numbers o f segments and elements obtained w i t h maxima o f 10.0 (10.0) 90.0 f o r p r o f i l e Tripsdale (TR). These r e s u l t s show t h a t f o r t h i s k i n d of data, obtained w i t h s h o r t measured lengths, very high c o e f f i c i e n t s are needed t o produce few-component p a r t i t i o n s ( b u t , once again, the user would have t o f a l l back on i n t e l l i g e n t guesswork i n choosing new values f o r rerunning the program). I t i s also c l e a r t h a t r e s u l t s are d i r e c t i o n - v a r i a n t : d i f f e r e n t numbers o f components may be produced by d i f f e r e n t d i r e c t i o n s of processing, although the d i f f e r e n c e s i n numbers are r e l a t i v e l y s m all.

8G and 8H focus on a s p e c i f i c case: segments f o r 50.0. The i n s t a b i l i t y o f the c o e f f i c i e n t of v a r i a t i o n leads t o many shor t gentle segments, p a r t i c u l a r l y on the upper p a r t of the convexity. I t i s also c l e a r how r e s u l t s are b a s i c a l l y an a r t e f a c t o f d i r e c t i o n o f processing: only 56 out o f 403 measured lengths are a l l o c a t e d t o e x a c t l y the same segment i n both cases. (The d i r e c t i o n - v a r i a n t character o f Young's method was f i r s t p o i n t e d out by Gerrard, 1974). Results f o r 90.0 which are given i n 81, are even worse: only 5 out of 403 lengths are a l l o c a t e d t o ex a c t l y the same segment.

Results f o r p r o f i l e TO are given i n 8J, which t e l l s a s i m i l a r s t o r y : while the numbers o f components are approximately equal f o r d i f f e r e n t d i r e c t i o n s of processing, the a c t u a l components may once again d i f f e r markedly. The example of segments at 50.0 i s p a r t i c u l a r l y s t r i k i n g .

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8P Results of best segments analysis and best elements analysis on TRa by d i r e c t i o n of processing

Maximum c o e f f i c i e n t of v a r i a t i o n

10.0 20.0 30.0 40.0 50.0 60.0 70.0 80.0 90.0

# segments Base-crest Crest-base

284 208 139 95 57 35 25 21 11

284 210 130 100 64

35 26

15 8

# elements Base-crest Crest-base

379 357 339 313 296 274 262 236 224

379

356

340

315

294

274

260

230

216

(#measured angles = 403)

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8G Results of best segments analysis on TR: maximum c o e f f i c i e n t o f v a r i a t i o n = 50.0

Base-crest

1,4 5 6 7 8,9 10 11,16 17 18 19 20 21 22 23 24,25 26 27 28,29 30 31 32,35 36,40 4 l 42,45 ^6,47 48 49 ,50 51,53 54,65 66 67,68 69,72 73,89 90,91 92 93 ,98 99 100,102 103,115 116,120

121,161 162 163 164 165,167 168 169,217 218,222 223,224 225 226,228 229 230 231,242 243,311 312 313,403

# segments = 57

Crest-base

1,4 5 6 7 8,9 10 11 12,16 17 18 19 20 21 22 23 24,25 26 27 28,29 30 31 32,35 36,40 41 42,45 46,47 48 49,50 51,53 54 55 56,65 66,67 68 69 70,74 75,90 91 92 93,98 99 100,103 104 105,114 115,129 130,131 132,134 135

136,169 170 171,175 176 177,182 183 184 185,226 227 228,229 230 231 232,255 256,257 258,347 348,403

# segments = 6 4

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81

Results of best segments analysis on TR: maximum c o e f f i c i e n t o f v a r i a t i o n = 90.0

Base-crest 1,18 19 20 21,22 23 24,25 26 27

28,30 31,70 71,404

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1,25 26 27 28,30 31 32,116 117,164 165,403

^segments = 8

Ordered segment lengths

Base-crest

1,1,1,1,1,2,2,3,18,40,334 loq = 1 med = 2 upq = 18

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1,1,1,3,25,48,85,239 loq = 1 med = 14 upq = 66.5

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8J

Results of best segments analysis and best elements analysis on TO

Maximum c o e f f i c i e n t of v a r i a t i o n #segments #elements

Base-crest Crest-base Base-crest Crest-10.0 157 155 291 291 20.0 55 58 267 266 30.0 22 22 253 253 40 .0 9 10 2 38 238 50.0 5 1 223 225 60.0 1 1 200 200 70.0 1 1 182 187 80.0 1 1 169 170

(^measured angles = 309)

Case o f c o e f f i c i e n t maximum = 50.0

Base-crest Actual c o e f f i c i e n t Segment

1,272 41.41 273 0.0 274 0.0 275 0.0 276,309 41 .50

Crest-base Segment

1,309 42.79

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While the whole p r o f i l e can be accepted as a s i n g l e segment at 42 . 79 , t h i s i s not revealed by base-crest processing which divid e s the p r o f i l e i n t o f i v e segments, and thus f a i l s t o f i n d the optimum.

Recently Parsons (1977) discussed a p p l i c a t i o n s o f Young's methods (and elaborations o f them) to a large amount of p r o f i l e data. For a c r i t i q u e o f t h i s paper and a r e p l y , see Cox (1978) and Parsons (1978) . A f u r t h e r paper (Abrahams and Parsons, 1977) i s open t o s i m i l a r o b j e c t i o n s .

8.4.5 Assessment of e x i s t i n g methods

This survey has concentrated on Ongley's and Young's methods as the most popular among geomorphologists. These methods may now be considered i n the l i g h t of the p r i n c i p l e s proposed i n Ch. 8.3 above.

The most s t r i k i n g f e a t u r e i s the ad hoc and a r b i t r a r y character of e x i s t i n g methods, which have b a s i c a l l y been developed i n ignorance of the large body o f work on numerical c l a s s i f i c a t i o n . Both Ongley's and Young's methods may be h i g h l y d i r e c t i o n - v a r i a n t , which seems impossible to j u s t i f y .

I n n e i t h e r case i s i t easy to vary the number of components, since the parameters c o n t r o l l e d are r e l a t e d t o the number of components i n a complex and unknown manner. Nor does e i t h e r method allow comparison of between-component and within-component v a r i a t i o n . The problem of s e n s i t i v i t y t o measured length has been l a r g e l y ignored, although i t

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would be s t r a i g h t f o r w a r d t o apply these methods t o aggregated data.

Hence no e x i s t i n g method can be regarded as s a t i s f a c t o r y . A t t e n t i o n now turns t o a method drawn from numerical c l a s s i f i c a t i o n , which was o r i g i n a t e d by Fisher (1958) ,

generalised by Hartigan (1975) , and which produces ' a d d i t i v e e r r o r p a r t i t i o n s ' . This method i s discussed i n some d e t a i l before being a p p l i e d to p r o f i l e data from the f i e l d area.

8.5 A d d i t i v e e r r o r p a r t i t i o n s i n p r i n c i p l e

Fisher (1958) proposed a weighted least-squares c r i t e r i o n f o r p a r t i t i o n i n g a series u of length n i n t o k contiguous subseries

minimise

Here w^ i s a weight, u^ i s the average of the subseries which includes u^, and mini m i s a t i o n i s over the ( ^ . j ) possible p a r t i t i o n s . This c r i t e r i o n i s one of a family of least-squares c r i t e r i a which also includes c r i t e r i a f o r vector-valued u and piecewise f u n c t i o n a l approximation. Various members o f t h i s f a m i l y have been used i n geology and palaeoecology, mainly f o r s t r a t i g r a p h i c a l zonation (Gordon and B i r k s , 1972; Gordon, 1973; Hawkins and Merriam, 1973, 1975).

Hartigan (1975, Ch .6) considered a d d i t i v e e r r o r c r i t e r i a o f the general form

k minimise

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Here dj i s a diameter f o r subseries j , and minimisation i s over the possible p a r t i t i o n s . Hartigan mentioned, as s p e c i a l cases o f dj

d . = jvalue - median| dj = Z! I value - me an j 2

where summation i s over values o f u which belong t o subseries j. Clea r l y both these can be seen as members of the fa m i l y

d j = S | value - t y p i c a l v a l u e | ^

I n p r i n c i p l e the choice of diameter might be j u s t i f i e d by considering c o n d i t i o n a l p r o b a b i l i t y d i s t r i b u t i o n s f o r each subseries (Hartigan, 1975, 135 ) , each of the form

pr (value o f u j l o c a l parameter) C l a s s i c a l r e s u l t s carry over so t h a t p = 1 i s the maximum l i k e l i h o o d procedure f o r double exponential (Laplace) d i s t r i b u t i o n s , and p = 2 t h a t f o r normal (Gaussian) d i s t r i b u t i o n s , assuming independent observations. However, such c o n d i t i o n a l p r o b a b i l i t y d i s t r i b u t i o n s are i n p r a c t i c e unknown (the task of p r o f i l e analysis being i n essence t o estimate l o c a l parameters); they may not f o l l o w any c l a s s i c a l d i s t r i b u t i o n s ( r e s u l t s f o r marginal frequency d i s t r i b u t i o n s discussed i n Ch. 6.2 above show a broad general tendency t o long-tailedness); the form o f d i s t r i b u t i o n s may vary from subseries t o subseries; and mutual independence of observations from each d i s t r i b u t i o n may w e l l be a very str o n g assumption (even though weaker than mutual independence of a l l observations, discussed i n Ch. 7 above).

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Such important reservations aside, a d d i t i v e e r r o r p a r t i t i o n methods produce optimal components f o r k = l , 2 , 3 « . . .

The number o f components i s c o n t r o l l e d d i r e c t l y , and both generalised and d e t a i l e d p a r t i t i o n s may be produced i n a s i n g l e program run. Component a t t r i b u t e s ( t y p i c a l value, v a r i a t i o n , boundaries) are of course o f great i n t e r e s t , w h ile p a r t i t i o n s as a whole can also be compared f o r d i f f e r e n t values of k, and f o r d i f f e r e n t p r o f i l e s , p e r h a p s by considering the measure

Finding the optimal p a r t i t i o n i s n o n t r i v i a l computationally

can be tacked by a dynamic programming a l g o r i t h m (Bellman and Dreyfus, 1962; H a r t i g a n , 1975). Such an a l g o r i t h m w i l l always f i n d an optimum, although i t may not be unique. This raises the question whether d i f f e r e n t d i r e c t i o n s o f processing would f i n d d i f f e r e n t optima w i t h equal values of the o b j e c t i v e f u n c t i o n . I t i s conjectured here t h a t f o r real-valued o b j e c t i v e f u n c t i o n s o f the k i n d considered below, t h i s i s i n p r i n c i p l e an event o f p r o b a b i l i t y zero ( i n the exact sense of the expression) and d i r e c t i o n - i n v a r i a n c e can thus be assumed i n practice.. This conjecture i s supported by Hartigan (personal communication, 1977).

K

71 J

= V say.

given the combinatorial explosion of ( n - 1 k - 1 ) . However, i t

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8.6 A d d i t i v e e r r o r p a r t i t i o n s i n p r a c t i c e

A d d i t i v e e r r o r p a r t i t i o n s were produced f o r curvature series u = A9 using the least-squares c r i t e r i o n

minimise s ( \3u

1 =

f o r a l l eleven h i l l s l o p e p r o f i l e s , f o r k = 1(1)10.

The r e s u l t s showed a stro n g tendency f o r the method to y i e l d a large number of very short components. For each p a r t i t i o n of n i n t o k, the k components can be ordered numerically by t h e i r lengths ( i . e . numbers o f observations i n c l u d e d ) . This can be repeated f o r each p r o f i l e , and order s t a t i s t i c s and fun c t i o n s o f them c a l c u l a t e d over a l l p r o f i l e s . Medians and midspreads over p r o f i l e s of ordered component lengths are given i n 8K.

There i s a c l e a r p a t t e r n i n these r e s u l t s . Least-squares p a r t i t i o n s p i c k up a large amount of l o c a l roughness, 'bumps' and ' r u t s ' , and declare many such features t o be components. This seems undesirable. Accordingly i t was decided t o smooth the data before p a r t i t i o n . The o r i g i n a l data are angles 8 which are d i f f e r e n c e d to produce curvatures AB • I f smoother series are d e s i r e d , then there are various p o s s i b i l i t i e s :

( i ) smooth angles, then compute curvatures ( i i ) do not change angles, but smooth curvatures ( i i i ) smooth angles, then compute and smooth

curvatures.

The f i r s t p o s s i b i l i t y seems the most s t r a i g h t f o r w a r d and was t h e r e f o r e adopted.

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N a t u r a l l y many smoothing methods are possible t o produce smoothed s e r i e s v from o r i g i n a l s e r i e s u. Linear smoothers o f the general form

where £w ._ = I and values I are those i n the neighbourhood I 1

o f i , have been f r e q u e n t l y used i n geography and other d i s c i p l i n e s (e.g. Holloway, 1958; Rayner, 1971, 65-7^;

C h a t f i e l d , 1975, 17-20; Tobler, 1975; see P i t t y , 1969,

45-9 f o r a h i l l s l o p e example). Nonlinear smoothers have a t t r a c t e d much less a t t e n t i o n but are both simple and advantageous (Beaton and Tukey, 1974; Velleman, 1977;

McNeil, 1977, Ch. 6; Tukey, 1977, Chs. 7 and 1 6 ) .

The simplest nonlinear smoothers are running medians: the g e n t l e s t i s o f length 3 whereby

v i = med ( u ^ , u i s u i + 1 ) ,

w i t h appropriate end-value r u l e s , here

V i = med ( u ^ , U 2 )

v n = med (u n_-L, u^)

Running medians are a t t r a c t i v e as r e s i s t a n t or robust smoothers which are less s e n s i t i v e t o bumps and r u t s than comparable running means (moving averages). (For a contemporary i n t r o d u c t i o n t o resistance and robustness, see M o s t e l l e r and Tukey, 1977, Ch. 10 ) .

Nonlinear smoothing can be fol l o w e d by l i n e a r smoothing, f o r example by Hanning (Blackman and Tukey, 1959, 171 and references on nonlinear smoothing j u s t c i t e d )

- 282 -

V- - - i - UL 1

+ -J" U- + 4 - ^ - ^ . 1 t - l 2 I + 1 + I

Again end-values must be t a c k l e d i n some way: here

i 3 l 1 3 *

Some p r e l i m i n a r y experiments with running median of

length 3 a followed by Hanning (3H f o r s h o r t ) , i n d i c a t e d

that i t was too g e n t l e . The procedure was thus repeated,

hence the smoother 3H3H. Smoothed angles, rough angles

(rough = data - smooth), and curvatures derived from

smooth angles were computed and p l o t t e d i n each case.

This p a r t i c u l a r smoother i s f a i r l y c o n s e r v a t i v e : i t

s t i l l l eaves much l o c a l v a r i a b i l i t y , as can be seen

from a comparison of o r i g i n a l and smooth curvatures f o r

p r o f i l e TR i n 8L and 8M, and i n other p l o t s f o r the

remaining p r o f i l e s . The smoother removes some 69 to f8%

of v a r i a t i o n on a l i n e a r s c a l e (8N).

Additive e r r o r p a r t i t i o n s were produced fo r these

r e l a t i v e l y smooth curvature s e r i e s . The d i s t r i b u t i o n of

component lengths (80) shows t h a t a strong tendency to

produce a multitude of very short components p e r s i s t s

despite smoothing. There i s , however, a broad tendency

f o r the s h o r t e s t components to grow s l i g h t l y longer, while

some (but by no means a l l ) of the longest components

co n t r a c t i n length.

With t h i s problem i n mind, the component breaks, can

be examined ( 8 P ) . P a r t i t i o n s are given f o r d i f f e r i n g

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8N

V a r i a b i l i t y of curvature s e r i e s before and a f t e r smoothing

P r o f i l e s t d % reduction midspread % reduction

BO 5.6 1.30 77 5.5 1.4 75 ST 4.8 1.14 76 4.5 1.1 76

PR 3.7 0.95 74 3.5 0.9 74

PA 1.26 71 4.0 1.0 75 FA 4.8 1.29 73 5.0 1.1 78

LA 3.8 0.96 75 3.0 0. 8 73 TR 6.1 1.60 74 5.0 1.2 76

AR 4.3 1. 34 69 4.5 1.1 76

TA 5.7 1.65 71 6.0 1.5 75 HO 9-4 2.88 69 7.0 2.2 69 TO 7.0 1. 82 74 7.7 2.2 71

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8P Component bounds

BO #curvatures = 393 k 2 386 3 382 386 4 382 386 390 5 381 385 386 390 6 298 299 382 386 390 7 298 299 381 385 386 390 8 295 297 299 381 385 386 390 9 285 287 292 298 299 382 386 390

10 285 287 292 298 299 381 385 386 390

ST #curvatures = 452 k 2 1,45 3 43 45 4 39 43 45 5 43 45 429 433 6 39 43 45 429 433 7 39 43 45 429 434 440 8 39 43 45 296 298 429 433 9 39 43 45 296 298 303 429 433

10 39 43 45 296 298 304 306 429 433

PR #curvatures = k 2 3 4 5 174 177 6 174 177 7 174 177 8 159 161 174 177 9 159 161 174 177

10 159 161 174 177

264 264 265

261 264 265 264 265

261 264 265 261 263 264 265 261 264 265 261 263 264 265 261 263 264 265 266

PA #curvatures = 420 k 2 4 392 5 6 7 24 26 8 24 26 9 2 4 24 26 10 2 4 24 26

415 415 418 415 418

408 410 416 418 408 410 415 416 4l8 408 410 416 418 408 410 415 *J16 418 408 410 416 418 408 ^10 415 416 4l8

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8P (cont inued)

PA #curvatures = 505

k 2 489 3 38 41 4 38 41 43 5 38 41 43 489 6 38 41 43 483 489 7 38 41 43 483 489 491 8 38 41 43 483 489 9 34 38 41 43 483 489

10 34 38 41 43 483 489 491

497 500 497 500 497 500

LA #curvatures = 468

K 2 3 65 66 4 373 376 5 65 66 6 65 66 373 376 7 65 66 219 221 8 65 66 373 376 9 65 66 219 221 373 376

10 65 66 219 221 373 376

459

459 462

459 462 459 462 459 462

TR tc u r v a t u r e s = 402

k 2 394 3 251 253 4 251 253 379 5 251 253 379 387 6 251 253 353 355 387 7 251 253 353 355 379 387 8 251 253 353 355 359 379 387 9 251 253 342 344 353 355 379 387

10 251 253 314 342 344 353 355 379 387

AR #curvatures = 362

k 2 357 } 357 360 4 320 357 360 5 320 3511 357 360 6 320 351 353 357 360 7 340 345 350 353 357 360 8 320 340 345 350 353 357 360 9 320 327 340 345 350 353 357 360 10 163 165 320 340 345 350 353 357 360

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TA #curvatures = k 2 3 411 418 4 423 5 411 418 6 411 418 7 411 418 8 411 418 427 9 411 418

10 411 418

HO #curvatures =

8P (continued)

566

560 443 447 443 447 443 448 455 443 448 455 560 443 448 455 560

428 431 443 448 455 560 428 431 443 448 455 458 560

413 k 2 131 3 128 130 4 124 127 130 5 118 124 127 130 6 118 124 127 130 155 7 118 124 127 130 155 175 8 118 124 127 130 152 154 175 9 118 124 127 128 130 152 154 175

10 118 124 127 128 129 130 152 154 175

TO #curvatures = 318 k 2 16 3 270 271 4 270 271 300 5 270 271 300 303 6 16 270 271 300 303 7 116 118 270 271 300 303 8 116 118 123 270 271 300 303 9 16 116 118 123 270 271 300 303 10 28 30 116 118 123 270 271 300 303

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numbers of components i n a format which allows comparison of the e f f e c t s o f vary i n g the l e v e l of g e n e r a l i s a t i o n . I t seems l i k e l y t h a t only genuine breaks w i l l be maintained as the number o f components v a r i e s : breaks which appear and disappear r e l a t i v e l y q u i c k l y are l i k e l y t o be spurious. Moreover, very short components w i l l be disregarded as l a c k i n g i n geomorphological i n t e r e s t .

BO i s probably best regarded as an upper component (1,380) w i t h an i r r e g u l a r middle s e c t i o n (286,299)- and a r e l a t i v e l y short basal component (381,393). On ST, the breaks around 43 are caused by a t r a c k , w h i l e the very sho r t components around 300 are also a t t r i b u t a b l e t o purely l o c a l i r r e g u l a r i t y . There i s a good case f o r i d e n t i f y i n g a basal component (430,452) and a long upper component (1,429). S i m i l a r l y on PR breaks i n midslope may be discounted and a break made near the base: (1,264) and (265*271). Yet again on PA possible breaks near the crest do not seem t o have s u b s t a n t i a l s i g n i f i c a n c e and the c l e a r e s t break i s between a long upper (1,408) and a sho r t lower component (408,420).

I n the case of PA the breaks near the crest are caused by a d i t c h and once again the basal p a r t i s best d i s t i n g u i s h e d from an upper p a r t , say (1,489) and (490,505). On LA, there are b a s i c a l l y f o u r possible breaks: around 65, around 220, around 376, and around 459. The f i r s t three correspond t o l o c a l i r r e g u l a r i t y , whereas the l a s t can be d i s t i n g u i s h e d as the s t a r t of a short basal component (460,468).

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On TR three components may be d i s t i n g u i s h e d : a long convexity (1,252), a s h o r t e r convexity (253,353) and an i r r e g u l a r basal component (354,402). AR i s best taken as an upper component (1,320) together w i t h an i r r e g u l a r basal component (321,362). On TA an upper component ( l , 4 l l ) i s evident. To t h i s may be added an intermediate (and i r r e g u l a r ) crags component (412,455), a long lower component (456,560) and a short basal component (561,566).

HO i s a complicated case. The most appropriate d i v i s i o n would seem t o be i n t o an upper component (1,118), an i r r e g u l a r crags component (119,175)* and a t h i r d component reaching to the base, (176, 413). F i n a l l y on TO the most important break i s at 270. A basal component may be d i s t i n g u i s h e d (271,308) i n c l u d i n g two minor i r r e g u l a r i t i e s . Breaks around 118 are a t t r i b u t a b l e t o a t r a c k w h i l e t h a t a t 16 i s not p e r s i s t e n t : t h i s leaves a long upper component (1,270).

The component bounds accepted as o f probable geomorphological s i g n i f i c a n c e are l i s t e d i n 8Q, together w i t h u n d e rlying g e o l o g i c a l formations repeated from 6L. The most obvious common f e a t u r e , observed f o r every p r o f i l e except HO, i s a d i s t i n c t , r e l a t i v e l y s h o r t and o f t e n i r r e g u l a r basal component. Apart from t h a t , there i s a broad but by no means p e r f e c t r e l a t i o n s h i p between the number o f components accepted and the complexity o f the geology. BO, ST, PR, PA, PA and LA, a l l w i t h i n the Lower O o l i t e , are a l l accepted as one-component slopes above t h e i r basal components. TR, TA and HO (although not AR),

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8Q Component bounds and geology

P r o f i l e Geology Component bounds BO 1 1,380 381,393 ST 1,429 430,452

> d e l PR \ 1,264 265,271 PA J 1,408 409,420

PA 1 1,489 490,505 T g l s / d e l

LA J 1,459 460,468

TR 1 1,252 253,353 354,402 \ d e l / u l i

AR J 1,320 321,362

TA g l s / d e l / u l i 1,411 412,455 456,560 561,566

HO d e l / d o g / u l i / m l i 1,118 119,175 176,413

TO l c g / o x f / k e l / c o r / d e l 1,270 271,318

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extending from Lower O o l i t e t o L i a s , a l l have appreciably more complex forms. On the other hand TO, which extends across f i v e mapped formations i n the Lower and Middle O o l i t e s , i s accepted as one component above i t s basal component.

These r e s u l t s suggest t h a t d i v e r s i t y of und e r l y i n g s t r a t a i s not i n e v i t a b l y associated w i t h complexity of h i l l s l o p e forms i n the sense of a m u l t i p l i c i t y of d i s t i n c t components. Examination of the v a r i a t i o n of within-component mean of square V w i t h k helps t o complete the p i c t u r e (8R). The ranking of p r o f i l e s by V i s r e l a t e d to geology, but much of the v a r i a b i l i t y i s on a microscale and does not f i n d expression at component s c a l e , t h a t i s , between components. (This conclusion i s strengthened when i t i s remembered t h a t o r i g i n a l data have been smoothed before computing c u r v a t u r e s ) .

8.7 Summary and discussion

This chapter includes three d i s t i n c t c o n t r i b u t i o n s t o the theory and p r a c t i c e of p r o f i l e a n a l y s i s .

( i ) An extended f o r m u l a t i o n embeds the problem w i t h i n : data analysis (8 .1 ) and geomorphology ( 8 . 2 ) ; g u i d i n g p r i n c i p l e s are suggested ( 8 . 3 ) .

( i i ) A c r i t i q u e of e x i s t i n g methods shows t h a t none are r e a l l y acceptable ( 8 . 4 ) .

( i i i ) Methods f o r producing a d d i t i v e e r r o r p a r t i t i o n s are a t t r a c t i v e a l t e r n a t i v e s ( 8 . 5 ) . I n p r a c t i c e , however,

- 29 4 -

8r

W i t h i n component mean square V f o r various p a r t i t i o n s

k BO ST PR PA PA LA TR AR TA HO TO 1 1 .70 1.31 0.91 1.59 1.68 0.92 2.55 1.79 2. 73 8.28 3. 30 2 1.50 1.28 0.87 1.27 1.59 0 .90 2.52 1.33 2. 63 8.19 3. 21

3 1.32 1.20 0.75 1.16 1.42 0. 86 2.20 1.15 2. 45 7.07 3. 00 4 1.22 1.13 0.70 1.10 1.20 0 .84 2.17 1.07 2. 31 6.34 2. 89 5 1.18 1.09 0.67 1.01 1.11 0. 80 1.95 1.01 2. 12 5-70 2. 66

6 1.13 1.03 0.63 0.98 1.03 0.78 1 .90 0.96 2. 03 5 .26 2. 57 7 1.10 1.01 0 .60 0.92 0.96 0.75 1.72 0.91 1. 93 4.95 2. 38 8 1.08 0.95 0.58 0.89 0 .90 0.72 1.67 0.86 1. 87 4.76 2. 29 9 1 .04 0 . 91 0.56 0.83 0.86 0.69 1.58 0.83 1. 81 4.60 2. 19

10 1.01 0.87 0.54 0.79 0.82 0.67 1.53 0. 80 1. 77 4.46 2. 12

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l e a s t squares p a r t i t i o n s of curvature series need to be preceded by smoothing of o r i g i n a l data. Components i d e n t i f i e d on f i e l d p r o f i l e s allow some inferences about the r e l a t i o n s h i p between p r o f i l e morphometry and u n d e r l y i n g geology (8.6).

The method o f p r o f i l e analysis adopted here, a combination of n o n l i n e a r smoothing and a d d i t i v e e r r o r p a r t i t i o n , i s r a t h e r a r b i t r a r y and perhaps unduly complex. I t can only be regarded as an i n t e r i m s o l u t i o n . Future work would be best d i r e c t e d at these two f a m i l i e s o f methods - n o n l i n e a r smoothing and a d d i t i v e e r r o r p a r t i t i o n -probably w i t h the aim o f e l i m i n a t i n g one f o r the sake of s i m p l i c i t y .

I t has been argued above t h a t o p t i m a l i t y c r i t e r i a f o r p a r t i t i o n s need t o be discussed i n r e l a t i o n t o the appropriate decency assumptions about the i d e a l character of data. The case of Hartigan's a d d i t i v e e r r o r p a r t i t i o n s i s i n s t r u c t i v e i n t h i s respect. I t can be shown t h e o r e t i c a l l y t h a t o p t imal subseries diameter depends on the c o n d i t i o n a l d i s t r i b u t i o n of subseries values, but i t i s also c l e a r e m p i r i c a l l y t h a t t h i s may w e l l vary, both w i t h i n and between p r o f i l e s . Hence no s i n g l e diameter can be optimal.

I n such a s i t u a t i o n i t i s important t o consider adaptive methods of p r o f i l e analysis i n which the components are. d i s t i n g u i s h e d by procedures which depend on the p r o p e r t i e s o f the data. I r o n i c a l l y enough, apparently s u i t a b l e adaptive methods have only received prominence since the work reported here was undertaken, notably

- 296 -

adaptive methods of r e s i s t a n t / r o b u s t e s t i m a t i o n ( M o s t e l l e r and Tukey, 1977), and of nonlinear smoothing (McNeil, 1977;

Tukey, 1977). Future work w i l l examine these as possible bases f o r p r o f i l e analysis methods.

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8.8 Notation

c curvature d i n ordinary d e r i v a t i v e d subseries diameter

H 0 n u l l hypothesis

i s u b s c r i p t I s u b s c r i p t

J s u b s c r i p t k number o f components 1 measured length n number of observations

P power s arc l e n g t h u, u value of s e r i e s , average value V value o f series V w i t h i n component v a r i a t i o n

w weight X h o r i z o n t a l coordinate z v e r t i c a l coordinate

i n t e r c e p t parameter slope parameter

A d i f f e r e n c e operator e s t o c h a s t i c e r r o r

e angle summation operator

# number of i given

l I modulus

- 298 -

Mnemonics

loq med pr s t d upq

lower q u a r t i l e median p r o b a b i l i t y standard d e v i a t i o n upper q u a r t i l e

- 299 -

Chapter 9

FITTING CONTINUOUS CURVES TO HILLSLOPES

The Saturnian and the S i r i a n exhausted themselves i n conjectures upon t h i s s u b j e c t , and a f t e r abundance of argumentation equally ingenious and u n c e r t a i n , were obliged t o r e t u r n t o matters o f f a c t .

V o l t a i r e , Micromegas, Ch. I I .

9.1 The general s i t u a t i o n

9.2 S p e c i f i c a t i o n and e s t i m a t i o n : f i r s t approximation

9.3 S p e c i f i c a t i o n and e s t i m a t i o n : f u r t h e r approximations

9.4 D e t e r m i n i s t i c e s t i m a t i o n o f the Kirkby parameter

9.5 Checking

9.6 Results f o r f i e l d p r o f i l e s

9.7 Summary and discussion

9.8 Notation

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9.1 The general s i t u a t i o n

I t i s c l e a r t h a t v i s u a l e v a l u a t i o n o f model p r e d i c t i o n s ('the model p r o f i l e s seem f a i r l y r e a l i s t i c 1 ) cannot be accepted as a s a t i s f a c t o r y means of model t e s t i n g : mere 'eyeb a l l i n g ' leaves too much scope f o r s u b j e c t i v e and a r b i t r a r y judgements. Model p r o f i l e s should be f i t t e d t o ac t u a l p r o f i l e s s t a t i s t i c a l l y , and goodness-of-fit assessed q u a n t i t a t i v e l y . This much seems almost i n d i s p u t a b l e , yet the many models which have been proposed i n the l i t e r a t u r e have received g r e a t l y v a r y i n g amounts o f s t a t i s t i c a l t e s t i n g . Some, indeed, have received none at a l l . Moreover, model f i t t i n g has o f t e n been characterised by a c a v a l i e r disregard f o r the s t a t i s t i c a l d i f f i c u l t i e s which a r i s e i n the process (see, f o r example, Cox, 1975, on Woods, 197^ and Cox, 1977a on B u l l , 1975). This chapter provides a systematic discussion of f i t t i n g t i m e - i n v a r i a n t continuous d e t e r m i n i s t i c models t o h i l l s l o p e p r o f i l e s , i l l u s t r a t e d by the d e r i v a t i o n and a p p l i c a t i o n of f i t t i n g procedures f o r a model f u n c t i o n o r i g i n a l l y obtained by Kirkby (1971).

A framework f o r discussion i s given i n 9A. Model f i t t i n g i s here regarded as a three-stage process, as i s common i n modelling l i t e r a t u r e ( c f . Matalas and Maddock, 1976).

( i ) S p e c i f i c a t i o n : the form of the model i s decided ( i i ) E s t i m a t i o n : the parameters are estimated ( i i i ) Checking: the re s i d u a l s are analysed.

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9A H i l l s l o p e model f i t t i n g as a three-stage process

D e t e r m i n i s t i c model f u n c t i o n

Decide on v a r i a b l e s , known constants, unknown parameters

Choose estimators o f parameters Decide on s t o c h a s t i c e r r o r s t r u c t u r e

Choose measure of discrepancy

Use l i n e a r i s i n g transformation? Errors i n variables?

Autocorrelated errors?

Estimate parameters by minimising discrepancy ( a n a l y t i c a l s o l u t i o n or search algorithm)

Comparison of estimates Lack of f i t measures

S p a t i a l d i s t r i b u t i o n of r e s i d u a l s

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9.2 S p e c i f i c a t i o n and e s t i m a t i o n : f i r s t approximation

The f u n c t i o n

was f i r s t obtained by Kirkby (1971) as an approximate c h a r a c t e r i s t i c form s o l u t i o n t o a p a r t i c u l a r c o n t i n u i t y equation. I t i s also an exact constant form s o l u t i o n t o the same c o n t i n u i t y equation ( c f . Ch. 3.3.15, 3.3.21 above). Moreover, i t i s r e l a t e d t o the power func t i o n s used as e m p i r i c a l s t a t i c models by Hack and Goodiett (1960) and others ( c f . Ch. 3.2.2 and 3B above), k i s here termed the Kirkby parameter.

The c e n t r a l question i s the f o l l o w i n g : How can such a f u n c t i o n be f i t t e d t o an a c t u a l h i l l s l o p e p r o f i l e or a component o f such a p r o f i l e ? (The g e n e r a l i s a t i o n t o cover components i s s t r a i g h t f o r w a r d . Henceforth 'base' and 'cres t ' imply base and crest o f p r o f i l e or component).

A f i r s t task i n s p e c i f i c a t i o n i s t o d i s t i n g u i s h between v a r i a b l e s , known constants and unknown parameters. I n t h i s case z and x are v a r i a b l e s , z^ and x^ are known constants and k i s an unknown parameter. Further a l g e b r a i c s i m p l i f i c a t i o n i s possible by s c a l i n g coordinates t o dimensionless v a r i a b l e s . I f we w r i t e

r e l a t i v e f a l l = v = z d " z

k = (

r e l a t i v e distance = u = x

- 303 -

Then v = u k

I t i s now necessary t o make three r e l a t e d d ecisions: (a) To choose estimators of the parameters (b) To sp e c i f y the s t r u c t u r e of s t o c h a s t i c e r r o r s (c) To choose a measure of discrepancy between model

and data.

(a) An estimator i s a procedure f o r e s t i m a t i n g a parameter; w i t h any p a r t i c u l a r set of data i t produces an estimate, an a c t u a l number. The p r o p e r t i e s of estimators are w e l l discussed by Bard (197*0, P l a c k e t t (1971) and Silvey (1970), among many others. I n p r a c t i c e f o u r p r o p e r t i e s are p a r t i c u l a r l y important (Bard, 1974, 44):

( i ) Small bias On average the estimator should produce a value as near as possible t o the a c t u a l value.

( i i ) Small variance ( e f f i c i e n c y ) The spread of estimat e s around the average estimate should be as small as p o s s i b l e .

( i i i ) Robustness The estimator should be s t a b l e under s l i g h t changes i n the p r o b a b i l i t y d i s t r i b u t i o n of s t o c h a s t i c e r r o r s : i n p a r t i c u l a r , i t should not be thrown out by the occurrence of o u t l y i n g or w i l d observations.

( i v ) Computability I t should be easy t o c a l c u l a t e .

(b) A model i s o f t e n w r i t t e n i n standard reduced form (Bard, 1974, 26): dependent = d e t e r m i n i s t i c f u n c t i o n of v a r i a b l e ( s ) independent v a r i a b l e ( s ) + s t o c h a s t i c e r r o r ( s )

and parameter(s)

- 304 -

Why are s t o c h a s t i c e r r o r s included i n a model? The sto c h a s t i c terms capture the remainder of the v a r i a t i o n not captured by the d e t e r m i n i s t i c f u n c t i o n ( G i l b e r t , 1973). This remainder w i l l r e f l e c t one or more o f the f o l l o w i n g :

( i ) the i n f l u e n c e of va r i a b l e s not included i n the analysis ( i i ) i n a p p r o p r i a t e choice of model f u n c t i o n ( i i i ) r e a l but random e f f e c t s ( i v ) measurement e r r o r (v) i n t e r a c t i o n among the other f o u r

(Mark and Church, 1977, 71)

I n the example of t h i s chapter a f i r s t approximation i s t o w r i t e

v = u k + 6 where 6 i s s t o c h a s t i c e r r o r . The f o l l o w i n g assumptions about 6 are standard : zero mean E( t ^ ) = o constant variance _ or homoscedasticity : E ( 6 ^ ) = a constant uncor r e l a t e d : E (€^€-.) = 0 i * j

(c) F i t t i n g the model may us u a l l y be viewed as minimising some measure of the discrepancy between the model and the data (Nelder, 1975,7). The resid u a l s may be defined as a f u n c t i o n of the parameter k

e^ (k) = - u-f ; i = l , . . . , n The sum o f squared r e s i d u a l s i s then

Y\ ^ e,. , a f u n c t i o n o f k.

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Minimising the sum o f squared r e s i d u a l s t o estimate an unknown parameter i s known as ordinary l e a s t squares e s t i m a t i o n (OLS).

Why should t h i s p a r t i c u l a r measure of discrepancy be used f o r estimation? C l e a r l y i t i s necessary t o consider OLS i n the l i g h t of the c r i t e r i a emphasised by Bard (1974,4*0 small b i a s , small variance, robustness and c o m p u t a b i l i t y .

There i s an important r e s u l t f o r l i n e a r models, the Gauss-Markov theorem, which i s roughly t h a t i f the e r r o r s £ s a t i s f y the three standard assumptions i n (b) above, then the OLS estimators are unbiased, and have the smallest variance of any l i n e a r unbiased estimator. (For f u l l e r statements, see Wonnacott and Wonnacott, 1970, 48-51 or S i l v e y , 1970, 51-4). Hence there are some t h e o r e t i c a l grounds f o r expecting small bias and small variance.

OLS i s not robust under a l l circumstances, and i s well-known t o be unstable i n the presence of w i l d observations. Consequently many a l t e r n a t i v e procedures have been proposed (e.g. M o s t e l l e r and Tukey, 1977; McNeil, 1977). However, the h i l l s l o p e s considered here are well-behaved i n the sense t h a t bumps and r u t s are microscale f e a t u r e s . Hence only least-squares c r i t e r i a w i l l be considered f o r the discrepancy f u n c t i o n . This has two major advantages: they are r e l a t i v e l y easy t o compute and they are f a i r l y w e l l understood i n p r i n c i p l e .

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Returning t o the Kirkby model the sum of squared r e s i d u a l s may be w r i t t e n

This q u a n t i t y i s nonlinear i n k and there i s thus no closed-form expression f o r the OLS estimator of k. This problem could be t a c k l e d by using a search a l g o r i t h m t o f i n d the minimum value of the discrepancy and hence an estimate of k ( c f . Bard, 1971*, Chs. 5 and 6, and Chambers, 1977, Ch. 6, on search a l g o r i t h m s ) . I t can also be broached by a f a i r l y s t r a i g h t f o r w a r d t r i c k at the expense o f some complications. I n the next s e c t i o n t h i s t r i c k (a l i n e a r i s i n g t r a n s f o r m a t i o n ) w i l l be discussed together w i t h ways of overcoming two f u r t h e r problems ( e r r o r s i n v a r i a b l e s and a u t o c o r r e l a t e d e r r o r s ) .

9.3 S p e c i f i c a t i o n and e s t i m a t i o n : f u r t h e r approximations 9.3-1 L i n e a r i s i n g transformations

The d e t e r m i n i s t i c f u n c t i o n v = u k

can be reexpressed i n l o g a r i t h m i c terms I n v = k I n u

(Here and below n a t u r a l logarithms are used). I f i t i s supposed t h a t a d d i t i v e s t o c h a s t i c e r r o r s perturb t h i s d e t e r m i n i s t i c f u n c t i o n then

I n v = k I n u + whence r e s i d u a l s may be defined

e^(k) = I n v^ - k I n u^

fc\X 1 c 1*

- 307 -

and the sum of squared r e s i d u a l s formed

Ze*(le) = Z ( U v. - k In a.)"

The l i m i t s of summation have not been s p e c i f i e d i n t h i s expression because the use of a l o g a r i t h m i c t r a n s f o r m a t i o n brings a minor problem i n i t s wake. Near the c r e s t of a slope r e l a t i v e f a l l may be i d e n t i c a l l y zero, and hence the logarithm may be indeterminate. At the base r e l a t i v e f a l l and r e l a t i v e distance are i d e n t i c a l l y one. I f measured lengths o f zero gradient occur at the crest they should be omitted from the c a l c u l a t i o n , and an equal number o f measured lengths omitted symmetrically at the base. Otherwise a l l summations w i l l be ^ . Indices w i l l be omitted from the expressions below, and i t should be understood t h a t the l i m i t s of summation are determined by t h i s procedure.

The s u b s t i t u t i o n v/ = I n v and u' = I n u i n the model above y i e l d the obvious l i n e a r model

v = k u + £ w i t h OLS discrepancy ( o r sum of squared r e s i d u a l s )

Z(v' - ku')2 The a n a l y t i c a l d e r i v a t i o n of the OLS estimator w i l l

now be given. Expanding the term i n parentheses

For a minimum i t i s s u f f i c i e n t t h a t

^ [ Z C v ' - k a ' ) 1 ] =0

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whence

-2Su.V + Ik Su,'1 = O

_ £ U u.. (n v

The use o f a l i n e a r i s i n g t r a n s f o r m a t i o n i s f r e q u e n t l y recommended: the problems are less f r e q u e n t l y emphasised. The account given here draws on Wonnacott and Wonnacott (1970, 91-8), Johnston (1972, 27-53), Goldf e l d and Quandt (1972, Ch. 5) and Bard (197^, 78-80).

The l i n e a r model

£Y\ V = k L*\ ^ + €'

corresponds t o the model

w i t h m u l t i p l i c a t i v e e r r o r s . While i t leads t o a closed-form es t i m a t o r , the assumption of s t o c h a s t i c e r r o r s which are m u l t i p l i c a t i v e i n the o r i g i n a l metric needs substantive j u s t i f i c a t i o n . I t imp l i e s h e t e r o s c e d a s t i c i t y (unequal variances) together w i t h some other problems (Goldf e l d and Quandt, 1972, 136). As these authors wrote: 'In s p i t e of the p o s s i b i l i t y t h a t there might be a p r i o r i reasons f o r s p e c i f y i n g the e r r o r term t o be o f a p a r t i c u l a r type, i n most cases the m u l t i p l i c a t i v e form seems t o be chosen f o r i t s computational convenience. The two basic or pure a l t e r n a t i v e s . . . are r a r e l y contrasted.' Moreover, the data ought t o be allowed t o reveal which of the two hypotheses about e r r o r s i s acceptable (Goldf e l d and Quandt, 1972, 137).

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Goldf e l d and Quandt proposed a generalised model i n c o r p o r a t i n g both a d d i t i v e and m u l t i p l i c a t i v e e r r o r terms. However, a simpler method of overcoming t h i s d i f f i c u l t y seems more appropriate f o r the h i l l s l o p e model case. Following a suggestion by Bard (1974, 79) a l i n e a r i s i n g t r a n s f o r m a t i o n i s used together w i t h weighted l e a s t squares as a way of t a c k l i n g the h e t e r o s c e d a s t i c i t y which i s assumed t o be associated w i t h the t r a n s f o r m a t i o n . The connection i s given by a theorem which gives the r e l a t i o n s h i p between the variances of two random v a r i a b l e s (s and t , say)

v/or 0) « var ( s ) [ l ^ f

(see, e.g., P l a c k e t t , 1971, 59-60).

I f an a d d i t i v e homoscedastic model i s a p p r o p r i a t e , then

Vox £ v | a ) = VCMT ( e )

but i f we transform by logarithms

\IOJr

because

1 ( U v) - i Since i n t h i s s i t u a t i o n the variance i s not constant,

i t i s best t o allow f o r t h i s and weight each r e s i d u a l . The WLS discrepancy (sum o f weighted squared r e s i d u a l s ) f o r a l i n e a r model

v' « k t e' w i t h weights w i s

/2

- 310 -

For a minimum i t i s s u f f i c i e n t t h a t

0

whence

|c = Swu/v'/Zwu.'2

I n the h i l l s lope case, c o n d i t i o n a l variance of the 1 2

sto c h a s t i c e r r o r s i s proportional t o (—) . I t i s l o g i c a l to weight squared r e s i d u a l s i n v e r s e l y by variance, t o discount i n h e r e n t l y more v a r i a b l e f l u c t u a t i o n s . Hence

p w = v and the WLS estimator i s

9. 3- 2 Errors i n va r i a b l e s

I t has been t a c i t l y assumed so f a r t h a t v i s a dependent v a r i a b l e , which i s erro r - p r o n e , and t h a t u i s an independent v a r i a b l e , which i s e r r o r - f r e e . I n geomorphological terms, however, there i s no j u s t i f i c a t i o n at a l l f o r any such asymmetric d i s t i n c t i o n , v i s not a 'response' t o ' f a c t o r ' u any more than u i s a 'response' t o ' f a c t o r ' v ( t o use the ex c e l l e n t terminology of Tukey, 1977, 125-6). Nor i s u h e l d at f i x e d values t o see the r e s u l t i n g change i n v. Both v and u must t h e r e f o r e be regarded as subject t o s t o c h a s t i c f l u c t u a t i o n , a s i t u a t i o n knownas 'errors i n v a r i a b l e s ' .

2 v2- O u) 1

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A v a r i e t y of methods has been devised f o r t h i s s i t u a t i o n . A thorough review i s given by Moran (1971). I n the g e o l o g i c a l and geographical l i t e r a t u r e the problem has been discussed by McCammon (1973), T i l l (1973), Mark and Church (1977), Kuhry and Marcus (1977), and Mark and Peucker (1978), among others. McCammon (1973) suggested minimising the discrepancy

where e^ and &2 a r e r e s i d u a l distances measured perpendicular to h o r i z o n t a l and v e r t i c a l axes, and w. and Wj are corresponding weights. McCammon gave d e t a i l s o f programs implementing a minimi s a t i o n a l g o r i t h m . The main problem w i t h t h i s method i s the need f o r s p e c i f y i n g weights beforehand, which r e a l l y requires d e t a i l e d knowledge of e r r o r s t r u c t u r e .

T i l l (1973) c r i t i c i s e d the use o f standard l i n e a r regression i n geomorphological s i t u a t i o n s where both var i a b l e s are subject t o e r r o r . He recommended i n s t e a d the use of the reduced major a x i s , the l i n e which b i s e c t s the angle between the two standard regression l i n e s . This i s the c o r r e c t s o l u t i o n i f the s t o c h a s t i c e r r o r s p e r t u r b i n g the two v a r i a b l e s have equal variances. T i l l reworked some examples o f f i t t i n g power fu n c t i o n s t o g l a c i a l v a l l e y s i d e p r o f i l e s given by Doornkamp and King (1971). He f i t t e d a reduced major axis t o the l o g a r i t h m i c a l l y transformed data, but using the assumption of m u l t i p l i c a t i v e e r r o r s and wit h o u t commenting on obviously a u t o c o r r e l a t e d r e s i d u a l s . The use of reduced major axes i s c l e a r l y not

- 312 -

a general s o l u t i o n t o t h i s problem given the strong assumption of equal variances.

Mark and Church (1977), discussing the problem of e r r o r s i n v a r i a b l e s , c r i t i c i s e d the misuse of regression i n e arth science, e s p e c i a l l y i n geomorphometry. (See also Mark and Peucker, 1978, on geographical a p p l i c a t i o n s ) . They reviewed s o l u t i o n s appropriate i n d i f f e r e n t circumstances. Most emphasis i s placed on e s t i m a t i o n procedures which can be used when the r a t i o of e r r o r variances i s known. The main disadvantage of such methods i s t h a t they r e q u i r e considerable knowledge about each v a r i a b l e . Kuhry and Marcus (1977) recommended a covariance r a t i o method which requires observations on a t h i r d v a r i a b l e , which i s not possible i n t h i s case.

None of these s o l u t i o n s seems acceptable f o r the Kirkby model, and so the only p o s s i b i l i t y i s t o estimate parameters f o r p o l a r s i t u a t i o n s ( ( i ) v a'.response, u a f a c t o r ( i i ) u a response, v a f a c t o r ) and consider the v a r i a t i o n i n r e s u l t s ( c f . Moran, 1971, 251).

By symmetry OLS and WLS estimators can be derived f o r the case i n which u i s regarded as dependent and v as independent.

The OLS estimator of 1/k i n the m u l t i p l i c a t i v e model

- (i/k) Lw v + e'

i s Y> LYV v In u. 2 ( ( * v)

- 313 -

The WLS estimator of 1/k i n the a d d i t i v e model i s

I f i t i s assumed t h a t estimate o f k = l / ( e s t i m a t e of 1/k)

then these estimators may be compared w i t h others proposed.

9.3-3 Autocorrelated e r r o r s

One of the standard assumptions about s t o c h a s t i c e r r o r s £• i s t h a t p a i r s of terms are uncorr e l a t e d

Hence one possible problem i s a u t o c o r r e l a t i o n among e r r o r terms, e s p e c i a l l y i f data are time or space s e r i e s . The problems which a r i s e when e r r o r terms are a u t o c o r r e l a t e d , and methods f o r overcoming these problems, have received much a t t e n t i o n , e s p e c i a l l y i n econometrics where l i n e a r models are o f t e n f i t t e d t o time s e r i e s (e.g. Wonnacott and Wonnacott, 1970, 136-45; Johnston, 1972, Ch. 8; Stewart, 1976, 137-^6). This work has r e c e n t l y been extended t o s p a t i a l s e r i e s by s t a t i s t i c a l geographers ( C l i f f and Ord, 1973, Ch. 5; M a r t i n , 197^).

Johnston (1972, 246-9) o u t l i n e d the consequences o f using OLS estimators on a l i n e a r model when e r r o r terms are au t o c o r r e l a t e d . F i r s t l y , estimators of the parameters have large variances. Secondly, standard s i g n i f i c a n c e t e s t s are i n v a l i d . T h i r d l y , p r e d i c t i o n using the model i s i n e f f i c i e n t .

E 1 J

= 0

- 314 -

Such problems may be s u f f i c i e n t l y serious t o warrant the use of other methods as a l t e r n a t i v e s t o OLS i f there are grounds f o r suspecting a u t o c o r r e l a t e d e r r o r s . I f the model i s being a p p l i e d t o time s e r i e s , then a f i r s t order autoregressive (Markov) scheme may be appropriate f o r the e r r o r s

where

a constant

This Markov model assumes u n i l a t e r a l i n f l u e n c e s : the past i s assumed t o a f f e c t the present, but not vice versa. I f the model i s being a p p l i e d t o space series then b i l a t e r a l or m u l t i l a t e r a l i n f l u e n c e s must be allowed. With a f u r t h e r g e n e r a l i s a t i o n t o allow the p o s s i b i l i t y of unequal w e i g h t i n g , the f o r m u l a t i o n of C l i f f and Ord (1973, 90) i s obtained:

The b i l a t e r a l case i s more appropriate f o r h i l l s l o p e p r o f i l e s ,

I f p i s unknown, then there are two p o s s i b i l i t i e s : i t can be estimated i t e r a t i v e l y from the data, or a value can be assumed a p r i o r i . OLS i m p l i c i t l y assumes |?= 0; the p o l a r p o s s i b i l i t y i s t o assume p= 1 ( M a r t i n , 1974). However, the assumption ^= 1 b r i n g s some t h e o r e t i c a l problems i n i t s wake, even f o r the l i n e a r model and time serie s case ( c f . Wonnacott and Wonnacott, 1970, 141; Johnston, 1972, 245; Stewart, 1976, 145-6). Moreover, the estimators used by C l i f f and Ord (1973) and Martin (1974)

- 315 -

would need f u r t h e r adjustment f o r n o n l i n e a r i t y and e r r o r s i n v a r i a b l e s . F i n a l l y , such estimators are r a t h e r unstable numerically because they are based on products and squares of l o c a l d i f f e r e n c e s which tend t o be very small i n the case of h i l l s l o p e s e r i e s .

Since estimator f o r m u l a t i o n i s d i f f i c u l t i n t h i s case, and since no attempt w i l l be made at e i t h e r p r e d i c t i o n or s i g n i f i c a n c e t e s t i n g , a t t e n t i o n t o a u t o c o r r e l a t i o n w i l l here be confined t o i n s p e c t i n g a u t o c o r r e l a t i o n f u n c t i o n s of r e s i d u a l s .

9 . 4 D e t e r m i n i s t i c e s t i m a t i o n of the Kirkby parameter

The case of the model f u n c t i o n o r i g i n a l l y derived by Kirkby ( 1971) i s unusual because i t i s also possible t o estimate the Kirkby parameter k w i t h o u t any reference t o s t o c h a s t i c e r r o r s . This s e c t i o n i s an intermezzo developing t h i s p o i n t .

Kennedy ( 1 9 6 7 , 22) introduced a 'height-length i n t e g r a l f o r h i l l s l o p e p r o f i l e s w h i c h was defined by a g r a p h i c a l example. This example was repeated by Chorley and Kennedy ( 1 9 7 1 , 5*0 accompanied by an a l g e b r a i c d e f i n i t i o n which i s i n f a c t both meaningless and i n c o r r e c t . For r e l a t i v e f a l l v and r e l a t i v e distance u, a proper d e f i n i t i o n o f the i n t e g r a l (here c a l l e d the Kennedy i n t e g r a l ) i s

0 The i n t e g r a l i s a measure of the shape of a h i l l s l o p e p r o f i l e

) ( l - v ) d k K , say

- 316 -

0 < K < 1 ; K = 0.5 f o r s t r a i g h t slopes, K>0.5 f o r convex, K<^0.5 f o r concave.

Kennedy (personal communication, 1977) computed values of K g r a p h i c a l l y . I t i s also possible t o compute K d i r e c t l y from r e l a t i v e coordinates u and v. P u t t i n g y = 1 - v, the i n t e g r a l i s the sum o f n t r a p e z i a :

K = E ( --L** )( a,- ac-,) where

Jo = 1 - a o = 0

As Chorley and Kennedy (1971, 290) h i n t e d , there i s a simple r e l a t i o n s h i p between Kennedy i n t e g r a l K and Kirkby parameter k. Given t h a t

K = j o - * ) *»• and v = u k

then i

r° i k + l V

kTi k+1 and i n v e r s e l y

f - K

Since the appropriate e r r o r s t r u c t u r e f o r Kirkby's model i s not obvious, such a simple d e t e r m i n i s t i c estimator i s h i g h l y a t t r a c t i v e .

- 317 -

9.5 Checking

9 . 5 . 1 General remarks

Each method of e s t i m a t i o n y i e l d s an estimate of the parameter, a set of f i t t e d values f o r the 'dependent v a r i a b l e ' and a set of estimated r e s i d u a l s .

r e s i d u a l = observed - f i t t e d

These r e s u l t s must be analysed c a r e f u l l y t o check the adequacy o f the model. The basic approach i s simple: 'a good f i t does not prove t h a t the model i s c o r r e c t . . . a lack of f i t c o n s t i t u t e s s t r o n g grounds f o r r e j e c t i n g , or at l e a s t amending the model' (Bard, 1 9 7 ^ , 1 9 8 ) .

9 . 5 . 2 Comparison of estimators

I n e a r l i e r sections several estimators have been derived f o r the Kirkby parameter k, i n c l u d i n g one w i t h o u t any reference t o s t o c h a s t i c e r r o r s . Since i t i s never c e r t a i n i n p r a c t i c e which assumptions are most a p p r o p r i a t e , i t seems best t o employ a l l the e s t i m a t o r s , and t o compare the r e s u l t s . The und e r l y i n g p r i n c i p l e i s t h a t great v a r i a b i l i t y between estimators i n d i c a t e s an inadequate model, wh i l e conversely i f remainder terms are a l l s m a l l , i t should matter r e l a t i v e l y l i t t l e which assumptions are invoked.

9 . 5 . 3 Analysis o f r e s i d u a l s

The set of r e s i d u a l s deserves t o be studied i n d e t a i l (McNeil, 1977; Tukey, 1 9 7 7 ) . I n analysing r e s i d u a l s , i t i s best t o supplement numerical i n v e s t i g a t i o n w i t h g r a p h i c a l d i s p l a y .

- 318 -

Measures o f o v e r a l l lack o f f i t may r e a d i l y be defined. Three are used here: root mean square r e s i d u a l , midspread of res i d u a l s and range of r e s i d u a l s .

The s p a t i a l d i s t r i b u t i o n o f re s i d u a l s i s worth c o n s i d e r a t i o n , f o r as C l i f f and Ord (1973, 71) r e p o r t e d , good a s p a t i a l f i t and good s p a t i a l f i t are not nec e s s a r i l y associated. Residuals should be p l o t t e d i n s e r i a l order and t h e i r a u t o c o r r e l a t i o n p r o p e r t i e s i n v e s t i g a t e d .

9.5.4 Against s i g n i f i c a n c e t e s t i n g

Kirkby (1976b) and Moon (1977) have regarded the assignment o f s i g n i f i c a n c e l e v e l s t o f i t t e d models as an important aspect of model checking. However, t h i s approach seems t o be both unnecessary and problematic, f o r three reasons.

( i ) I n d i v i d u a l h i l l s l o p e p r o f i l e s can be t r e a t e d on t h e i r own m e r i t s , without any reference t o h y p o t h e t i c a l sets o f approximately i d e n t i c a l p r o f i l e s o f which they are supposedly r e p r e s e n t a t i v e .

( i i ) Those s i g n i f i c a n c e t e s t s which at f i r s t glance appear appropriate t u r n out on close r i n s p e c t i o n t o be in a p p r o p r i a t e f o r the h i l l s l o p e case. For example, the Durbin-Watson t e s t f o r a u t o c o r r e l a t e d r e s i d u a l s i s appropriate only f o r a model w i t h an i n t e r c e p t term and w i t h s t o c h a s t i c e r r o r s f o l l o w i n g a u n i l a t e r a l Markov property ( c f . Wonnacott and Wonnacott, 1970, 142-3; Johnston, 1972, 250-2; Stewart, 1976, 147-50).

- 319 -

( i i i ) I n hypothesis t e s t i n g of the standard k i n d , we decide between n u l l and a l t e r n a t i v e hypotheses w i t h s p e c i f i e d p r o b a b i l i t i e s o f e r r o r . The value of such a procedure i s very much i n doubt, and i s a matter of considerable controversy among s t a t i s t i c i a n s (Edwards, 1969; B a r n e t t , 1973; Cox and Hinkley, 1974), although i t i s r a r e l y questioned i n geography (but c f . Cox and Anderson, 1978). Naive s i g n i f i c a n c e t e s t i n g reduces model checking t o a binary decision based on one number (which may be w i l d l y inaccurate) and a n u l l hypothesis (which may be t o t a l l y i r r e l e v a n t ) ( c f . Cox, 1977a on B u l l , 1975). I t i s b e t t e r to base any decisions on the i n d i c a t i o n s provided by a l l the model r e s u l t s .

9.6 Results f o r f i e l d p r o f i l e s

The Kirkby parameter k was estimated by the f i v e estimators o u t l i n e d above (see 9B) on the fourteen components i d e n t i f i e d i n Ch. 8.6 above which were based on 100 or more observations (see 8Q and 9C). For each estimate, r e s i d u a l s were c a l c u l a t e d of the form

k • i v i - u£ I = 1,...,n The frequency d i s t r i b u t i o n of r e s i d u a l s i s here summarised by min, l o q , upq, max, range, midspread and r o o t mean square (rms). A u t o c o r r e l a t i o n f u n c t i o n s were c a l c u l a t e d f o r each r e s i d u a l series up t o a maximum l a g determined by the r u l e of thumb given i n Ch. 7.5 above (the smaller of 50 and f l o o r ( n / 4 ) ) . 9D shows the number of angles i n each component; the Kennedy i n t e g r a l ; the set of estimates; summary measures f o r r e s i d u a l d i s t r i b u t i o n s (zero and decimal p o i n t e l i d e d ) ;

- 320 -

9B Estimators of the Kirkby parameter

I d e n t i f i e r Assumptions and d e f i n i t i o n

KMU Errors m u l t i p l i c a t i v e ; u c o n t r o l l i n g f a c t o r (9-3.1)

' £ (U u.f KAU Errors a d d i t i v e ; u c o n t r o l l i n g f a c t o r (9.3.1)

E v1 (U a) 1

KMV Errors m u l t i p l i c a t i v e ; v c o n t r o l l i n g f a c t o r (9.3.2)

£ (U vf

KAV Errors a d d i t i v e ; v c o n t r o l l i n g f a c t o r (9.3.2.)

Z (Cn vf u,a U v U u,

KKEN Uses t h e o r e t i c a l r e l a t i o n between Kirkby parameter and Kennedy i n t e g r a l (9.4)

d i s c r e t e version of 0

- 321 -

9C Components f i t t e d by Kirkby curves

P r o f i l e Indices Dimensions (m)

b d

(deg) e

BO 1,380 573 56 5.5 ST 1,429 647 75 6 .6

PR 1,264 398 47 6.8

PA 1,408 614 73 6.8

PA 1,489 732 103 8 . 0

LA 1,459 687 109 9 .0

TR 3 ,250 375 24 3.7 TR 253 ,353 144 45 17.5 AR 1,320 479 75 8.9 TA 6,406 602 78 7.3 TA 456,560 147 56 20.8

HO 2,117 173 27 9 . 0

HO 176,413 345 96 15.5 TO 1,270 383 136 19.6

- 322 -

9D Results of f i t t i n g Kirkby curves

BO (1,380) # 380 K = 0.618

Estimate min loq upq

K e s i a u j

max ais range midspr rms a u t o c o r r e l a t i o r

KMU 1.579 -035 -008 002 012 047 010 012 l a g 44 0.124 KAU 1.619 -027 -003 007 018 045 010 010 la g 38 0.132 KMV 1.588 -033 -006 003 013 047 010 012 l a g 43 0.115 KAV 1.612 -029 -003 006 017 046 010 010 l a g 39 0.131 KKEN 1.615 -028 -003 007 017 045 010 010 la g 39 0.123

ST (1,429) #429 K = 0.516

Residuals Estimate min loq upq max range midspr rms au t o.c or re.l at i on

KMU 1.196 -018 000 053 070 087 053 039 [ l a g 50 0.641] KAU 1.054 -044 -028 016 02 4 068 044 024 [ l a g 50 0.6531 KMV 1.201 -017 001 054 071 088 053 040 [ l a g 50 0.641] KAV 1.092 -036 -020 025 036 073 045 024 [ l a g 50 0.648] KKEN 1.066 -042 -026 019 027 O69 044 023 [ l a g 50 0.651]

PR (1,264) # 264 K = 0.572 •» U - U \JL \A. C

Estimate min l o q upq max range midspr rms autocor r e l a t i o r

KMU 1.255 -040 -027 -001 016 056 027 021 l a g 30 0.153 KAU 1. 324 -024 -016 008 034 058 024 014 l a g 31 0.131 KMV 1.259 -039 -027 -000 017 056 026 020 l a g 31 0.108 KAV 1.329 -024 -015 009 035 059 024 014 l a g 31 0.137 KKEN 1.335 -023 -014 010 036 060 024 014 l a g 31 0.144

- 323 -

PA (1,408)

9D (continued)

# 408 K = 0.618 Residuals

Estimate min loq upq max range midspr rms autocor.relatior

KMU 1.445 -085 -041 -010 009 095 031 037 [ l a g 50 0.626] KAU 1.572 -073 -020 010 025 098 029 029 [ l a g 50 0.653] KMV 1.455 -084 -039 -008 O i l 095 031 036 [ l a g 50 0.629] KAV 1.556 -075 -022 007 023 097 029 029 [ l a g 50 0.652] KKEN 1.618 -069 -012 019 036 105 031 029 [ l a g 50 0.653]

FA (1,489) # 489 K = 0.625 Residuals

Estimate min loq upq max range midspr rms a u t o c o r r e l a t i o r

KMU 1.446 -093 -053 -016 003 096 037 043 [ l a g 50 0.759] KAU 1.604 -082 -025 012 025 107 037 032 [ l a g 50 0.796] KMV 1.458 -092 -051 -014 004 096 037 042 [ l a g 50 0.764] KAV 1.593 -082 -027 010 022 105 037 033 [ l a g 50 0.795] KKEN 1.664 -078 -017 025 038 116 041 033 [ l a g 50 0.796]

LA (1,459) # 459 K = 0.606 Residuals

Estimate min lo q upq max range midspr rms aut 0 c or re 1 at i or

KMU 1.389 -082 -039 -007 006 088 032 034 [ l a g 50 0.483] KAU 1.538 -045 -006 012 030 075 018 018 Hag 50 0.329] KMV 1.401 -079 -036 -005 006 O85 032 032 [ l a g 50 0.472] KAV 1.563 -039 -003 015 036 075 018 018 [ l a g 50 0.309] KKEN 1.540 -045 -006 012 030 075 018 018 [ l a g 50 0.328]

- 324 -

9D (continued)

TR (3,250) # 248 K = 0. 652

Estimate min loq upq max range midspr rms a u t o c o r r e l a t i o r

KMU 1.491 -115 -089 -008 012 127 08J 068 l a g 41 0.134 KAU 1.885 -033 -016 017 051 084 033 020 l a g 25 0.158 KMV 1.529 -106 -080 -005 016 122 075 062 l a g 40 0.151 KAV 1.876 -034 -018 016 050 084 034 020 l a g 26 0.162 KKEN 1.877 -034 -018 016 050 085 034 020 l a g 26 0.160

TR (253,353) # 101 K = 0. 510

Estimate min lo q upq max range midspr rms a u t o c o r r e l a t i o n

KMU 0.982 -123 -072 041 062 185 112 062 [ l a g 25 0.324] KAU 0.948 -128 -076 028 049 177 104 062 [ l a g 25 0.313] KMV 1.003 -120 -070 047 069 189 117 062 [ l a g 25 0.328] KAV 1.047 -113 -059 062 085 197 121 064 [ l a g 25 0.330] KKEN 1.043 -113 -060 061 083 197 121 064 [ l a g 25 0.330]

AR (1,320) # 320 K = 0. 572

Estimate min loq upq max range midspr rms a u t o c o r r e l a t i o n

KMU 1.351 -064 -031 027 083 147 059 042 [ l a g 50 0.272] KAU 1. 322 -072 -039 023 077 148 062 043 [ l a g 50 0.285] KMV 1. 360 -061 -029 029 085 147 058 042 [ l a g 50 0.267] KAV 1. 434 -042 -013 044 102 144 057 044 [ l a g 50 0.238] KKEN 1.338 -067 -034 025 080 148 060 042 [ l a g 50 0.278]

- 325 -

9D (continued)

TA (6,406) # 4 0 1 K = 0.622 Residuals

Estimate min loq upq max range midspr rms a u t o c o r r e l a t i o n

KMU 1.640 -051 -018 018 058 109 036 028 [ l a g 50 0.563] KAU 1.610 -057 -022 014 052 110 035 02 8 [ l a g 50 0.565] KMV 1.688 -043 -013 024 066 109 038 029 [ l a g 50 0.562] KAV 1.762 -031 -008 038 079 109 045 034 [ l a g 50 0.563] KKEN 1.648 -050 -017 019 059 109 036 028 [ l a g 50 0.563]

TA (456,560) # 105 K = 0. 522

Estimate min loq upq max range midspr rms a u t o c o r r e l a t i o n

KMU 0.999 -084 -039 002 016 100 040 036 l a g 22 0.245 KAU 1.062 -068 -026 016 036 105 042 030 l a g 25 0.219 KMV 1.001 -084 -038 002 016 100 041 035 la g 22 0.249 KAV 1.086 -063 -024 021 044 106 045 030 la g 25 0.246 KKEN 1.093 -061 -024 02 3 046 107 046 030 l a g 26 0.215

HO (2 ,117) # 116 K = 0. 603

Estimate min loq upq max range midspr rms aut 0 cor re 1 at i on

KMU 1.330 -113 -059 002 016 129 060 050 la g 23 0.239 KAU 1. 482 -079 -025 024 049 128 049 036 l a g 24 0.234 KMV 1.335 -112 -057 002 018 129 060 050 l a g 24 0.196 KAV 1.487 -078 -024 025 050 128 049 036 l a g 24 0.234 KKEN 1.518 -072 -020 030 056 127 051 036 l a g 24 0.235

- 326 -

9D (continued)

HO (176,413) # 238 K = 0.407 Residuals

Estimate min loq upq max range midspr rms au t o.c or re l a t i on

KMU 0.730 -014 003 030 043 057 027 021 l a g 32 0.126 KAU 0.691 -025 -012 019 032 057 031 016 l a g 40 0.151 KMV 0.733 -014 004 031 043 057 027 022 l a g 31 0.157 KAV 0.687 -027 -013 018 031 058 031 016 l a g 41 0.158 KKEN 0.686 -027 -013 018 031 058 031 016 l a g 41 0.161

TO ( 1 ,270) # 270 K = 0. 431

Estimate min loq upq max range midspr rms. .autocorrelation

KMU 0.995 -017 036 096 135 153 059 080 l a g 42 0.127 KAU 0.744 -077 -014 016 030 107 031 031 l a g 45 0.140 KMV 1.029 -013 042 108 148 160 066 089 l a g 42 0.150 KAV 0.760 -071 -008 020 038 109 028 030 l a g 43 0.154 KKEN 0.757 -073 -009 019 036 109 028 031 l a g 44 0.128

- 327 -

and a u t o c o r r e l a t i o n c h a r a c t e r i s t i c s , summarised by the f i r s t value f a l l i n g on or below the upper bound of the 0.01 confidence i n t e r v a l f o r n u l l a u t o c o r r e l a t i o n , or f a i l i n g t h a t , by the value f o r the maximum l a g c a l c u l a t e d .

I n i n t e r p r e t i n g these r e s u l t s , i t i s necessary t o e s t a b l i s h which estimates are best f o r each component, and then t o consider the geomorphological i m p l i c a t i o n s o f the best estimates. I n doing t h i s the f i g u r e s of 9D are u s e f u l l y supplemented by p l o t s o f r e s i d u a l series i n s p a t i a l order.

The estimators were ordered from best t o worst f o r each component, using the c r i t e r i a ( i ) low rms ( i i ) low midspread ( i i i ) low range ( i v ) low a u t o c o r r e l a t i o n (v) symmetry of r e s i d u a l d i s t r i b u t i o n s about zero. Ties on any c r i t e r i o n were resolved by i n v o k i n g c r i t e r i a lower i n t h i s l i s t . The c r i t e r i a used here, and t h e i r o r d e r i n g i n t h i s l i s t j a r e a r b i t r a r y t o some e x t e n t , but i t w i l l be seen t h a t the r e s u l t s are s u f f i c i e n t l y c l e a r c u t t o remove serious doubts about such a r b i t r a r i n e s s . The choice o f rms r e s i d u a l as the most important c r i t e r i o n stems from the general approach adopted here o f using l e a s t squares estimators. N a t u r a l l y any monotonic t r a n s f o r m a t i o n of ro o t mean square r e s i d u a l (e.g. t o mean square r e s i d u a l ) would give the same ordering.

Estimators are thus ordered i n 9E which gives estimators and rms f o r each estimator and component. 9F

- 328 -

Estimators ordered by performance w i t h estimate and rms

Rank BO (1,380) KAU

1.619 0.010

KKEN 1.615 0.010

KAV 1.612 0.010

KMV 1.588 0.012

KMU 1.579 0.012

ST (1,429) KKEN 1.066 0.023

KAU 1.054 0.024

KAV 1.092 0.024

KMU 1.196 0.039

KMV 1.201 0.040

PR (1,264) KAU 1. 324 0.014

KAV 1.329 0.014

KKEN 1.335 0.014

KMV 1.259 0.020

KMU 1.255 0.021

PA (1,408) KAV 1.556 0.029

KAU 1.572 0.029

KKEN 1.618 0.029

KMV 1.455 0.036

KMU 1.445 0.037

PA (1,489) KAU 1.604 0.032

KAV 1.593 0.033

KKEN 1.664 0.033

KMV 1.458 0.042

KMU 1.446 0.043

LA (1,459) KAV 1.563 0.018

KKEN 1.540 0.018

KAU 1.538 0.018

KMV 1.401 0.032

KMU 1.389 0.034

TR (3,250) KAU 1.885 0.020

KAV 1.876 0.020

KKEN 1.877 0.020

KMV 1.529 0.062

KMU 1.491 0.068

TR (253,353) KAU 0.948 0.062

KMU 0.982 0.062

KMV 1.003 0.062

KKEN 1.043 0.064

KAV 1.047 0.064

- 329 -

9E (continued)

Rank AR (1,320)

1 KMV 1.360 0.042

2 KMU 1.351 0.042

3 KKEN 1.338 0.042

4 KAU

1. 322 0.043

5 KAV 1.434 0.044

TA (6,406) KAU 1.610 0.028

KMU 1.640 0.028

KKEN 1.648 0.028

KMV 1.688 0.029

KAV 1.762 0.034

TA (456,560) KAU 1.062 0.030

KAV 1.086 0.030

KKEN 1.093 0.030

KMV 1.001 0.035

KMU 0.999 0.036

HO (2,117) KAV 1.487 0.036

KAU 1.482 0.036

KKEN 1.518 0.036

KMU 1.330 0.050

KMV 1.335 0.050

HO (176,413) KAU 0.691 0.016

KAV 0.687 0.016

KKEN 0.686 0.016

KMU 0.730 0.021

KMV 0.733 0.022

TO (1,270) KAV 0.760 0.030

KKEN 0.757 0.031

KAU 0.744 0.031

KMU 0.995 0.080

KMV 1.029 0.089

- 330 -

and 9G give f u r t h e r views of estimator performance. 9F shows the absolute performance

(rms f o r t h i s e s timator - rms f o r best e s t i m a t o r ) > 0

and 9G the r e l a t i v e performance

rms f o r t h i s e s timator > 1

rms f o r best estimator

Good estimators score low on each measure.

I t i s c l e a r that i t can make a great d i f f e r e n c e which

est i m a t o r i s employed. KMU and KMV are sometimes very

poor performers: the assumption of h e t e r o s c e d a s t i c e r r o r s

i s u s u a l l y unwarranted. KAV g e n e r a l l y does w e l l , while KKEN

and KAU are the two best e s t i m a t o r s . The good performance

of KKEN i s p l e a s i n g , s i n c e i t does not re.quire any recourse

to ideas of s t o c h a s t i c e r r o r s . However, KAU appears to

perform marginally b e t t e r o v e r a l l .

T his a n a l y s i s of estimator performance leans h e a v i l y

on rms r e s i d u a l as a numerical summary. I t would c l e a r l y

be p o s s i b l e to base the a n a l y s i s on other c r i t e r i a . A

second approach t r i e d was the use of

t h i s midspread - min midspread

and of

t h i s midspread/min midspread

as analogues to the measures of performance given i n 9F and

9G which are based on rms. The r e s u l t s , not reported here

but r e a d i l y obtainable from 9D, support those already given,

but show a c l e a r edge f o r KAU over KKEN as the best o v e r a l l

e stimator.

- 331 -

Absolute performance f o r each estimator on each component

Quantity t a b u l a t e d i s (rms r e s i d u a l f o r t h i s estimator on t h i s component - rms r e s i d u a l f o r best estimator on t h i s component) x 1000

KMU KAU KMV KAV KKEN

BO ( [1,380) 2 0 2 0 0

ST ( [1,429) 16 1 17 1 0

PR ( [1,264) 7 0 6 0 0

PA ( [1,408) 8 0 7 0 0

PA ( [1,489) 11 0 10 1 1

LA 1 [1,^59) 16 0 14 0 0

TR ( [3,250) 48 0 42 0 0

TR ( [253,353) 0 0 0 2 2

AR 1 :i,320) 0 1 0 2 0

TA ( [6,406) 0 0 1 6 0

TA ( [456,560) 6 0 5 0 0

HO ( [2,117) 14 0 14 0 0

HO 1 [176,413) 5 0 6 0 0

TO ( [1,270) 50 1 59 0 1

max 50 1 59 6 2

upq 16 0 14 l 0

med 8 0 7 0 0

loq 2 0 2 0 0

min 0 0 0 0 0

- 332 -

9G R e l a t i v e performance f o r each estimator on each component

Quantity t a b u l a t e d i s

rms r e s i d u a l f o r t h i s estimator on t h i s component

rms r e s i d u a l f o r best estimator on t h i s component

KMU KAU KMV KAV KKEN

BO ( 1,380) 1.20 1.00 1.20 1.00 1.00

ST ( 1,129) 1.70 1.04 1.74 1.04 1.00

PR ( 1,264) 1.50 1.00 1. 43 1.00 1.00

PA ( 1,408) 1.28 1.00 1.24 1.00 1.00

FA ( 1,489) 1.34 1.00 1.31 1.03 1.03

LA ( [1,459) 1.89 1.00 1.78 1.00 1.00

TR ( 3,250) 3.40 1.00 3 .10 1.00 1.00

TR ( .253,353) 1 .00 1.00 1.00 1.03 1.03

AR ( '1,320) 1 .00 1.02 1.00 1.05 1.00

TA ( r6,4o6) 1.00 1.00 1.04 1.21 1.00

TA ( [456,560) 1.20 1.00 1.17 1.00 1.00

HO ( [2,117) 1.39 1.00 1.39 1.00 1.00

HO ( [176,413) 1.31 1.00 1.38 1.00 1.00

TO < :i,270) 2.67 1.03 2.97 1.00 1.03

max 3.40 1.04 3 .10 1.21 1.03

upq 1.70 1.00 1.74 1.03 1.00

med 1.33 1.00 1.35 1.00 1.00

loq 1.20 1.00 1.17 1.00 1.00

min 1.00 1.00 1.00 1.00 1.00

- 333 -

9H gives a f u r t h e r p i c t u r e of v a r i a t i o n among e s t i m a t o r s . The quantity d i s p l a y e d i s the d i f f e r e n c e between the parameter estimate f o r a given estimator and that f o r the best estimator fo r a given component, as given i n 9E. KMU and KMV are again i n d i c t e d by t h e i r poor performance and KAU once more emerges as b e t t e r than i t s nearer competitors KKEN and KAV.

I t i s now appropriate to draw together the best estimates

(as defined using the l i s t of c r i t e r i a above), rms r e s i d u a l s

and Kennedy i n t e g r a l s (91 )• At t h i s stage some concreteness

i s a l s o i n t r o d u c e d : m u l t i p l y i n g rms r e s i d u a l by component

height gives a dimensioned measure of l a c k of f i t . I t i s

now h e l p f u l a l s o to l i s t data on slopes above and below f i t t e d

components, omitting t r i v i a l l y s h o rt s e c t i o n s (9J)- I n 91

and 9J components have been ordered by value of Kirkby

parameter k. The fourteen components may be considered i n

three groups. The l a r g e s t group contains upper c o n v e x i t i e s

fo r TR, BO, TA, FA, LA, PA, HO, AR, PR, and ST. The

remaining groups are approximately s t r a i g h t midslopes f o r

TA and TR and c o n c a v i t i e s f o r TO and HO.

How are these f i t t e d components to be i n t e r p r e t e d ? At

a minimum, the family of power functions y i e l d s d e s c r i p t i v e

summaries of the h i l l s lope p r o f i l e s . This minimum approach

resembles t h a t of Hack and Goodlett (1960), although they

employed g r a p h i c a l e s t i m a t i o n , and paid no a t t e n t i o n to

e i t h e r s p e c i f i c a t i o n or checking. The value of the power

fun c t i o n approach i s s i m p l i c i t y , or more p r e c i s e l y , parsimony

- 334 -

9H V a r i a b i l i t y i n parameter estimates f o r each component

Quantity tabulated i s

(parameter estimate f o r t h i s e s timator - parameter estimate

f o r best e s t i m a t o r ) x 1000

KMU KAU KMV KAV KKEN

BO (1,380) -40 0 -31 -7 -4 ST (1,429) 130 -12 135 26 0 PR (1,264) -69 0 -65 5 11 PA (1,408) -111 16 -101 0 62 PA (1,489) -158 0 -146 -11 60

LA (1,459) -174 -25 -162 0 -23 TR (3,250) -394 0 -356 -9 -8 TR (253,353 34 0 55 99 95

AR (1,320) -9 -38 0 74 -22 TA (6,406) 30 0 78 162 38 TA(456,560) -63 0 -61 24 31 HO (2,117) -157 -5 -152 0 31 HO (176,413) 39 0 42 -4 -5 TO (1,270) 235 -16 269 0 -3

max 235 16 269 162 95

upq 34 0 55 26 38 med -52 0 -46 0 6 loq -157 -12 -146 -4 -5 min -394 -38 -356 -11 -23

- 335 -

9I Summary of best r e s u l t s f o r f i t t e d components

TR (3 , 2 5 0 )

k

1 .885

K

0 .652

rms

0 .020

rms ( i n metres)

0.5 BO (1 , 3 8 0 ) 1 .619 0 .618 0.010 0.6

TA (6 , 4 0 6 ) 1 .610 0 .622 0 .028 2.2

FA (1 , ^ 8 9 ) 1 .604 0 .625 0 .032 3.3 LA (1 , 4 5 9 ) 1 .563 0 .606 0 .018 2.0

PA (1,408) 1.556 0 .618 0.029 2.1

HO (2,117) 1.487 0.603 0.036 1.0

AR (1,320) 1. 360 0.572 0.042 3.1 PR (1,264) 1.324 0.572 0.014 0.7 ST (1,429) 1.066 0.516 0.023 1.7 TA (456,560) 1.062 0.522 0.030 1.7 TR (253,353) 0.948 0.510 0.062 2.8

TO (1,270) 0.760 0.431 0.030 4.1

HO (176,413) 0.691 0.407 0.016 1.5

- 336 -

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- 337 -

and i n t e l l i g i b i l i t y . Each component i s s p e c i f i e d by two known constants ( t o t a l height and t o t a l length) and one parameter estimate ( r e a d i l y i n t e r p r e t e d as a measure of convexity or c o n c a v i t y ) . The t h e o r e t i c a l r e l a t i o n s h i p between the Kirkby parameter and the Kennedy i n t e g r a l , supported i n p r a c t i c e by the g e n e r a l l y good performance of KKEN, a l s o aids i n t e r p r e t a t i o n .

One obvious p i t f a l l here i s the 'magic number syndrome 1

(Cox and Anderson, 1978) whereby a s i n g l e - v a l u e d c h a r a c t e r i s a t i o n

i s sought which i n some mysterious way captures a l l the

information i n a s e t of data. Such a tendency f o r the l u r e

of s i m p l i c i t y to triumph over the many-sidedness of

phenomena has been i d e n t i f i e d by d i s c e r n i n g i n v e s t i g a t o r s

i n s e v e r a l f i e l d s ( c f . Evans, 1972, 21-2 on general

geomorphometry; MacArthur, 1972, 197-8 on d i v e r s i t y

measures i n ecology; P h i l i p , 1974, 268 on s o i l p h y s i c s ;

Medawar, 1977, 13 on demography, economics and psychology;

Holton, 1978, 207-9 on s c i e n c e assessment). 'Magic number

syndrome' i s suggested as an i r o n i c term f o r t h i s p i t f a l l :

i t s a p p l i c a b i l i t y to cases i n the s o c i a l or environmental

s c i e n c e s need not be l i m i t e d by the f a c t t h a t 'magic numbers'

are genuine e n t i t i e s i n atomic p h y s i c s .

This p i t f a l l i s e a s i l y avoided. Residuals and t h e i r

summaries provide a r e a d i l y i n t e l l i g i b l e p i c t u r e of the

inadequacy of a s i n g l e - v a l u e d c h a r a c t e r i s a t i o n , s i n c e they

show how f a r the one-parameter model y i e l d s an accurate

r e c o n s t r u c t i o n of the dimensionless p r o f i l e . Hence we have

a measure of the cost of a simple c h a r a c t e r i s a t i o n .

- 338 -

This minimum d e s c r i p t i v e approach y i e l d s composite

model-based summaries of f i e l d p r o f i l e s (9K). This l i s t i s

based on r e s u l t s presented e a r l i e r , together with i n t e r p r e t a t i o n s

of the p l o t s of r e s i d u a l s i n s p a t i a l order. Each power

function i s c o n s t r a i n e d by d e f i n i t i o n to pass e x a c t l y through

the c r e s t and base of each component, and so r e s i d u a l s tend

to zero as endpoints are approached. However, t h e i r

behaviour over the length of the component i s a guide to

the adequacy of the model which supplements the summary lack-'-

o f - f i t measures. S e v e r a l changes of r e s i d u a l s i g n (from

p o s i t i v e to n e g a t i v e , or v i c e v e r s a ) w i t h i n a component

i n d i c a t e s m a l l - s c a l e r e s i d u a l v a r i a t i o n which may happily

be averaged out. On the other hand, simple s t r u c t u r e ,

e s p e c i a l l y i n the form of a changeover from r e s i d u a l s of

one s i g n near the c r e s t to r e s i d u a l s of the opposite s i g n

near the base, throws some doubt on the a c c e p t a b i l i t y of

the model: i n such c a s e s , concavo-convex (or convexo-

concave) d e v i a t i o n s from the model suggest t h a t two d i s t i n c t

components might have been combined. Such simple 'two regime'

r e s i d u a l s t r u c t u r e s occur on 6 out of 14 f i t t e d components,

on ST (1,429), FA (1,489), TR ; (253 , 3 5 3 ) , AR (1,320), TA

(456, 560) and HO (2,117). They are noted i n 9K. However,

despite the f a c t t h a t these 6 components tend to have higher

l a c k - o f - f i t (rms) than the other eig h t components, the r a t h e r

low values obtained seem to j u s t i f y keeping the models as

they stand, and thus regarding the r e s i d u a l v a r i a t i o n as

secondary.

- 339 -

9K Composite model-based summary of f i e l d p r o f i l e s

BO Upper p a r t (1,380) a long gentle convexity ( z ^ = 56m,

x b = 573m, 9 = 5.5°, k = 1.619) with gentle f l u c t u a t i o n s

(rms = 0.6m); lower p a r t (381,394) a short s t e e p e r b a s a l

slope (z,, = 5m, x = 20m, 8 = 13.1°) u b

ST Upper p a r t (1,429) a long gentle convexity ( z d = 75m,

x^ = 647m, 8 = 6 . 6 ° , k = 1.066) with moderate f l u c t u a t i o n s

(rms = 1.7m, r e s i d u a l s show changeover from negative near

c r e s t to p o s i t i v e near base of component); lower p a r t

(430,453) a short steeper b a s a l slope ( z d = 8m, x^ = 35m,

e = 1 2 . 5 0 )

PR Upper p a r t (1,264) a long gentle convexity (z = 47m, d

x b = 398m, 0 = 6 . 8 ° , k = 1.324) with gentle f l u c t u a t i o n s (rms = 0.7m); lower p a r t (265,272) a short steeper b a s a l

slope (Zrf = 3m, x^ = 12m, 8 = 1 2 . 6 ° ) u b PA Upper p a r t (1,408) a long gentle convexity ( z d = 73m,

x^ = 6l4m, 0 = 6 . 6 ° , k = 1.556) with moderate f l u c t u a t i o n s b (rms = 2.1m); lower p a r t (409,421) a short steeper b a s a l

slope ( z a = 10m, x b = 16m, 9 = 32.3°)

PA Upper p a r t (1,489) a long gentle convexity ( z d = 103m,

xfa = 732m, 9 = 8.0°, k = 1.604) with moderate f l u c t u a t i o n s

(rms = 3.3m, r e s i d u a l s show changeover from p o s i t i v e near

c r e s t to negative near base of component)^ lower p a r t

(490,506) a short steeper b a s a l slope ( z ^ = 12m, x^ = 22m,

0 = 29.0°)

- 340 -

9K (continued)

LA Upper pa r t (1,459) a long gentle convexity ( z ^ = 109m,

x b = 687m, 9 = 9.0°, k = 1 . 563) with moderate f l u c t u a t i o n s

(rms = 2.0m); lower p a r t (460,469) a short steeper b a s a l

slope ( z , = 4m, x,_ = 15m, 8= 1 3 . 8 ° ) a b TR Upper pa r t (3,250) a long gentle convexity ( z d = 24m,

x b = 375m, 8 = 3.7°, k = 1 . 885 ) with gentle f l u c t u a t i o n s

(rms = 0.5m); middle p a r t (253,353) an approximately

s t r a i g h t steep slope ( z ^ = 45m, x^ = 145m, 6 = 1 7 . 5 ° ,

k = 0.948) with moderate f l u c t u a t i o n s (rms = 2.8m,:

r e s i d u a l s mostly p o s i t i v e near c r e s t , negative towards

b a s e ) ; lower p a r t (354,403) a s h o r t e r s t e e p e r b a s a l

slope ( z d = 31m, x f e = 68m, 9 = 2 4 . 8 ° )

AR Upper p a r t (1,320) a long gentle convexity ( z ^ = 75m,

x D = 479m, 9 = 8 . 9 ° , k = 1.360) with moderate f l u c t u a t i o n s

(rms = 3.1m, r e s i d u a l s mostly negative near c r e s t ,

p o s i t i v e towards b a s e ) ; lower p a r t (321,363) a s h o r t e r

s t e e p e r b a s a l slope ( z d = 26m, x f e = 58m, 0 = 23.7°)

TA Upper p a r t (6,4o6) a long gentle convexity (z = 78m, _ d

x^ = 602m, 8 = 7 . 3 ° , k = 1.610) with moderate f l u c t u a t i o n s (rms = 2.2m); a s h o r t e r steeper slope (407,455) (z = 36m,

_ d x f e = 64m,0=29.2°); an approximately s t r a i g h t slope (456,560)

( z d = 56m, x f e = 147m, 9 = 2 0 . 8 ° , k = 1.062) with moderate

f l u c t u a t i o n s (rms = 1.7m, r e s i d u a l s show changeover from

p o s i t i v e near c r e s t to negative near b a s e ) ; lower p a r t

(561 ,567) short s t e e p e r b a s a l slope ( z d = 5m, x^ = 9m,

9 = 26.2°)

- 341 -

9K (continued)

HO Upper p a r t (2,117) a long gentle convexity (z = 27m, _ d x b = 173m, 9 = 9.0°, k = 1.487) with gentle f l u c t u a t i o n s

(rms = 1.0m, r e s i d u a l s p o s i t i v e near c r e s t , mostly

negative near b a s e ) ; middle p a r t (118,175) a steeper

slope ( x b = 75m, z d = 33m, 0 = 23.6°); lower p a r t a

steep concavity (x. = 345m, z j = 96m, Q = 19.6°, k = 0.691) D a

with moderate f l u c t u a t i o n s (rms = 1.5m)

TO Upper p a r t (1,270) a long steep concavity ( z d = 136m,

x b = 383m, G = 19-6°, k = O.76O) with l a r g e f l u c t u a t i o n s

(rms = 4.1m); lower p a r t a s h o r t e r g e n t l e r b a s a l slope

( z d = 17m, x b = 56m, 0= 16.7°)

- 342 -

9L and 9M are examples of r e s i d u a l p l o t s f o r AR (1,320) and KMV, and f o r BO (1,380) and KAU.

D i f f i c u l t i e s a r i s e when we go beyond a minimum

d e s c r i p t i v e approach, and consider i n t e r p r e t i n g the power

functions as s o l u t i o n s to a dynamic model i n which sediment

f l u x i s a power function of distance and gradient ( c f .

Ch. 3 .3.15, 3.3.21 above). 9N shows the values of the

Kirkby parameter which correspond to process exponents

given by Kirkby (1971, 21). A l l the c o n v e x i t i e s are l e s s

convex than the k = 2 p r e d i c t e d f o r the s i n g l e process of

s o i l creep. However, t h i s p a r t i c u l a r p r e d i c t i o n i s not very

w e l l founded, and many process s t u d i e s of s o i l creep have

found gradient to be a weak c o n t r o l of creep r a t e s (e.g.

Anderson, 1977). Moreover, values l e s s than 2 seem

compatible with the combination of s o i l creep and other

processes which almost c e r t a i n l y moulded these c o n v e x i t i e s .

This suggestion i s merely i n t u i t i v e , and no a n a l y t i c a l

r e s u l t s have been produced f o r a p b l y g e n e t i c g e n e r a l i s a t i o n

of the model i n question. The ten upper c o n v e x i t i e s vary

considerably i n Kirkby parameter from I . 8 8 5 to 1.066 (91)

which cannot be explained e a s i l y . I n f a c t t h i s i n d i c a t i o n

of v a r y i n g convexity i s not obviously r e l a t e d e i t h e r to

geology (no c l e a r d i f f e r e n c e between FA, LA and TA with

c r e s t s on g l s , or Grey Limestone S e r i e s , and the other

seven i n t h i s group, with c r e s t s on d e l , or D e l t a i c Beds),

or to p r o f i l e dimensions (no c l e a r r e l a t i o n s h i p with z d ,

x. or G ).

- 343 -

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- 344 -

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9N T h e o r e t i c a l p r e d i c t i o n s f o r Kirkby parameter

Process Distance exponent Gradient exponent

s o i l creep r a i n s p l a s h

s o i l wash

r i v e r s

0

0

1.3-1.7

2-3

1

1-2

1.3-2

3

2

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0.46-0.85

0.33-0.67

Source: A f t e r Kirkby (1971,21)

- 346 -

The other f o u r components are the approximately s t r a i g h t mid slopes of TA and TR and the concavities of TO and HO. Here the dynamic model approach breaks down, f o r these are steeper slopes ( 6 = 20 . 8 , 17.5, 19.6, 15.5 degrees) and one or more kinds o f threshold-dependent f a i l u r e processes has probably been operative i n each case. Without any i n v e s t i g a t i o n of r e g o l i t h p r o p e r t i e s i t would be rash indeed t o hazard a process i n t e r p r e t a t i o n of three components. V a r i a t i o n s i n l i t h o l o g y may also be more important i n these cases.

F i n a l l y , some remarks are i n order on the f i e l d p r o f i l e s i n t e r p r e t e d as combinations of components (9K). A l l but two include basal slopes s h o r t e r and steeper than the components above; these components were not f i t t e d t o models because they were so s h o r t . 90 gives an overview of basal slopes. I t shows how they are mostly secondary features on the scale of the h i l l s l o p e p r o f i l e , and how they e x h i b i t considerable v a r i a t i o n i n form. Whether such basal slopes may be a t t r i b u t e d t o r e j u v e n a t i o n remains an open question.

9 .7 Summary and discussion

( i ) Model p r e d i c t i o n s should be f i t t e d t o a c t u a l h i l l s l o p e p r o f i l e s s t a t i s t i c a l l y , and goodness-of-fit assessed q u a n t i t a t i v e l y . Model f i t t i n g may be regarded as a three - stage process, i n v o l v i n g s p e c i f i c a t i o n , e s t i m a t i o n and checking (9.1).

- 347 -

90 Basal slopes

*b 0 BO 20 5 13.1

ST 35 8 12.5

PR 12 3 12.6

PA 16 10 32.3

PA 22 12 29.0

LA 15 13.8

TR 68 31 2k. 8

AR 58 26 23.7

TA 9 5 26.2

HO 3^5 96 19.6

TO 56 17 16.7

max 3^5 96 32.3

upq 58 26 26.2

med 29 11 21.7

l o q 15 5 13.1

min 9 3 12.5

steeper, s h o r t e r than above

g e n t l e r , longer than above g e n t l e r , s h o r t e r than above

- 348 -

( i i ) I n the case of a model f u n c t i o n derived by Kirkby ( 197D, a f i r s t approximation i n s p e c i f i c a t i o n and e s t i m a t i o n leads t o a sum o f squared r e s i d u a l s discrepancy which i s n o n l i n e a r i n the parameter t o be estimated. A l i n e a r i s i n g t r a n s f o r m a t i o n leads t o a closed-form e s t i m a t o r , but may induce h e t e r o s c e d a s t i c i t y , which can be t a c k l e d by weighting r e s i d u a l s . Since both v a r i a b l e s i n the model are e r r o r - p r o n e , estimators should be developed f o r the p o l a r s i t u a t i o n s i n which each v a r i a b l e i s regarded as a response. Although a u t o c o r r e l a t e d e r r o r s may w e l l be present, no e s t i m a t i o n procedure i s known f o r t h i s model which takes account of a u t o c o r r e l a t i o n i n a s a t i s f a c t o r y manner. The Kirkby parameter can also be estimated d e t e r m i n i s t i c a l l y as a f u n c t i o n of the Kennedy i n t e g r a l (9.2, 9.3, 9.4).

( i i i ) Checking the model should be based on a v a r i e t y of estimators since i t i s never c e r t a i n i n p r a c t i c e which assumptions are appropriate. Lack of f i t measures, the s p a t i a l p a t t e r n o f r e s i d u a l s , and t h e i r a u t o c o r r e l a t i o n p r o p e r t i e s a l l deserve a t t e n t i o n . An approach t o model checking based on s i g n i f i c a n c e t e s t i n g seems both unnecessary and problematic (9.5).

( i v ) Five estimators were employed on fourteen components based on 100 or more observations. Residuals were c a l c u l a t e d i n each case, and extremes, q u a r t i l e s , range, root mean square, midspread and a u t o c o r r e l a t i o n f u n c t i o n o f r e s i d u a l s were der i v e d , supplemented by p l o t s o f r e s i d u a l series i n s p a t i a l order. The best estimates were i d e n t i f i e d f o r each component

- 349 -

mainly on the basis of root mean square r e s i d u a l , although other c r i t e r i a were considered as w e l l . At the same time performance of the i n d i v i d u a l estimators was reviewed, i n d i c a t i n g t h a t KMU and KMV were the poorest, w h i l e KAU i s b e t t e r than i t s nearer competitors KAV and KKEN.

The Kirkby power f u n c t i o n model can be used i n a minimum d e s c r i p t i v e approach, a t t r a c t i v e because of i t s parsimony and i n t e l l i g i b i l i t y . The 'magic number syndrome' i s r e a d i l y avoided, since the cost of a sin g l e - v a l u e d c h a r a c t e r i s a t i o n i s considered. A more ambitious approach i n which power func t i o n s are regarded as dynamic model s o l u t i o n s runs i n t o g r eater d i f f i c u l t i e s , l a r g e l y because of p o lygenetic development and the presumed operation o f f a i l u r e processes on steeper slopes.

The viewpoint i n t h i s chapter i s t h a t an appropriate methodology f o r model f i t t i n g must be developed before the important e m p i r i c a l issue of i d e n t i f y i n g adequate models can be t a c k l e d i n a s a t i s f a c t o r y manner. The one-parameter model used here i n exa m p l i f y i n g an approach t o model f i t t i n g i s a t t r a c t i v e as a candidate d e s c r i p t i v e and explanatory model, y e t i s s u f f i c i e n t l y simple t o serve as a s t a r t i n g p o i n t i n developing such a methodology. The r e s u l t s i n t h i s chapter support t h i s p o i n t o f view: even i n such a r e l a t i v e l y simple case t e c h n i c a l issues are not t r i v i a l i n any sense. I t does matter which estimator i s used; s t a t i s t i c a l theory does provide guidance over procedures; and the a t t i t u d e t h a t r e s i d u a l s should be i n v e s t i g a t e d pays

- 350 -

dividends i n p r a c t i c e . Some important t e c h n i c a l questions remain f o r consideration. The r e l a t i v e value o f robust-r e s i s t a n t f i t t i n g procedures has not been evaluated. More complex models would need t o be f i t t e d by o p t i m i s a t i o n algorithms. Simulation models would req u i r e s e n s i t i v i t y analyses i n a d d i t i o n t o some degree of parameter e s t i m a t i o n .

The rough adequacy of the power f u n c t i o n model f o r the components f i t t e d here i s almost guaranteed by the f a c t t h a t component bounds have been i d e n t i f i e d by p r o f i l e a n a l y s i s . This procedure may appear t o possess an element of c i r c u l a r i t y , although i t seems t o make l i t t l e sense t o f i t a model f o r a component t o anything but a component. There might be some value i n f i t t i n g power func t i o n s t o e n t i r e p r o f i l e s t o consider the adequacy of such crude models, but t h i s has not been attempted here. A f i n a l p o i n t i s t h a t since the p r o f i l e analysis methods used here are regarded as only i n t e r i m , new analyses would req u i r e new model f i t s .

- 351 -

9.8 Notation

d i n ordinary d e r i v a t i v e or i n i n t e g r a l e r e s i d u a l E expectation operator i index j index k Kirkby parameter K Kennedy i n t e g r a l n number of observations s random v a r i a b l e t random v a r i a b l e u r e l a t i v e distance u' I n u v r e l a t i v e f a l l v I n v w weight x h o r i z o n t a l coordinate x^ slope l e n g t h y 1 - v z v e r t i c a l coordinate z^ slope height £ s t o c h a s t i c e r r o r fc' " "

I I V

9 average angle ^ autoregressive parameter S summation operator

f i n p a r t i a l d e r i v a t i v e i n t e g r a l number

- 352 -

Mnemonics

loq lower q u a r t i l e max maximum med median min minimum OLS ordinary l e a s t squares rms root mean square upq upper q u a r t i l e var variance

- 353 -

Chapter 10

CONCLUSIONS

"When you say ' h i l l * " , the Queen i n t e r r u p t e d , " I could show you h i l l s , i n comparison w i t h which you'd c a l l t h a t a v a l l e y . "

Lewis C a r r o l l , Through the Looking-Glass and what A l i c e found t h e r e , Ch. I I .

10.1 Retrospect

10.2 Prospect

- 354 -

10.1 Retrospect

' H i l l s l o p e p r o f i l e morphometry' i s taken i n t h i s t h e s i s i n a wide sense t o include mathematical models and methods of sampling, measurement and data analysis used i n the study o f h i l l s l o p e p r o f i l e s . The aims o f the work reported here were set out i n Ch. 1.2, but may be repeated here f o r convenience.

( i ) To provide c r i t i c a l and comprehensive reviews of work i n modelling and data analysis i n h i l l s l o p e morphometry, concentrating p a r t i c u l a r l y on continuous models (Chs. 2 , 3) and p r o f i l e analysis (Ch. 8 ) .

( i i ) To place procedures on a f i r m mathematical basis and t o evaluate t h e i r p r a c t i c a l u t i l i t y , c o ncentrating p a r t i c u l a r l y on a u t o c o r r e l a t i o n analysis (Ch. 7 ) , p r o f i l e analysis (Ch. 8) and model curve f i t t i n g (Ch. 9 ) .

( i i i ) To consider the e m p i r i c a l r o l e of h i l l s l o p e morphometry i n geomorphology, which e n t a i l s c o n s i d e r a t i o n of the hypotheses put forward i n the l i t e r a t u r e f o r the f i e l d area (Ch. 4) and analysis of the i m p l i c a t i o n s of morphometric r e s u l t s f o r these hypotheses (Chs. 6,

8, 9 ) ,

( i v ) To set the f i e l d of h i l l s l o p e morphometry w i t h i n a methodological and t h e o r e t i c a l context, p a r t i c u l a r l y by r e l a t i n g ideas on continuous models t o geomorphological t h e o r y , the philosophy o f science, and the methodology o f the n a t u r a l and environmental sciences (Ch. 2 ) .

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The emphases o f t h i s t h e s i s are t h e r e f o r e on forms, r a t h e r than processes or development, and on methodological, t h e o r e t i c a l and t e c h n i c a l problems r a t h e r than e m p i r i c a l problems.

Major achievements and conclusions may now be summarised.

Models ( s i m p l i f i e d formal representations) of h i l l s l o p e p r o f i l e systems are brought together i n a c r i t i c a l and comprehensive review, presented i n a u n i f i e d n o t a t i o n . Fundamental p h i l o s o p h i c a l issues and major geomorphological problems a r i s i n g i n modelling are considered and modelling approaches expounded through a c l a s s i f i c a t i o n based on f i v e dichotomies (static/dynamic, d e t e r m i n i s t i c / s t o c h a s t i c , phenomeno-l o g i c a l / r e p r e s e n t a t i o n a l , a n a l y t i c a l / s i m u l a t i o n , d i s c r e t e / c o n t i n u o u s ) . Many f u n c t i o n a l forms have been suggested f o r s t a t i c continuous models, p a r t i c u l a r l y power series polynomial, power f u n c t i o n and exponential f u n c t i o n . Dynamic continuous models have been based on the assumption t h a t h i l l s l o p e s may be t r e a t e d as s e l f ­m odifying geometric systems: independent v a r i a t i o n s i n c l i m a t e , hydrology and s o i l p r o p e r t i e s have received l i t t l e a t t e n t i o n . Processes of f a i l u r e and s o l u t i o n have been r e l a t i v e l y neglected. The important ideas o f constant form and s t a b i l i t y p r o p e r t i e s , used by J e f f r e y s i n 1918, have been rediscovered since 1972. Extensions

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from modelling p r o f i l e s t o modelling surfaces face d i f f i c u l t i e s i n t a c k l i n g plan curvature e f f e c t s , endpoint behaviour and stream network development. Ideas o f s p a t i a l s t o c h a s t i c processes have f a i l e d t o y i e l d important new i n s i g h t s . S i m u l a t i o n , p r e f e r a b l y accompanied by s e n s i t i v i t y analyses, i s valuable f o r handling thresholds and frequency d i s t r i b u t i o n s . As knowledge about processes improves, the onus i s i n c r e a s i n g l y on modellers t o produce r e p r e s e n t a t i o n a l models (Chs. 2 and 3 ) .

P r o f i l e data t o serve as examples were c o l l e c t e d i n a 100 km 2 square centred on B i l s d a l e i n the North York Moors using a pantometer. Geomorphological i n t e r p r e t a t i o n s p r e v i o u s l y enunciated f o r t h i s f i e l d area include theses of profound l i t h o l o g i c a l i n f l u e n c e , p o l y c y c l i c denudation h i s t o r y , p r o g l a c i a l lake overflow channels and profound c r y o n i v a l i n f l u e n c e . I d e n t i f y i n g scale v a r i a t i o n s emerges as a fundamental task f o r a morphometric approach (Chs. 4 and 5 ) .

P r o f i l e dimensions and p r o f i l e shapes allow a f o u r ­f o l d grouping of measured p r o f i l e s which i s c l o s e l y r e l a t e d to v a r i a t i o n s o f bedrock geology. Angle and curvature frequency d i s t r i b u t i o n s are summarised using quantile-based measures which are considered appropriate f o r data c o n t a i n i n g o u t l i e r s . Median and midspread of angle can be r e l a t e d t o the f o u r f o l d grouping. A novel method of s p a t i a l averaging and d i f f e r e n c i n g shows c l e a r l y the scale v a r i a t i o n of extreme values (Ch. 6 ) .

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A u t o c o r r e l a t i o n analysis o f h i l l s l o p e s e r i e s i s a r e l a t i v e l y new technique: i t s usefulness has not so f a r been examined s y s t e m a t i c a l l y . Angle s e r i e s are s t r o n g l y a u t o c o r r e l a t e d , whereas curvature series a u t o c o r r e l a t i o n s dampen much more r e a d i l y . However, angle a u t o c o r r e l a t i o n s r e f l e c t not surface roughness but o v e r a l l p r o f i l e shape which can be measured more d i r e c t l y i n other ways. The Pearson analogue estimator i s recommended and i t i s emphasised t h a t n o n s t a t i o n a r i t y i s an important problem i n p r a c t i c e (Ch. 7 ) .

P r o f i l e analysis (the d i v i s i o n of a p r o f i l e i n t o d i s c r e t e components) i s a long e s t a b l i s h e d approach i n h i l l s l o p e geomorphology, but i t has not been widely recognised as a numerical c l a s s i f i c a t i o n problem and i t has usually been based on an u n c r i t i c a l l y h e l d a t o m i s t i c view o f the landscape. The problem i s formulated more c a r e f u l l y and embedded more deeply i n data analysis and geomorphology. Methods p r e v i o u s l y proposed by Ahnert (1970c), Ongley (1970), P i t t y (1970) and Young (1971)

are shown t o be unacceptable. A method based on a d d i t i v e e r r o r p a r t i t i o n and nonlinear smoothing i s presented as an i n t e r i m a l t e r n a t i v e . Results are i n t e r p r e t e d w i t h reference t o bedrock geology. S a t i s f a c t o r y methods of p r o f i l e analysis must n e c e s s a r i l y be adaptive (Ch. 8 ) .

F i t t i n g continuous models t o h i l l s l o p e p r o f i l e s (or components) may be regarded as a sequence of s p e c i f i c a t i o n ,

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e s t i m a t i o n and checking. A one-parameter power f u n c t i o n due t o Kirkby (1971) i s used as an example. Problems of l i n e a r i s i n g t r a n s f o r m a t i o n s , e r r o r s i n va r i a b l e s and aut o c o r r e l a t e d e r r o r s are considered c a r e f u l l y when developing s t o c h a s t i c estimators. A d e t e r m i n i s t i c e s t i m a t o r may be employed which i s a transform o f an i n t e g r a l proposed by Kennedy (1967). Model checking should be based not on s i g n i f i c a n c e t e s t i n g but on lack of f i t measures, s p a t i a l p a t t e r n of re s i d u a l s and a u t o c o r r e l a t i o n p r o p e r t i e s of r e s i d u a l s , a l l compared f o r d i f f e r e n t estimators. Results w i t h f i e l d data allow i d e n t i f i c a t i o n not only of best estimates o f the Kirkby parameter but also of good estimators. The power f u n c t i o n model can be used i n a minimum d e s c r i p t i v e approach but attempts a t process i n t e r p r e t a t i o n r a i s e g r e a t e r d i f f i c u l t i e s . Important t e c h n i c a l problems remain, however, before the e m p i r i c a l issue of i d e n t i f y i n g adequate models can be t a c k l e d i n a s a t i s f a c t o r y manner (Ch. 9 ) .

10. 2 Prospect

I t has o f t e n been point e d out t h a t research r a r e l y produces f i n a l s o l u t i o n s t o problems: new problems and new formulations of problems a r i s e from conceptual and e m p i r i c a l i n q u i r i e s . The work reported here includes c o n t r i b u t i o n s t o various geomorphological problems, methodological, t h e o r e t i c a l , t e c h n i c a l and e m p i r i c a l . I t i s appropriate t o close by i n d i c a t i n g some o f the major problems which remain outstanding and which w i l l be attacked i n f u t u r e research.

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The development o f landscapes i s now recognised t o be a much more complex a f f a i r than was i m p l i e d by the c y c l i c a l schemes which dominated English-language geomorphology u n t i l the 1960s. The key ideas which are c u r r e n t l y being found u s e f u l include those o f e q u i l i b r i u m s t a t e s , feedback loops, t h r e s h o l d c o n d i t i o n s , system s t a b i l i t y and response, and i n t e r m i t t e n c e o f events. There i s , however, great scope f o r f u r t h e r work on these concepts. Ideas of e q u i l i b r i u m , f o r example, are at present the subject o f considerable confusion, disagreement and i n e x a c t i t u d e , and i t i s important t h a t rigorous views are more widely adopted. Notions o f feedback and th r e s h o l d - t o give a second example - have earned many a passing aside but r e l a t i v e l y l i t t l e d e t a i l e d a n a l y s i s .

Scale v a r i a t i o n s of morphometric p r o p e r t i e s continue to deserve a t t e n t i o n i f only because geomorphology i s concerned w i t h forms at a range o f s p a t i a l scales. (This can hardly be a b a n a l i t y when many contemporary i n t e r p r e t a t i o n s f a i l t o specify r e l e v a n t scales: witness various hypotheses of l i t h o l o g i c a l and c r y o n i v a l i n f l u e n c e ) . I n the case o f h i l l s l o p e p r o f i l e s t u d i e s , i t remains unclear how f a r morphometric r e s u l t s are a r t e f a c t s o f the measured length used i n p r o f i l e survey. I t would be naive indeed t o seek the b e s t , s t i l l less the c o r r e c t , measured l e n g t h . A b e t t e r s t r a t e g y i s t o use a r e l a t i v e l y short measured l e n g t h and d i f f e r e n t l e v e l s of g e n e r a l i s a t i o n (e.g. through averaging and d i f f e r e n c i n g , smoothing or p r o f i l e a n a l y s i s ) , and thus t o i d e n t i f y p r o p e r t i e s which are s e n s i t i v e or i n s e n s i t i v e t o scale over the range o f i n t e r e s t .

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The d i v i s i o n of a h i l l s l o p e p r o f i l e i n t o d i s c r e t e components remains problematic: no method proposed i s r e a l l y s a t i s f a c t o r y . There can be l i t t l e p o i n t i n e m p i r i c a l a p p l i c a t i o n s o f published methods u n t i l b e t t e r a l t e r n a t i v e s have been developed. I t does seem t h a t any method w i l l have t o be adaptive t o be acceptable, whereas those published are t i e d up w i t h r e s t r i c t i v e decency assumptions about the i d e a l character of p r o f i l e data. A d d i t i v e e r r o r p a r t i t i o n and n o n l i n e a r smoothing are a t t r a c t i v e f a m i l i e s of methods which merit f u r t h e r examination.

Future work i n h i l l s l o p e p r o f i l e morphometry should draw upon r e s i s t a n t methods of data a n a l y s i s . Slavish adherence t o moment-based measures and least-squares e s t i m a t i o n would prove very l i m i t i n g . F i t t i n g complicated models w i l l r e q u i r e recourse t o search algorithms f o r discrepancy m i n i m i s a t i o n , while s i m u l a t i o n modelling w i l l need t o be supplemented by s e n s i t i v i t y analyses. I n these instances, as i n general, good work i n h i l l s l o p e p r o f i l e morphometry w i l l flow from a p p l i c a t i o n s o f the best techniques a v a i l a b l e , and not from an i n t r o v e r t e d muddling through w i t h ad hoc procedures.

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- 403 -

APPENDICES

- 404 -

Appendix I H i l l s lope p r o f i l e data

This appendix gives data f o r each measured p r o f i l e . E x h i b i t s IA t o IK show measured angles i n crest t o base order. Units are degrees.. The f o l l o w i n g supplementary data are given below f o r each p r o f i l e :

( i ) Notes on veg e t a t i o n , n a t u r a l and a r t i f i c i a l f e atures. Numbers given index measured angles.

( i i ) Notes on the basal stream: estimated ' b a n k f u l l w i d t h ' (average of about 5 measurements) and remarks on stream d e t r i t u s .

B o n f i e l d (BO) ( i ) 1 - 8 Heather

9 - 2 1 Eroding peat, some heather 22 - 387 Heather

388 - 394 Mixed v e g e t a t i o n , stream bank

( i i ) bfw 1.0m

some boulders about 500mm, many cobbles 50 - 100mm.

Stonymoor (ST)

1 - 42 Heather

43 - 48 Track 49 - 72 Heather

73 - 79 Track 80 - 202 Heather

203 - 2 81 Recovering burnt heather 282 - 299 Heather

- 4 0 5 -

i

o i n i n o o o i n i n o i n i n i r . o o i n o i n o o i r i n i n o i n

*•> p- >*• <3 o *o -o oo rg r~ <o -r m r- «\i m m f\j co c >o in >o

ooooinoiAir , in ino i fMrio ir>ooooc50o^^ —i ro m ro tr> r<j >»• >r *o ro oo o <JO in •«*• p» r\j ro p- oo r\i ro «\J co

i—i o m o t i i n c j i n o o o o i n K M n o o o i n o o i n o i n ^ o m r \ j -»r m «o- m m rsj >o o m -o ro m o -r ro o o> ro oo -H

o o o m iTi o o in m o o m m o o m mu" > in m o o mm 0 -o o f\J >o f\i ro in ro f\) r\j \o m ro ro oo —• p- -o o»in P - o» •-4 - - I — l ( N | 1 l

o o o o o o o o m o m o m o m o i n m o o i n o i n o m in - rsj o ro •>$• °o r~ •-• «o ro %o r- r«- oo P» CM C\J m in rg «a-m o o o u ^ o o o m o o o o o m o o o i n o o m o o o «r >j- m <• p- r- u ro m in in rg >o -* i/> ro r~ CT< r- <o o ro

o o m o m i n m o o o m o o o i n o o o m i n o o m i n m ro rsj ro m o ro m p- in sr o OJ u i/> »j- —• ^ p- oo o o ^ •-< m

o ~* ro <M —« co m P - m r\j r>-1*» r- o in m o> >r %f cr» r-»

o m m o m m m u m i n o o m o i n o u o o o m o m o o —< (\i in o rg >r r\i u» <• sr »-« >o m u i m -o r\j o> u» co v n eo

u " » t n o m o m o m o o o m o o o m u > o o o o o m o m

i\j ro -* o ro f\i r- p- co <r —• o» —' o —»ro p- >o r- —• w»—

ramooooommmomomominiriOoooooo

m m O ro o ro r<j <r %t r\j —• p- & p*. r\j o O* ^ co o ro r- O -* i i—i- >-t>r

m i n o o m o o i n o m m o o o m o o i n m i n o o i r . m o

<\jir>m—<op-'0!Mr\imomror*-mocO'Tmr~»op-co"^ I I | —<—<—' m

i n m m o i n o m o m o o o o i n o m m o o o o m o o o >r (M o p» ro P"- <o m p». t\j >o - r»- t\i o ro <N ro °o o m u» rsj cr- »H

t—I

o o o m m m o m o o o m m m o o o i n o o o o o m o

ootNi—<rg—<m NJ.-ONJ- o«^-<'"r>mmin J*-'r>coin-o-o—IO —» H N

o i n ^ o o o m m o o o m - i D ^ o o o a O i n K M n o o

N - 4 - *

fl'flo^aojdjo^in n ^ ' j a ^ j j i n ' o o o m i n

Oi-OiOr»—^-H-^u^-Oi"* -i) ~*~T M-*JW\ -T 0O-nuw>J»v/»'-U I I I

- 406 -

o i n i n i n o i n i n i n o o i n i n o o i n o i n o o i n i n o o i n i n i n o o oo ,j- co <3 m >•>• o oo rsj m o m -o ro r- r»- in —> m f»- m NU in m oo m ro

o o o t n o c u ^ i n i n o o o i n o o o o i n i n o o i n o o i n o o i n vOv)- o»r\ir>u»p*int^C3sroo>or*-ooo<vj~Hinooir\rw>o>cin>or^'*» i •—i i —<t—i —« •—i

O en rvj CO m in ."M f- *u r- <t> >r o «o f»-i*- o~> °J < f- vr co <\l I i—l-J t-4r—t

m in o o m m m m m o o m in in o m o m in m o in o o u I m in o

I Loooooooommmooooomomomoooooom

— i —! | CM

o o o o i n o o m o m i f M n i n o i n o o o o o p m o m m m o i n . - j <\j oo —4 co ,o - vu r\i o"> OO r*» «r pg co r»- —. «y r» m a> o f"~ m r- «-<

o m m o m m m i n o o u ' M n o o m i n o m o o o o o o o m m o m m o ou r> >o r>- »y <~< nj r» r~ m m IT cj cr* r- »o o m in r» >o m m r«-

I i-ti-l —' ,~4

o i n m o o m o m o o o m o o i n u ^ m c J i n o o o o m o o o o c\i m m - J in oo r- co o o- r*\ >o m «-« o ao -o oo >o r > - o oo «o r- co

P—4 •—* w—4 w~i •—• p—<

i n m m o m o o o O o o m o o o u > m m o o o o o o o o o m >t m co •*- >a- co -j- r~ w <• -o r»- >r in d • — I f i o f~» OD m >o oo m in ao m

in <n in o o o o in m o m m u m m in o m m o o m m i* > o o in m m r , »j- u \ in vo w* tf^ r« —* (M >r m o <M w r~ rn m !>» *JO m m r -

— i —4 . — I —4 i n i n v n i n m o m u M n o m m o m o o o o o m m i n o m o o m m

. . , . • • . « . . • . • • . • • . • . • • . • • • • « w <\j%r rnor^inini^cu^ooo>roin--<oi^'in^-incooinr--o>or*'a> ^}

bo o o i n m i n ^ o m o i n o m i A O O O O O o o o m o m i n m o o i n C rovon'-no^m oo,'-<(NOvO>ooocoh-<or-—toooooomNor r -tMO^ _<

<D

OintnOou"\ inoinoinooi f>Oinininir i00oinoininir \00 ^ inrg^ oou^^-^^o^o^r^ C7>u^u^>tcof^r^(Yif\i(»»<ootnoo',-—<cj(\; to

I — ( —H to m i n i n c j o i n m m o o o o m m o o m m o u o o o o o m o o m »B

co -M —J —4 —

o o o n-n no .3r>mo . 'amomoommmo ^ o ^ ^ ^ m o

—4 —4 —( H

— « - 4 —< —< - < — i r \ j

- 407 -

300 - 301 Gully 302 - 328 Heather

329 - 339 Recovering burnt heather 340 - 431 Heather 432 - 453 Mixed vegetation

( i i ) bfw 1.4m

cobbles, small boulders up t o 900mm.

Proddale (PR)

1 - 122 Mixed low heath

12 3 - 130 Eroding peat

131 - 163 Mixed low heath 164 - 167 Eroding peat 168 - 172 Mixed low heath

173 - 176 Eroding peat

177 - 184 Mixed low heath

185 - 193 Eroding peat 194 - 241 Vegetation, some boulders 242 - 272 Juncus, Sphagnum, heath

( i i ) bfw 0.8m

a few stones about 200mm.

Parci G i l l (PA)

1 - 4 Heath

5 - 7 D i t c h 8 - 27 Heath

28 - 29 Old g u l l y 30 - 288 Heath

289 - 296 Track and t r a c k banks

297 - 374 Heath, many boulders

375 - 421 Bracken, banks

- 408 -

mm ino in in inoin o o inin i n o o o

CM rvj «r m <• »t <5 °o oo o o »o >0 r«-u% >o

m m o o o m o o o m m m o o o o o ITS m m m r- r- >o o o o* >j- in o o P* O n

o o o m o o o o » » » m m o o m m o m

«f -m m «r «u cu r-m w o «o co C7> °J o

o o o o o m m o m o o o o o o m o

pn «o o r*- o m f- m «J o I s- U > r- f- P*- co m

o o m m o m o o o o o m o m m m m

o i n o i n i n i n o m o i n o o i n i n o i n i n ^•>oinro~<incJN'\i-*oo^-inO'-*iN»p»r>j

— I — I • — l r - 1 01

oinininininininmininooinoom rnd>s-ininf**in-<oow in coo in -rn

m in o o m in in o o in m in in in m o in

m «o in t»- r»- —< o» o en >a- «o <\i p» <o ^

o m o m o o m o m m o o o o o m o xt^vi-vrvrm-o'-^-o—<»rmcrico—icoc^

— i — i in in in in in in o in m o o o m o o in m

i n o o o i n m o m o m i n i n o m o o m (T|s3-%j->OsOrricoOO--'mr\jr~corg—<in

m m o o i n o o m i n o o o i n o m o i n m m m «r m r - n p>-co vo tn f~ NO P->o co

m o o m o m m o m o o o m m m o m

o i n o m o w o o o o o o m o i f t o i n

—<rnMin<T>--y o o ^ ' j ^ n r - i ^ r n i -* -* -4

O ' 3 ' j a i 3m.n 3 0 ' J am-nm o^n-n

- 409 -

in \r\ mm o IT . o o o o in C J o in o in o in in o o in o C J m o

^ ^ H »-« •-« ro L i o o o o o o i n o i n o ^ i f M i M M f t i r i O o i r . o o i n o i f t i n

>j- o to in m oo r\j p- GO o> P» p- p» ro —* ro <*; o ro y> co r«- -a- »o *o —t HHHPl

o o in o <-> o in m o o o o in in in O in in in in O O o o o m ro —« f\j p- m o «H »J- »r m p- o ro -O >0 m oj ro p- m m *•* »o

—i —«—

o m m o m o o o f m i n c j i n o o o i n o o m m i n m o m o

-H r- ro >a- p- r- ro -o P- • o m i/> r» -O V O> r- P~ 00 -o -o o r\i c\j r- m

I

m o m o o o o o i n m o o o o o i n o m u " \ i n o m o m m m

«j- >}• m rsj ro m co in >o <3 co r» >r eo 00 vo so 00 m co —' co m rsj in

m o m o o m o o i r . o m i n i n m O L ' ^ m o o m o o m o o o

-J- f\i in r- <J- sj- rg rsj ra p- in O sr r~ O -O >!• -O <o r- p» —' P» *f ro u*

m o o m o m m o o o o i n m o m m o m o m m o c j c j o o

in m ,j- ro rsj ro >r o °si 00 -o O ro -0 r-. co I s- m 00 >o a> v »0 >*• r-

i r \ o i n o o o m o o o m m o o o o o m o i n c 3 0 i n m o m

0 ro -3" co >r m p. p- o ro -st —• o ro >r O in -o 00 o fM ro r-1 #—1 ^ »rf _• —it—<_<<\j

m m m i n o o o i n o m m o m m m o m o o o m o m o o o r- i i n >o in rsj in ro ro o ^ in 0s r\j «o 0» o r>-co v0 tp >i" tr> p-I I I — I • - « i -H —1—JIVJ

o m m m o m i n i n i n o o m i n m o o o o o o o i r i i n m i n m

<j f\j in <r ro «r CM *3" <o m >o co «o m co o o co >o u o» cr> p» I

o m m o o o o m o o m m i n o o o o o o m i n o m o o i n r\j r\i rg in o -r co m >o >o m in P- P» m m »r ,o O in m -T O» ro —< I >-< — r\j

m o o i n i n m m o o o o o m o m m c j i . o o m o o o i n o o o

ro «4- o o »J" m <M »o -o o co in oo m •>*• >o m in in o « 3 rs j uo ^ -1 I f> r H - J f \ J

o i n i r \ O o o i n m o i n o o o i r i i n m i n o m o o o m o o » n o

• >r co f M o co «*" m co co «r <\J p- «r <o «o m P» in <3* <M > o

m o u m o m i n i n m i n i n m o t J o o O i n i ' M n m o o o m m o

I N >r •*> i-o in i-o <£\ -o P - vO P» vj* JO T .-o —1 >- p» •. o -* .> m

o ^ n ^ o r \ n i n ^ o o j * i O O o o ^ 3 ^ Q i / \ o .n'runbo > r ^ o u r v - Q — < J CP3 » r - o . - j r r i n i

—4 —^ —< —4 ."O >n n ^ a o in • o n m o m n o m >n j\ rt"v » o u «n o o.« x\ n '_i;->•) —«un r.\j ->qr w j > i n >J j»iO'*,r~ u» jm>T —< •om

- 410 -

( i i ) b f w 1.7m

b o u l d e r s up t o l m , much s m a l l e r m a t e r i a l .

Fangdale (FA)

1 - 41 H e a t h , some r o c k a t s u r f a c e

42 - 46 D i t c h

47 - 98 H e a t h , some r o c k a t s u r f a c e

99 - 124 Bracken

125 - 127 Heath

128 - 132 Track . .

134 - 287 Heath

288 - 469 Moss, b r a c k e n , r u s h

470 - 506 Bracken

( i i ) b f w 1.0m

some b o u l d e r s l - 2 m , m o s t l y s m a l l e r m a t e r i a l .

L a d h i l l (LA)

( i ) 1 - 12 Heath

13 Pa th

14 - 67 Heath

68 - 71 Tumulus

72 - 110 Heath

111 - 114 Tumulus

115 - 206 Heath

207 Pa th

208 - 224 B r a c k e n , s tones a t s u r f a c e

225 " 338 P a s t u r e

339 - 342 Up t o and away f r o m w a l l

343 - 378 P a s t u r e

379 - 381 Track and l i n e o f w a l l

- 411 -

o i n i n i n o i n i n i n i n o i r i O O i n i n o i n i n c o o o o i r i O O i n i n o i n o

O O f- ir\ •>! "-1 m o ra f\i v co —| <v* >o co co -o m r\j —too ou cr r*- f\j •—4 t—« «-4i-H •—< r-4 v*4t-4 —4 f \ J CO

m o o o cn o in in in o in in o in o o m o in. in o m o O o m m o o o o m c\j ~* sQ o P> ^- rn —i r- rvj r- -o <M oo o» «•*• o* I S * O r» co r- *-« co P- f\i m o in

I r-H _ I _ 4 <—l i_4 riHKIN

O i A O O i n u o o o i f i o o o i a o o m o o o i A Q O O u i o u o o i n o

•T r\j —i rr» >o »0 r- oo oo m —< co ~-t oo r» r» r- r- sj» r- «o oo co ro —* o co o

I

u o m o o o o m m o t n o u \ o u i o o o o o o o i n o o i n u o m m i f N cj sf- <r »i- —< P- —* •>* «r vo o f"> —• u» ao o» o m >o ~i co r~ u > co r~ in CT> «O in o» m o i n o o o o m m o o o i n m i n m o o o o m u ^ o m o i n m o o m o r» .j- 00 <M nj 00 r-t m in <o o co r» m co ih (\i rvi o in o «o -J" o> o «r o m r» o

—4 _ i —* 1—1 ^-(t—if—

O m m m i n i n o o m o o o o m i n m o o o o o o o o m i n o o m i n m 00 >r o co in «M m >o >j- in >o 00 >t» o co co ro o co m co co CP in o 00 njrouco 1 •—) ^ , !•—if\j m

iTioooinomooooomomoooomoooomminmmomm

rvj o i/ 1 in t\i >r \0 co \o r- —' o co r- «\J co in 10 u> co r- o> >o *o m —• CM O> ^ I S " • * <J I —i—4 —( r-l _ | f-TSI^J- i—l

o o u ^ o o o o o i r \ o o o i n i n i n i n i n o i n i n o i n o o i n o o o u ^ t J O C » >o m u» o >r o 00 I s - in o «3- r\j —«o ou r- >o «*• O >S" " >C 0 <*> co o -r w ro 1*-

i r i W i n i n u i o i ^ O i n o u > o i o o i r i i f t o o o o u i n i A O O i f i o w o o u i or^^orgin,^r ,i>f^r'ioay^<i^uju^r^^uvrvu^oocrin^'Hvoo>roosr

I —4 1—I —< — I — < - H l \ J f \ | t \ |

ouooooooi/nriiAoooo^imnooOLoooooi^inuoiniri

r\io rn >T «x in I T cy rg<M r~ 00 «•>r< 1 0 o o» <i >0 O <Mnj r*- no^rncy

i n o m i n o m o m o o o o m m o ^ o i n o o i n o O i n o o m m m m o u

s O fo o <\| o 00 p- so 1*- o o 00 NO f*- <NI m <n >r m co in «o P- m r*- I M —< —1

r-« _< —ir\ jrgr\ j

o m i n o o o i n o o i n o o o m o o m o i n m o i n i n o o o m m m m m o

r-i—<NOmr--p<-oo,oo r -jr>-i»>. r>.o ^•»-ir»-inu,o,cooopr>inm 0 0 o c j o o o o o i n c j o o o o o m i n i n u ^ o o m o m i n m o m m o i n o o

m rn ci m in >o o &o r - - j - o>> »o m o in m o <M O 0 O f\j co'M <M < — • • — • <—4 —ti—1 ii—<(\ir\j

in m c> in o o o in in in m o o o m m in o o m in in in o in o in in <_> o o o

—<rsi,"\j,- o -'Ni - g O ^ M or-0"0>j-o.n-<mo.tjaJ»J*-«MOro-o4-—<

o o o m .n r\ o ^1 n m o m o m • n; n m o x» o ;n n *n 'n o m o m m o

- 4 - 4 - 4 -4-4.-H-4'M"g

- 412 -

OOu^irii^ooo^Ou^^trirvoiriTiOOOOin^ooi^oiriO

m m r\i\r «r u —* o r— CJ -o co <\J O f\i m m o o ^ cr r\j in m MO m in

o ir> irv o ir\ o LH in c ir> in IT. o iri iri iT'IA ir\ IT. ir\ c IT* o o iri in o o

•-i m vj- m m >o <\j •© m ro »J" v o m r- «o cr ni CM m o cr f» in m —i r- cr INJi—••—lp-4 M » H _ I I—Ii-< I—• i—I •—i

o m o o o a m o o m « ' \ o m i n m m o m o i f > o m o o o u > m « - » o

cr rg (\i in m > NJ r«j in >r m in cp m CM «-i ro m o cr in o fM m fM f\ CM r- r» i—I ( \ J ~4 f H " H —• 1—I <-<cH • — i i - H » ^ " J

ooomotfMnomotnoooinmomoomoominooom

I

irmoooooooinouioooooomwoooooKioo >0 f i ro »u <3" m *-> m ir i ro >o rn >o o i m >*• ir co o cr o eo <- >r c\j cr sr inOOOOOlOOOOOOOOUlOOlAOlAlAOOOiniTufMriO r->r i^nir^^iAinuir-voo<)"^^^rn^coa)ocrvT~rf\i--<rNJco i n o i n i n o o o i n o i T > u > o i n i n o i n t f > o i n i r » o i n o i n u > o o i n i n r\jin m (Mm in co »*• «o o in o *N o -r fM o —* co »-i com m cr m o

o o o m m o o o in o o o in u o if1 m m o m u •• m in in o ih in in m ^rovjco^u^inr^in>tininco<5>ro^^f\joo^^O'^rn>3-OLnr\j

o o o o o o i r t o o o o o o o o o m o o o u u i o w o o o i f l

0 o <-> o in o o m m m m o m o o in o a m m m o m o m in o o m y» m m «o m ro in r\i *o *n MJ in <i i \» i-i «4> >}• CM —• wo o o r- to CM ( M ^ P - C M

o o o o o m o m o m m m o o o o m m i r M n o m o o o o m m o in r\j - H mm «r o> <N <T r- r- o m o o cr o JM O mcr u mro

| ^1 — ' — I — J - J » J _ l — ' r - l^ -<

m o o o o o m m m i n o m o o o c D o o o m m o m o o o O o o m

i n o o o m o o o m i n o o o o i n o m o o o o o o o o m o i n m o f\j m cr r» in m o co p» r<» p- o t\; rsj rn r-* o - H O f\j o —• —* o -o •r

o o o o o o o i f i o o o i n o i n o i n o i n o i n i n i n o o o m o m o o

tnr\ ^in^-omincr-o o-0'n v4-f T5-<o.,\j 0fs"-n—•.-o--."0:n>*-

O - n m o o o o o • ^ o m o m j ' N " 3 m . n . n m x » o ^ o m i n i n o m m

1 —4 —< —* —« "* —* —^ —< —*—^—< —^

- 413 -

382 - 465 Pas tu re

466 - 469 Trees

( i i ) b f w 3.3m

b o u l d e r s up t o 1.5m, m o s t l y s m a l l e r c o b b l e s , some f i n e s .

T r i p s d a l e (TR)

1 - 22 C a l l u n a

23 - 27 T r a c k

28 - 253 C a l l u n a

254 - 256 B r a c k e n , C a l l u n a

257 Scar 1.8m

258 - 263 B r a c k e n , C a l l u n a

264 - 283 Some b o u l d e r s l - 2 m

284 - 345 B r a c k e n , C a l l u n a

346 - 357 Some b o u l d e r s l - 2 m

358 - 384 B r a c k e n , C a l l u n a

385 - 403 Sparse wood , m i x e d g round v e g e t a t i o n ,

some b o u l d e r s

( i i ) b f w 7m

many b o u l d e r s up t o 2m, much s m a l l e r m a t e r i a l .

A r n s g i l l (AR)

( i ) 1 - 96 M i x e d low h e a t h , some r o c k a t s u r f a c e

97 - 98 T r a c k

99 - 117 Mosses, some ba re g round

118 - 160 Mosses, much ba re g r o u n d , s m a l l b o u l d e r s

161 - 262 B r a c k e n , medium b o u l d e r s

263 - 310 Mosses, medium b o u l d e r s

311 - 363 B r a c k e n , bank

( i i ) b f w 1 . 5m

b o u l d e r s up t o 1.5m, m o s t l y s m a l l m a t e r i a l

- m n -

inoooinoooin ino in in in in in intn ir iwin ino ino r\j (\i —t^M—t o -a- >o ro fvJ o o r>- <\j in >o m r- eo co r- %r rn

. o o i n o i n i n i n i n i n i n i n o o i n i n o o o u ^ o o o i n o o n i*- »r 1-1 »r o -o fM u» m p- «t u» <o u ro m u> o y f\i

j i r y o o i n i f l u o o u o i n i r i o o u \ o u u i o u i n o u w \ i n

l rj<—<r-< >-i>r —' -<r

o i n o i n i n i n o o i n o o c j i n o i n o i n u \ i n i n i n i n i n o o _• —i —< cj —< m r~ • M m m m o o o —• o —«"J in .j- « J m m

i—< _ U \ J I - H _ < p-< (\i•—(rn

o o o o u ^ o o m m m o L - i o m o m o o o m o m m o m

! ^•-i<\io^«orn»o>r»oinnf^o«9'oorgi^-»}-^Tr^or-<)in J •—*i—I ^ i — I i — I P )

' o o m m m o m o m o m m m o o m i n o i n i n o o m m m I • ^•inoor\j^n^H>j,>3->i-i^-CT»r-wi^">o^c\jc\ii*-»j-vOf>-'-'<J i | —i ( \J_4_ I —< >j-<NJ CVJ > J "

o o o o o o o o m o o m o o o m i n o i n m i n m i n o m • • • • • • • • • • • • • • • • • • • • • • • • •

—»r\j o I H u ro o —1 in o oo m m m r«- rn —i o» <\j rg o CM -* i | —• | r\i(\j.~(—if-i<\jf\j-<m 1 ooomoooooomoinuAommoomommmifi ! r\j'-<roo-Hm(\jr--rj^-inm^m>)-«* -f^om-^co^»or s-'-* j | l \ j C \ | - H * H f \ j i - l i - n j -

i o o o c j o o o m o ' m m o o o m i n o m i n m o o o i n o

| | i - l l \ J - ^ — ' | —IO|lNJ(N|l>J

! m o o o m i n m o o o o i r i o m i n m o m o o i n i n m i n m o sr <N i >J —< ci pg m rg in in »r w' o o o r~ m o u> n i —i o w

; i —i | >H «—imc\j—*<—' <N rviro i u i n o o m m o o m o m m i ' ^ o o o o o m m i n c j i n m m

(M o o m o o «M m in m <r w o «4- —• so oo in m m .-i in r\j ; i i —• rrif\i>-i—<-Hrr)i-if\j>r ) o o m m o o o m m o i n o o m o t j m m m m o o i n m m • ^rgo^i-*fNi 'HO <fNiin^-^-cooin vo- -inin>i-foinoooo i M l l —1 f\i^H—i (\imf\ifn i

—< m <N <\j o -o >o >r CJ , j m o ro «o a* o «r -r o o r- -o C\J — I r\j—i ,_i.~i(\l(\j<\jm

CO (U

ooomoomOommooooommommu-Mnuiin

.i • • • • • • b i —i —< —i —i —i , n M >J : n 52

, • t o

i ' • • • • t E

i ^

I >j- —< N-- -<( >r 1 - 1

- Hl5 -

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- m vf r- r> m u> o o co to <J* r- p- i*» o tr- »r co m m

inin o m o m o o o o in o m m O O m o o o m o •-i «r r- P- m r»- >o r- a- ro p» *o r- o r- «o inm m o co p-

^ i — I — * — « « J • — I . — | i — I f O

i n O i r \ u o i n i f | O o o u i o o o i / N O o o o i r \ ^ i o

•—i rsi rvi —* •—i «H i\im

O »Tl iTVlTi O in ITMT\ o O iTV O O IfV U A O C3 O in O oo <M -4- m >T ro I S * O —10 «M mir\ r< ro f\i n n O p- o

— J —I C\l _ l — I I ' i—* — x + -

uitfiinouiuMnoooininiriooinoooao^ FNJ r\i in n -J - >0 co p» -J1 co u \ ~* p- •-< o <f\ in P~ f i o

i n o o i n o i n o o o i n o o o o i n o o o i n o i n o i n mfor>-nvr i / \ooo- j n -H ' H ^turoOHO'-<(^Hr»

o o o o m m m i n m o o o o o o i n m m m i . ' ^ o m u

r-<4" vTP~mm^-vj"p-oos—<- 'inu,r '>o>r m i'O'H—< — t ^ - i —i rvirn—<

o o m ^ o m m i n . m o m o o o o o o o o o u ' M n o

r\j—(—i—I •—• i—< •—l(\J>J-

lnoOlnlnlntf <OinoinOinoinooinu"\ooinin

>r ci «u p- r- -o r* c/> c»> I M P~ *vi M V P» r* —' ir> co o —1 o

inouir \o ir>ui f i inu\uMr\ ir ioooouMnooau vu P»J <• r> i m «j- rvj o <o uu p- cu co P I o m >r «t> o o m m o

—t _ I - H - ^ . - < * - J _i.-J.-<r\j<*-

oooo^oooKioiniAiAOinini/iinoin^oo f\ivt-mr-r-yj-vU<—(> o—'O—< 3"r-rsj>i"—'-4-CPofNic

m m o i n m o o o m i n o m m m o m i n m m o i r i i n m co • • • • • • Q)

o>$-mo ino- i>f\Jmor-—foNCivOM-fr-^r^ H

o o o o m i n m o m m o ' - n o o m i n o m m m m o o § «r ro m <j-P-m »r o triro-^" in-^o >!••>»• coo—^vOi-im *d

inoinommtfAoooinu^ovnoinovnommoin 3 • • • • • • 03

r\j o r » ^ - M - t f loONN-*4- o-«'jio.7<-^mun aj —^ ^ - 1 r- ~ 4 » PSJ — 4 r-J 'NrO Q)

g t • • t » • • • • • • • •

• » . , » . 1H a«u \ r*. ^ r\ *r -u a <j m u">« N P» *» —• m >-w j»

- 4l6 -

Tarn Hole (TA)

( i ) 1 - 3 Eroding peat

4 - 2 0 Heather

21 - 37 Eroding peat

38 - 76 Heather

7 7 - 9 3 Eroding peat

94 - 150 Heather, bare ground, rock at surface

151 - 152 Boulder

153 - 169 Heather, bare ground, rock at surface

170 - 242 Heather, some boulders

243 - 281 Mosses

282 - 419 Bracken, boulders

420 - 431 Crag (Abney level estimate for angle)

432 - 526 Bracken, boulders in upper part

527 - 567 Open woodland, mixed ground vegetation

( i i ) bfw 3m

many sandstone boulders up to J>m in bed and banks,

clasts of other sizes down to shale fragments.

Hollow Bottom (HO)

1 - 13 Heather moor

14 - 17 Path

18 - 91 Heather moor

92 - 122 Pasture grasses, Juncus

123 - 131 On crags

132 - 135 Crag (Abney level estimate for angle)

136 - 173 Mixed moor vegetation, bedrock at 145

174 Shale exposure, boulders around

175 - 269 Shale spoil heaps, Juncus, bracken

- 417 -

<M «r co ~* eg m c »o r- r~ co rsi in co •>- m <\i ->o CM cuocj^fNjf\ir^oro.—'rvjo.—iro I | _ 4 - 4 _ i 1-4—4 i-i-J(\j»ri—ir\jr\i.-H.—4rO(\jro

o o o o o o i n o o o o i f i o o i r i o o o o i A O O i f \ o i n o o o o i f \ i r i o o o o f\i,—i «N ro cr< CP >r m -o co m •*/ »n r- »r co rvi CP ~« o i \ c » o * « - < - i o o o ( , i c o o ^ o o f r >

| —i rt^HH P - I -H»H<\jm—4f\jf\J-H tM-—ifO o o o i n o i n ^ o o u ^ L ^ ^ O i o o u i o u o o o o i n o i r i O o i n i n o o o i n i A o

—• ro - H o r» —• —' ro co co cj r~ co p- in o o *-> o tr o co r\j co corNJO(N»r*J(\Jincr,>rinc) | | •—I »—4 ^ > H > - l r H > H > 4 i—l » H •—tlf"\i—'PlCM-Hi—!•—IfO

I i T i i n ^ o o o i n i n i n u i f i i n i n o o i n o ^ o o o i n a o i A O O O O o o o i n o i n

vOiritNj.—40cprnc\j^co(\j.^<)r-rnr»oi^^^

I r-i . —i — i .-H-H-Hm rg^H—I—<CN>T ro

o i n o o o o o o o o i r . o o i n i n o i n o i n o o i n o o i n o o o o o i n i n o o u ' ^

m sr (M o m m —«ro ro vo —i * >© co r— o *r >o o p» o >o m —< m o cocrcp>0rof\JCPO i - 4 _ < - 4 _ 4 i — « - H — I — • • — I _ | . - l l / \ . - 4 r \ l - H i — l i — " . - J r v J C I

u ^ o o i n o o i n o i n o i n o i n o o o o o o i n o o o o i n i o o o o o o o o i n i n tvj p- C M >it ro I N J >r m co o -o m cp o —• co o m r- o m i^jr^i\ir\j>oorocrcoocr'>o<_)ov

• — I | | . - H I — « • - < • — ! • — i » H H H ^ i n rOi-j— i—"fO- ouoomoooininooomoinOLncjoominooooLnoinooomo ^cj*o^ro>o^i^r\iinr^incprop"p-cp o. o^ooo^oi** m om»cmmmrocp»o I | — I — I - H - H ——'rNj—i^- i^rNjtNj

L n o o o i n o o o i n o i n o o i n o o i n i n i n o o i n o o i n u - i o o o o o a \ i n o o

—J CP- —1 m ro in —i (M r- O —• <o r\j co CP cr —1 O >0 o o O O o cr <N O >0 C\J t\ icoorrom | •—<•—< - H — ! • — ( i — 4 ^ i — i t — ( i—imi\j(\ji"j-—rg^i>jrvj

o ^ o o i f t i n o i f t o i t i u i i n i n o o o i p o o o o o u i o ^ o o o i n o o o o o i n

^in^oinP-^vOsOsOrorvicpr^cPr^ocp ^rof\jcosOCP<N.'-00<f^"-^Lnroroop-| | —<•—• i—i i - l i—41—i - H i — • m O J I M ' . N f - - » « " M - H

i n o o o i i ^ i n o i r i a Q u o u i r i i n i n i r i o ^ i n i n i ^ o i n o o o c j o a u i ^ o u o o r\j u CM <\j •—! o !*• •—t —« f» u r« i o uu o o CP CP -u »0 co r— - H U »r>-i»jr\i>rr\iuj^roini—

i—ii—4— —4 i—i -H-H—4inrgr\jrsi—4f-4_4vj-r\ji-H o o o o ^ o o ^ i n o u i n i r \ o o o i n o o o i n o o i n i n o o o o i n o i n o c j v n i n co

• • • • • • • t o * i * 4 l * a * a « « * 4 ) * o t o 4 i « * * * « t i < < > <U

I —i t-4 _ 4 ,—4 —4 -H<-4 mixirorsjcM—4i-iro-^—* W)

^ o o ^ i A ^ u ^ i A i A ^ o o u i n o L " \ o i r i o i r i O o i n i n o u ' - ' o o i f \ o i f l O i A o

I I I - H - H - H —i —« — 4 — 4 — 4 l O % l ^ - ' - ^ - H — 4 O ~ 4 ' O

o-j -~<(Oin33-Ht , J>r •O'O-HiM .r —4 .n n !,r 1 0 T -O o gii>.\| .ucpmwn I I —4—4 —4 —4 -4] .H (\j :\j . T N - < —4-H \jNj-rsj

s i n o i x \ o o o o i n O i r \ i n o o o o c 3 C 3 o i n o ' n ' n i n o o o o o o i r i O o o i n u o i n •^ • -<r \ jO»t inro<J^CP^CPOOinr^OCPOOX^rOcocOoO(0 (O^^^OT I —4-Hi—4 1-4 -1—1 1-4 -H-Hir-.-HrOfMf^ifNJ—4(\jf\jr\i CU

M

moini o^oooooini/M^u>uuiooinifiOinoooooooooi/iu"\i/i o 3 ocr OCNJ-HOOCNJ ococT'o^in^i^uTrxicprsj^r^ror^^in &

I 1 i—4 —4—4 _i —mrOfOfM-^fN rsj-HfO 9> 5

H H

o.T|-H.o-J>!• •Uio-^"'^^—oj'r-'UP- jiTsjr^'jv*.-^i-* j«'jj-JOM>ru'>r ;>J O r 3 I | | -H —(-H —I -H —4 ^••O ' l lHH HH fl-HrO

- 418 -

0 o o in in u m o m o in o o m o m in in o m IT, m in o o in eo in r» cr <NJ no o nJ oo <-t •—• in to r - in sr o> •-' cr> C7» <J *0 00 1 —1 rod I miNJrofNi-—1 —4—4_4t\j_i —1

o o b o o m o o c o i r . i n o o o o m o o c m i n o o o i n o 00 o m o O «r o >o <M 00 «r (\i o> in m vj- *o -4 m o -4- m

—4—— U>r>~l —ir\jr«i —< •—4 — -4 >—1 • —4 —«

o u i o i / i u o o m o i r . o o o C i o O u i o o i i i o o o i n u i i n

•C" CO >0 CO O »J" CT- f\J O C7» *M O •>*• <M "J" —< O »Q C> >0 —1 00 >0 «T CO «—4 4-4_4(r\lT\ | M " f\J(\J —4>4rtHr-<—<—'w-i •—<

o i n i n o i n o o m o o u i n o o o m o i n i n o i n i n o o o o i incoincor~cja)rnvo^^o^or^Oi^C7>«or\iir»u«-i>o- J<ti

IN PH<" (\jrsjm«-l—( «-t4-4 4-l_4 4-4,—Ir.4—(>—« 1 i I i o m o u ^ o o o o o m o m m m o m o o m m o o m o o o I aocoQOLrir iLrvtniomr^ (Nja»sOvO'-< rMr^icjCMntNjr-r-i^r>-| —4—llNJ-J"—4- f\J(XJ—4—4(\4.-4r\jr\J —H— —4 . ] o o o m o o m m m o m m o o o o o o o o o o o o m o i c^cor-rntroroo^-—4<r j<o^co^>oc^cor«,iro»rc7>h-u»Q I —4^r\Jrrlt—4f\Jf\J—4 r-4 —• 44-4,-44—f—4 1 inoinQoniooooooouifiiAinQifioimnoouio j *ou*>r -4\yof\ic\i>oinm.-4>r vu<jfnr->a"r~—'rooovonjr-

i-4 ro,—it-4po_4(\ir«-)rvirfl^Hf\j t—n—i^j f—n—i—i 4-4_4 1 momoooo*nomooi->u>ominoou-\ooo<_)oi f i • I • • ( • • • • • • • • t « * 4 ) O t 4 > | 0 4 ) « * * 4 > « • «TvOrnul«n^<-r\j^r^Nor^vonjinr-^-r«.>oin4-'<0'- ,NOCoo

j 4—1 _4u^_i_ifr||4 )•—• r Nj^i r^^H_i^^_ l^,^4 _

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[ I —4 (\| :>J — I i-H CM— 4-4 4-4 1 uoopoifiooooouiomooomooinoooo

4-4-_l »0 I —4 ff\i-44-1 ,—llM4-1 p — ,-4—4 —I — 4-4(\J

i o o c j i n o o o m o o i r i c j o o m u m o m o o o m o o m

I inooin\rr-r>j^ inro>f i^(^4-<ornr^^30rninoooino(^i - J H 4-1—4 .^j—(<-.—'mr\j—t—irvj!—i-^ n j | r \ j

mo o o o m m m o o o c o o m o o o c j o i n i n o m m o m to • • • • • # , , , , « , , , , , , , , , , , , , , , d)

I 4-1—4—of\jr\jmcNjrvit\j—4—4i-i H - 4 4 H H I CM • bO

mir\00inu">u"*inooirMniriooininooinoooinoo °3 •j"O^OiTnn^<\jX)mrnr-r—irnin-j-r--r»-i<)'Nit^r--(Njfr)ro-'r) TJ

! —1 —ii-t B r O - H n M M ^ —I —4tNJi-i-44-t —4- CD U

1 o o o i n o i n o m o o i n o o i n i n m o o o i n o o o o i r i o 3 • , CO

, ."\jr~. o > r o j M . f i o - * -M 0'-< .N- J*o'N»^--rNjin' , , ,-of ,-co-o —4—4—4 | -\| | NT—lr*VN -4-NJ—4.NJ-4 ^ CD I E

I O O a m o i n o o o o o m o oo^oo o . n o i n o o o m | O , 'vjin-+i<f\<3 g>ru*>o"VJ~r<NJ-<r<>. 4*rn \j,-n j» | >-u»0'-o-ti oo* W ! H-lsT'NJ \ | vT-XJm --«-4 NJ N —*-4 —4—4 J JO ' J f lU4lO -3J3i f \ a4TJf l JWJOJOO 'Ju lOin ^ J O J I O - T JJ •. vj 4) JJ . \ j -4 jy-^•g4^0•,g•»»•^Jl/'S^.O•>l>•l!,* -W

-4—4^ r )T | | ffl.-^>J-4-4^'\J..\j-4-4-4 •r

- 419 -

270 - 275 Grasses

276 - 315 Bracken, grasses

316 - 397 Pasture g r a s s e s , Juncus

398 - 4l4 Pasture g r a s s e s , some t r e e s

( i i ) bfw 1.0m

boulders i n bank up to 2m, shale pebbles, intermediate

boulders.

T o d h i l l (TO)

1 - 36 Vaccinium, heath

37 - 120 Open c o n i f e r s , Vaccinium

121 - 129 Grass, t r a c k

130 - 141 Bracken

142 Path

143 - 175 Bracken

176 - 178 Path

179 - 240 Bracken

241 - 264 Bracken, heath

265 - 266 Path

267 - 273 Bracken, heath

274 - 275 Path

276 - 309 Bracken, heath

( i i ) bfw 2.4m

one boulder 2.7m, mostly s m a l l e r boulders below

lm, cobbles and pebbles.

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o o o i n o i n o i n o o i n o o i n o o o o u - i in^o,'-<-J"^ooco<Tnrrir-MJOr^cr,MJvj-m cMcncMncNjrvj-* • — ' ( \ i f \ i — 1 •-<<-4m

O O O O O I / l O ^ O O O O K l O Q O O O i n

oo o» o oo m —• o C M co *n o •>*• i*~ m C M r - ~*-f <_> r N j c n m c i c M C M f s c - ^ — * — ' — ' " - " ^ i (jouoouiooioououiOinoutnu

—< in oo oo oo m co >*• <-> oo p- W M J r~ o o m oo iVCOCMCnCMCM—4CM t \ j - * i - i « — — ( C O u M n i n o m o o m m o o m i n u - M n o o o i n

inr-i-icoot>Jr*-«Or^C7>'-imcMouu»'Cr~^-«>j-

c j inou^ominoocj inmomin inmmo C M C M o r» —! cr> oo o >}• tr> cr f*- C M - J " oo C M O O O O «-ir,im<Mf\i<M!-*—ii-H'-*-*-*cM-H — i \ j

o o m C3 m in in u m m m m o u i m o in o o m m o >j- oo co u» m w o ro r— r - r - >r — i C M ( J > O

C M C M m m r g C M — i - H ^ I - I ^ H - ^ — 4 i - i r i

m o m vO >t r~ C M -o m ro —' vu rvi c> tr> >f oo oo ~ - , r T M T | r \ j f \ j C M •—ir\jrncM>-ti-n—( •—• •—<

o o m o o i n o u n o m o c j o o o o o o m

C M C M -o «o ro — i co >J- C M co C M C M oo «o C» r>- C M C M

H f l m n c i i M i v | fMtsj^K—ICM — I - ^ _ I - * ( M

• CMCMl>JtMrM>M<-< I —*i—i 'M•-«—••—IPIIHHCM

uMTiOooomooooommomoom \u rn C M o> o C M u> >o - j - o o m C M C J m oo >o — i >-<rOCMf r(rOCM"H ,- ,' _' , _'CMCMCMr-lCMi-« — " C M

o c j u ^ o o m i n o o o m m o m o m o m m

( M r - rn os in r» cj» -st ••J" o oo oo r— C M o o r~ — • C M C M C M C M C M — • — • — * — 1 C M — * — I — « — ' — 1 — I - 1 — I

in o L'Mn in o m ir\ in o m m O c_> m o o o o

C M >o oo C M m »j- r\j ,j- m iri ro <st f— <o o * M </> •-' iri C M

—irMCMrMrOCMCM—( H-HfMfM-H—(CM—* —i r \ | C M o o o o o m o i n o o o o o i n o o i m n m o o o o o f i o o i f l O r o n o r - t M O f o r M O v t v r — i en ro rn C M m — i —4—• — « C M C M — i «-i —• — i —* — i —•

m o in o in \ j o o CJ o u <-> m o in m in in m o

f—i jN>-m.-n-(*oo^>*'r*-coin^-cr>'3>f o :<1 :N *Vriog "VJ—«—<—I'M-M—< - 4 — 1

I O i n O i n c a o m o o i n o m o m o i n i n o m m

M M;0'M.\J M ~* —(r\im'-< -H—< — i — * — i

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Append ix I I

B r i e f p rog ram d e s c r i p t i o n s

T h i s append ix i n c l u d e s n o t e s on v a r i o u s o r i g i n a l

computer programs used i n t h i s s t u d y . A l l were w r i t t e n i n

FORTRAN and r u n on t h e NUMAC sys tem a t t he U n i v e r s i t y o f

Durham. L i s t i n g s are a v a i l a b l e f r o m t h e a u t h o r .

PROFILE. Reads c h a r a c t e r s t r i n g h e a d e r , number o f

ang le s and measured a n g l e s e r i e s ( c r e s t f i r s t ) . Measured

l e n g t h g i v e n as c o n s t a n t i n DATA s t a t e m e n t .

C a l c u l a t e s p r o f i l e c o o r d i n a t e s and average ang le

and produces d i m e n s i o n l e s s p r o f i l e p l o t and a n g l e s e r i e s

p l o t .

AVEDIFF. Reads c h a r a c t e r s t r i n g h e a d e r , number o f

d a t a v a l u e s , d a t a s e r i e s , number o f s u b s e r i e s l e n g t h s ,

o u t p u t d e v i c e numbers and s u b s e r i e s l e n g t h s .

For each s u b s e r i e s l e n g t h , s u b s e r i e s averages and

d i f f e r e n c e s p roduced i n s e r i a l o r d e r . Then each s e r i e s

s o r t e d t o n u m e r i c a l o r d e r , median and q u a r t i l e s p r o d u c e d ,

f r e q u e n c y d i s t r i b u t i o n c o m p i l e d and h i s t o g r a m p l o t t e d .

Repeated f o r a b s o l u t e v a l u e s .

S u b r o u t i n e s f o r median and q u a r t i l e s based on

s u b r o u t i n e s i n Andrews e t a l . ( 1 9 7 2 , Append ix 1 1 ) . S u b r o u t i n e

f o r s o r t i n g based on code i n Day ( 1 9 7 2 , 7 2 - 3 ) . S u b r o u t i n e

f o r h i s t o g r a m p l o t t i n g based l o o s e l y on s u b r o u t i n e i n

Dav i s ( 1 9 7 3 , 2 2 9 ) .

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AUTO. Reads c h a r a c t e r s t r i n g h e a d e r , number o f d a t a v a l u e s and d a t a s e r i e s . C a l c u l a t e s a u t o c o r r e l a t i o n f u n c t i o n f o r l ags up t o s m a l l e r o f 5 0 , f l o o r (number o f v a l u e s A ) , t o g e t h e r w i t h 0 .05 and 0 . 0 1 c o n f i d e n c e l i m i t s . P l o t s f u n c t i o n a g a i n s t l a g . Repeats f o r f i r s t d i f f e r e n c e s o f d a t a .

S u b r o u t i n e f o r c a l c u l a t i o n i n two v e r s i o n s , f o r Pearson

and a b b r e v i a t e d e s t i m a t o r s . S u b r o u t i n e f o r p l o t t i n g based

l o o s e l y on s u b r o u t i n e i n Davis ( 1 9 7 3 , 2 2 9 ) .

0NGLEY. Reads c h a r a c t e r s t r i n g h e a d e r , number o f a n g l e s ,

measured ang le s e r i e s and t o l e r a n c e . W r i t e s d e t a i l s o f

a c c e p t a b l e segments , i n c l u d i n g r e g r e s s i o n e q u a t i o n , r e s i d u a l s ,

l e n g t h o f r e g r e s s i o n l i n e , r e s i d u a l sum and average r e s i d u a l .

Based on p rog ram by Ongley ( 1 9 7 0 ) , r e v i s e d t o a l l o w

ang le i n p u t ; s e v e r a l m i n o r m o d i f i c a t i o n s .

YOUNG. Reads c h a r a c t e r s t r i n g h e a d e r , number o f a n g l e s ,

measured ang le s e r i e s and maximum c o e f f i c i e n t s o f v a r i a t i o n .

W r i t e s d e t a i l s o f a c c e p t a b l e segments , e lements and u n i t s .

Based on p rog ram SLOPEUNITS by A. Young , U n i v e r s i t y o f

East A n g l i a , k i n d l y s u p p l i e d by t h e a u t h o r . R e w r i t t e n t o

remove I C L FORTRAN f e a t u r e s ; s e v e r a l m i n o r m o d i f i c a t i o n s .

FISHER. Reads c h a r a c t e r s t r i n g h e a d e r , number o f

ang les and measured ang le s e r i e s . C a l c u l a t e s f i r s t d i f f e r e n c e s

and computes l e a s t squares p a r t i t i o n s o f d i f f e r e n c e s e r i e s

i n t o 1 ( 1 ) 10 components . W r i t e s i n d i c e s , l e n g t h , mean,

s t a n d a r d d e v i a t i o n o f each component , and w i t h i n - c o m p o n e n t

e r r o r f o r each p a r t i t i o n .

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Based i n p a r t on s u b r o u t i n e s g i v e n by H a r t i g a n ( 1 9 7 5 ,

1 4 1 - 2 ) .

SLOTH. Reads c h a r a c t e r s t r i n g h e a d e r , number o f ang les

and measured ang le s e r i e s . Computes and p l o t s c u r v a t u r e s ,

3H3H smooth and r o u g h a n g l e s , and c u r v a t u r e s f r o m 3H3H

smooth a n g l e s . W r i t e s smooth c u r v a t u r e s .

K I R P I T . Reads c h a r a c t e r s t r i n g h e a d e r , number o f measured

a n g l e s , measured a n g l e s e r i e s and t e r m i n a l i n d i c e s o f

s u b s e r i e s t o be f i t t e d . P i t s K i r k b y cu rve t o s u b s e r i e s

u s i n g d i f f e r e n t e s t i m a t o r s . For each e s t i m a t o r , w r i t e s

e s t i m a t e ; mean square and r o o t mean square r e s i d u a l ;

o r d e r e d r e s i d u a l s ; q u a r t i l e s , m i d s p r e a d , ex t remes and range

o f r e s i d u a l s ; and a u t o c o r r e l a t i o n f u n c t i o n o f r e s i d u a l s .

P l o t s r e s i d u a l s i n s p a t i a l o r d e r .