Environmental Modelling

Environmental ModellingFinding Simplicity in Complexity

Editors

John Wainwrightand

Mark Mulligan

Environmental Monitoring and Modelling Research Group,Department of Geography,

King’s College London,Strand

London WC2R 2LSUK

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Library of Congress Cataloging-in-Publication Data

Environmental modelling : finding simplicity in complexity / editors, John Wainwright andMark Mulligan.

p. cm.Includes bibliographical references and index.ISBN 0-471-49617-0 (acid-free paper) – ISBN 0-471-49618-9 (pbk. : acid-free paper)1. Environmental sciences – Mathematical models. I. Wainwright, John, 1967-II.

Mulligan, Mark, Dr.

GE45.M37E593 2004628–dc22

2003062751

British Library Cataloguing in Publication Data

A catalogue record for this book is available from the British Library

ISBN 0-471-49617-0 (Cloth)ISBN 0-471-49618-9 (Paper)

Typeset in 9/11pt Times by Laserwords Private Limited, Chennai, IndiaPrinted and bound in Great Britain by Antony Rowe Ltd, Chippenham, WiltshireThis book is printed on acid-free paper responsibly manufactured from sustainable forestryin which at least two trees are planted for each one used for paper production.

For my parents, Betty and John, without whose quiet support over theyears none of this would be possible. (JW)

To my parents, David and Filomena, who taught (and teach) me so muchand my son/daughter-to-be whom I meet for the first time in a few weeks

(hopefully after this book is complete). (MM)

Contents

List of contributors xvii

Preface xxi

Introduction 1John Wainwright and Mark Mulligan

1 Introduction 12 Why model the environment? 13 Why simplicity and complexity? 14 How to use this book 25 The book’s website 4

References 4

Part I Modelling and Model Building 5

1 Modelling and Model Building 7Mark Mulligan and John Wainwright

1.1 The role of modelling in environmental research 71.1.1 The nature of research 71.1.2 A model for environmental research 71.1.3 The nature of modelling 81.1.4 Researching environmental systems 8

1.2 Models with a purpose (the purpose of modelling) 101.2.1 Models with a poorly defined purpose 12

1.3 Types of model 131.4 Model structure and formulation 15

1.4.1 Systems 151.4.2 Assumptions 151.4.3 Setting the boundaries 161.4.4 Conceptualizing the system 171.4.5 Model building 171.4.6 Modelling in graphical model-building tools: using SIMILE 181.4.7 Modelling in spreadsheets: using Excel 201.4.8 Modelling in high-level modelling languages: using PCRASTER 221.4.9 Hints and warnings for model building 28

viii Contents

1.5 Development of numerical algorithms 291.5.1 Defining algorithms 291.5.2 Formalizing environmental systems 291.5.3 Analytical models 341.5.4 Algorithm appropriateness 351.5.5 Simple iterative methods 381.5.6 More versatile solution techniques 40

1.6 Model parameterization, calibration and validation 511.6.1 Chickens, eggs, models and parameters? 511.6.2 Defining the sampling strategy 521.6.3 What happens when the parameters don’t work? 541.6.4 Calibration and its limitations 541.6.5 Testing models 551.6.6 Measurements of model goodness-of-fit 56

1.7 Sensitivity analysis and its role 581.8 Errors and uncertainty 59

1.8.1 Error 591.8.2 Reporting error 661.8.3 From error to uncertainty 661.8.4 Coming to terms with error 67

1.9 Conclusion 67References 68

Part II The State of the Art in Environmental Modelling 75

2 Climate and Climate-System Modelling 77L.D. Danny Harvey

2.1 The complexity 772.2 Finding the simplicity 79

2.2.1 Models of the atmosphere and oceans 792.2.2 Models of the carbon cycle 812.2.3 Models of atmospheric chemistry and aerosols 832.2.4 Models of ice sheets 842.2.5 The roles of simple and complex climate-system models 84

2.3 The research frontier 852.4 Case study 86

References 89

3 Soil and Hillslope Hydrology 93Andrew Baird

3.1 It’s downhill from here 933.2 Easy does it: the laboratory slope 93

3.2.1 Plane simple: overland flow on Hillslope 1 933.2.2 Getting a head: soil-water/ground-water flow in Hillslope 2 95

3.3 Variety is the spice of life: are real slopes too hot to handle? 973.3.1 Little or large?: Complexity and scale 973.3.2 A fitting end: the physical veracity of models 993.3.3 Perceptual models and reality: ‘Of course, it’s those bloody macropores again!’ 100

Contents ix

3.4 Eenie, meenie, minie, mo: choosing models and identifying processes 102Acknowledgements 103References 103

4 Modelling Catchment Hydrology 107Mark Mulligan

4.1 Introduction: connectance in hydrology 1074.2 The complexity 108

4.2.1 What are catchments? 1084.2.2 Representing the flow of water in landscapes 1084.2.3 The hydrologically significant properties of catchments 1134.2.4 A brief review of catchment hydrological modelling 1134.2.5 Physically based models 1144.2.6 Conceptual and empirical models 1154.2.7 Recent developments 116

4.3 The simplicity 1174.3.1 Simplifying spatial complexity 1174.3.2 Simplifying temporal complexity 1174.3.3 Simplifying process complexity 1174.3.4 Concluding remarks 118

References 119

5 Modelling Fluvial Processes and Interactions 123Katerina Michaelides and John Wainwright

5.1 Introduction 1235.2 The complexity 123

5.2.1 Form and process complexity 1235.2.2 System complexity 126

5.3 Finding the simplicity 1265.3.1 Mathematical approaches 1275.3.2 Numerical modelling of fluvial interactions 1295.3.3 Physical modelling of fluvial processes and interactions 131

5.4 Case study 1: modelling an extreme storm event 1355.5 Case study 2: holistic modelling of the fluvial system 1365.6 The research frontier 138

Acknowledgements 138References 138

6 Modelling the Ecology of Plants 143Colin P. Osborne

6.1 The complexity 1436.2 Finding the simplicity 144

6.2.1 A question of scale 1446.2.2 What are the alternatives for modelling plant function? 1456.2.3 Applying a mechanistic approach 146

6.3 The research frontier 1486.4 Case study 1486.5 Conclusion 152

Acknowledgements 152References 152

x Contents

7 Spatial Population Models for Animals 157George L.W. Perry and Nick R. Bond

7.1 The complexity: introduction 1577.2 Finding the simplicity: thoughts on modelling spatial ecological systems 158

7.2.1 Space, spatial heterogeneity and ecology 1587.2.2 Three approaches to spatially explicit ecological modelling 1587.2.3 Top-down or bottom-up? 162

7.3 The research frontier: marrying theory and practice 1627.4 Case study: dispersal dynamics in stream ecosystems 163

7.4.1 The problem 1637.4.2 The model 1647.4.3 The question 1647.4.4 Results 165

7.5 Conclusion 165Acknowledgements 167References 167

8 Ecosystem Modelling: Vegetation and Disturbance 171Stefano Mazzoleni, Francisco Rego, Francesco Giannino and Colin Legg

8.1 The system complexity: effects of disturbance on vegetation dynamics 1718.1.1 Fire 1718.1.2 Grazing 1728.1.3 Fire and grazing interactions 172

8.2 Model simplification: simulation of plant growth under grazing and after fire 1738.3 New developments in ecosystem modelling 1768.4 Interactions of fire and grazing on plant competition: field experiment and modelling

applications 1778.4.1 A case study 1778.4.2 A model exercise 177

8.5 Conclusion 183Acknowledgements 183References 183

9 Erosion and Sediment Transport 187John N. Quinton

9.1 The complexity 1879.2 Finding the simplicity 189

9.2.1 Parameter uncertainty 1899.2.2 What are the natural levels of uncertainty? 1909.2.3 Do simple models predict erosion any better than more complex ones? 190

9.3 Finding simplicity 1919.4 The research frontier 1929.5 Case study 192

Note 195References 195

10 Modelling Slope Instability 197Andrew Collison and James Griffiths

10.1 The complexity 197

Contents xi

10.2 Finding the simplicity 19810.2.1 One-dimensional models 19810.2.2 Two-dimensional models 19910.2.3 Three-dimensional models 20010.2.4 Slope instability by flow 20110.2.5 Modelling external triggers 20110.2.6 Earthquakes 202

10.3 The research frontier 20210.4 Case study 203

10.4.1 The problem 20310.4.2 The modelling approach 20310.4.3 Validation 20410.4.4 Results 205

10.5 Conclusion 207References 207

11 Finding Simplicity in Complexity in Biogeochemical Modelling 211Hordur V. Haraldsson and Harald U. Sverdrup

11.1 Introduction to models 21111.2 Dare to simplify 21211.3 Sorting 21411.4 The basic path 21611.5 The process 21611.6 Biogeochemical models 21711.7 Conclusion 221

References 222

12 Modelling Human Decision-Making 225John Wainwright and Mark Mulligan

12.1 Introduction 22512.2 The human mind 22612.3 Modelling populations 22812.4 Modelling decisions 229

12.4.1 Agent-based modelling 22912.4.2 Economics models 23412.4.3 Game theory 23512.4.4 Scenario-based approaches 23712.4.5 Integrated analysis 239

12.5 Never mind the quantities, feel the breadth 23912.6 Perspectives 241

References 242

13 Modelling Land-Use Change 245Eric F. Lambin

13.1 The complexity 24513.1.1 The nature of land-use change 24513.1.2 Causes of land-use changes 245

13.2 Finding the simplicity 24613.2.1 Empirical-statistical models 246

xii Contents

13.2.2 Stochastic models 24713.2.3 Optimization models 24713.2.4 Dynamic (process-based) simulation models 248

13.3 The research frontier 24913.3.1 Addressing the scale issue 25013.3.2 Modelling land-use intensification 25013.3.3 Integrating temporal heterogeneity 250

13.4 Case study 25113.4.1 The problem 25113.4.2 Model structure 251

References 253

Part III Models for Management 255

14 Models in Policy Formulation and Assessment: The WadBOS Decision-Support System 257Guy Engelen

14.1 Introduction 25714.2 Functions of WadBOS 25814.3 Decision-support systems 25814.4 Building the integrated model 259

14.4.1 Knowledge acquisition and systems analysis 25914.4.2 Modelling and integration of models 26014.4.3 Technical integration 261

14.5 The integrated WadBOS model 26214.5.1 The economic submodel 26214.5.2 Landscape 26414.5.3 The ecological subsystem 265

14.6 The toolbase 26614.7 The database 26714.8 The user interface 26814.9 Conclusion 269

Acknowledgements 270References 271

15 Decision-Support Systems for Managing Water Resources 273Sophia Burke

15.1 Introduction 27315.2 Why are DSS needed? 27315.3 Design of a decision-support system 27415.4 Conclusion 27615.5 DSS on the web 276

References 276

16 Soil Erosion and Conservation 277Mark A. Nearing

16.1 The problem 27716.2 The approaches 27916.3 The contributions of modelling 281

16.3.1 Potential changes in rainfall erosivity in the USA during the twenty-first century 282

Contents xiii

16.3.2 Effects of precipitation intensity changes versus number of days of rainfall 28416.4 Lessons and implications 287

Acknowledgements 288Note 288References 288

17 Modelling in Forest Management 291Mark J. Twery

17.1 The issue 29117.2 The approaches 292

17.2.1 The empirical approach 29217.2.2 The mechanistic approach 29217.2.3 The knowledge-based approach 292

17.3 The contribution of modelling 29317.3.1 Models of the forest system 29317.3.2 Models of human responses and interactions 29417.3.3 Integrating techniques 295

17.4 Lessons and implications 29717.4.1 Models can be useful 29717.4.2 Goals matter 29717.4.3 People need to understand trade-offs 297

Note 298References 298

18 Stability and Instability in the Management of Mediterranean Desertification 303John B. Thornes

18.1 Introduction 30318.2 Basic propositions 30418.3 Complex interactions 306

18.3.1 Spatial variability 30918.3.2 Temporal variability 310

18.4 Climate gradient and climate change 31118.5 Implications 31318.6 Plants 31318.7 Conclusion 314

References 314

Part IV Current and Future Developments 317

19 Scaling Issues in Environmental Modelling 319Xiaoyang Zhang, Nick A. Drake and John Wainwright

19.1 Introduction 31919.2 Scale and scaling 320

19.2.1 Meanings of scale in environmental modelling 32019.2.2 Scaling 320

19.3 Causes of scaling problems 32219.4 Scaling issues of input parameters and possible solutions 322

19.4.1 Change of parameters with scale 32219.4.2 Methods of scaling parameters 323

xiv Contents

19.5 Methodology for scaling physically based models 32419.5.1 Incompatibilities between scales 32419.5.2 Methods for upscaling environmental models 32519.5.3 Approaches for downscaling climate models 328

19.6 Scaling land-surface parameters for a soil-erosion model: a case study 32819.6.1 Sensitivity of both topographic slope and vegetation cover to erosion 32819.6.2 A fractal method for scaling topographic slope 32919.6.3 A frequency-distribution function for scaling vegetation cover 32919.6.4 Upscaled soil-erosion models 329

19.7 Conclusion 332References 332

20 Environmental Applications of Computational Fluid Dynamics 335Nigel G. Wright and Christopher J. Baker

20.1 Introduction 33520.2 CFD fundamentals 335

20.2.1 Overview 33520.2.2 Equations of motion 33620.2.3 Grid structure 33620.2.4 Discretization and solution methods 33820.2.5 The turbulence-closure problem and turbulence models 33820.2.6 Boundary conditions 33920.2.7 Post-processing 33920.2.8 Validation and verification 339

20.3 Applications of CFD in environmental modelling 33920.3.1 Hydraulic applications 33920.3.2 Atmospheric applications 342

20.4 Conclusion 345References 347

21 Self-Organization and Cellular Automata Models 349David Favis-Mortlock

21.1 Introduction 34921.1.1 The ever-decreasing simplicity of models? 349

21.2 Self-organization in complex systems 35121.2.1 Deterministic chaos and fractals 35121.2.2 Early work on self-organizing systems 35221.2.3 Attributes of self-organizing systems 35221.2.4 The counter-intuitive universality of self-organization 354

21.3 Cellular automaton models 35521.3.1 Self-organization on a cellular grid 35621.3.2 Kinds of CA models 35621.3.3 Computational constraints to CA modelling 35621.3.4 Modelling self-organization: the problem of context and boundaries 35621.3.5 Terminology: self-organization and cellular automata 35721.3.6 Geomorphological applications of CA models 357

21.4 Case study: modelling rill initiation and growth 35821.4.1 The RillGrow 1 model 35821.4.2 The RillGrow 2 model 359

Contents xv

21.5 Conclusion 363Acknowledgements 365Notes 366References 367

22 Data-Based Mechanistic Modelling and the Simplification of Environmental Systems 371Peter C. Young, Arun Chotai and Keith J. Beven

22.1 Introduction 37122.2 Philosophies of modelling 37222.3 Statistical identification, estimation and validation 373

22.3.1 Structure and order identification 37322.3.2 Estimation (optimization) 37322.3.3 Conditional validation 373

22.4 Data-based mechanistic (DBM) modelling 37422.5 The statistical tools of DBM modelling 37622.6 Practical examples 376

22.6.1 A linear example: modelling solute transport 37622.6.2 A nonlinear example: rainfall-flow modelling 380

22.7 The evaluation of large deterministic simulation models 38322.8 Conclusion 385

Notes 386References 386

23 Pointers for the Future 389John Wainwright and Mark Mulligan

23.1 What have we learned? 38923.1.1 Explanation 38923.1.2 Qualitative issues 39123.1.3 Reductionism, holism and self-organized systems 39223.1.4 How should we model? 39223.1.5 Modelling methodology 39323.1.6 Process 39323.1.7 Modelling in an integrated methodology 39323.1.8 Data issues 39423.1.9 Modelling and policy 39523.1.10 What is modelling for? 39523.1.11 Moral and ethical questions 395

23.2 Research directions 39523.3 Is it possible to find simplicity in complexity? 396

References 396

Index 397

List of Contributors

Andrew Baird, Department of Geography, University of Sheffield, Sheffield, S10 2TN, UK.http://www.shef.ac.uk/geography/staff/baird andrew.html

Chris J. Baker, School of Engineering, Mechanical Engineering, The University of Birmingham, Edgbaston,Birmingham, B15 2TT, UK.http://www.eng.bham.ac.uk/civil/people/bakercj.htm

Keith J. Beven, Centre for Research on Environmental Systems and Statistics, Institute of Environmental andNatural Sciences, Lancaster University, Lancaster LA1 4YQ, UK.http://www.es.lancs.ac.uk/hfdg/kjb.html

Nick R. Bond, School of Biological Sciences, Monash University (Clayton Campus), Victoria 3800, Australia.http://biolsci.dbs.monash.edu.au/directory/labs/fellows/bond/

Sophia Burke, Environmental Monitoring and Modelling Research Group, Department of Geography, King’sCollege London, Strand, London WC2R 2LS, UK.http://www.kcl.ac.uk/kis/schools/hums/geog/smb.htm

Arun Chotai, Centre for Research on Environmental Systems and Statistics, Institute of Environmental andNatural Sciences, Lancaster University, Lancaster LA1 4YQ, UK.http://www.es.lancs.ac.uk/cres/staff/achotai/

Andrew Collison, Senior Associate, Philip Williams and Associates, 720 California St, San Francisco, CA94108, USA.http://www.pwa-ltd.com

Nick A. Drake, Environmental Monitoring and Modelling Research Group, Department of Geography, King’sCollege London, Strand, London WC2R 2LS, UK.http://www.kcl.ac.uk/kis/schools/hums/geog/nd.htm

Guy Engelen, Research Institute for Knowledge Systems bv, P.O. Box 463, 6200 AL Maastricht, The Netherlands.http://www.riks.nl/Projects/WadBOS

David Favis-Mortlock, School of Geography, Queen’s University Belfast, Belfast BT7 1NN, NorthernIreland, UK.http://www.qub.ac.uk/geog

Francesco Giannino, Facolta di Agraria, Universita di Napoli ‘Federico II’, Portici (NA), Italy.http://www.ecoap.unina.it

James Griffiths, River Regimes Section, Centre for Ecology and Hydrology, Wallingford OX10 1BB, UK.http://www.nwl.ac.uk/ih/

xviii List of Contributors

Hordur V. Haraldsson, Unit of Biogeochemistry, Chemical Engineering, Lund University, Box 124, 221 00Lund, Sweden.http://www2.chemeng.lth.se/staff/hordur/index.shtml

L.D. Danny Harvey, Department of Geography, University of Toronto, 100 St. George Street, Toronto, Ontario,M5S 3G3 Canada.http://www.geog.utoronto.ca/info/faculty/Harvey.htm

Eric F. Lambin, Department of Geography, University of Louvain, 3, place Pasteur, B-1348Louvain-la-Neuve, Belgium.http://www.geo.ucl.ac.be/Recherche/Teledetection/index.html

Colin Legg, School of GeoSciences, The University of Edinburgh, Darwin Building, King’s Buildings, MayfieldRoad, Edinburgh EH9 3JU, Scotland, UK.http://www.geos.ed.ac.uk/contacts/homes/clegg/

Stefano Mazzoleni, Facolta di Agraria, Universita di Napoli ‘Federico II’, Portici (NA), Italy.http://www.ecoap.unina.it

Katerina Michaelides, School of Geographical Sciences, University of Bristol, University Road, Bristol, BS81SS, UK.http://www.ggy.bris.ac.uk/staff/staff michaelides.htm

Mark Mulligan, Environmental Monitoring and Modelling Research Group, Department of Geography, King’sCollege London, Strand, London WC2R 2LS, UK.http://www.kcl.ac.uk/kis/schools/hums/geog/mm.htm

Mark A. Nearing, United States Department of Agriculture, Agricultural Research Service, Southwest WatershedResearch Center, 2000 E. Allen Road, Tucson, AZ 85719, USA.http://www.tucson.ars.ag.gov/

Colin P. Osborne, Department of Animal and Plant Sciences, University of Sheffield, Sheffield, S10 2TN, UK.http://www.shef.ac.uk/aps/staff-colin-osborne.html

George L.W. Perry, Environmental Monitoring and Modelling Research Group, Department of Geography, King’sCollege London, Strand, London WC2R 2LS, UK.http://www.kcl.ac.uk/kis/schools/hums/geog/gp.htm

John N. Quinton, Cranfield University, Silsoe, Bedford MK45 4DT, UK. (Now at Department of EnvironmentalScience, Institute of Environmental and Natural Sciences, Lancaster University, Lancaster LA1 4YQ, UK.)http://www.es.lancs.ac.uk/people/johnq/

Francisco Rego, Centro de Ecologia Aplicada Prof. Baeta Neves, Instituto Superior de Agronomia, Tapada daAjuda 1349-017, Lisbon, Portugal.http://www.isa.utl.pt/ceabn

Harald U. Sverdrup, Unit of Biogeochemistry, Chemical Engineering, Lund University, Box 124, 221 00Lund, Sweden.http://www2.chemeng.lth.se/staff/harald/

John B. Thornes, Environmental Monitoring and Modelling Research Group, Department of Geography, King’sCollege London, Strand, London WC2R 2LS, UK.rthornes@btinternet.co.uk

List of Contributors xix

Mark J. Twery, Research Forester, USDA Forest Service, Northeastern Research Station, Aiken Forestry SciencesLaboratory, 705 Spear Street, PO Box 968, Burlington, VT 05402-0968, USA.http://www.fs.fed.us/ne/burlington

John Wainwright, Environmental Monitoring and Modelling Research Group, Department of Geography, King’sCollege London, Strand, London WC2R 2LS, UK.http://www.kcl.ac.uk/kis/schools/hums/geog/jw.htm

Nigel G. Wright, School of Civil Engineering, The University of Nottingham, University Park, Nottingham NG72RD, UK.http://www.nottingham.ac.uk/%7Eevzngw/

Peter C. Young, Centre for Research on Environmental Systems and Statistics, Systems and Control Group,Institute of Environmental and Natural Sciences, Lancaster University, Lancaster LA1 4YQ, UK; and CRES,Australian National University, Canberra, Australia.http://www.es.lancs.ac.uk/cres/staff/pyoung/

Xiaoyang Zhang, Department of Geography, Boston University, 675 Commonwealth Avenue, Boston, MA02215, USA.http://crsa.bu.edu/∼zhang

Preface

Attempting to understand the world around us has beena fascination for millennia. It is said to be part of thehuman condition. The development of the numericalmodels, which are largely the focus of this book, isa logical development of earlier descriptive tools usedto analyse the environment such as drawings, classifica-tions and maps. Models should be seen as a complementto other techniques used to arrive at an understanding,and they also, we believe uniquely, provide an importantmeans of testing our understanding. This understandingis never complete, as we will see in many examplesin the following pages. This statement is meant to berealistic rather than critical. By maintaining a healthyscepticism about our results and continuing to test andre-evaluate them, we strive to achieve a progressivelybetter knowledge of the way the world works. Mod-elling should be carried out alongside field and labora-tory studies and cannot exist without them. We wouldtherefore encourage all environmental scientists not tobuild up artificial barriers between ‘modellers’ and ‘non-modellers’. Such a viewpoint benefits no-one. It maybe true that the peculiarities of mathematical notationand technical methods in modelling form a vocabularywhich is difficult to penetrate for some but we believethat the fundamental basis of modelling is one which,like fieldwork and laboratory experimentation, can beused by any scientist who, as they would in the field orthe laboratory, might work with others, more specialistin a particular technique to break this language barrier.

Complexity is an issue that is gaining much attentionin the field of modelling. Some see new ways of tacklingthe modelling of highly diverse problems (the economy,wars, landscape evolution) within a common framework.Whether this optimism will go the way of other attemptsto unify scientific methods remains to be seen. Ourapproach here has been to present as many ways aspossible to deal with environmental complexity, and toencourage readers to make comparisons across theseapproaches and between different disciplines. If a unifiedscience of the environment does exist, it will onlybe achieved by working across traditional disciplinary

boundaries to find common ways of arriving at simpleunderstandings. Often the simplest tools are the mosteffective and reliable, as anyone working in the field inremote locations will tell you!

We have tried to avoid the sensationalism of plac-ing the book in the context of any ongoing envi-ronmental ‘catastrophe’. However, the fact cannot beignored that many environmental modelling researchprogrammes are funded within the realms of work onpotential impacts on the environment, particularly dueto anthropic climate and land-use change. Indeed, themodelling approach – and particularly its propensity tobe used in forecasting – has done much to bring poten-tial environmental problems to light. It is impossible tosay with any certainty as yet whether the alarm has beenraised early enough and indeed which alarms are ringingloudest. Many models have been developed to evaluatewhat the optimal means of human interaction with theenvironment are, given the conflicting needs of differentgroups. Unfortunately, in many cases, the results of suchmodels are often used to take environmental exploita-tion ‘to the limit’ that the environment will accept,if not beyond. Given the propensity for environmentsto drift and vary over time and our uncertain knowl-edge about complex, non-linear systems with thresh-old behaviour, we would argue that this is clearly notthe right approach, and encourage modellers to ensurethat their results are not misused. One of the valuesof modelling, especially within the context of decision-support systems (see Chapter 14) is that non-modellersand indeed non-scientists can use them. They can thusconvey the opinion of the scientist and the thrust of sci-entific knowledge with the scientist absent. This givesmodellers and scientists contributing to models (poten-tially) great influence over the decision-making process(where the political constraints to this process are notparamount). With this influence comes a great responsi-bility for the modeller to ensure that the models used areboth accurate and comprehensive in terms of the drivingforces and affected factors and that these models are not

xxii Preface

applied out of context or in ways for which they werenot designed.

This book has developed from our work in envi-ronmental modelling as part of the EnvironmentalMonitoring and Modelling Research Group in theDepartment of Geography, King’s College London. Itowes a great debt to the supportive research atmospherewe have found there, and not least to John Thornes whoinitiated the group over a decade ago. We are particu-larly pleased to be able to include a contribution fromhim (Chapter 18) relating to his more recent work inmodelling land-degradation processes. We would alsolike to thank Andy Baird (Chapter 3), whose thought-provoking chapter on modelling in his book Ecohy-drology (co-edited with Wilby) and the workshop fromwhich it was derived provided one of the major stim-uli for putting this overview together. Of course, thestrength of this book rests on all the contributions, andwe would like to thank all of the authors for providing

excellent overviews of their work and the state-of-the-art in their various fields, some at very short notice. Wehope we have been able to do justice to your work.

We would also like to thank the numerous individualswho generously gave their time and expertise to assist inthe review of the chapters in the book. Roma Beaumontre-drew a number of the figures in her usual cheerfulmanner. A number of the ideas presented have beentested on our students at King’s over the last fewyears – we would like to thank them all for their inputs.Finally, we would like to thank Keily Larkins and SallyWilkinson at John Wiley and Sons for bearing with usthrough the delays and helping out throughout the longprocess of putting this book together.

John Wainwright and Mark MulliganLondon

December 2002

Introduction

JOHN WAINWRIGHT AND MARK MULLIGAN

1 INTRODUCTION

We start in this introduction to provide a prologue forwhat follows (possibly following a tradition for books oncomplex systems, after Bar-Yam, 1997). The aim hereis to provide a brief general rationale for the contentsand approach taken within the book.

In one sense, everything in this book arises from theinvention of the zero. Without this Hindu-Arabic inven-tion, none of the mathematical manipulations required toformulate the relationships inherent within environmen-tal processes would be possible. This point illustrates theneed to develop abstract ideas and apply them. Abstrac-tion is a fundamental part of the modelling process.

In another sense, we are never starting our investiga-tions from zero. By the very definition of the environ-ment as that which surrounds us, we always approach itwith a number (nonzero!) of preconceptions. It is impor-tant not to let them get in the way of what we aretrying to achieve. Our aim is to demonstrate how thesepreconceptions can be changed and applied to providea fuller understanding of the processes that mould theworld around us.

2 WHY MODEL THE ENVIRONMENT?

The context for much environmental modelling atpresent is the concern relating to human-induced cli-mate change. Similarly, work is frequently carried outto evaluate the impacts of land degradation due to humanimpact. Such application-driven investigations providean important means by which scientists can interactwith and influence policy at local, regional, national andinternational levels. Models can be a means of ensur-ing environmental protection, as long as we are careful

about how the results are used (Oreskes et al., 1994;Rayner and Malone, 1998; Sarewitz and Pielke, 1999;Bair, 2001).

On the other hand, we may use models to develop ourunderstanding of the processes that form the environ-ment around us. As noted by Richards (1990), processesare not observable features, but their effects and out-comes are. In geomorphology, this is essentially thedebate that attempts to link process to form (Richardset al., 1997). Models can thus be used to evaluatewhether the effects and outcomes are reproducible fromthe current knowledge of the processes. This approachis not straightforward, as it is often difficult to evaluatewhether process or parameter estimates are incorrect,but it does at least provide a basis for investigation.

Of course, understanding-driven and applications-driven approaches are not mutually exclusive. It is notpossible (at least consistently) to be successful in thelatter without being successful in the former. We followup these themes in much more detail in Chapter 1.

3 WHY SIMPLICITY AND COMPLEXITY?

In his short story ‘The Library of Babel’, Borges (1970)describes a library made up of a potentially infinitenumber of hexagonal rooms containing books thatcontain every permissible combination of letters andthus information about everything (or alternatively, asingle book of infinitely thin pages, each one openingout into further pages of text). The library is a model ofthe Universe – but is it a useful one? Borges describesthe endless searches for the book that might be the‘catalogue of catalogues’! Are our attempts to modelthe environment a similarly fruitless endeavour?

Environmental Modelling: Finding Simplicity in Complexity. Edited by J. Wainwright and M. Mulligan 2004 John Wiley & Sons, Ltd ISBNs: 0-471-49617-0 (HB); 0-471-49618-9 (PB)

2 John Wainwright and Mark Mulligan

Compare the definition by Grand (2000: 140): ‘Some-thing is complex if it contains a great deal of informationthat has a high utility, while something that contains a lotof useless or meaningless information is simply compli-cated.’ The environment, by this definition, is somethingthat may initially appear complicated. Our aim is to ren-der it merely complex! Any explanation, whether it is aqualitative description or a numerical simulation, is anattempt to use a model to achieve this aim. Althoughwe will focus almost exclusively on numerical models,these models are themselves based on conceptual mod-els that may be more-or-less complex (see discussionsin Chapters 1 and 11). One of the main questions under-lying this book is whether simple models are adequateexplanations of complex phenomena. Can (or should)we include Ockham’s Razor as one of the principal ele-ments in our modeller’s toolkit?

Bar-Yam (1997) points out that a dictionary defini-tion of complex means ‘consisting of interconnectedor interwoven parts’. ‘Loosely speaking, the complex-ity of a system is the amount of information neededin order to describe it’ (ibid.: 12). The most com-plex systems are totally random, in that they cannotbe described in shorter terms than by representing thesystem itself (Casti, 1994) – for this reason, Borges’Library of Babel is not a good model of the Universe,unless it is assumed that the Universe is totally random(or alternatively that the library is the Universe!). Com-plex systems will also exhibit emergent behaviour (Bar-Yam, 1997), in that characteristics of the whole aredeveloped (emerge) from interactions of their compo-nents in a nonapparent way. For example, the propertiesof water are not obvious from those of its constituentcomponents, hydrogen and oxygen molecules. Riversemerge from the interaction of discrete quantities ofwater (ultimately from raindrops) and oceans from theinteraction of rivers, so emergent phenomena may oper-ate on a number of scales.

The optimal model is one that contains sufficientcomplexity to explain phenomena, but no more. Thisstatement can be thought of as an information-theoryrewording of Ockham’s Razor. Because there is adefinite cost to obtaining information about a system,for example by collecting field data (see discussion inChapter 1 and elsewhere), there is a cost benefit todeveloping such an optimal model. In research termsthere is a clear benefit because the simplest model willnot require the clutter of complications that make itdifficult to work with, and often difficult to evaluate (seethe discussion of the Davisian cycle by Bishop, 1975,for a geomorphological example).

Opinions differ, however, on how to achieve this opti-mal model. The traditional view is essentially a reduc-tionist one. The elements of the system are analysed andonly those that are thought to be important in explainingthe observed phenomena are retained within the model.Often this approach leads to increasingly complex (orpossibly even complicated) models where additionalprocess descriptions and corresponding parameters andvariables are added. Generally, the law of diminishingreturns applies to the extra benefit of additional vari-ables in explaining observed variance. The modellingapproach in this case is one of deciding what levelof simplicity in model structure is required relative tothe overall costs and the explanation or understand-ing achieved.

By contrast, a more holistic viewpoint is emerging.Its proponents suggest that the repetition of simplesets of rules or local interactions can produce thefeatures of complex systems. Bak (1997), for example,demonstrates how simple models of sand piles canexplain the size of and occurrence of avalanches onthe pile, and how this approach relates to a series ofother phenomena. Bar-Yam (1997) provides a thoroughoverview of techniques that can be used in this wayto investigate complex systems. The limits of theseapproaches have tended to be related to computingpower, as applications to real-world systems requirethe repetition of very large numbers of calculations.A possible advantage of this sort of approach is thatit depends less on the interaction and interpretationsof the modeller, in that emergence occurs through theinteractions on a local scale. In most systems, theselocal interactions are more realistic representations ofthe process than the reductionist approach that tends tobe conceptualized so that distant, disconnected featuresact together. The reductionist approach therefore tendsto constrain the sorts of behaviour that can be producedby the model because of the constraints imposed by theconceptual structure of the model.

In our opinion, both approaches offer valuable meansof approaching an understanding of environmental sys-tems. The implementation and application of bothare described through this book. The two differentapproaches may be best suited for different types ofapplication in environmental models given the currentstate of the art. Thus, the presentations in this book willcontribute to the debate and ultimately provide the basisfor stronger environmental models.

4 HOW TO USE THIS BOOK

We do not propose here to teach you how to suckeggs (nor give scope for endless POMO discussion),

Introduction 3

but would like to offer some guidance based on theway we have structured the chapters. This book isdivided into four parts. We do not anticipate that manyreaders will want (or need) to read it from coverto cover in one go. Instead, the different elementscan be largely understood and followed separately, inalmost any order. Part I provides an introduction tomodelling approaches in general, with a specific focuson issues that commonly arise in dealing with theenvironment. We have attempted to cover the processof model development from initial concepts, throughalgorithm development and numerical techniques, toapplication and testing. Using the information providedin this chapter, you should be able to put together yourown models. We have presented it as a single, largechapter, but the headings within it should allow simplenavigation to the sections that are more relevant to you.

The twelve chapters of Part II form the core of thebook, presenting a state of the art of environmentalmodels in a number of fields. The authors of thesechapters were invited to contribute their own viewpointsof current progress in their specialist areas using a seriesof common themes. However, we have not forced theresulting chapters back into a common format as thiswould have restricted the individuality of the differentcontributions and denied the fact that different topicsmight require different approaches. As much as wewould have liked, the coverage here is by no meanscomplete and we acknowledge that there are gaps in thematerial here. In part, this is due to space limitationsand in part due to time limits on authors’ contributions.We make no apology for the emphasis on hydrologyand ecology in this part, not least because these are theareas that interest us most. However, we would alsoargue that these models are often the basis for otherinvestigations and thus are relevant to a wide rangeof fields. For any particular application, you may findbuilding blocks of relevance to your own interests acrossa range of different chapters here. Furthermore, it hasbecome increasingly obvious to us while editing thebook that there are a number of common themes andproblems being tackled in environmental modelling thatare currently being developed in parallel behind differentdisciplinary boundaries. One conclusion that we havecome to is that if you cannot find a specific answer to amodelling problem relative to a particular type of model,then a look at the literature of a different disciplinecan often provide answers. Even more importantly, thiscan lead to the demonstration of different problemsand new ways of dealing with issues. Cross-fertilizationof modelling studies will lead to the development ofstronger breeds of models!

In Part III, the focus moves to model applications. Weinvited a number of practitioners to give their viewpointson how models can or should be used in their particularfield of expertise. These chapters bring to light thedifferent needs of models in a policy or managementcontext and demonstrate how these needs might bedifferent from those in a pure research context. This isanother way in which modellers need to interface withthe real world, and one that is often forgotten.

Part IV deals with a number of current approaches inmodelling: approaches that we believe are fundamentalto developing strong models in the future. Again, theinclusion of subjects here is less than complete, althoughsome appropriate material on error, spatial models andvalidation is covered in Part I. However, we hope thispart gives at least a flavour of the new methods beingdeveloped in a number of areas of modelling. In general,the examples used are relevant across a wide rangeof disciplines. One of the original reviewers of thisbook asked how we could possibly deal with futuredevelopments. In one sense this objection is correct, inthe sense that we do not possess a crystal ball (and wouldprobably not be writing this at all if we did!). In another,it forgets the fact that many developments in modellingawait the technology to catch up for their successfulconclusion. For example, the detailed spatial modelsof today are only possible because of the exponentialgrowth in processing power over the past few decades.Fortunately the human mind is always one step aheadin posing more difficult questions. Whether this is agood thing is a question addressed at a number of pointsthrough the book!

Finally, a brief word about equations. Because the bookis aimed at a range of audiences, we have tried to keepit as user-friendly as possible. In Parts II, III and IV weasked the contributors to present their ideas and resultswith the minimum of equations. In Part I, we decidedthat it was not possible to get the ideas across without theuse of equations, but we have tried to explain as much aspossible from first principles as space permits. Sooner orlater, anyone wanting to build their own model will needto use these methods anyway. If you are unfamiliar withtext including equations, we would simply like to passon the following advice of the distinguished professor ofmathematics and physics, Roger Penrose:

If you are a reader who finds any formula intimidating(and most people do), then I recommend a procedureI normally adopt myself when such an offending linepresents itself. The procedure is, more or less, toignore that line completely and to skip over to thenext actual line of text! Well, not exactly this; one

4 John Wainwright and Mark Mulligan

should spare the poor formula a perusing, rather thana comprehending glance, and then press onwards.After a little, if armed with new confidence, one mayreturn to that neglected formula and try to pick outsome salient features. The text itself may be helpful inletting one know what is important and what can besafely ignored about it. If not, then do not be afraid toleave a formula behind altogether.

(Penrose, 1989: vi)

5 THE BOOK’S WEBSITE

As a companion to the book, we have developed arelated website to provide more information, links,examples and illustrations that are difficult to incorporatehere (at least without having a CD in the back ofthe book that would tend to fall out annoyingly!).The structure of the site follows that of the book,and allows easy access to the materials relating toeach of the specific chapters. The URL for the siteis www.kcl.ac.uk/envmod. We will endeavour to keepthe links and information as up to date as possibleto provide a resource for students and researchersof environmental modelling. Please let us know ifsomething does not work and, equally importantly, ifyou know of exciting new information and models towhich we can provide links.

REFERENCES

Bair, E. (2001) Models in the courtroom, in M.G. Andersonand P.D. Bates (eds) Model Validation: Perspectives in

Hydrological Science, John Wiley & Sons, Chichester,57–76.

Bak, P. (1997) How Nature Works: The Science of Self-Organized Criticality, Oxford University Press, Oxford.

Bar-Yam, Y. (1997) Dynamics of Complex Systems, PerseusBooks, Reading, MA.

Bishop, P. (1975) Popper’s principle of falsifiability and theirrefutability of the Davisian cycle, Professional Geographer32, 310–315.

Borges, J.L. (1970) Labyrinths, Penguin Books, Harmon-dsworth.

Casti, J.L. (1994) Complexification: Explaining a ParadoxicalWorld Through the Science of Surprise, Abacus, London.

Grand, S. (2000) Creation: Life and How to Make It, Phoenix,London.

Oreskes, N., Shrader-Frechette, K. and Bellitz, K. (1994) Veri-fication, validation and confirmation of numerical models inthe Earth Sciences, Science 263, 641–646.

Penrose, R. (1989) The Emperor’s New Mind, Oxford Univer-sity Press, Oxford.

Rayner, S. and Malone, E.L. (1998) Human Choice and Cli-mate Change, Batelle Press, Columbus, OH.

Richards, K.S. (1990) ‘Real’ geomorphology, Earth SurfaceProcesses and Landforms 15, 195–197.

Richards, K.S., Brooks, S.M., Clifford, N., Harris, T. andLowe, S. (1997) Theory, measurement and testing in ‘real’geomorphology and physical geography, in D.R. Stoddart(ed.) Process and Form in Geomorphology, Routledge, Lon-don, 265–292.

Sarewitz, D. and Pielke Jr, R.A. (1999) Prediction in scienceand society, Technology in Society 21, 121–133.

Part I

Modelling and Model Building

1

Modelling and Model Building

MARK MULLIGAN AND JOHN WAINWRIGHT

Modelling is like sin. Once you begin with one formof it you are pushed to others. In fact, as with sin,once you begin with one form you ought to considerother forms. . . . But unlike sin – or at any rate unlikesin as a moral purist conceives of it – modelling isthe best reaction to the situation in which we findourselves. Given the meagreness of our intelligencein comparison with the complexity and subtlety ofnature, if we want to say things which are true, aswell as things which are useful and things which aretestable, then we had better relate our bids for truth,application and testability in some fairly sophisticatedways. This is what modelling does.

(Morton and Suarez, 2001: 14)

1.1 THE ROLE OF MODELLING INENVIRONMENTAL RESEARCH

1.1.1 The nature of research

Research is a means of improvement through under-standing. This improvement may be personal, but it mayalso be tied to broader human development. We mayhope to improve human health and well-being throughresearch into diseases such as cancer and heart disease.We may wish to improve the design of bridges or aircraftthrough research in materials science, which provideslighter, stronger, longer-lasting or cheaper bridge struc-tures (in terms of building and of maintenance). Wemay wish to produce more or better crops with feweradverse impacts on the environment through research inbiotechnology. In all of these cases, research providesin the first instance better understanding of how thingsare and how they work, which can then contribute to the

improvement or optimization of these systems throughthe development of new techniques, processes, materialsand protocols.

Research is traditionally carried out through theaccumulation of observations of systems and systembehaviour under ‘natural’ circumstances and duringexperimental manipulation. These observations providethe evidence upon which hypotheses can be generatedabout the structure and operation (function) of thesystems. These hypotheses can be tested against newobservations and, where they prove to be reliabledescriptors of the system or system behaviour, thenthey may eventually gain recognition as tested theoryor general law.

The conditions, which are required to facilitateresearch, include:

1. a means of observation and comparative observation(measurement);

2. a means of controlling or forcing aspects of thesystem (experimentation);

3. an understanding of previous research and the stateof knowledge (context);

4. a means of cross-referencing and connecting threadsof 1, 2 and 3 (imagination).

1.1.2 A model for environmental research

What do we mean by the term model? A model isan abstraction of reality. This abstraction represents acomplex reality in the simplest way that is adequatefor the purpose of the modelling. The best model isalways that which achieves the greatest realism (mea-sured objectively as agreement between model outputs

Environmental Modelling: Finding Simplicity in Complexity. Edited by J. Wainwright and M. Mulligan 2004 John Wiley & Sons, Ltd ISBNs: 0-471-49617-0 (HB); 0-471-49618-9 (PB)

8 Mark Mulligan and John Wainwright

and real-world observations, or less objectively as theprocess insight gained from the model) with the leastparameter complexity and the least model complexity.

Parsimony (using no more complex a model orrepresentation of reality than is absolutely necessary) hasbeen a guiding principle in scientific investigations sinceAristotle who claimed: ‘It is the mark of an instructedmind to rest satisfied with the degree of precision whichthe nature of the subject permits and not to seek anexactness where only an approximation of the truth ispossible’ though it was particularly strong in medievaltimes and was enunciated then by William of Ockham,in his famous ‘razor’ (Lark, 2001). Newton stated it asthe first of his principles for fruitful scientific researchin Principia as: ‘We are to admit no more causes ofnatural things than such as are both true and sufficientto explain their appearances.’

Parsimony is a prerequisite for scientific explanation,not an indication that nature operates on the basis ofparsimonious principles. It is an important principle infields as far apart as taxonomy and biochemistry andis fundamental to likelihood and Bayesian approachesof statistical inference. In a modelling context, a par-simonious model is usually the one with the greatestexplanation or predictive power and the least parametersor process complexity. It is a particularly important prin-ciple in modelling since our ability to model complexityis much greater than our ability to provide the data toparameterize, calibrate and validate those same mod-els. Scientific explanations must be both relevant andtestable. Unvalidated models are no better than untestedhypotheses. If the application of the principle of parsi-mony facilitates validation, then it also facilitates utilityof models.

1.1.3 The nature of modelling

Modelling is not an alternative to observation but,under certain circumstances, can be a powerful toolin understanding observations and in developing andtesting theory. Observation will always be closer totruth and must remain the most important componentof scientific investigation. Klemes (1997: 48) describesthe forces at work in putting the modelling ‘cart’ beforethe observation ‘horse’ as is sometimes apparent inmodelling studies:

It is easier and more fun to play with a computer thanto face the rigors of fieldwork especially hydrologicfieldwork, which is usually most intensive during themost adverse conditions. It is faster to get a resultby modeling than through acquisition and analysis of

more data, which suits managers and politicians aswell as staff scientists and professors to whom it meansmore publications per unit time and thus an easierpassage of the hurdles of annual evaluations andother paper-counting rituals. And it is more glamorousto polish mathematical equations (even bad ones) inthe office than muddied boots (even good ones) inthe field.

(Klemes, 1997: 48)

A model is an abstraction of a real system, it is asimplification in which only those components whichare seen to be significant to the problem at handare represented in the model. In this, a model takesinfluence from aspects of the real system and aspectsof the modeller’s perception of the system and itsimportance to the problem at hand. Modelling supportsin the conceptualization and exploration of the behaviourof objects or processes and their interaction as ameans of better understanding these and generatinghypotheses concerning them. Modelling also supportsthe development of (numerical) experiments in whichhypotheses can be tested and outcomes predicted. Inscience, understanding is the goal and models serve astools towards that end (Baker, 1998).

Cross and Moscardini (1985: 22) describe modellingas ‘an art with a rational basis which requires the use ofcommon sense at least as much as mathematical exper-tise’. Modelling is described as an art because it involvesexperience and intuition as well as the development of aset of (mathematical) skills. Cross and Moscardini arguethat intuition and the resulting insight are the factorswhich distinguish good modellers from mediocre ones.Intuition (or imagination) cannot be taught and comesfrom the experience of designing, building and usingmodels. Tackling some of the modelling problems pre-sented on the website which complements this book willhelp in this.

1.1.4 Researching environmental systems

Modelling has grown significantly as a research activitysince the 1950s, reflecting conceptual developmentsin the modelling techniques themselves, technologicaldevelopments in computation, scientific developmentsin response to the increased need to study systems(especially environmental ones) in an integrated manner,and an increased demand for extrapolation (especiallyprediction) in space and time.

Modelling has become one of the most power-ful tools in the workshop of environmental scien-tists who are charged with better understanding the

Modelling and Model Building 9

interactions between the environment, ecosystems andthe populations of humans and other animals. Thisunderstanding is increasingly important in environ-mental stewardship (monitoring and management) andthe development of increasingly sustainable means ofhuman dependency on environmental systems.

Environmental systems are, of course, the same sys-tems as those studied by physicists, chemists and biolo-gists but the level of abstraction of the environmentalscientist is very different from many of these scien-tists. Whereas a physicist might study the behaviour ofgases, liquids or solids under controlled conditions oftemperature or pressure and a chemist might study theinteraction of molecules in aqueous solution, a biologistmust integrate what we know from these sciences tounderstand how a cell – or a plant or an animal – livesand functions. The environmental scientist or geogra-pher or ecologist approaches their science at a muchgreater level of abstraction in which physical and chem-ical ‘laws’ provide the rule base for understanding theinteraction between living organisms and their nonliv-ing environments, the characteristics of each and theprocesses through which each functions.

Integrated environmental systems are different inmany ways from the isolated objects of study inphysics and chemistry though the integrated study ofthe environment cannot take place without the buildingblocks provided by research in physics and chemistry.The systems studied by environmental scientists arecharacteristically:

• Large-scale, long-term. Though the environmentalscientist may only study a small time-scale and space-scale slice of the system, this slice invariably fitswithin the context of a system that has evolved overhundreds, thousands or millions of years and whichwill continue to evolve into the future. It is also aslice that takes in material and energy from a hierar-chy of neighbours from the local, through regional, toglobal scale. It is this context which provides muchof the complexity of environmental systems com-pared with the much more reductionist systems of thetraditional ‘hard’ sciences. To the environmental sci-entist, models are a means of integrating across timeand through space in order to understand how thesecontexts determine the nature and functioning of thesystem under study.

• Multicomponent. Environmental scientists rarely havethe good fortune of studying a single componentof their system in isolation. Most questions askedof environmental scientists require the understand-ing of interactions between multiple living (biotic)

and nonliving (abiotic) systems and their interaction.Complexity increases greatly as the number of com-ponents increases, where their interactions are alsotaken into account. Since the human mind has someconsiderable difficulty in dealing with chains ofcausality with more than a few links, to an environ-mental scientist models are an important means ofbreaking systems into intellectually manageable com-ponents and combining them and making explicit theinteractions between them.

• Nonlaboratory controllable. The luxury of controlledconditions under which to test the impact of individualforcing factors on the behaviour of the study systemis very rarely available to environmental scientists.Very few environmental systems can be re-built inthe laboratory (laboratory-based physical modelling)with an appropriate level of sophistication to ade-quately represent them. Taking the laboratory to thefield (field-based physical modelling) is an alterna-tive, as has been shown by the Free AtmosphereCO2 Enrichment (FACE) experiments (Hall, 2001),BIOSPHERE 2 (Cohn, 2002) and a range of otherenvironmental manipulation experiments. Field-basedphysical models are very limited in the degree ofcontrol available to the scientist because of the enor-mous expense associated with this. They are also verylimited in the scale at which they can be applied,again because of expense and engineering limita-tions. So, the fact remains that, at the scale at whichenvironmental scientists work, their systems remaineffectively noncontrollable with only small compo-nents capable of undergoing controlled experiments.However, some do argue that the environment itselfis one large laboratory, which is sustaining global-scale experiments through, for example, greenhousegas emissions (Govindasamy et al., 2003). These arenot the kind of experiments that enable us to predict(since they are real-time) nor which help us, in theshort term at least, to better interact with or managethe environment (notwithstanding the moral implica-tions of this activity!). Models provide an inexpensivelaboratory in which mathematical descriptions of sys-tems and processes can be forced in a controlled way.

• Multiscale, multidisciplinary. Environmental systemsare multiscale with environmental scientists need-ing to understand or experiment at scales from theatom through the molecule to the cell, organism orobject, population of objects, community or landscapethrough to the ecosystem and beyond. This presenceof multiple scales means that environmental scien-tists are rarely just environmental scientists, they maybe physicists, chemists, physical chemists, engineers,

10 Mark Mulligan and John Wainwright

biologists, botanists, zoologists, anthropologists, pop-ulation geographers, physical geographers, ecologists,social geographers, political scientists, lawyers, envi-ronmental economists or indeed environmental scien-tists in their training but who later apply themselvesto environmental science. Environmental science isthus an interdisciplinary science which cuts acrossthe traditional boundaries of academic research.Tackling contemporary environmental problems ofteninvolves large multidisciplinary (and often multina-tional) teams working together on different aspects ofthe system. Modelling provides an integrative frame-work in which these disparate disciplines can workon individual aspects of the research problem andsupply a module for integration within the mod-elling framework. Disciplinary and national bound-aries, research ‘cultures’ and research ‘languages’ arethus no barrier.

• Multivariate, nonlinear and complex. It goes withoutsaying that integrated systems such as those handledby environmental scientists are multivariate and as aresult the relationships between individual variablesare often nonlinear and complex. Models providea means of deconstructing the complexity of envi-ronmental systems and, through experimentation, ofunderstanding the univariate contribution to multivari-ate complexity.

In addition to these properties of environmental systems,the rationale behind much research in environmentalsystems is often a practical or applied one such thatresearch in environmental science also has to incorporatethe following needs:

• The need to look into the future. Environmentalresearch often involves extrapolation into the futurein order to understand the impacts of some currentstate or process. Such prediction is difficult, not leastbecause predictions can only be tested in real time.Models are very often used as a tool for integrationof understanding over time and thus are well suitedfor prediction and postdiction. As with any means ofpredicting the future, the prediction is only as goodas the information and understanding upon which itis based. While this understanding may be sufficientwhere one is working within process domains thathave already been experienced during the period inwhich the understanding was developed, when futureconditions cross a process domain, the reality may bequite different to the expectation.

• The need to understand the impact of events thathave not happened (yet). Environmental research

very often concerns developing scenarios for changeand understanding the impacts of these scenarios onsystems upon which humans depend. These changesmay be developmental such as the building of houses,industrial units, bridges, ports or golf courses and thusrequire environmental impact assessments (EIAs).Alternatively, they may be more abstract events suchas climate change or land use and cover change(LUCC). In either case where models have beendeveloped on the basis of process understandingor a knowledge of the response of similar sys-tems to similar or analogous change, they are oftenused as a means of understanding the impact ofexpected events.

• The need to understand the impacts of human beha-viour. With global human populations continuing toincrease and per capita resource use high and increas-ing in the developed world and low but increasing inmuch of the developing world, the need to achieverenewable and nonrenewable resource use that can besustained into the distant future becomes more andmore pressing. Better understanding the impacts ofhuman resource use (fishing, forestry, hunting, agri-culture, mining) on the environment and its abilityto sustain these resources is thus an increasing thrustof environmental research. Models, for many of thereasons outlined above, are often employed to inves-tigate the enhancement and degradation of resourcesthrough human impact.

• The need to understand the impacts on human beha-viour. With human population levels so high andconcentrated and with per capita resource needs sohigh and sites of production so disparate from sitesof consumption, human society is increasingly sen-sitive to environmental change. Where environmen-tal change affects resource supply, resource demandor the ease and cost of resource transportation, theimpact on human populations is likely to be high.Therefore understanding the nature of variation andchange in environmental systems and the feedback ofhuman impacts on the environment to human popula-tions is increasingly important. Environmental scienceincreasingly needs to be a supplier of reliable fore-casts and understanding to the world of human healthand welfare, development, politics and peacekeeping.

1.2 MODELS WITH A PURPOSE (THE PURPOSEOF MODELLING)

Modelling is thus the canvas of scientists on whichthey can develop and test ideas, put a number ofideas together and view the outcome, integrate and

Modelling and Model Building 11

communicate those ideas to others. Models can play oneor more of many roles, but they are usually developedwith one or two roles specifically in mind. The type ofmodel built will, in some way, restrict the uses to whichthe model may be put. The following seven headingsoutline the purposes to which models are usually put:

1. As an aid to research. Models are now a fairly com-monplace component of research activities. Throughtheir use in assisting understanding, in simulation, asa virtual laboratory, as an integrator across disciplinesand as a product and means of communicating ideas,models are an aid to research activities. Models alsofacilitate observation and theory. For example, under-standing the sensitivity of model output to parame-ter uncertainty can guide field data-collection activi-ties. Moreover, models allow us to infer informationabout unmeasurable or expensively measured propertiesthrough modelling them from more readily measuredvariables that are related in some way to the variableof interest.2. As a tool for understanding. Models are a tool forunderstanding because they allow (indeed, require)abstraction and formalization of the scientific conceptsbeing developed, because they help tie together ideasthat would otherwise remain separate and because theyallow exploration of the outcomes of particular ‘experi-ments’ in the form of parameter changes. In building amodel, one is usually forced to think very clearly aboutthe system under investigation and in using a model oneusually learns more about the system than was obviousduring model construction.3. As a tool for simulation and prediction. Though thereare benefits to be gained from building models, their realvalue becomes apparent when they are extensively usedfor system simulation and/or prediction. Simulation withmodels allows one to integrate the effects of simple pro-cesses over complex spaces (or complex processes oversimple spaces) and to cumulate the effects of those sameprocesses (and their variation) over time. This integra-tion and cumulation can lead to the prediction of systembehaviour outside the time or space domain for whichdata are available. This integration and cumulation areof value in converting a knowledge or hypothesis of pro-cess into an understanding of the outcome of this processover time and space – something that is very difficultto pursue objectively without modelling. Models arethus extensively employed in extrapolation beyond mea-sured times and spaces, whether that means prediction(forecasting) or postdiction (hindcasting) or near-termcasting (nowcasting) as is common in meteorology andhydrology. Prediction using models is also a commonly

used means of making better any data that we do havethrough the interpolation of data at points in which wehave no samples, for example, through inverse distanceweighting (IDW) or kriging techniques. Furthermore, anunderstanding of processes can help us to model highresolution data from lower resolution data as is commonin climate model downscaling and weather generation(Bardossy, 1997; see also Chapters 2 and 19).4. As a virtual laboratory. Models can also be ratherinexpensive, low-hazard and space-saving laboratoriesin which a good understanding of processes can supportmodel experiments. This approach can be particularlyimportant where the building of hardware laboratories(or hardware models) would be too expensive, toohazardous or not possible (in the case of climate-systemexperiments, for example). Of course, the outcomeof any model experiment is only as good as theunderstanding summarized within the model and thususing models as laboratories can be a risky businesscompared with hardware experimentation. The morephysically based the model (i.e. the more based onproven physical principles), the better in this regardand indeed the most common applications of models aslaboratories are in intensively studied fields in which thephysics are fairly well understood such as computationalfluid dynamics (CFD) or in areas where a laboratorycould never be built to do the job (climate-systemmodelling, global vegetation modelling).5. As an integrator within and between disciplines. Aswe will see in Chapters 14 and 15, models also have theability to integrate the work of many research groupsinto a single working product which can summarize theunderstanding gained as well as, and sometimes muchbetter than, traditional paper-based publications. Under-standing environmental processes at the level of detailrequired to contribute to the management of chang-ing environments requires intensive specialization byindividual scientists at the same time as the need toapproach environmental research in an increasingly mul-tidisciplinary way. These two can be quite incompati-ble. Because of these two requirements, and because offunding pressures in this direction, scientific researchis increasingly a collaborative process whereby largegrants fund integrated analysis of a particular envi-ronmental issue by tens or hundreds of researchersfrom different scientific fields, departments, institutions,countries and continents working together and hav-ing to produce useful and consensus outcomes, some-times after only three years. This approach of big sci-ence is particularly clear if we look at the scientificapproach of the large UN conventions: climate change,biological diversity, desertification and is also evident

12 Mark Mulligan and John Wainwright

in the increasingly numerous authorship on individualscientific papers.

Where archaeologists work with hydrologists work-ing with climate scientists working with ecologists andpolitical scientists, the formal language of mathematicsand the formal data and syntax requirements of modelscan provide a very useful language of communication.Where the research group can define a very clear pictureof the system under study and its subcomponents, eachcontributing group can be held responsible for the pro-duction of algorithms and data for a number of subcom-ponents and a team of scientific integrators is chargedwith the task of making sure all the subcomponents ormodules work together at the end of the day. A teamof technical integrators is then charged with making thismathematical construct operate in software. In this waythe knowledge and data gained by each group are tightlyintegrated where the worth of the sum becomes muchmore than the worth of its individual parts.6. As a research product. Just as publications, websites,data sets, inventions and patents are valid researchproducts, so are models, particularly when they canbe used by others and thus either provide the basisfor further research or act as a tool in practicalenvironmental problem solving or consultancy. Equally,models can carry forward entrenched ideas and canset the agenda for future research, even if the modelsthemselves have not been demonstrated to be sound.The power of models as research products can be seenin the wealth of low cost publicly available modelsespecially on the online repositories of models heldat the CAMASE Register of Agro-ecosystems Models(http://www.bib.wau.nl/camase/srch-cms.html) and theRegister of Ecological Models (http://eco.wiz.uni-kassel.de/ecobas.html). Furthermore, a year-by-yearanalysis of the number of English language academicresearch papers using one prominent, publicly availablehydrological model (TOPMODEL) indicates the amountof research that can stem from models. Accordingto ISI, from 1991 to 2002 inclusive, some 143scientific papers were published using TOPMODEL,amounting to more than 20 per year from 1997 to2002 (this figure therefore does not include the rashof papers in conference proceedings and other nonpeer-reviewed publications). Models can also be veryexpensive ‘inventions’, marketed to specialist marketsin government, consultancy and academia, sometimespaying all the research costs required to produce them,often paying part of the costs.7. As a means of communicating science and the resultsof science. To write up science as an academic paper,in most cases, confines it to a small readership and to

fewer still users. To add the science as a componentto a working model can increase its use outside theresearch group that developed it. In this way, models canmake science and the results of science more accessibleboth for research and for education. Models can bemuch more effective communicators of science because,unlike the written word, they can be interactive andtheir representation of results is very often graphical ormoving graphical. If a picture saves a thousand words,then a movie may save a million and in this wayvery complex science can be hidden and its outcomescommunicated easily (but see the discussion below onthe disadvantages in this approach).

Points 1–4 above can be applied to most models while5, 6 and 7 apply particularly to models that are interface-centred and focused on use by end users who are not themodellers themselves. In environmental science thesetypes of model are usually applied within the context ofpolicy and may be called ‘policy models’ that can beused by policy advisers during the decision-making orpolicy-making process. They thus support decisions andcould also be called decision-support systems (DSS: seeChapters 14 and 15) and which perform the same taskas do policy documents, which communicate researchresults. The hope is that policy models are capableof doing this better, particularly in the context ofthe reality of scientific uncertainty compared with themyth that policy can be confidently based on exact,hard science (Sarewitz, 1996). The worry is that theextra uncertainties involved in modelling processes(parameter uncertainty, model uncertainty, predictionuncertainty) mean that although models may be goodcommunicators, what they have to communicate can berather weak, and, worse still, these weaknesses may notbe apparent to the user of the model output who maysee them as prophecy, to the detriment of both scienceand policy. There is no clear boundary between policymodels and nonpolicy (scientific models) but policymodels or DSS in general tend to focus on 5, 6 and7 much more than purely scientific models, which areusually only used by the model builder and few others.We will see later how the requirements of research andpolicy models differ.

1.2.1 Models with a poorly defined purpose

We have seen why modelling is important and howmodels may be used but before building or using amodel we must clearly define its purpose. There areno generic models and models without a purpose aremodels that will find little use, or worse, if they do find

Modelling and Model Building 13

use, they will often be inappropriate for the task at hand.In defining the purpose of a model, one must first clearlydefine the purpose of the research: what is the problembeing addressed?; what are the processes involved?; whoare the stakeholders?; what is the physical boundary ofthe problem and what flows cross it from outside of themodelling domain?; over which timescale should theproblem be addressed?; and what are the appropriatelevels of detail in time and space?

Here, the research question is the horse and the modelthe cart. The focus of all activities should be to answerthe research question, not necessarily to parameterizeor produce a better model. Once the research hasbeen defined, one must ask whether modelling is theappropriate tool to use and, if so, what type of model.Then follows the process of abstracting reality anddefining the model itself.

1.3 TYPES OF MODEL

Models are by no means a new tool in the scientists’toolbox. Environmental scientists have used spatial mod-els of populations, environments, infrastructures, geolo-gies and geographies in the form of maps and drawingsfor as long as science itself. Maps and drawings areabstractions of the form of nature in the same waythat models are (usually) abstractions of the process ofnature. Mathematics has its origins in the ancient Orientwhere it developed as a practical science to enable agri-culture and agricultural engineering through the devel-opment of a usable calendar, a system of mensurationand tools for surveying and design. With the ancientGreeks, mathematics became more abstract and focusedmuch more on deduction and reasoning. Mathematicalmodels have been developed since the origin of mathe-matics, but there was a significant increase in modellingactivity since the development of calculus by Newtonand Leibniz working independently in the second halfof the seventeenth century. Cross and Moscardini (1985)define three ages of modelling: (1) the ‘Genius Age’;(2) the ‘Transitional Age’; and (3) the ‘ContemporaryAge’. The ‘Genius Age’ followed the development ofcalculus and is characterized by the development ofmodels of complex physical phenomena such as thoseof gravitation by Newton, of electromagnetic waves byClerk Maxwell (of our own university) and of relativityby Einstein. Modelling in the ‘Genius Age’ was alwayslimited by the need to produce analytical solutions tothe set of equations developed. Cross and Moscardini’s‘Transitional Age’ was initiated by the availability ofmechanical and then electromechanical aids to arith-metic but these devices were expensive, difficult to use

and slow. The development of increasingly inexpensivecomputer power, a bank of numerical techniques thatcan yield accurate solutions to most equation sets andthe softening of the human–computer communicationbarrier through the development of personal comput-ers (PCs) and high-level programming languages havemoved the ‘Transitional Age’ into the ‘ContemporaryAge’. Thus, modern numerical computer models can beseen rather simply as the continuation of the relation-ship between science and mathematics with the greatersophistication afforded by advances in the power ofcomputer input, processing and display. No-one knowswhat will come next but we make some educatedguesses in the last chapters of this book.

Models can be classified hierarchically. The two top-level model types are the mathematical models andthe physical or hardware models (not to be confusedwith physically based, mathematical models). Hardwaremodels are scaled-down versions of real-world situa-tions and are used where mathematical models wouldbe too complex, too uncertain or not possible because oflack of knowledge. Examples include laboratory chan-nel flumes, wind tunnels, free atmosphere CO2 enrich-ment apparatus, rhizotrons and lysimeters, and the BIO-SPHERE 2 laboratory (Cohn, 2002). Many instrumentsare also hardware models which allow control of someenvironmental conditions and the measurement of theresponse of some system to these controls. The Parkin-son leaf chamber which forms the basis of most leafphotosynthesis systems is a good example. The cham-ber is a small chamber (usually 5 cm3) which is clampedonto a leaf in vivo and controls the input humidityand CO2 concentration and measures the output ofthe same so that photosynthesis and transpiration canbe measured.

Hardware models are usually small-scale comparedwith the systems which they simulate and their resultsmay be prone to uncertainty resulting from scale effectswhich have to be balanced against the increased costof larger-scale hardware models. Hardware models arealso expensive. The 1.27 ha BIOSPHERE 2 (BIO-SPHERE 1 is, apparently, the Earth) experiment in theSanta Catalina Hills, near Oracle, Arizona, USA, is nowused to test how tropical forest, ocean and desert ecosys-tems might respond to rising CO2 and climate changeand cost some US$150 million (UK-based readers maywant to visit the Eden project near St Austell, Corn-wall, to get an impression of a similar physical model[www.edenproject.com]). Most physical models are con-siderably less ambitious and thus much cheaper butthis does, nevertheless, give an indication of the kindof costs that are required to ‘simulate’ ecosystems in

14 Mark Mulligan and John Wainwright

hardware. Hardware models give a degree of controlon the systems under investigation, in the case of BIO-SPHERE 2, CO2, water and nutrient inputs can be con-trolled. There is, however, always the problem that thehardware representation of a process is only as good asour understanding of that process and our ability to repli-cate it: BIOSPHERE 2 cannot simulate storms with highwinds or damaging downpours or the diversity of soilcomplexes that exist in the rainforest which it simulates.This may render the BIOSPHERE 2 rainforest responseto climate change rather simplistic compared with thefield reality. Furthermore, the structure of the hardwaremodel may also interfere with natural processes in theenvironment, for example, the glass windows of BIO-SPHERE 2 cut out some 50% of the incoming light.Because of the cost involved, it is also usually diffi-cult to replicate hardware experiments: there is only oneBIOSPHERE 2 and laboratories usually only have onewind tunnel or channel flume. Nevertheless hardwaremodels couple the scientific rigour of observation withthe controllability of mathematical modelling. For manyapplications, it is only logistic difficulties and cost whichkeep them as a relatively uncommon approach to mod-elling: physical models require a great deal more set-upand maintenance costs than software and data.

Mathematical models are much more common andrepresent states and rates of change according to for-mally expressed mathematical rules. Mathematical mod-els can range from simple equations through to complexsoftware codes applying many equations and rules overtime and space discretizations. One can further definemathematical models into different types but most mod-els are actually mixtures of many types or are transi-tional between types. One might separate mathematicalmodels into empirical, conceptual or physically based:

• Empirical models describe observed behaviourbetween variables on the basis of observations aloneand say nothing of process. They are usually thesimplest mathematical function, which adequatelyfits the observed relationship between variables. Nophysical laws or assumptions about the relationshipsbetween variables are required. Empirical modelshave high predictive power but low explanatory depth,they are thus rather specific to the conditions underwhich data were collected and cannot be generalizedeasily for application to other conditions (i.e. othercatchments, other forests, other latitudes).

• Conceptual models explain the same behaviour onthe basis of preconceived notions of how the sys-tem works in addition to the parameter values,which describe the observed relationship between the

variables. A conceptual model of hillslope hydrologymay separately model processes of surface runoff,subsurface matrix quickflow and subsurface pipeflow(see Chapter 4). While all these models are empiri-cal, their separation incorporates some process under-standing. Conceptual models have slightly greaterexplanatory depth but are as nongeneralizable as theempirical models which make them up.

• Physically based models should be derived deduc-tively from established physical principles andproduce results that are consistent with observa-tions (Beven, 2002) but in reality physically basedmodels often do one of these but rarely both. In gen-eral use, there is a continuum of models that fallsbroadly under the heading of physically based, butthat might include some level of empirical generaliza-tion in order to allow them to operate at an appropriateenvironmental scale, or to fill gaps where the physicsis not known. Process models emphasize the impor-tance of the processes transforming input to output.In some respects, this tradition may arise from thelink between the study of process and form withinthe discipline of geomorphology. Similarly, ecolo-gists often talk of mechanistic models. It may bethat the choice of terminology (relating to physics ormechanics) represents a desire to suggest the scienceunderlying the model has a sound basis. Physicallybased models tend to have good explanatory depth(i.e. it is possible to interrogate the model to findout exactly why, in process terms, an outcome is asit is). On the other hand, physically based modelsare characterized by low predictive power: they oftendo not agree with observations. This lack of agree-ment often demonstrates a poor understanding of thephysics of the system. These models thus often needto be calibrated against observations (see Chapter 4,for example). One has to think carefully about theexplanatory depth of a model that does not replicatethe observed reality well (Beven, 2001). Where theyare not highly calibrated to observed data and if theyare appropriately and flexibly structured, physicallybased models offer a greater degree of generality thanempirical models.

According to the level of process detail and understand-ing within the model, it may be termed black box orwhite box. In a black box model, only the input andoutput are known and no details on the processes whichtransform input to output are specified, and the trans-formation is simulated as a parameter or parametersdefining the relationship of output to input. On the otherhand, in a white box model, all elements of the physical

Modelling and Model Building 15

processes transforming input to output are known andspecified. There are very few systems for which whitebox models can be built and so most models, being amixture of physically based and empirical approaches,fall in between white and black to form various shadesof grey boxes. Empirical models are usually closer to theblack box end of the spectrum while physically basedmodels fall in between this and the white box, dependingon their detail and the extent to which they are calibratedto observed data.

There are no universally accepted typologies of mod-els and, given the diversity of approaches apparent ineven a single model code in the multi-process mod-els, which are increasingly common, there is little pointin specifying one. Nevertheless, it is useful to under-stand the properties according to which models maybe classified. We have already discussed the differenttypes of model (empirical, conceptual, physically based).Models can be further subdivided according to howthe equations are integrated (either analytically solv-ing the model equations as differential equations ornumerically solving them within a computer as differ-ence equations, see this chapter). Further subdivisioncan take place according to the mathematics of themodel, for example, whether the equations are deter-ministic, that is, a single set of inputs always pro-duces one (and the same) output. In the alternative,stochastic approach, a single set of inputs can producevery different outputs according to some random pro-cesses within the model. Up to this point most mod-els are still mixtures of many of these types, thoughtwo further properties are still to be specified. Modelsare of different spatial types. Lumped models simu-late a (potentially) spatially heterogeneous environmentas a single – lumped – value. Semi-distributed modelsmay have multiple lumps representing clearly identifi-able units such as catchments. Distributed models breakspace into discrete units, usually square cells (rasters) ortriangular irregular networks (TINs, e.g. Goodrich et al.,1991) or irregular objects. The spatial realm of a modelmay be one-dimensional, two-dimensional (sometimeswithin the context of a geographical information systemor GIS) and, sometimes, three-dimensional. All mod-els are lumped at the scale of the cell or triangle andbecause the sophistication of modelling techniques isway ahead of the sophistication of measurement tech-niques, data limitations mean that most distributed mod-els use lumped data. Finally, one has to consider themanner in which the model handles time. Static modelsexclude time whereas dynamic ones include it explic-itly. A summary of the potential means of classifyingmodels is given below:

Conceptual type: empirical, conceptual, physicallybased or mixed

Integration type: analytical, numerical or mixedMathematical type: deterministic or stochastic or mixedSpatial type: lumped, semi-distributed, distributed,

GIS, 2D, 3D or mixedTemporal type: static, dynamic or mixed.

1.4 MODEL STRUCTURE AND FORMULATION

1.4.1 Systems

Contemporary mathematical modelling owes much tosystems thinking and systems analysis. A system is aset of inter-related components and the relationshipsbetween them. Systems analysis is the ‘study of the com-position and functioning of systems’ (Huggett, 1980). Inpractice, systems analysis involves the breaking downor modularization of complexity into simple manage-able subsystems connected by flows of causality, matter,energy or information. The purpose of systems analy-sis is to make complex systems more easily understood.Systems usually comprise of compartments or stores thatrepresent quantities such as height (m), mass (kg), vol-ume (m3), temperature (◦C), annual evaporation (mm)and which are added to or subtracted from by flows orfluxes such as height increment (m a−1), weight gain (kga−1), volume increment (m3 a−1) and evaporation (mmmonth−1). Further details on systems analysis can befound in Chorley and Kennedy (1971), Huggett (1980)and Hardisty et al. (1993).

In modelling a variable is a value that changes freelyin time and space (a compartment or flow) and a statevariable is one which represents a state (compartment).A constant is an entity that does not vary with the systemunder study, for example, acceleration due to gravity isa constant in most Earth-based environmental models(but not in geophysics models looking at gravitationalanomalies, for example). A parameter is a value whichis constant in the case concerned but may vary from caseto case where a case can represent a different model runor different grid cells or objects within the same model.

1.4.2 Assumptions

In order to abstract a model from reality, a set of assump-tions has to be made. Some of these assumptions willbe wrong and known to be so but are necessary forthe process of abstraction. The key to successful mod-elling is to know which assumptions are likely to bewrong and to ensure that they are not important forthe purpose for which the model is intended. Further,

16 Mark Mulligan and John Wainwright

one should only use the model for that purpose andensure that no-one else uses the model for purposeswhich render incorrect assumptions significant or cor-rect assumptions invalid. The value of a model dependstotally on the validity and scope of these assumptions.These assumptions must be well understood and explic-itly stated with reference to the conditions under whichthey are valid and, more importantly, the conditionsunder which they are invalidated. Abstraction shouldalways be guided by the principle of parsimony. Perrinet al. (2001) indicate that simple models (with few opti-mized parameters) can achieve almost as high a levelof performance as more complex models in terms ofsimulating their target variable. Although the additionof model parameters and more detailed process descrip-tion may have benefits from a theoretical point of view,they are unlikely to add greatly to model predictivecapability even if substantial data resources are avail-able to keep the uncertainty in these parameters to aminimum (de Wit and Pebesma, 2001). The greater thenumber of parameters, the greater the likelihood thatthey will be cross-correlated and that each extra param-eter will add relatively little to the explanation (van derPerk, 1997). Perrin et al. (2001) concur with Steefel andvan Cappellen (1998) who indicate, for models withequal performance, that the best model is the simplestone. Simplicity must be strived for, but not at the cost ofmodel performance. In this way building models is bestachieved by starting with the simplest possible structureand gradually and accurately increasing the complex-ity as needed to improve model performance (see Nashand Sutcliffe, 1970). Figures 11.3–11.5 (in Chapter 11)indicate the manner in which model performance, modelcomplexity, the costs (in time and resources) of modelbuilding and the uncertainty of results can be related.

1.4.3 Setting the boundaries

We have now defined the problem and agreed thatmodelling is part of the solution. Further, we know whatthe purpose of the model is and can thus define whatkind of model is appropriate. The next task is to beginbuilding the model. We must first set the boundariesin time and space to identify which times and spaceswill be modelled and which must be supplied as data.We call these data the boundary conditions for data,representing processes outside the spatial domain of themodel and the initial conditions for data, representingprocesses internal to the model spatial domain butexternal (before) the model temporal domain. Modelresults, from those of planetary rotation (Del Genio,1996), through river flooding (e.g. Bates and Anderson,

1996) to moth distributions (Wilder, 2001) are usuallysensitive to the initial and boundary conditions sothese must be carefully specified. In the case of ageneral circulation model of the atmosphere (GCM),the boundary is usually the top of the atmosphere andthus one of the variables which must be supplied as aboundary condition, because it is not modelled withinthe GCM, is the incoming solar radiation flux. It isimportant to mention that boundary conditions mayexist outside the conceptual space of a model even ifthey are inside the physical space of the same. Forexample, until recently, global vegetation cover wasnot simulated within GCMs and thus, despite the factthat it resides within the spatial domain of GCMs, ithad to be specified as a boundary condition for alltime steps (and also as an initial condition for thefirst time step). Nowadays many GCMs incorporate aninteractive vegetation modelling scheme so this is nolonger necessary. GCMs have to be supplied with aninitial condition sea surface temperature (SST) field foreach sea surface grid cell (GCMs are usually 3D rasterdistributed) for time zero after which the model willcalculate the SST for each timestep.

Let us consider a simple model of soil erosion. Theobjective of the model is for us to understand more ofthe relative importance of climatic, landscape and plantfactors on soil erosion. Specifically, we will simulatewash erosion (E, variable over space and time) which issimulated on the basis of runoff (Q, variable over spaceand time), slope gradient (S, variable over space constantover time), vegetation cover (V , state variable overspace and time) and the erodibility (K , variable overspace constant in time) of the soil and three parameters,m,n and i. Wash erosion is soil erosion by water runningacross the land surface and, after Thornes (1990) themodel can be expressed as:

E = kQmSne−iV (1.1)

whereE = erosion (mm month−1)k = soil erodibility

Q = overland flow (mm month−1)m = flow power coefficient (1.66)S = tangent of slope (mm−1)n = slope constant (2.0)V = vegetation cover (%)i = vegetation erosion exponential function

(dimensionless).

In this way, soil erosion is a function of the erodibilityof the soil (usually controlled by its organic mattercontent, structure, texture and moisture), the ‘stream

Modelling and Model Building 17

power’ of runoff (Qm), the slope angle effect (Sn) andthe protection afforded by vegetation cover (e−0.07V ).Let us say that it is a distributed model runningover 25-metre raster cells and at a time resolutionof one month for 50 years. Erosion and runoff arefluxes (flows), vegetation, slope gradient and erodibilityare states (compartments). Since the model is of soilerosion, it does not simulate vegetation growth norrunoff generation, hence these are outside the boundaryof the model and they must be specified for eachtimestep as a boundary condition. No initial conditionsneed be specified since the only state variable, vegetationcover, is already specified as a boundary condition. If wewere to calculate the change in soil thickness (Z, statevariable) according to erosion, then we must specify aninitial condition for Z.

1.4.4 Conceptualizing the system

The understanding gained from being forced to ratio-nalize one’s conceptual view of a process or systemand quantify the influence of each major factor is oftenthe single most important benefit of building a math-ematical model (Cross and Moscardini, 1985). If themodel results match reality, then this conceptual viewis supported and, if not, then the conceptualization maybe, partly or wholly, at fault and should be reviewed.Different people will produce quite different concep-tualizations of the same system depending upon theirown background and experience: to a climate scientista forest is a surface cover interacting with the atmo-sphere and land surface and affecting processes suchas the CO2 concentration of the atmosphere and theenergy and water fluxes across the atmospheric boundarylayer. To a forester a forest is a mass of wood-producingtrees of different ages, sizes and monetary values. Crossand Moscardini (1985) specify five stages of mathemat-ical modelling: problem identification, gestation, modelbuilding, simulation and pay-off. These stages are takensequentially, although one may move back to earlierstages at any time as needed.

Problem identification is a fairly obvious first stage.If the problem is not properly identified, then it willnot be possible to arrive at a solution through mod-elling or any other means. The gestation stage is animportant, though often neglected, stage consisting ofthe gathering of background information, amassing anunderstanding of the system under investigation, separat-ing relevant information from irrelevant and becomingfamiliar with the whole context of the problem. Thetwo substages of gestation may be considered as mod-ularization and reviewing the science. Modularization

breaks the problem down into solvable chunks in themanner of systems analysis. A clear mental map of theprocesses involved will help very much in the separa-tion of big complex processes into families of small andsolvable self-contained modules. Once modularized, theexisting science and understanding of each process mustbe reviewed, allowing abstraction of the relevant fromthe irrelevant. On departing from this stage the mod-eller should have a good conceptual understanding ofthe problem, its context and how it will be solved. Thisis where the process of abstraction is most important andthe modeller’s intuition as to what is and is not impor-tant will be most valuable. Having devised an acceptableconceptual framework and a suitable data subset, theformulation of the mathematical model is usually fairlystraightforward. The process of model building incorpo-rates a further three substages: developing the modules,testing the modules and verifying the modules. Devel-oping the modules will often involve some re-use ofexisting models, some routine model development andsome flashes of inspiration. It is important that, evenat this early stage, the modules are tested so that thesolution developed has a reasonable chance of produc-ing consistent results and not, for example, negativemasses, humidities or heights. In addition to defining themodel itself, one will also have to, at this stage, givesome thought to the method of solution of the modelequations. Before detailing the numerical methods avail-able for this, we will outline in some detail the practicalaspects of putting a simple model together.

1.4.5 Model building

Model building may consist of stringing together setsof equations to which an analytical solution will bederived, but more likely these days it will involve com-partmentalization of the problem and its specification aseither compartments and flows within a graphical modelbuilding environment such as STELLA (http://hps-inc.com), VENSIM (http://www.vensim.com), Power-Sim (http://www.powersim.com), ModelMaker (http://www.cherwell.com), SIMULINK, the graphical mod-elling environment of MATLAB or SIMILE (http://www.simulistics.com) or as routines and procedures ina high level computer programming language such asBASIC, FORTRAN, Pascal, C++ or Java, or a custommodelling language such as PCRASTER (http://www.pcraster.nl).

Some modellers prefer to build their models graph-ically by adding compartments and flows, linkingthem with dependencies and entering the appropri-ate equations into the relevant compartments, flows

18 Mark Mulligan and John Wainwright

or variables. Indeed, this approach is fairly closeto systems analysis and the way that many non-programmers conceptualize their models. Others, usu-ally those who are well initiated in the art of com-puter programming, prefer to construct their modelsin code, or indeed even in the pseudo-code whichmodern spreadsheet programs allow for the solutionof equations as reviewed extensively by Hardisty et al.(1994). By way of an introduction to model build-ing let us look back on the soil-erosion equation weintroduced a few pages back and examine the man-ner in which this could be constructed in (a) a graph-ical model building program, in this case SIMILEwhich is produced by the Edinburgh-based simulis-tics.com and is currently distributed free of charge foreducational use from http://www.simulistics.com; (b) aspreadsheet program, in this case Microsoft Excel, avail-able from http://www.microsoft.com; and (c) a spatialmodel building language called PCRASTER producedby the University of Utrecht in The Netherlands anddistributed free of charge for educational use fromhttp://www.pcraster.nl. For simplicity we will keep themodel simple and nonspatial in the SIMILE and Excelimplementation and make it spatial in the PCRASTERimplementation.

Though the graphical user interface (GUI) and syntaxof these specific software tools will not be identical toany others that you may find, the basic principles ofworking with them will be similar and so this exerciseshould also prepare you for work using other tools.There is not space here to enter into the complexities ofhigh-level programming syntax but suffice to say that,for a modeller, knowing a programming language – andthey all share the same basic constructs – is a veryuseful but not indispensable skill to have mastered.Coding in a high-level language does allow moreefficient models to be developed in terms of the timethey take to run (excluding BASIC and Java from theabove list because they interpret commands just likePCRASTER) and the memory they occupy. Coding alsosidesteps the limitations of software tools by giving theprogrammer full, unlimited access to the computer’sfunctions.

1.4.6 Modelling in graphical model-building tools:using SIMILE

SIMILE is a new addition to the suite of graphicalsystems-modelling tools and is attractive because of itslow cost, comprehensive online documentation and pow-erful modelling capabilities. The construction of a modelin SIMILE is achieved first through the specification

of compartments and the flow variables between themand subsequently by the specification of the parametersthat affect the flow variables and the direction of theinfluence they have. The SIMILE interface is as shownin Figure 1.1. The modelling canvas is initially blankand the modeller adds compartments, flows, parameters,variables and influences using the toolbar short cuts.These are then given a label and are populated with therelevant values or equations through the equation barabove the model canvas, which becomes active when acompartment, flow or variable is clicked with the com-puter mouse or other pointing device. The model shownon the canvas is our soil-erosion model but do not looktoo closely at this yet as we will now attempt to con-struct it.

The first step is to define the compartment or statevariable. In this case of soil erosion, this could be soilthickness so we will label it as such. Note that thesymbols that SIMILE produces are standard systems-analysis symbols. Now let’s add the flow (Figure 1.2).This is, of course, soil erosion that is a flow out ofthe soil-thickness compartment and not into it because,for simplicity, we are only simulating erosion and notdeposition. The flow is given the label E and an influ-ence arrow must be drawn from E to the compartmentwhich represents soil thickness since erosion affects soilthickness (note that we use a bold symbol to denotethe parameter as used in the model formalization com-pared to the italicized symbol when discussing themodel form in an equation). Note that until all influ-ences and equations are fully specified, the flow andits associated arrows remain red. On full parameteri-zation all model components will be black in colour.We cannot now specify the equation for erosion untilwe have fully specified all of the parameters and vari-ables that influence it. We therefore add the variables:Q, k, s, m, n, i and V (Figure 1.3) and draw influencearrows from each of them to soil erosion (Figure 1.4)since they all affect soil erosion. We can now spec-ify the parameter values of each of Q, k, s, m, n, iand V either as (fixed) parameters or as variables whichare time dependent and either calculated in SIMILE orread from an input file or is the outcome of anotherequation or submodel in SIMILE. To keep things sim-ple we will enter these values as constants, Q = 100(mm month−1), k = 0.2, s = 0.5, m = 1.66, n = 2.0,i = −0.07 and V = 30(%). Finally, we can click onthe flow, E and enter the equation (from Equation 1.1),which determines the value of E as a function of Q, k, s,m, n, i, V (see the equation bar in Figure 1.1). Note thatin the SIMILE syntax we only enter the right-hand sideof the equals sign in any equation and so Equation 1.1

Modelling and Model Building 19

Figure 1.1 Interface of the SIMILE modelling software

Soil thickness E

Figure 1.2 Adding the flow component to the SIMILEmodel developed in the text

Soil thickness

Qk s

E

m n V

i

Figure 1.3 The SIMILE erosion model with the variables(Q, k, s, m, n, i, V) added

becomes k*pow(Q,m)*pow(s,n)*exp(i*V) with pow(x)representing to the power of x and exp(x) representing

e (the base of natural logarithms) to the power of x. Wemust also specify the initial condition for soil thickness,in mm because E is also in mm. Let’s say the initialthickness is 10 m (10000 mm). All that remains is tobuild or compile the model allowing SIMILE to convertthis graphic representation to a pseudo-code which isthen interpreted into machine code at runtime.

To run the model we need to specify the number andfrequency of timesteps in the runtime environment andthen run the model. Output is logged for each timestepand the user can examine the model output variable byvariable, graphically or as text. If checked and working,this model can be wrapped and used as a componentof a larger, more complex model. Many of the graph-ical modelling tools also have helpers or wizards forperforming model calibration, optimization and sensi-tivity analysis. The great advantage of graphical model-building tools is the rapidity with which they can belearned and the ease with which even very complex sys-tems can be represented. Their disadvantages are thatthey are generally expensive and it can be rather dif-ficult to do more advanced modelling since they are

20 Mark Mulligan and John Wainwright

Figure 1.4 Influences between the variables and the flow for the SIMILE soil-erosion model

rather more constrained than modelling in code. If youwant to look into it further, the online help and tutorialfiles of SIMILE provide much greater detail on the morepowerful aspects of its functionality including its abilityto deal with many (spatial) instances of a model rep-resenting for example, individual hillslopes or plants orcatchments. You can download this erosion model fromthe companion website for this book.

1.4.7 Modelling in spreadsheets: using Excel

A very detailed training for modelling in Excel isgiven in Hardisty et al. (1994), so we only provide asimple example here. To build the same soil-erosionmodel in a spreadsheet such as Microsoft Excel requiresthe specification of the time interval and number oftimesteps upfront since these will be entered directly intothe first column of the spreadsheet. Open the spreadsheetand label column A as Timestep at position A:2 in thespreadsheet (column A, row 2). Then populate the next30 rows of column A with the numbers 1 to 30. Thiscan be easily achieved by adding the number 1 to A:3.At position A:4 type the following equation: =A3+1

and press enter. The spreadsheet will now calculatethe results of this equation which will appear as thenumber in A:4. We can now highlight cells A:4 throughto A:32 with our pointing device (left click and dragdown) and go to the menu item Edit and the Fill andthen Down (or Ctrl-D) to copy this equation to all thecells in the highlighted range. Note that in each cell,the cell identifier to the left of the plus sign in theequation (which for the active cell can be viewed inthe equation bar of Excel) is changed to represent thecell above it so that in A:4 it is A:3 but in A:5 it is A:4and so on. This is relative referencing which is a veryuseful feature of Excel as far as modelling is concernedbecause it allows us to define variables, i.e. parameterswhich change with time and time, which are usuallyrepresented along the rows of the spreadsheet or evenon another spreadsheet. Model instances representingdifferent locations or objects are usually representedacross the columns of the spreadsheet, as are thedifferent compartment, flow and variable equations.

If we want to avoid relative referencing in order tospecify a constant or parameter (rather than a variable),

Modelling and Model Building 21

we can enter it into a particular cell and use absolutereferencing to access it from elsewhere on the spread-sheet. In absolute referencing a $ must be placed infront of the identifier which we do not want to changewhen the equation is filled down or across. To spec-ify a parameter which remains constant in time (downthe spreadsheet) we would place the $ before the rowidentifier, for example, =A$3+1, to specify a parame-ter which remains constant across a number of modelinstances (across the spreadsheet) we place the $ beforethe column identifier, =$A3+1. To specify a constantthat changes nowhere on the spreadsheet, then we use forexample =$A$3+1. This form always refers to a singleposition on the spreadsheet, whenever the equation isfilled to.

Going back to our soil-erosion model, we have anumber of model parameters which will not changein time or space for this implementation and thesecan be sensibly placed in row 1, which we have leftuncluttered for this purpose. We can label each parameterusing the cell to the left of it and enter its value

directly in the cell. So let’s label Q, k, s, m, n,i and V and give them appropriate values, Q = 100(mm month−1), k = 0.2, s = 0.5, m = 1.66, n = 2.0,i = −0.07 and V = 30(%). See Figure 1.5 for the layout.Now that we have specified all of the parameters wecan enter the soil erosion equation in cell B:3 (weshould place the label Soil Erosion (mm month−1) inB:2). We will use absolute referencing to ensure thatour parameters are properly referenced for all timesteps.In column D we can now specify soil thickness inmetres so let us label C:2 with ‘Soil Thickness (m)’and specify an initial condition for soil thickness, of say10 m, in cell D:2. In the syntax of Excel Equation 1.1becomes =E$1*(C$1ˆI$1)*(G$1ˆK$1)*EXP(M$1*O$1)where E$1 holds the value of k, C$1 holds the valueof Q, I$1 holds the value of m, G$1 holds the valueof s, K$1 holds the value of n, M$1 holds the valueof i and O$1 holds the value of V. In Excel a caret(ˆ) represents ‘to the power of’ and EXP(x) representse (the base of natural logarithms) to the power of x.On pressing enter, this equation will produce a value of

Figure 1.5 Layout of the soil-erosion model in Excel

22 Mark Mulligan and John Wainwright

12.7923 (mm month−1) of erosion. If we fill this downall 30 months then we find that the value remains thesame for all months since no inputs vary over time.We can now update the soil thickness for each timestepby removing the soil eroded each timestep from thesoil thickness at the beginning of that timestep. To dothis type =D2-(B3/1000) into D:3, where D:2 is theinitial soil thickness and B:3 is the soil erosion duringthe first timestep. This produces a soil thickness at theend of the first month of 9.987 metres. Fill this downall 30 months and you will see the soil thickness after30 months has fallen to 9.616 metres. Note that, becauseyou have used relative referencing here, each month thenew soil thickness is arrived at by taking the soil erosionfor the month from the soil thickness at the end of theprevious month (the row above). Why not now add acouple of line graphs indicating the evolution of soilerosion and soil thickness over time as in Figure 1.5?So, we have a model but not a very interesting one. Soilerosion is a constant 12.79 mm month−1 and as a resultsoil thickness decreases constantly over time (note thatfor illustration purposes this rate is very high and wouldrapidly lead to total soil impoverishment unless it werereplaced by weathering at least at the same rate). We cannow change the values of the model parameters to viewthe effect on soil erosion and soil thickness. Changingtheir value causes a shift in the magnitude of soil erosionand soil thickness but no change in its temporal pattern.

We might make the model a little more interesting byspecifying Q as a variable over time. We can do this bycreating a new column of Qs for each timestep. Put thelabel ‘Qvar’ in cell E:2 and in E:3 type =rand()*100and fill this down to E:32. This will generate a randomQ between 0 and 100 for each timestep. Now go backto the erosion equation in B:3 and replace the referenceto C$1, the old Q with a relative reference to your newQvar at E:3. Highlight B3:B32 and fill the equationdown. You will see immediately from the updated graphsthat erosion is now much more variable in response tothe changing Q and, as a result, soil thickness changesat different rates from month to month. We can alsoadd a feedback process to the model by decreasingthe erodibility as soil erosion proceeds. This feedbackreflects the field observation that soil erosion takesaway the fine particles first and leads to the armouringof soils thus reducing further erosion. To do this wewill first reduce the soil thickness to 0.5 metres atD:2. We will then specify a new variable k in columnF. Label F:2 with the label ‘k(var)’ and in F:3 add=E$1*(D2/D$2). This modification will ensure that assoil thickness reaches zero so will erodibility, whichis handy because it will also ensure that your model

does not produce negative erosion if left to run forlong enough. Erodibility will decrease linearly with theratio of current to original soil thickness. It is importantto note that the erodibility at time t is calculated onthe basis on the soil thickness at the end of timestept − 1. For the first timestep that means that erodibilityis calculated on the basis of the initial soil thickness. Ifthis were not the case, then when we come to updatesoil erosion at B:3 by changing the reference to k fromE$1 to F:3, we would produce a circularity because soilerosion at t1 would depend upon erodibility at time t1which would, in turn, depend upon soil erosion at timet1 and so the calculations would be circular and couldnot be solved using Excel (see the section on iterativemethods below). When you have changed the referenceto k at B:3 from E$1 to F:3 and filled this down toB:32, you should notice that the rate of soil erosion,though still very much affected by the random rate ofrunoff, declines over time in response to the decliningerodibility. This will be particularly apparent if youextend the time series by filling down A32 through toF32 until they reach A92 through to F92 (you will haveto re-draw your graphs to see this). On the other hand,you could simply increase the lid on the random runoffgeneration to 200 at cell E:3 and fill this down. Theincreased rate of erosion will mean that the negativefeedback of decreased erodibility kicks in sooner. Sonow the model is showing some interesting dynamics,you could go on to make the vegetation cover respondto soil thickness and/or make the slope angle respondto erosion. You now have the basics of modelling inspreadsheets.

The great advantage of spreadsheet modelling isafforded by the ease with which equations can beentered and basic models created and integrated withdata, which usually reside in spreadsheets anyway.Furthermore, modelling in spreadsheets allows veryrapid production and review of results and very rapidanalysis of model sensitivity to parameter changes. It istherefore very useful, at least at the exploratory stage ofmodel development. Most computer users will have usedspreadsheet software and it is fairly widely availablesince it is not specialist software like the graphicalmodel building tools or the modelling languages. Youcan download this model from the companion websitefor this book.

1.4.8 Modelling in high-level modelling languages:using PCRASTER

‘High-level’ in this context does not mean that ahigh level of detailed knowledge is required. High-level computer languages are those which are closest

Modelling and Model Building 23

to human language compared with low-level languageswhich are closer to machine code. High-level languagesdefine a rigid syntax in which the user is able to developsoftware or a string of calculations without knowing toomuch about how the computer actually performs thecalculations or functions that will be done. Examplesof high level computer programming languages includeBASIC, Pascal, FORTRAN, C++ and Java.

A number of researchers have designed generic com-puter modelling languages for the development of spatialand nonspatial models (van Deursen, 1995; Wesselinget al., 1996; Maxwell and Costanza, 1997; Karssenberget al., 2000). None has been universally accepted andmany have not been realized in software. The flexibil-ity required of a generic modelling language is so greatthat few codes are robust enough to handle all poten-tial applications. PCRASTER (http://www.pcraster.nl) isa good example of a high-level, spatial modelling lan-guage. Instead of working with parameters, constants andvariables which point to single numbers, PCRASTERparameters, constants and variables point to raster gridmaps which may contain categorical data, ordinal values,real numbers, a Boolean (true or false) value or one ofa number of more specialist types of data. PCRASTERis a pseudo-code interpreter, which provides the mod-eller with high-level functions for cartographic modelling(GIS) and dynamic modelling (spatial modelling overtime). These functions can be invoked at the MS-DOScommand line to perform simple calculations or in modelscript files for the integration of dynamic spatial mod-els. A number of utilities are available for map and timeseries display, import and export and map attribute con-version. A PCRASTER model consists of a simple ASCIItext file in which commands are listed in the sequencein which calculation is required. PCRASTER reads andinterprets this file line by line at runtime to realize themodel simulation. A PCRASTER model consists of the

following sections: binding, areamap, timer, initial anddynamic. In the binding section the user specifies a setof variable names that will be used in the model scriptand the corresponding disk file names from which thedata can be read or written, in the format:

VariableName=mapname.map; for a variable whichpoints to a single map and always reads or writes tothat map.

VariableName=mapname; for a variable which pointsto a different map each timestep, in the first timestepthe map is given the suffix mapname0.001, in thesecond timestep mapname0.002 and so on, with zerosbeing used to fill the file name to eight characters andalso to fill the unused parts of the file extension.

VariableName=timeseriesname.tss; for a variablewhich reads from or to columns in a timeseries file.

VariableName=tablename.tbl; for a variable whichreads or writes to a data table or matrix.

Note that like many programming and scripting lan-guages PCRASTER is case sensitive. You must beconsistent with the usage of lower and upper case inyour scripts.

In the areamap section the user specifies a clonemap of Boolean type that identifies the area for cal-culations with a TRUE (1) and the area in which nocalculations will be performed with a FALSE (0). In thetimer section the user specifies the initial timestep, thetime interval and the number of timesteps. In the initialsection the user specifies any initial conditions (calcu-lations that will be performed only once at the startof the simulation). The bulk of a PCRASTER modelusually falls within the dynamic section which speci-fies calculations that will be performed in sequence forevery timestep. The text below represents a PCRASTERversion of the soil-erosion model.

# Erosion Model.binding#inputDem=dem.map;#outputErosion=eros;#time series and tablesErosTss=Erosion.tss;OflowTss=oflow.tss;

#constantsm=1.66;n=2.0;

24 Mark Mulligan and John Wainwright

e=2.718;Q=100;i=-0.07;k=0.2;V=30;areamapclone.map;timer1 92 1;initialSlopeDeg=scalar(slope(Dem));#slope tangentThickness=scalar(10);dynamicreport Erosion=k*(Q**m)*(SlopeDeg**n)*(e**(i*V);#(mm)report ErosTss=maptotal(Erosion);Thickness=Thickness-(Erosion/1000);#(m)

Note that in this very simple model the only spatiallyvariable quantity is slope angle (SlopeDeg), which isderived from the digital elevation model (Dem) usingthe PCRASTER slope() command which performs awindow operation to define the slope on the basis ofthe change in altitude between each cell and its eightcontiguous neighbours (see Chapter 4). All other quan-tities are defined as constants. Even the soil thickness,which is specified here as an initial condition equal to100, is constant across the map though, as a state vari-able, it will vary over time. Equation 1.1 in PCRASTERsyntax becomes:

report Erosion= k∗(Q∗∗m)∗(SlopeDeg∗∗n)

∗(e∗∗(i∗V); #(mm)

where ‘report’ is the keyword which ensures that theresults of the calculation are written to the output mapon disk, ** represents ‘to the power of’ and * repre-sents ‘multiplied by’. All PCRASTER script commandlines (except the section headings) must end in a :..The # (hash) symbol indicates that all material to theright of the hash and on the same line is a commentwhich is not to be interpreted by PCRASTER. Thespatial functionality of PCRASTER is used in the map-total function that, in this case, sums erosion across thewhole map for output as a single value per timestepin a time series file. The final line of the code updatesthe soil thickness according to the erosion which hastaken place. It might seem unusual to have a state-ment such as Thickness=Thickness-(Erosion/1000);because the same variable is both sides of the equalssign. This is in fact a fairly common construct inprogramming and modelling languages and is equiv-alent to Thicknesst = Thicknesst−1 – (Erosiont/1000);

in other words, it allows passage of informationbetween timesteps. In order that soil thickness is neverallowed to become negative we should use the syntaxThickness=max(Thickness-(Erosion/1000),0);.

Again this is not a particularly interesting model.When supplied with a DEM, in this case for a catchmentin Spain, this model will produce a series of 92 maps oferosion, which will vary over space according to slopeangle but will be constant in time (Figure 1.6). To makethe model more interesting we may, for example, usesome of the functionality of PCRASTER to (a) makethe spatial distribution of Q more realistic; (b) producea temporally variable Q; and (c) vary soil thicknessaccording to landscape and provide a negative feedbackto soil erosion to represent armouring. In order to dis-tribute Q spatially we must define the local drainagedirection (LDD) and then specify a precipitation andan infiltration rate and calculate the difference betweenthem (the runoff) for each timestep. Further, we mustaccumulate this runoff down the stream network orLDD. The cumulated runoff will represent a much morerealistic situation than a uniform runoff. In the initialsection we use the PCRASTER function LDDCREATEto calculate the drainage direction for each cell basedon the steepest drop to its eight contiguous neigh-bours (the so-called D8 algorithm; see O’Callaghan andMark, 1984, or Endreny and Wood, 2001). The syn-tax is: Ldd=lddcreate(Dem, elevation, outflowdepth,corevolume, corearea, catchmentprecipitation) wherethe five latter parameters are thresholds used in theremoval of local pits that can be produced by inaccura-cies in the production of the DEM and can be safelyset to 1e31 each here since the DEM to be used is

Modelling and Model Building 25

Figure 1.6 Initial results of the soil-erosion model implemented in PCRASTER. Reproduced with permission (see vanDeursen 1995)

sound. The resulting LDD map has a drainage direc-tion for each cell and this produces a clear drainagenetwork (Figure 1.7). We can ensure that the map iswritten to disk by reporting it and adding a refer-ence in the binding section. We can now specify arainfall and soil infiltration rate, which could be spa-tially variable but, for simplicity, we will use a lumpedapproach. The infiltration rate can be added to the

binding section as a constant: Infil=50;#(mm/month)but in order to allow temporal variation we willspecify the rainfall as a time series which we mayhave prepared from meteorological station data. In thebinding section rainfall is defined as a time seriesusing Rainfall=rainfall.tss;#(mm/month) and in thedynamic model section the rainfall is read from the rain-fall series using the command timeinputscalar (because

26 Mark Mulligan and John Wainwright

Figure 1.7 Drainage direction for each cell in the PCRASTER soil-erosion model and its corresponding altitude.Reproduced with permission (see van Deursen 1995)

rainfall is a scalar quantity that is a continuous, real, asopposed to integer, number). The command which mustbe added to the dynamic section before erosion is cal-culated is Rainfallmm=timeinputscalar(Rainfall,1);.The number 1 indicates that the first column of therainfall file is read and applied to all cells, but a mapof nominal values could have equally been suppliedhere allowing the reading of different columns of thetime series file for different cells or areas of the map.As in the spreadsheet, time passes down the rows oftime series files. The format for time series files isgiven in the PCRASTER online documentation. Wecan now specify the infiltration excess for each cell as:InfilExcess=max(Rainfallmm-Infil,0);. The max state-ment ensures that InfilExcess is never less than zero.In order to cumulate this excess along the flow net-work we use the accuflux command in the form: reportQ=accuflux(Ldd, InfilExcess);. Now remove (or com-ment out using #) the reference to Q as a constant andthe new Q will be distributed along the local drainagenetwork. We can now specify soil thickness as a spatiallyvariable entity instead of a constant by commenting outits definition as a constant in the initial section and

by specifying it as a function of landscape propertiesinstead. Saulnier et al. (1997) model soil thickness as adecreasing function of slope on the basis of greater ero-sion on steep slopes and greater deposition on shallowones. Here we will approximate a distribution of soilthickness according to a compound topographic index,the so-called TOPMODEL wetness index (WI), whichis the ratio of upslope area to local slope angle andtherefore expresses the potential for soil accumulationas well as erosion. We can calculate the WI in the initialsection as WI=accuflux(Ldd,25*25)/SlopeDeg; where25 is the width of the individual raster grid cells (inmetres). We can scale soil thickness in a simple way asThickness=(WI/mapmaximum(WI))*MaxThickness;where MaxThickness is the maximum potential soilthickness. Finally, we can replicate the armouring algo-rithm we developed in the spreadsheet model by stor-ing the initial soil thickness using InitialThickness=Thickness; in the initial section, removing the definitionof erodibility (k) as a constant in the binding section andreplacing it with the following definition in the dynamicsection of the model script: k = 0.2*(Thickness/Initial-Thickness);.

Modelling and Model Building 27

We now have a more sophisticated model in whicherosion will respond to passing rainfall events over timeand spatially to the cumulation of infiltration excessdown the spatially variable stream network. Further-more, the soil erosion will vary with slope gradient anda negative feedback mechanism reduces erosion as itprogresses to simulate surface armouring. The modelis still very simple and could be made more realis-tic in many ways (by making the vegetation respondto soil moisture and soil thickness, by making slope

angle responsive to erosion, and so on). However,even this simple model can enhance our understand-ing of the spatial patterns of soil erosion. Figure 1.8shows (a) the DEM; (b) the calculated slope angle;(c) the natural log of soil erosion for month 4 (72.4 mmrainfall); and (d) the natural log of soil erosion formonth 32 (53.8 mm rainfall). The scale used is loga-rithmic in order that the spatial pattern can be morereadily seen. The final PCRASTER code is given inthe box:

# Erosion Model.binding#inputDem=dem.map;Rainfall=rainfall.tss;#(mm/month)#outputErosion=eros;Ldd=ldd.map;#time series and tablesErosTss=Erosion.tss;OflowTss=oflow.tss;#constantsm=1.66;n=2.0;e=2.718;#Q=100;i=-0.07;#k=0.2;V=30;Infil=50;#(mm/month)areamapclone.map;timer1 92 1;initialSlopeDeg=scalar(slope(Dem));#slope tangentWI=accuflux(Ldd,25*25)/SlopeDeg;Thickness=(WI/mapmaximum(WI))*10;InitialThickness=Thickness;report Ldd=lddcreate(dem.map,1e31,1e31,1e31,1e31);

dynamick=0.2*(Thickness/InitialThickness);Rainfallmm=timeinputscalar(Rainfall,1);InfilExcess=max(Rainfallmm-Infil,0);report Q=accuflux(Ldd,InfilExcess);Erosion=k*(Q**m)*(SlopeDeg**n)*(e**(i*V));#(mm)report ErosTss=maptotal(Erosion);Thickness=max(Thickness-(Erosion/1000),0);#(m)

28 Mark Mulligan and John Wainwright

Figure 1.8 Final PCRASTER soil-erosion model results showing: (a) the DEM; (b) the calculated slope angle; (c) thenatural log of soil erosion for month 4 (72.4 mm rainfall); and (d) the natural log of soil erosion for month 32(53.8 mm rainfall). Reproduced with permission (see van Deursen 1995)

You can download this model from the companionwebsite for this book.

The main advantage of scripting languages likePCRASTER is in their power and flexibility and in therapidity with which even complex operations can be per-formed through the application of pre-coded high-levelfunctions. The main disadvantage is that such languagesrequire a considerable effort to learn and have steeplearning curves, but, once mastered, significant mod-elling feats are relatively easy. Scripting languages arenot, however, as flexible and powerful as high-levelcomputer languages but these in turn can require largeprogramming marathons to achieve the same as a fewlines in a well-crafted scripting language. High-levelcomputer languages offer the advantages of flexibilityto implement new procedures and the ability to opti-mize complex procedures to run as quickly as possi-ble. For detailed climate (Chapter 2) or computationalfluid dynamics (Chapter 20) modelling, such optimiza-tion (perhaps by running a model in parallel on a numberof processors at once) is an absolute necessity so thatresults may be obtained within a reasonable time frame.Even in general applications, a compiled language (e.g.FORTRAN, Pascal or C++) will be noticeably faster thanan interpreted one (e.g. PCRASTER, BASIC and usu-ally Java).

So, we have used three modelling approaches toproduce essentially the same model. Each approach hasits advantages and disadvantages and you will probablyhave made up your mind already which approach youfeel more comfortable with for the kind of modellingthat you wish to do. Whichever technique you use tobuild models, it is critically important to keep focusedon the objective and not to fall into the technology trapthat forces all of your mental power and effort intojumping the technological hurdles that may be needed toachieve a set of calculations and as a result leaves littleenergy and time after the model building stage for itstesting, validation and application to solving the problemat hand. A simpler model which is thoroughly testedand intelligently applied will always be better than atechnically advanced model without testing or sufficientapplication.

1.4.9 Hints and warnings for model building

Cross and Moscardini (1985) give a series of hintsand warnings intended to assist novice modellers inbuilding useful models. This list is expanded hereaccording to our own environmentally biased perspec-tive on modelling.

Modelling and Model Building 29

1. Remember that all models are partial and willnever represent the entire system. Models are neverfinished, they always evolve as your understand-ing of the system improves (often with thanks tothe previous model). In this way all models arewrong (Sterman, 2002)! Nevertheless, it is impor-tant to draw some clear lines in model developmentwhich represent ‘finished’ models which can betested thoroughly and intensively used.

2. Models are usually built for a specific purpose sobe careful of yourself or others using them forpurposes other than those originally intended. Thisissue is particularly relevant where models are givena potentially wide user base through distribution viathe Internet.

3. Do not fall in love with your model (it will not loveyou back!).

4. Take all model results with a grain of salt unlessthose specific results have been evaluated in someway.

5. Do not distort reality or the measurement of realityto fit with the model, however convincing yourconceptualization of reality appears to be.

6. Reject properly discredited models but learn fromtheir mistakes in the development of new ones.

7. Do not extrapolate beyond the region of validityof your model assumptions, however powerful itmakes you feel!

8. Keep ever present the distinction between modelsand reality. The sensitivity of a model to climatechange is not the sensitivity of reality to climatechange. If you write about ‘model sensitivity’ thendo not omit the word ‘model’ (Baker, 2000).

9. Be flexible and willing to modify your modelas the need arises. Note that this is modify notexpand upon.

10. Keep the objectives ever present by continuallyasking ‘What is it that I am trying to do?’

11. As in all aspects of science, keep the model honestand realistic despite any short-term financial orstatus gain that may seem to accrue from doingthe opposite.

1.5 DEVELOPMENT OF NUMERICALALGORITHMS

1.5.1 Defining algorithms

An algorithm is the procedure or set of rules that we useto solve a particular problem. Deciding on the algorithmto use in any particular model can be thought about intwo stages. The first has already been covered in the

previous section – that of conceptualizing the systemand developing a mental model of the system. Thesecond is the conversion of this conceptualization to aform that can be calculated. It may be that, at this stage,the problem as formalized is not calculable, and we mustreturn to the former stage and try to re-evaluate anddescribe the system in question, so that an approximationcan be arrived at without compromising the reliabilityof the model results.

1.5.2 Formalizing environmental systems

Environmental models are focused upon change. Thischange may be of properties in time or in space.Increasingly, models are being developed where bothtemporal and spatial variations are evaluated, so weneed to have techniques that can assess both of thesechanges. The branch of mathematics that allows us toassess change is known as calculus, initially derivedin the seventeenth century independently by Newtonand Leibniz. A basic understanding of calculus isfundamental for all modelling, so we will provide abrief overview here. More detailed information can befound in numerous books, some of the more accessiblebeing Bostock and Chandler (1981), De Sapio (1976),Ferrar (1967), Friedman (1968), Haeussler and Paul(1988), Lang (1978) and Thompson (1946).

Consider the case where we are interested in the trans-port of coarse sediment through a river. To investigatethe process we have tagged rocks with radio transmit-ters that allow us to record the location of the rocksthrough time (e.g. Schmidt and Ergenzinger, 1992). InFigure 1.9 we have plotted the position along the riverchannel of a rock over a period of 24 hours. Theseresults can also be plotted as a graph showing thelocation in space and time of the rock (Figure 1.9b).This graph can also be used to define the change inposition through time. The change in position throughtime is effectively the velocity of the rock as it moves.Remember that velocity is the change in distance overthe time to move that distance. Over the 24 hours, therock moved a total of 8.57 m, so that its average veloc-

ity was8.57 m

24 h= 0.36 m h−1. Notice, however, that the

rock was not moving at a constant velocity through theday, as the graph is not a straight line. We can use thesame technique to show how the velocity of the particlevaried over time. In the second hour, the particle moved0.72 m, so its velocity was 0.72 m h−1. Fourteen hourslater, the velocity was 0.22 m h−1 (Figure 1.9c). Thesevelocities are simply the slope of the graph at succes-sive points in time. We now have enough information

30 Mark Mulligan and John Wainwright

(a)

(c)

(e)

(d)

0123456789

0 6 12 18 24

Time h

Time h

Dis

tanc

e m

oved

m

Dis

tanc

e m

oved

m0.72 m

1 h

1 h 0.22 m

9876543210

12 18 2460(b)

0

0.5

1

1.5

2

2.5

3

0 5 10 15 20Time h

Vel

ocity

m h

−1

Independent variable x

Dep

ende

nt v

aria

ble

y

y = f (x)

Figure 1.9 (a) Map showing position along a stream channel of a tagged rock over a period of 24 hours; (b) Graphshowing the location in time and space of the tagged rock as shown in Figure 1.9a. The line interpolates the positionof the rock at times between those measured; (c) Graph showing how the variation in the velocity of the measuredrock can be calculated as the local slope of the graph; (d) Graph showing relationship between the two variables as acontinuous function. The thick lines show examples of the tangent of the slope at specific points; (e) Graph showing thevelocity of the rock particle through time. The line shows the continuous change in velocity calculated using the formulady

dx= y ′ = 0.875x−0.5, obtained by differentiation, whereas the points show the measured points at hourly intervals. Note

the major difference in the early part of the graph, where the velocity is rapidly changing, due to the fact that the hourlymeasurements are based on an average rather than continuous measurements

to model the movement of the rock at a daily andhourly timestep.

However, if we need to model movements at shortertime intervals, we need to have some method forestimating the velocity of the rock, and thus the slope ofthe line, at these shorter intervals. To do this, we may

be able to use a function that shows us how the twovariables are related, either empirically or from physicalprinciples. Typically, this function may be written inthe form:

y = f (x) (1.2)

Modelling and Model Building 31

In this case, we find empirically (for example, usingregression analysis) that the relationship has a sim-ple form:

y = 1.75x0.5 (1.3)

where the variable y is the distance moved in metresand x is the time in hours. The velocity of the particleis still the slope at each point on the graph; we can nowestimate these slopes continuously from our function,as the tangent to the curve at each successive point(Figure 1.9d). To calculate this tangent, we need to workout the derivative of the function, which has the standardform for power-law relationships such as Equation 1.3:

dy

dx= nxn−1 (1.4)

How this standard form is obtained is shown inFigure 1.10.

The derivativedy

dx(usually said as ‘dy by dx’) may

also be written as y ′ or f′(x). (The difference betweenthese is essentially that the former is Newton’s notation,whereas the latter is that of Leibniz.) Applying this ruleto the example of Equation 1.3, we have:

dy

dx= y ′ = 0.875x−0.5 (1.5)

Thus, we can now calculate the instantaneous velocity ofour rock at any point in time. The process of calculatingthe derivative is known as differentiation. More complexfunctions require different rules for differentiation, andthese can be found in the standard texts cited above andon the book’s website.

Environmental models may also commonly use termsrelating to acceleration or diffusion. Acceleration isthe change in velocity through time, so we can alsocalculate it using differentiation. This time, we plotvelocity against time (Figure 1.9e) and find the tangentat a point by differentiating this curve. Relative to ouroriginal measurement, we have calculated:

ddy

dx

dx= d2y

dx2= f′′(x) = y ′′ (1.6)

This term is called the second derivative. The standardform for power-law relationships means that we simplyneed to repeat the operation in Equation 1.4 twice tocalculate the specific value of the second derivative:

d2y

dx2= n(n − 1)xn−2 (1.7)

Thus, our rock particle is accelerating at a rate of y ′′ =−0.5 × 0.875x−1.5 = −0.4375x−1.5 m h−1, or in otherwords it is consistently decelerating.

Thornes (1985, 1988) uses derivatives in modellingthe interaction between vegetation and erosion to showcases of stability in the landscape. Given a functionalrelationship for vegetation change, it is possible to findthe conditions where vegetation cover becomes constant.

This is simply wheredV

dt= 0, where V is the vegetation

amount and t is time. This example is examined furtherin Chapter 18. Another use for derivatives is in theprocess of optimization to find the optimal values ofparameters in a model. These optimal values may fall atmaximum or minimum points of the graph showing therelationship. At these points, the following conditionsfor the first and second derivatives will hold:

Maximum:dy

dx= 0

d2y

dx2< 0 (1.8a)

Minimum:dy

dx= 0

d2y

dx2> 0 (1.8b)

Care must, however, be taken that the graph is notsimply at a local plateau, or inflexion point, where:

Inflexion:dy

dx= 0

d2y

dx2= 0. (1.8c)

Integration is the opposite process to differentiation.It allows us to calculate the sum of a property overspace or time. If we plot the function that representsthe relationship we are interested in, then the sum ofthis property is the area under the curve. To see whythis is so, consider a function that relates the change indepth of water along a section of river (Figure 1.11).If the depth (h) is constant along section of length y,then the sum of the values of y is simply y · h. In otherwords, it is the area under water along the section lengthin which we are interested – in this case representedby a simple rectangle. If, however, the depth increaseslinearly along the section, then the area under the curvewill be a trapezium with area 0.5(h1 + h2) · y, whereh1 is the depth at the start of the reach and h2 isthe depth at the end. We often use this approach tocalculate integrals where the shape of the relationshipis complex, or known only from data at certain points.The ‘trapezium rule’ involves dividing the curve or datainto a set of known intervals, calculating the area of thetrapezium that is a simple representation of each interval,and summing the total area. This approach is best whenwe know the change between points is linear, or thedistance between points is sufficiently small. Note thatwe are not restricted to integration in one dimension.If we integrated the area of water by the width of the

32 Mark Mulligan and John Wainwright

Deriving the Standard Form for Differential Calculus

Let us take, as an example, the function y = x2 + 2x. Intabular and graphical forms, this function is:

x = y =0 01 32 83 154 245 35. . . . . .

0 420

6 8 10

20

40

60

80

100

120

y

x

A line joining any two points on the curve is known as achord. In the graph below, the line PQ is a chord.

y = f(x)

P

Q

A B

C

y

x

The slope of this line is:

gradient of PQ = QC

PC

To define the gradient at a point, let the length of PC in thegraph above be h (an arbitrary length). If the point C is (2,8), then the value of the coordinates of point Q are given bythe function, by inserting the x value of Q, which is (2 + h).The y value at Q is then:

y = (2 + h)2 + 2(2 + h)

which is 8 + 6h + h2. The distance QC is then 6h + h2 (the8 disappears because it is the value of y at C) and the

gradient of the chord PQ is:

gradient of PQ = QC

PC= 6h + h2

h= 6 + h

To find the value of the gradient at the point P, we let thevalue of h become vanishingly small, so that the points Pand Q are in the same location. In other words, the valueof h approaches zero. This process is known as taking the

limit and is written: hlim 0. Thus, we have:

gradient at P = h 0 (6 + h) = 6

In more general terms, if P is any point (x, y), then Q isgiven by the function as (x + h, (x + h)2 + 2(x + h)), or byexpansion of terms (x + h, x2 + 2xh + h2 + 2x + 2h) andthe gradient of the chord PQ is:

gradient of PQ = 2xh + h2 + 2h

h= 2x + h + 2

In this case, the value of y at P (i.e. x2 + 2x) disappearswhen calculating the distance from Q, leaving the remainingterms (i.e. 2xh + h2 + 2h). The gradient at P then becomes:

gradient at P = h 0 (2x + h + 2) = 2x + 2

which we can verify by substitution, so that if x = 2, thegradient at P is 6. The derivative of y = x2 + 2x is thus

demonstrated as beingdy

dx= 2x + 2, which is the same as

given by the standard form in the main text (i.e. we applydy

dx= nxn −1 to each of the terms which contain an x in

the function).It is more usual to use the incremental notation as

illustrated below:

A B x

y

CP(x,y)dx

dy

Q (x + dx,y + dy)

y = f(x)

where dx means the increment in x and dy the increment iny. We can then re-write:

gradient of PQ = QC

PC= dy

dx

dx

and by definition:

dy

dx= 0 = dy

dx

lim

lim

lim

Figure 1.10 Derivation of the standard form of differentiation

Modelling and Model Building 33

Distance downstream

Dep

th o

f flo

w

(a)

(b)

(c)

(d)

h

Length of reach section, y

Area = y • h

Distance downstream

Dep

th o

f flo

w

Length of reach section, y

h1

h1

h2

Area = 0.5 (h1+ h2) • y

Distance downstreamDistance across stream

Dep

th o

f flo

w

Distance downstream

Dep

th o

f flo

w

5

Ai = 0.5 (hi + hi + 1) d

A1 A2 A3 A4 A5 A6 A7 A8 A9 A10 A11 A12 A13 A14 A15 A16

h2 h3 h4 h5h6

h7h8 h9

h10h11

h12

h13

h14h15

h16 h17

Length of reach section, y

d

Total area = ∑ Ai

16

i =1

Figure 1.11 Derivation of integration in one and twodimensions using the example of changing flow depthalong a river section: (a) the case where the depth isconstant along the reach of interest; (b) linear change ofdepth along the reach; (c) use of the trapezium rule for acomplex function representing change in water depth; and(d) integrating by the width at each point to give the volumeof water in the reach

channel, then we would have a means of calculating thevolume of water.

The trapezium rule approximates the integral in caseswhere change is nonlinear. It is also possible to calculateintegrals directly if we know the form of the function.The relationship between differentiation and integrationcan be seen when considering the simple power-lawrelationship considered above as a standard form. Thesymbol ∫ is used to mean ‘integral of’:∫

nxn−1 dx = xn + c (1.9a)

or ∫xn dx = 1

n + 1xn+1 + c (1.9b)

Thus, practically as well as conceptually, integration isthe reverse of differentiation. The value c is knownas the constant of integration, and arises because thederivative of a constant is zero. Other standard functionscan be integrated by reading tables of derivatives in thereverse direction. The dx in Equation 1.9 simply tells uswhich variable we are integrating over. It does not meanthat we multiply by the change in x. The general form ofan integral is ∫ f(x) dx. In more complicated functions,there may be other variables as well as constants, so it isvital to know which of the variables we are considering.

∫ is known more specifically as the indefinite integral.It is used to define the general form of the integral weare interested in. Often, we will want to calculate thevalue of the integral over a specific range of values, asfor example in the case above where we were interestedin a specific reach of a river, rather than the entire lengthof the river. In this case, we write

∫ b

af(x) dx where a

is the starting value of x that we want to consider andb is the end value that we want to consider. To solvea definite integral, first derive the indefinite integral forthe relationship. In the case of our rock moving in theriver channel, let us consider the sum of its velocitiesby calculating the integral of Equation 1.4a:∫

0.875x−0.5 dx = 0.8751

−0.5 + 1x−0.5+1 + c

= 1.75x0.5 (1.10)

Notice two things here: first, the constant of integrationhas disappeared because by definition the distancemoved at the start of the observations (i.e. when x = 0)was zero. Second, the result we have obtained is thesame as Equation 1.3. In other words, the sum of thevelocities of our particle movement is the same asthe total distance moved. Obtaining this relationshipin reverse should further emphasize the relationship

34 Mark Mulligan and John Wainwright

between integration and differentiation. The secondstage of calculation involves substituting the start andend values into the indefinite integral. Third, we subtractthe starting value from the end value to give us the finalanswer. Over the whole 24-hour period of observation,the rock therefore moved:

∫ 24

00.875x−0.5 dx = (1.75 · 240.5) − (1.75 · 00.5)

= 8.57 m (1.11)

which agrees with our initial observations. Confirm foryourself that the rock moved 6.06 m during the first 12hours and 2.51 m in the second 12 hours (note the totaldistance moved when calculated this way); between 14hours and 17 hours 30 minutes it moved 0.77 m. Whathappens if you calculate the distance moved betweenhours 0 and 6, 6 and 12, 12 and 18 and 18 and 24? Whatwould happen to the total distance moved if you usedsmaller intervals to calculate? Note that the constant ofintegration always vanishes when calculating a definite

integral, whether we know its value or not, as wesubtract it at b from itself at a.

1.5.3 Analytical models

It is possible to use developments of the approachesoutlined above to provide solutions to specific problemsunder specific sets of conditions. These we would termanalytical solutions. Consider the simple population-growth model (Burden and Faires, 1997):

Nt+1 = Nt + λNt (1.12)

This model states that the size of the population N atthe next timestep we consider (the t + 1 subscript) isgiven by the current population, Nt , plus a growth ratethat is a linear function of the current population. Theparameter λ represents the net growth rate (number ofbirths minus deaths) over the time interval. Versions ofthis model are commonly used for simulating vegeta-tion growth and animal (including human) populations(see Chapters 6–8 and 12). How does this model fit

Figure 1.12 Derivation of the analytical solution to the population equation (Equation 1.13)

Modelling and Model Building 35

into what we have just been saying about differentialequations? The model is expressing change in the pop-ulation, so we can also specify how the population ischanging. In this case, it is the second part on the right-hand side of Equation 1.11 that represents the change,so that we can also say:

dN

dt= λN (1.13)

The solution (see Figure 1.12) to this equation is:

Nt = N0 eλt (1.14)

where N0 is the initial population. The solution showsthat populations with a linear growth rate will growin size exponentially through time. In reality, this(Malthusian) assumption only tends to be valid overshort time periods because of resource constraints tosustained linear growth rates. We will consider how tomake the model more valid below.

However, in many cases, environmental systemsmay be described by functions for which there areno analytical solutions. In other words, we cannotuse algebra to reformulate the problem into a finalform. Alternatively, we may be interested in spatialor temporal variability in our initial and/or boundaryconditions that cannot be described in a functional form.For example, in rainfall-runoff modelling, we have todeal with rainfall rates that vary through space andtime and irregular topography that is not a simplefunction of distance along a slope. In these cases, weneed to develop alternative approaches that allow us toapproximate solutions to the systems we are trying tomodel. These approaches require numerical solutions.Numerical solutions typically involve approximationand iterative attempts to provide an answer to theproblem in hand. The use of iterative approaches isconsidered in more detail below.

1.5.4 Algorithm appropriateness

In most systems, there will be more than one way of rep-resenting the system components and their interlinkages.At the most fundamental level, we can contrast blackbox approaches with grey box or white box approaches.How do we evaluate whether an algorithm is appropri-ate?

One set of criteria to evaluate for appropriateness is tojudge in terms of process representation. If we considerthe example of population growth, we have alreadyseen what is probably the simplest possible model, inEquations 1.12 to 1.14. The population is considered as

a single, simple unity, and one parameter is used tocontrol how it changes. This parameter is meaningful,in that it represents the difference between births anddeaths through time, but assumes that this value isconstant. In other words, there is always the sameproportion of births and deaths relative to the size of thepopulation. This assumption is reasonable on average,but might not be appropriate in all circumstances. Wehave also noted that the model predicts exponentialgrowth, which is usually only true for short timeperiods, for example, because increased competition forresources means that continued growth is unsustainable.We might want to account for these limitations bymodifying the model as follows:

dN

dt= β

(1 − N

Nmax

)N − θN (1.15)

where β is the birth rate in individuals per unit time,θ is the death rate in individuals per unit time andNmax is the maximum number of individuals that canexist in the specific environment. Nmax is often calledthe carrying capacity of the environment. In the firstpart of this model, we have essentially added a compo-

nent,

(1 − N

Nmax

), which says that as the population

approaches the carrying capacity, and thus the ratioN

Nmaxapproaches one, the birth rate is multiplied by an

increasingly small number, and so the rate of increasegets smaller. This effect is illustrated in Figure 1.13.This S-shaped curve is known as the logistic growthcurve. Note that in the second part of this model, wehave simply assumed that death rates are again a con-stant proportion of the population. In terms of processrepresentation, we may again feel that this is inadequate,because it fails to deal with different ways in whichmembers of the population may die – for example, dueto old age, disease or accidents – and want to add extracomponents to account for these processes and how theymight change with population or resources. These com-ponents may be simple functions, or may depend on thepopulation size or density. Alternatively, we may wantto account for apparently random elements in these pro-cesses by using stochastic functions to represent them(see below).

One assumption we have still incorporated implicitlyin this model is that the population grows in anisolated environment, without competition from otherpopulations. For example, it is common to investigatepopulation changes in terms of separate populations ofpredators and prey. These investigations form the basisof the Lotka–Volterra population models. These models

36 Mark Mulligan and John Wainwright

0 50 100 150 200 250 300Time years

0 50 100 150 200 250 300Time years

0100200300400500600700800900

1000

Pop

ulat

ion

Nmax

PreyPredator

(a)

0102030405060708090

100

Pop

ulat

ion

(b)

Figure 1.13 (a) Graph showing population simulatedusing Equation 1.12, with a limitation to a carrying capacityof 1000 individuals. The birth rate β is 0.05 individuals peryear in this example; (b) Graph showing population simu-lated using Equation 1.16, with βH = 0.05 individuals peryear, p = 0.01 individuals per predator per prey per year,βH = 0.002 individuals per year per prey eaten per yearand θ = 0.06 individuals per year

link the populations of a prey species, H , and a predatorspecies, P , as:

dH

dt= βHH − pHP (1.16a)

dP

dt= βP HP − θP (1.16b)

where βH is the birth rate of the prey species, p is therate of predation, βP is the birth rate of the predatorper prey eaten, and θ is the death rate of the predatorspecies. Figure 1.13b shows an example of the outputof this model, showing cycles of increase and decreaseof the prey and predator species. Note that in thisformulation the death rate of the predator is given(compare Equation 1.15) but death of the prey is onlyconsidered to occur through predation. This model isstill a simple one that assumes an isolated environmentand an interaction of two species, but can still providecomplex results.

All of these population models still make the assump-tion that ‘population’ is a useful level of generaliza-tion. It may, however, be more appropriate to deal withthe population number as an emergent property of theactions of the individuals that make it up. In other words,we can assess the birth, death and interaction of individ-uals within an environment, and derive the populationas the sum of all remaining individuals. Certainly, this isa more realistic model representation of what is actuallyhappening. Berec (2002) provides an overview of theapproach and shows the relationship between the aver-age outputs of individual-based models and those basedon population statistics. Pascual and Levin (1999) useda cellular automata approach (see below) to representthe predator–prey model and show how spatial patternsare important in considering outcomes. Bousquet et al.(2001) provide a more specific example of the hunting ofantelope in Cameroon, showing that spatial patterns oftrap sites are important as well as cooperation betweenhunters. Other overviews can be found in Epstein andAxtell (1996) and Ginot et al. (2002).

Another useful example of different process represen-tations is the ways that river flows are modelled. In thesimplest case, we might take a highly empirical form thatrelates the amount of rainfall in the catchment to the bulkor peak outflow from the catchment (see the review oflumped catchment models by Blackie and Eeles, 1985).This approach will allow us to make basic estimationsof available water resources or of the potential for flood-ing. However, it is highly dependent on data availabilityand only works for a single point in the catchment. If wewant to evaluate how flows change locally, for example,to investigate flooding in particular areas, or to under-stand how the flow dynamics interact with the channelshape, or how they control sediment movement, thenclearly this level of simplification is not useful.

Flows are most often simulated using the St Venantapproximations (named after the French hydrologistJean-Claude Barre de St Venant) and first formulated in1848. These approximations state that water flows arecontrolled by the continuity (mass balance) equation:

∂h

∂t+ h

∂u

∂x+ u

∂h

∂x− q = 0 (1.17a)

and a conservation of momentum equation made up ofthe following components:

1

g

∂u

∂t+ u

g

∂u

∂x+ ∂h

∂x− (S − Sf ) = 0

(1) (2) (3) (4)

(1.17b)

where h is depth of flow (m), u is flow velocity (m s−1), tis time (s), x is distance downstream (m), q is unit lateral

Modelling and Model Building 37

inflow (m s−1), g is acceleration due to gravity (m s−2),S is the slope of the channel bed (m m−1) and Sf isthe slope of the water slope (also known as the frictionslope) (m m−1). If we use all of these components(1–4), then the model is known as a dynamic waveapproximation (Graf and Altinakar, 1998; Singh, 1996;Dingman, 1984). Component (1) reflects the inertia ofthe flow; (2) the gravity effect on the flow; (3) thepressure effect; and (4) the friction effect. However, itmay be possible to ignore the inertial term because itsvalue is so small. In this case we omit the component(1) and produce the diffusive wave approximation:

u

g

∂u

∂x+ ∂h

∂x− (S − Sf ) = 0 (1.18a)

which, when combined with the continuity equation1.15a, gives the diffusion-wave approximation:

∂h

∂t+ u

∂h

∂x− D

∂2h

∂x2− q = 0 (1.18b)

where D is the diffusion coefficient (equal toQ

2wSwhere Q is the discharge [m3 s−1] and w the flow width[m]). This approach may be valid under conditions ofnegligible convective acceleration. In other cases wecan omit the diffusive component of the flow, so thatwe assume the flows are simply controlled by a singlerelationship between depth and discharge. In this case,components 2 and 3 become negligible, S = Sf and:

∂h

∂t+ u

∂h

∂x− q = 0 (1.19)

which is known as the kinematic wave approximation.This approximation is valid particularly in overlandflows on hillslopes (see below) and where waves inthe flow are of low amplitude relative to their wave-length. Where there are backwater effects, as at trib-utary junctions or upstream of lakes and estuaries, theassumptions of the kinematic wave approximation breakdown (Dingman, 1984) so that diffusive or dynamicwave algorithms should be used in these cases. Singh(1996) contains numerous references to and examplesof the use of these equations. Further details are alsogiven in Chapter 5.

In all these cases, we have simply made one-dimensional approximations to the flow process. In otherwords, we assume that it is valid to model average con-ditions at a series of cross-sections down the channel. Ifwe want to estimate flooding, then this approach may bevery reasonable (e.g. Bates and De Roo, 2000). How-ever, we may want to know how conditions change

across a cross-section, or as a continuum along a chan-nel. These approaches would be better for looking at theeffects of flows on structures, for example, bridge sup-ports, or on how the flow interacts with the morphology.In these cases, approaches using more complex, compu-tational fluid dynamics (CFD) models will usually bemore appropriate. Such models are covered in detail inChapter 20.

Overall, we can summarize that the process represen-tation is not a single definition, and different criteria willbe used depending on our modelling objectives. Bothbottom-up and top-down approaches may be useful formodelling the same system in different conditions. Wecan see how this difference can lead to different levelsof complexity in terms of model components, equationsrepresenting their interactions, and expectations in termsof outputs.

Second, we might evaluate algorithm appropriatenessin terms of computational resources. With the rapidlyincreasing rate of progress of microprocessor design andreduction of costs of microprocessors, memory and datastorage, it is possible to design and implement larger andmore complex models on desktop computers. Moore’s‘law’ of increasing computer power is based on the factthat more transistors can be crammed into successivelysmaller spaces on integrated circuits, allowing them torun progressively faster. In the past 35 years, transistordensity has doubled about once every 18 months, allow-ing the corresponding exponential increase of model size(number of points simulated) and complexity (Voller andPorte-Agel, 2002). The design of supercomputers andparallel processing, where a number of calculations canbe carried out simultaneously, is also proceeding apace.One criterion we might use is the time taken to exe-cute the model. This time will include that required toread in all of the data and parameters required, carryout the calculations in our algorithm, and output all ofthe results. Algorithms are often evaluated in terms ofthe number of operations required, as this can often berelated to the processing speed (measured in FLOPS orfloating-point operations per second – modern machinesare capable of megaflops [millions] or gigaflops [thou-sands of millions]). Modern computers are designedto be able to make calculations on the several mem-ory addresses involved in a floating-point calculationat once.

For example, if we consider the population-growthmodels from this section, the models based on grosspopulation characteristics have substantially fewer cal-culations than those based on individuals. Therefore,if either algorithm is appropriate from a process pointof view, but we need the calculations to be carried

38 Mark Mulligan and John Wainwright

out quickly (for example, if we then use these calcu-lations to drive another part of a larger model) or ifcomputer time has an associated cost which we needto minimize, then it would be better to choose thepopulation-based approach. A second criterion in termsof computational resources is that of available mem-ory while the model is running or data storage forinputs and outputs. Many environmental models arespatially distributed, requiring a large amount of spa-tial information to be stored to carry out calculations.This information is usually held in dynamic (volatile)memory, which is relatively expensive, although muchless so than even a few years ago. If parts of the spa-tial information need to be read from and stored backonto other media, such as hard drives, model perfor-mance suffers significantly. Thus, it may be necessaryto design large spatial models to store only absolutelynecessary information, and to access other informationas and when necessary. Specific techniques are cur-rently being designed to implement these approaches.For example, in the analysis of large digital elevationmodels (DEMs), Arge et al. (2002) demonstrated thata parallel approach to flow routing is between twiceand ten times as fast as a serial approach for largedatasets (they used examples with 13 500 × 18 200 and11 000 × 25 500 cells). It is also capable of process-ing larger datasets (e.g. 33 454 × 31 866 cells, occupy-ing 2 Gb of memory) that are not possible to evalu-ate using a serial approach due to machine-memorylimitations.

Third, we need to consider our models in rela-tion to available data. Data are required as basicinputs, or parameters, and for use in testing the resultsthat are obtained. It may be better to use avail-able data rather than have to produce estimates orlarge-scale interpolations or extrapolations from thosethat are available. On the other hand, this restrictionshould not always be too strongly applied, as mod-els can often be used in the design of field data-collection campaigns by suggesting where critical mea-surements could be made. Increasingly, techniques arebecoming available for relating model results at coarsescale to field measurements, which are often takenat points. These techniques are investigated further inChapter 19.

1.5.5 Simple iterative methods

The most straightforward numerical approaches involvesimple repetition or iteration of a calculation to evaluatehow it changes through time. As an example, we willuse a simple ‘bucket’ storage model of how a lake level

will behave through time. The model is illustrated inFigure 1.14a and is described by the equation:

dh

dt= r + i − e − kh (1.20a)

and the inequality:

0 ≤ h ≤ hmax (1.20b)

In other words, the depth in the lake changes eachyear by an amount equal to the rainfall, r (in m a−1) plusthe river and surface inflow, i (in m a−1), less the annualevaporation from the lake, e (in m a−1) less the seepagefrom the base of the lake, which is a linear function ofthe depth of water, h (in m) in the lake. This function is

Rainfall

Evaporation

Seepage

Overflow

Inflow

Currentwater depth

Maximumwater depth

05

101520253035404550

0 20 40 60 80 100 120 140 160 180 200Time years

Dep

th m

abc

(b)

(a)

Figure 1.14 (a) Schematic diagram showing the represen-tation of a lake water balance as a ‘bucket’ model with aseries of inputs: rainfall and inflow from rivers; and out-puts: evaporation, seepage and overflow once the maximumdepth of the lake has been reached; and (b) graph showingthe results of the ‘bucket’ model representing lake depthchanges. Curve (a) represents rainfall of 1 m a−1, an inflowrate of 0.8 m a−1, evaporation rate of 0.05 m a−1 and seep-age intensity of 0.05 a−1; curve (b) has a lower inflow rateof 0.2 m a−1, and higher evaporation rate of 1 m a−1, con-ditions perhaps reflecting higher temperatures than in (a);curve (c) has a higher inflow of 2 m a−1, compared to (a),perhaps representing vegetation removal, or the existenceof a larger catchment area. In case (c), overflow from thelake occurs because the maximum depth threshold of 50 mis reached

Modelling and Model Building 39

controlled by a rate coefficient k, which is related to thepermeability of the lake bed. There is a maximum levelto which the lake can be filled, hmax , in m, above whichoverflow occurs. Similarly, the level cannot go belowthe ground surface. We need to use a numerical solutionbecause of the inequalities constraining the minimumand maximum depths. To calculate iteratively, we usethe following steps:

1. Define initial lake level ht=0.

2. Calculate the annual change in leveldh

dtusing

Equation 1.20a.3. Calculate the new level ht+1 by adding the annual

change value.4. If ht+1 > hmax (Equation 1.20b), then calculate the

overflow as ht+1 − hmax , and constrain ht+1 = hmax .Alternatively, if ht+1 < 0, constrain ht+1 = 0.

5. Go back to step (2) and repeat until the simulationtime is complete.

Figure 1.14b illustrates the results for simple scenariosof environmental change, or for comparing lakes in dif-ferent geomorphic settings. Similar sorts of models havebeen used for lake-level studies in palaeoenvironmentalreconstructions (see discussion in Bradley, 1999). Thistype of storage model is also used in studies of hillslopehydrology, where the bucket is used to represent theability of soil to hold water (e.g. Dunin, 1976). Notethat this method is not necessarily the most accurateiterative solution, as it employs a backward-differencecalculation (see below). More accurate solutions can beachieved using Runge–Kutta methods (see Burden andFaires, 1997; Engeln-Mullges and Uhlig, 1996).

Another example of where we would want to useiterative solutions is where we cannot calculate a closedalgebraic form for a relationship that might be used inthe model. For example, many erosion models use thesettling velocity of particles to predict the amount ofdeposition that will occur (e.g. Beuselinck et al., 2002;Morgan et al., 1998). The settling velocity states that asediment particle will move downwards (settle) througha body of water at a rate given by Stokes’ Law:

v2s = 4

3

g

(ρs

ρ− 1

)d

CD

(1.21)

where vs is the settling velocity (m s−1), g is accelerationdue to gravity (m s−2), ρs is the density of the particle(kg m−3), ρ is the density of water (kg m−3), d is theparticle diameter (m) and CD is the drag coefficientof the particle. As noted by Woolhiser et al. (1990),

the drag coefficient is a function of the Reynoldsnumber (Re) of the particle, a dimensionless relationshipshowing the balance between the gravitational andviscous forces on the particle:

CD = 24

Re+ 3√

Re+ 0.34 (1.22)

However, the Reynolds number is itself a function ofthe particle settling velocity:

Re = vsd

ν(1.23)

where ν is the kinematic viscosity of the water (m2 s−1).Now we have the settling velocity on both sides of theequation, and no way of algebraically rearranging theequation to provide a simple solution in terms of vs .

We can use a set of techniques for finding roots ofequations (e.g. Burden and Faires, 1997) to tackle thisproblem. If we have an equation in the form:

f(x) = 0 (1.24)

then a root-solving technique allows us to find thevalue of x for which the function gives the result ofzero. This approach can be implemented by combiningEquations 1.21 to 1.23 and rearranging them to give:

v2s

24

vsd

ν

+ 3√vsd

ν

+ 0.34

− 4

3g

(ρs

ρ− 1

)d = 0.

(1.25)

The bisection method is a very robust technique forfinding roots, as it will always converge to a solution.One problem that we will find with a number ofnumerical techniques is that they can fail to find thecorrect solution, or may even provide incorrect solutionsto the model.

Bisection works by choosing an interval in which thesolution is known to lie. Estimates are made of the lowerand the upper value of this interval. This choice caneasily be checked by calculating the function with thespecific values. We can find the root by successivelyhalving the interval, and comparing the midpoint withthe two end points to find which subinterval the root lies.The computational steps are as follows (Figure 1.15):

1. Set an iteration counter i equal to 1.2. Calculate the midpoint mi of the lower (li) and upper

(ui) estimates of the interval:

mi = 12 (li + ui)

40 Mark Mulligan and John Wainwright

0 0.05 0.1 0.15 0.2 0.25 0.3

-

l = l1

u = u1

vs m s−1

l1 m1

l2

u1

−0.03

−0.02

−0.01

0

0.01

0.02

0.03

0.04

f(u1)

f(l1)

f(v s

)

s

u2m2

l3 u3m3

l4 u4m4

Figure 1.15 Graph to show how the bisection technique can be used to solve a root numerically by taking an intervalabout the solution, and successively arriving at the solution, s, by subdivision of the interval it is known to be in. Thelower value of the interval is called l and the upper value u in this diagram. The numbered subscripts show the successiveiterations of these values. The function shown is Equation 1.25, used to calculate the settling velocity of a 1-mm diametersediment particle with a density of 2500 kg m−3

3. Calculate f(mi). If the value is zero, then the root hasbeen found. Otherwise . . .

4. Compare f(mi ) with f(li). If they have the same sign,then the root must lie between mi and ui , so setli+1 = mi and ui+1 = ui . If they have different signs,then the root must lie between li and mi , so setli+1 = li and ui+1 = mi .

5. If the interval between li+1 and ui+1 is acceptablysmall (i.e. less than a predefined tolerance, ε), thenaccept the midpoint as the root.

6. Otherwise, increment the value of i. If i is withina predefined limit, return to step 2 and repeatthe process.

The bisection method is useful because it always con-verges to a solution. It is thus useful even where quitesophisticated techniques will fail. The disadvantages ofthe technique are that it is very slow in convergingand that in the process we can be close to a goodapproximation at an intermediate stage in the processwithout realizing it. The efficiency of the algorithmcan be assessed because the number of iterations (n)required to achieve a result with a given toleranceε is:

n = log2ui − li

ε. (1.26)

The Newton–Raphson technique is far more efficient inthat it converges much more rapidly. However, problemswith the Newton–Raphson technique can arise (a) if theinitial estimate is too far away from the root; (b) ifthe function has a local maximum or minimum; or(c) in the case of some functions that can cause thealgorithm to enter an infinite loop without arriving ata solution. Numerous other techniques for root solution,together with discussion of their respective advantagesand limitations, can be found in Burden and Faires(1997) and Press et al. (1992).

1.5.6 More versatile solution techniques

In this section, we consider a range of techniques thatcan be used to solve environmental models where ana-lytical and simple iterative approaches are not appropri-ate. There are many variants of these techniques avail-able, but space here only permits a brief overview. Inall these cases, we assume that a model formalizationis available in the form of a differential equation, start-ing with straightforward examples, but then consideringmore commonly used and more complicated forms.

1.5.6.1 Finite differences

The finite difference approach assumes that a solu-tion to a differential equation can be arrived at by

Modelling and Model Building 41

Af(x − h)

P

B

x − h x + hx

f(x) f(x + h)

y=

f(x

)

Figure 1.16 Graph showing the derivation of the finitedifference technique for finding a solution to a functionat point x. The distance h is the ‘finite difference’ of thename, and the entire function can be solved by repeating thecalculations with x located at different places along the axis

approximation at a number of known, fixed points.These points are separated from each other by a known,fixed distance – the finite difference of the name of thetechnique (as compared with the infinitesimal distancesused to calculate derivatives). In a simple case, we canconsider how the technique works by looking at thefunction in Figure 1.16. To evaluate the rate of changein our model at point x, we need to find the slope of the

function f(x)

(i.e.

dy

dx

)at this point. If the distance (h)

between our calculation points is small and/or the func-tion does not change too dramatically, then the slope isbest estimated by the line AB, in other words, the slopebetween the values of the function at x + h and x – h.

This estimate is called the central difference approx-imation, and is given by the relationship:

f′(x) ≈ f(x + h) − f(x – h)

2h(1.27)

Similarly, the line PB is also an estimate of theslope at P. This estimate is the forward differenceapproximation with:

f′(x) ≈ f(x + h) − f(x)

h(1.28)

In the same way, the line AP is also an estimate,called the backward difference approximation, with:

f′(x) ≈ f(x) − f(x − h)

h(1.29)

It is also possible to define the second derivative atx using a similar approach. The central difference

approximation is:

f′′(x) ≈ f(x + h) − 2f(x) + f(x − h)

h2(1.30)

Note that in talking about the finite difference approach,the term approximation is fundamental (observe the useof ≈ meaning ‘approximately equal to’ in Equations1.27 to 1.30 and in general through the rest of thissection). We will see in a later section that evaluating theerrors inherent in models is also fundamental. The errorinvolved in these approximations can be assessed usinga technique called Taylor expansion. For the centraldifference approximations the error is related to valuesof h2f′′(x) + h3f′′′(x) + h4f′′′′(x) + . . .. This sequence iscommonly written as O(h2), which means ‘values ofthe order h2 and greater’. Because the value of thehigher order derivatives rapidly approaches zero, theapproximation is reasonable. Similarly, the error in thebackward difference approximations is O(h). Clearlythis is a much poorer approximation, but is valid in caseswhere the values change less rapidly, or if such error iswithin the bounds of acceptability.

We will commonly come across the case where thevariable we are looking at is a function of more than onevariable, e.g. y = f(x, t), so that the variable changesthrough space and time. In this case it is useful to thinkof the value on a regularly spaced grid, as shown inFigure 1.17. Our finite differences are given by distancesh, which is the increment in the x direction, and k, theincrement in the t direction. We define index parametersto describe points on the grid – i in the x direction andj in the t direction – which have integer values. Wewould then denote the value at any point on the gridP(ih, jk ) as:

yP = y(ih, jk) = yi,j (1.31)

The central difference approximation of∂y

∂tat P is:

∂y

∂t≈ yi,j+1 − yi,j−1

2k(1.32)

while the central difference approximation of∂y

∂xat P is:

∂y

∂x≈ yi+1,j − yi−1,j

2h(1.33)

Other derivations are given in Figure 1.17. Note the

change in notation fromdy

dxto

∂y

∂x. The first case, with

the normal d, denotes an ordinary differential equation(ODE); that is, the change we are interested in relatesto one variable only. The curly ∂ denotes a partial

42 Mark Mulligan and John Wainwright

P(ih, jk)

i, j

i, j −1

i − 1, j i + 1, j

i, j + 1

t

jk

ih

k = dt

h = dxx

Approximations of at P:

Backwardyi,j − yi,j − − −−1

k

Centralyi,j+1 yi,j 1

2k

Forwardyi,j+1 yi,j

k

Approximations of at P:

Backwardyi,j yi 1,j

h

Centralyi+1,j yi 1,j

2h

Forwardyi+1,j yi,j

h

Central difference approximation of2y

t2at P:

2y

t2

yi,j+1 2yi,j + yi,j 1

k2.

Central difference approximation of2y

x2at P:

2y

x2

yi+1,j 2yi,j + yi 1,j

h2.

− −

− −

− −

− −−

t

t

y

y

x y

x y

x y

x y

t y

t y

Figure 1.17 Finite difference approximations to partial differential equations

differential equation (PDE). Change in this exampleis partially related to two different variables – x andt – at the same time. There are three different types ofpartial differential equation – hyperbolic, parabolic andelliptic – as shown in Figure 1.18.

We will commonly want to solve two forms ofproblem. The first occurs as above where one of thevariables we are looking at is time, and we want to solvechange through time from a set of given conditions,known as initial values. These conditions will usuallybe prescribed for t = t0. On our grid, the solution wouldappear as in Figure 1.19.

This is called an initial value problem. Note that wealso have to specify the values immediately beyond our

grid so we can make statements about the solution of theproblem at the edges. These values are called boundaryconditions, and may be specific values, or functions ofvalues at the edges of the grid.

A less frequent problem involves two (or more)variables which do not include time and require thesolution of the function in terms of the variableat a single point in time over a finite are. Thevariables are commonly spatial coordinates, but maydefine another parameter space. A heat-flow problemmight be a common example of the former – eitherin two or three dimensions. In this case our gridappears as in Figure 1.20, and we have a boundary-value problem.

Modelling and Model Building 43

Partial Differential Equations

PDEs come in three different varieties, and the solution methods for each vary. The three types are:

1. hyperbolic – for which the one-dimensional wave equation is the prototype form:

∂2u

∂t2= v2 ∂2u

∂x2

where v is the velocity of wave propagation.2. parabolic – for which the diffusion equation is the prototype form:

∂u

∂t= ∂

∂tD

∂u

∂x

where D is the diffusion rate.3. elliptic – for which the Poisson equation is the prototype form:

∂2u

∂t2+ ∂2u

∂y2= r(x, y)

where r is a given source term. If r = 0, this is called Laplace’s equation.

Figure 1.18 Types of differential equation

t

t = 0x

Initial values

Boundaryconditions

Solution through time

Figure 1.19 Illustration of an initial value problem. Givenstarting conditions at time t = 0 and boundary conditionsbeyond the domain of interest, solution is progressivelycarried out for each spatial location at successive timeintervals on the finite difference grid

The most straightforward way of solving a finite dif-ference approximation to a partial differential equationis to use the backward difference technique (also calledEuler’s method in some texts). As an example, considerthe kinematic wave approximation to the flow of waterover a hillslope on which runoff is being generated at a

y

x

Boundaryvalues

Single solution atinterior grid nodes

Figure 1.20 Illustration of a boundary-value problem. Asingle (steady-state) solution is calculated across the grid,given known boundary values or boundary conditions

known rate:∂d

∂t+ v

∂d

∂x= e(x, t) (1.34)

where d is depth of water (L), t is time (T), v is flowvelocity (L T−1), x is horizontal distance (L) and e(x, t)

is the unit runoff rate at a given location and time (LT−1) (Singh, 1996: compare the general derivation of

44 Mark Mulligan and John Wainwright

Equation 1.19 above which is identical apart from thesource term has changed from q to e (x, t) to make itexplicitly relate to runoff from infiltration excess). Thisis an example of a hyperbolic equation. If we know theinitial condition:

d(x, 0) = g(x), x ≥ 0 (1.35a)

and the boundary condition

d(0, t) = b(t), t > 0 (1.35b)

then we can solve the equation using backward differ-ences. The backward difference expression for the firstterm is:

∂d

∂t≈ d(x, t) − d(x, t − k)

k+ O(k) (1.36)

and for the second term:

∂d

∂x≈ d(x, t − k) − d(x − h, t − k)

h+ O(h) (1.37)

Putting these together and ignoring the error terms, wehave a full solution, which is:

d(x, t) − d(x, t − k)

k

+ vd(x, t − k) − d(x − h, t − k)

h= e(x, t) (1.38)

As we know the runoff rate (possibly using a simplebucket model of infiltration as described above), and aretrying to solve for the new water depth, we rearrangethis equation to give:

d(x, t) = d(x, t − k) + kv

h{d(x, t − k)

− d(x − h, t − k)} + {e(x, t)k} (1.39)

Thus, to calculate the solution for a given surface,we define the grid of values in x and t , and carryout an iterative solution for x = ih, i = 1, 2, . . . , n, foreach value t = jk , j = 1, 2, . . . , N . The solution worksbecause at each time t , the solution depends only onknown values of d at the previous timestep t − k. Thesolution then follows from the initial condition. Thistype of solution is known as an explicit scheme. All thevalues needed for the calculations at a specific timestepare known explicitly because we are using a backwarddifference solution. Other details required for the fullsolution of this equation as applied to desert hillslopesare given in Scoging (1992) and Scoging et al. (1992).

Implications of its use are discussed in Parsons et al.(1997; see also Chapter 3).

An advantage of the backward difference schemegiven above is its simplicity, but it has two disad-vantages which can lead to problems in certain cir-cumstances. First, as noted above, the error terms arerelatively high, being O(h) + O(k). Second, the solu-tion can become unstable under specific conditions. Suchconditions are known as the Courant–Friedrichs–Lewystability criterion, which states that stable solutions willonly be produced if:

vk

h≤ 1 (1.40)

The practical implication of this is that for cases wherethe velocity, v, is high, the timestep k must be reduced tomaintain stability. Very short timesteps may mean thatthe computer time required becomes excessively highto solve the problem, and thus the algorithm may notbe appropriate in computer-resource terms, as discussedabove. Alternatively, the spatial resolution would haveto be compromised by increasing h, which may not beviable or useful in certain applications.

There are a number of other explicit techniques whichcan be used to solve similar equations, which improvethe accuracy of the solution by reducing the error terms.Alternatively, higher accuracy and reduced error may beobtained by using an implicit solution to the problem.In this case, there are several unknowns during the cal-culations at any one timestep. Solution of simultaneousequations and/or iterative calculations are thus requiredto arrive at a final solution. Further details can befound in a number of texts, such as Reece (1976), Smith(1985), Ames (1992) or Press et al. (1992).

Boundary conditions are a critical element of anynumerical solution. Basically, we must define whathappens at the edge of the solution space so that itinteracts realistically with the space around it. In somecases, the boundary conditions can be very simple. Forexample, when simulating water flows on hillslopes,the boundary at the top of a catchment will have nowater flux entering the solution domain because thereis nowhere for it to come from. At the base of thehillslope, flow may occur freely out of the system. Wemay set the gradient that controls the speed of this flowas equal to the gradient at the edge cell. Other ways ofcontrolling the behaviour at the edges of the solutiondomain are to define the values of boundary cellsas a function of time (Dirichelet boundary condition)or to specify the gradient at the boundary (Neumannboundary condition) (Press et al., 1992). For example,in our hillslope-hydrology case, a Dirichelet boundary

Modelling and Model Building 45

condition might be used if the channel flow is knownat the base of the slope as a function of time (perhapsusing an approximation such as the instantaneous unithydrograph [see Chapter 5]); the Neumann boundarycondition might be appropriate where there is a floodcontrol structure where the slope is fixed. Correctdefinition of boundary conditions is often fundamentalin controlling the correct solution of simulations, forexample by preventing the development of numericaloscillations or exponentially increasing values.

1.5.6.2 Finite elements

An alternative method for solving models of this typeis the finite element method. Rather than dividing thearea in which we are interested into a regular grid,finite elements allow us to use regular shapes (triangles,quadrilaterals) to represent the area of interest. It isalso possible to use cubes, cuboids and hexahedronsto represent volumes in three-dimensional models, andto have curved boundaries to the solution domain.Although it is possible to use a regular grid, it ismuch more common to vary the size of the elementsso that locations where change occurs rapidly arerepresented by much smaller areas. In this way, theeffects of errors in approximations may be minimized.This compares with the finite difference approach,where changes in grid size must take place uniformlyacross the same row or same column of the grid,allowing much less flexibility. A second advantagecomes with the way in which boundary conditionsare addressed. Because the boundary conditions aredirectly included in the formulation in the finite elementapproach, it is not necessary to approximate gradientsat the boundaries. On the other hand, finite elementcalculations can require much more computer memoryand computing power (see Smith and Griffiths, 1998,for example).

The finite element method is based on the fact that anyintegral over a specific range can be evaluated by addingthe integrals of subintervals making up that range (seethe discussion of definite integrals above) (Henwoodand Bonet, 1996). In one dimension, we have:∫ xn

x1

f(x) dx =∫ x2

x1

f(x) dx +∫ x3

x2

f(x) dx

+ · · · +∫ xn

xn−1

f(x) dx (1.41)

In other words, we can provide a solution over thewhole domain by splitting it into subdomains (elements)and using local solutions, or approximations to them, to

evaluate the whole. We need to solve the overall integralin a way that provides the best approximations to thelocal conditions. One way of doing this is to use theweighted residuals (R) of the function:∫

R · w dx = 0 (1.42)

where w are a series of weights, to provide a spatiallyaveraged best solution.

The simplest approach is to use approximations tolocal solutions of the elements using linear interpolation,using the ‘hat’ function (Figure 1.21). At any point i, thevalue of the hat function, Ni , using this approach is:

Ni =

0 if x ≤ xi−1

x − xi−1

xi − xi−1if xi−1 ≤ x ≤ xi

xi+1 − x

xi+1 − xi

if xi ≤ x ≤ xi+1

0 if x ≥ xi+1

(1.43)

We therefore have an approximation to function f(x) atany point within an element, based on known values atany point i as:

f(x) ≈ Nif(xi) + Ni+1f(xi+1) (1.44)

which is called the linear basis function for the approxi-mation. In the Galerkin approach to finite elements, theweights in Equation 1.42 are set as equal to the valuesof the basis function, i.e. w = Ni .

As an example, consider the case where we areinterested in the distribution of heat in a soil, for whichwe have a temperature measurement at the surface andat a known depth. Under steady-state conditions, thetemperature distribution will be described by a functionof the form:

∂

∂x

(−k

∂τ

∂x

)+ τ = 0 (1.45)

where τ is the temperature in ◦C and k the rate of heattransmission or thermal conductivity (W m−1 ◦C−1). Inthe finite element method, this function becomes theresidual term to be minimized, so that we insert thisequation into Equation 1.42:

∫ xn

0

{∂

∂x

(−k

∂τ

∂x

)w + τw

}dx = 0 (1.46)

Integration by parts (see Wilson and Kirkby, 1980) canbe used to transform this relationship into a set of

46 Mark Mulligan and John Wainwright

N1

N2

N3

N4

1 2 3 4x

1 2 3 4x

1 2 3 4x

1 2 3 4x

Ni

i − 1 x

(a)

(b)

(c)

i i + 1

dxdt

x = 0

kK11

t3t4

t2

t1K12 0 000

K21 K22 K23 00 K32 K33 K340 0 K43 K44

−

dxdt

x = 1

k

=

Figure 1.21 Solution method using finite elements: (a) hatfunction; (b) solution of by piece-wise interpolation usingsliding hat function; and (c) finite element matrix

ordinary differential equations:

∫ xn

0

(−k

dτ

dx

dw

dx+ τw

)dx =

[k

dτ

dxw

]xn

0

(1.47)

Next, we split the whole domain into a number of ele-ments – in this example, three – and apply the Galerkinestimate of the weights using the linear basis functionsand estimate the local value of τ using Equation 1.44.

Taking the element as having unit length, and multiply-

ing by a scaling factor

∣∣∣∣dx

dξ

∣∣∣∣, we have at node i:

τi

∫ 1

0

(k

dNi−1

dξ

dξ

dx

dNi

dξ

dξ

dx+ Ni−1Ni

) ∣∣∣∣dx

dξ

∣∣∣∣ dξ

=[k

dτ

dxNi

]1

0

(1.48)

where ξ (the Greek letter ‘xi’) refers to the fact that weare integrating along the element rather than the solutiondomain as a whole. For each node in the area of interest(Figure 1.21b) we now have a relationship based onthe values of Ni and Ni−1 that can be used to solve

within the elements. AsdNi−1

dξ= −1,

dNi−1

dξ= 1 and

dξ

dx= 3 (the number of elements), we can now define

directly the values of the integral used to multiply thevalues τi on the left-hand side of Equation 1.48. Thisprovides a matrix of values (Figure 1.21c) that is usuallyknown as the ‘stiffness’ matrix (because of the historicalorigins of the technique in structural analysis). It nowremains to solve the right-hand side of Equation 1.48by inserting the relevant weights. In all but the firstand last cases, these cancel, providing source and sinkterms at each of the boundaries of interest (the soilsurface and the location of the temperature probe inour case). Using our known boundary conditions atthese points (T1 is the temperature at the surface andT4 is the temperature at the probe), we produce a set ofsimultaneous equations:

τ1 = T1

K21τ1 + K22τ2 + K23τ3 = 0

K32τ2 + K33τ3 + K34τ4 = 0

τ4 = T4

(1.49)

that can be solved using standard techniques (see Presset al., 1992, for details). Finally, the values of τ1 canbe substituted into Equation 1.48 to give the heat fluxesinto the top of the soil and out of the base of the soil(Equation 1.49).

Finite elements are widely used in structural andengineering applications, including slope stability (seeChapter 10), fluid and heat-flow studies. More detailson the technique can be found in Henwood and Bonet(1996), Wait and Mitchell (1985), Smith and Griffiths(1998), Gershenfield (1999), Burden and Faires (1997)and Zienkiewicz and Taylor (2000), including informa-tion on how to model time-dependent conditions. Flow

Modelling and Model Building 47

modelling examples can be found in the articles byMunoz-Carpena et al. (1993, 1999), Motha and Wigham(1995) and Jaber and Mohtar (2002). Beven (1977) pro-vides a subsurface flow example as applied to hills-lope hydrology.

1.5.6.3 Finite volumes

The finite volume or control volume approach is sim-ilar to the finite element method in that it uses inte-gration over known areas, but differs in the solutionmethods used. Versteeg and Malalasekera (1995) pro-vide a very readable introduction to the technique, whichwe largely follow here. The area under study is firstdivided into a series of control volumes by locating anumber of nodes throughout it. The boundaries of thecontrol volumes are located midway between each pairof nodes (Figure 1.22). The process studied is inves-tigated by integrating its behaviour over the controlvolume as:∫ V

0f′(x)dV = A

i+ 12 δx

f(xi+ 1

2 δx) − A

i− 12 δx

f(xi− 1

2 δx)

(1.50)

which is the same as looking at the mass balance overthe control volume.

Using the soil heat-flow example of Equation 1.45again, the control-volume equation becomes:∫ V

0

∂

∂x

(−k

∂τ

∂x

)+ τ dV =

(−kA

dτ

dx

)i+ 1

2 δx

−(

−kAdτ

dx

)i− 1

2 δx

+ τ iV = 0 (1.51)

and we can use linear interpolation to calculate theappropriate values of k and finite differences to estimate

the values ofdτ

dxat the edges of the control volume.

dxwP

∆x = dxwe

dxPE

dxPe

dxWP

wW EP e

Figure 1.22 Discretization of the solution space in thefinite volume techniqueSource: Versteeg and Malalasekera, 1995

Assuming a uniform node spacing, these approxima-tions give:(

−kAdτ

dx

)i+ 1

2 δx

= −ki+1 + ki

2A

i+ 12 δx

(τi+1 − τi

0.5δx

)

(1.52a)(−kA

dτ

dx

)i− 1

2 δx

= −ki−1 + ki

2A

i− 12 δx

(τi − τi−1

0.5δx

)

(1.52b)

Substitution of these into Equation 1.51 gives:

− ki+1 + ki

2A

i+ 12 δx

(τi+1 − τi

0.5δx

)+ ki−1 + ki

2A

i− 12 δx(

τi − τi−1

0.5δx

)+ τiV = 0 (1.53)

Rearranging to give common terms in τi−1, τi and τi+1,we have:(ki+1 + ki

δxA

i+ 12 δx

+ ki−1 + ki

δxA

i− 12 δx

+ V

)τi

=(

ki−1 + ki

δxA

i− 12 δx

)τi−1 +

(ki+1 + ki

δxA

i+ 12 δx

)τi+1

(1.54)

Over the whole solution domain, this equation is againa set of simultaneous equations that can be solved inexactly the same ways as mentioned before.

Finite volume approaches are used commonly in com-putational fluid dynamics approaches (see Chapter 20).More detailed applications in this area can be foundin the books by Versteeg and Malalasekera (1995)and Ferziger and Peric (1999). Applications relatingto simpler flow models (such as the kinematic waveapproximation used in the finite difference example) canbe found in the articles by Gottardi and Venutelli (1997)and Di Giammarco et al. (1996).

1.5.6.4 Alternative approaches

There is increasing interest in models that do notdepend on numerical discretization as described in thesections above. Of these, the most common is thecellular automata approach. This method is dealt within detail in Chapter 21, so only a summary is givenhere. The advantages of cellular automata are that theycan represent discrete entities directly and can reproduceemergent properties of behaviour. However, becausethe number of entities simulated in realistic models ishigh, they can have a large computational load, although

48 Mark Mulligan and John Wainwright

special computer architectures that take advantage of thestructure of cellular automata have been used to speedthe process considerably (Gershenfield, 1999).

Cellular automata use a large number of cells or nodesthat can take a number of states. Each cell or node isprogressively updated using a fixed set of rules thatdepend only on local conditions. Thus at no stage isinformation on the whole solution space required. Themost commonly cited example is John Conway’s ‘Gameof Life’. A cell can either be ‘on’ or ‘off’, representinglife or no life in that location. There are three rules thatdetermine the state of a cell in the next iteration of themodel based on counting the number of occupied cellsin the immediate eight neighbours (Figure 1.23a):

1. ‘Birth’ – a cell will become alive if exactly threeneighbours are already alive.

2. ‘Stasis’ – a cell will remain alive if exactly twoneighbours are alive.

3. ‘Death’ – a cell will stop living under any otherconditions (0 or 1 neighbours reflecting isolation;or ≥ 4 neighbours reflecting overcrowding).

Compare these rules to the population models dis-cussed above. One of the interesting outcomes of thisapproach is that patterns can be generated that are self-reproducing.

Cellular automata have been used in a wide vari-ety of applications, in particular fluid flows. It is pos-sible for example to demonstrate the equivalence ofthe Navier–Stokes equations of fluid flow to cellularautomata approximations (Wolfram, 1986; Gershenfield,1999). More details of how to apply the method canbe found in Ladd (1995), Bar-Yam (2000) and Wolfram(2002). See Chapter 21 for a discussion of environmen-tal applications in more detail.

A further approach is the use of artificial neural net-works. An artificial neural network is a computer rep-resentation of the functioning of a human brain. It doesso by representing the neurons that are the fundamen-tal building blocks of the brain structure. A numberof inputs are passed through weighting functions rep-resenting synapses (Figure 1.24). The weighted valuesare summed and then affected by a local bias value.The result is passed to a function that decides whetherthe current neurone should be activated, and thus con-trols its output. The number of inputs and the numberof neurons and ways in which they are combined con-trol the function of the neural network. An importantfeature of neural networks is their ability to capture non-linear behaviour.

A common use of artificial neural networks isin rainfall-runoff modelling. A feed-forward model

structure is usually used, with a single hidden layer ofneurons (Figure 1.24). For example, Xu and Li (2002)used 29 input variables to estimate the next inflow intoa reservoir in Japan. These inputs were hourly averageupstream rainfall over seven hours, the seven hours ofprevious releases from two upstream reservoirs, the fourprevious hours of tributary inflow and the four previoushours of inflow into the reservoir. Xu and Li found rea-sonable predictions for small and medium-sized flows upto seven hours in advance. It is also possible to interpretthe weights on the different inputs and their combina-tions to suggest the relative importance of the differentinput parameters. Essentially, though the issues relatingto the use of artificial neural networks are those relatingto any data-based approach (see Chapter 22). There arelimits to what can be captured based on the data usedto train the neural network. It is unlikely that signif-icant numbers of extreme events will be available forforecasting of major events using this approach. Imrieet al. (2000) discuss these issues and suggest poten-tial improvements relating to restricting the values ofthe weighting function during the learning process, andensuring that the structure of the artificial neural networkis appropriate for all cases.

Haykin (1999), Bar-Yam (2000), Blum (1992), Wel-stead (1994) and Rogers (1997) provide extensivedetails of how to apply the approach. Other appli-cations in rainfall-runoff and flow modelling can befound in Hsu et al. (1995), Sajikumar and Thandav-eswara (1999), Shamseldin (1997) and Campolo et al.(1999). French et al. (1992) present an applicationto rainfall forecasting; Dekker et al. (2001) use theapproach in a forest transpiration model; Tang and Hsieh(2002) use it in atmospheric modelling; and De Falcoet al. (2002) have used it to predict volcanic eruptions.

Stochastic approaches can provide important simpli-fications, particularly when key elements of the systemare incompletely known. In a stochastic model, one ormore probability distributions are employed to repre-sent either deviations from known values of parametersor specific processes that have an apparently randombehaviour (at least at the scale of observation). As anexample of the former, consider the population-growthmodel discussed above in Equation 1.15. It has threeparameters – β the birth rate, θ the death rate and Nmax

the environmental carrying capacity. Measurements ofeach of these variables may show that they are not con-stant through time, but exhibit a natural variability. Birthand death rates may vary according to the incidence ofdisease, whereas the carrying capacity may vary accord-ing to climatic conditions (e.g. rainfall controlling cropproduction). Assuming that the deviations are normally

Modelling and Model Building 49

(a)

(b)

‘Birth’(three

neighbours)

‘Stasis’(two

neighbours)

‘Isolation death’(less than twoneighbours)

‘Overcrowding death’(more than three

neighbours)

Stable (bottom left) and unstable (becoming extinct: top right) forms

‘Blinkers’

‘Glider’

Figure 1.23 Conway’s ‘Game of Life’ cellular automata model: (a) rules; and (b) examples of how the rules operateon the local scale to produce stable, unstable and self-reproducing features

50 Mark Mulligan and John Wainwright

Nodeswith input

data

'Hidden' neuron layer

Output neurons

InputDendrites

Dendrites

Dendrites

Cell bodyCell body

Cell bodyAxon

Axon Axon

Neurons Computational unit

Output

(a)

(b)

Input

Output

Figure 1.24 (a) Structure of typical human neurons com-pared to that of a neuron used in an artificial neural net(from McLeod et al., 1998); and (b) example of an artifi-cial neural network model – a feed-forward model with asingle hidden layer of neurons

distributed, with standard deviations σβ, σθ and σNmax ,respectively, we could modify Equation 1.15 to give astochastic model with:

dN

dt= (β + n[σβ ])

(1 − N

Nmax + n[σNmax ]

)

N − (θ + n[σ θ ])N (1.55)

where n[σ ] is a normally distributed random numberwith mean 0 and standard deviation σ . As the valuesof σβ, σθ and σNmax increase, so does the variability inoutput. Example simulations are shown in Figure 1.25.The model is very sensitive to high variability incarrying capacity, where very variable environments areoften subject to population crashes.

The second type of stochastic model assumes thatthe best process description at a particular scale ofobservation is as a random process. This descriptionmay arise because there are too many different vari-ables to measure, or because it is impossible to pro-vide deterministic values for all of them because theyrequire very detailed and/or impractical measurements.

0

1000900800700600500400300200100

0 50 100 150 200 250 300Time years

Pop

ulat

ion

Nmax

Figure 1.25 Examples of ten stochastic simulations ofpopulation growth using Equation 1.55 (the stochastic ver-sion of Equation 1.15), with β = 0.05 individuals per year,σβ = 0.025 individuals per year, θ = 0.01 individuals peryear, σθ = 0.005 individuals per year, Nmax = 1000 indi-viduals and σNmax

= 100 individuals

This approach is commonly used when modelling cli-matic inputs into other models. For example, if the dailyrainfall input is required over 100 years for a site thatonly has a short data record, the stochastic characteris-tics of the data record can be used to provide a model.The rainfall process at this scale can be simply approxi-mated by stating that there is a distribution function forthe length of time without rain. This distribution func-tion is commonly a Poisson distribution, representingthe random occurrence of dry spells of a specific, dis-crete length of time. Once a day is defined as havingrainfall on it, a second distribution function is used todefine the amount of rainfall that is produced. Com-monly used distribution functions in this case are neg-ative exponential or gamma distributions, with a strongpositive skew representing the fact that most storms aresmall, with very few large events (see Wainwright et al.,1999a; Wilks and Wilby, 1999). Another approach tosimulating the length of dry spells is to use Markovtransition probabilities. These probabilities define thelikelihood that a dry day is followed by another dryday and that a wet day is followed by another wet day(only two probabilities are required for the four possi-ble combinations: if pdd is the probability that a dryday is followed by a dry day, then pdw = 1 − pdd is theprobability that a wet day follows a dry day; similarly,if pww is the probability that a wet day is followedby another wet day, then pwd = 1 − pww is the prob-ability that a dry day follows a wet day). The Markovapproach can be extended to define probabilities that adry day is followed by two dry days, or a wet day is

Modelling and Model Building 51

followed by two wet days and so on. This approachis useful if the climate system can be thought to have a‘memory’. For example, in regions dominated by frontalrainfall, it is likely that there will be periods of contin-uously wet weather, whereas thunderstorms in drylandsevents generally occur on particular days and changeatmospheric conditions such that a further thunderstormis unlikely. Note that it is still possible to interpretthe meaning of such stochastic models in a physicallymeaningful way. Further details on stochastic rainfallgenerators can be found in Wilks and Wilby (1999).Markov models are described in more detail in Collins(1975) and Guttorp (1995), with other applications insedimentology (e.g. Parks et al., 2000), mapping (e.g.Kovalevskaya and Pavlov, 2002) and ecology (e.g. Hillet al., 2002) among others.

Another commonly employed example of a stochas-tic model is related to sediment transport. Although thebalance of forces on individual particles is generallywell understood (e.g. Middleton and Southard, 1984),there are a number of factors that make it impracti-cal to measure all the necessary controls in an opera-tional way. For example, the likelihood of movementdepends on the way in which a stone sits on the bedof a channel. If there is a variable particle size (whichthere is in all but the best-controlled laboratory experi-ments), then the angle at which the stone sits will varyaccording to the size of the stones in the bed and theirorientation. Similarly, river flows exhibit turbulence,so that the force exerted on the stone is not exactlypredictable. Einstein (1942) defined an approach thatdefines the sediment-transport rate as a function of thetime periods over which sediment particles do not moveand a function of the travel distance once a particle startsmoving (see Raudkivi, 1990, for a derivation). Naden(1987) used this approach successfully to simulate thedevelopment of bedforms in gravel-bed rivers. A mod-ified version of this approach was used by Wainwrightet al. (1999b) to simulate the development of desertpavements.

It is a common misconception that stochastic modelswill always be faster than corresponding determinis-tic models. This situation is not necessarily the casebecause there is often a high computational overheadin generating good random numbers with equal likeli-hood that all values appear and that successive valuesare not correlated or repeated over short time inter-vals. In fact, it is impossible for a computer to generatea true random number. Most random-number genera-tors depend on arithmetic manipulations that ensure thatthere are cycles of numbers that repeat but that the num-ber over which they repeat is very large. The outputs are

called pseudo-random numbers. Generating and evalu-ating good pseudo-random numbers are dealt with indetail in Press et al. (1992) and Kloeden et al. (1994).Problems in certain packages are presented in Sawitzki(1994) and McCullough and Wilson (2002).

1.6 MODEL PARAMETERIZATION,CALIBRATION AND VALIDATION

1.6.1 Chickens, eggs, models and parameters?

Should a model be designed around available measure-ments or should data collection be carried out only oncethe model structure has been fully developed? Manyhardened modellers would specify the latter choice asthe most appropriate. The parameters that are requiredto carry out specific model applications are clearly bestdefined by the model structure that best represents theprocesses at work. Indeed, modelling can be used in thisway to design the larger research programme. Only bytaking the measurements that can demonstrate that theoperation of the model conforms to the ‘real world’ isit possible to decide whether we have truly understoodthe processes and their interactions.

However, actual model applications may not be sosimple. We may be interested in trying to reconstructpast environments, or the conditions that led to catas-trophic slope collapse or major flooding. In such cases,it is not possible to measure all of the parameters of amodel that has a reasonable process basis, as the condi-tions we are interested in no longer exist. In such cases,we may have to make reasonable guesses based on indi-rect evidence. The modelling procedure may be carriedout iteratively to investigate which of a number of recon-structions may be most feasible.

Our optimal model structure may also produce param-eters that it is not possible to measure in the field setting,especially at the scales in which they are representedin the model. The limitations may be due to cost, orthe lack of appropriate techniques. It may be necessaryto derive transfer functions from (surrogate) parametersthat are simpler to measure. For example, in the caseof infiltration into hillslopes, the most realistic resultsare generally obtained using rainfall simulation, as thisapproach best represents the process we are trying toparameterize (although simulated rain is never exactlythe same as real rain – see Wainwright et al., 2000,for implications). However, rainfall simulation is rela-tively difficult and expensive to carry out, and generallyrequires large volumes of water. It may not be feasi-ble to obtain or transport such quantities, particularly inremote locations – and most catchments contain some

52 Mark Mulligan and John Wainwright

remote locations. Thus, it may be better to parameter-ize using an alternative measurement such as cylinderinfiltration, or pedo-transfer functions that only requireinformation about soil texture. Such measurements maynot give exactly the same values as would occur underreal rainfall, so it may be necessary to use some formof calibration or tuning for such parameters to ensureagreement between model output and observations. Inextreme cases, it may be necessary to attempt to cali-brate the model parameter relative to a known outputif information is not available. We will return to theproblems with this approach later.

Parameterization is also costly. Work in the fieldrequires considerable investment of time and generallyalso money. Indeed, some sceptics suggest that theresearch focus on modelling is driven by the need tokeep costs down and PhDs finished within three years(Klemes, 1997). Equipment may also be expensive and,if it is providing a continuous monitored record, willneed periodic attention to download data and carry outrepairs. Therefore, it will generally never be possible toobtain as many measurements as might be desirable inany particular application. As a general rule of thumb,we should invest in parameter measurement accordingto how great an effect the parameter has on the modeloutput of interest. The magnitude of the effect ofparameters on model output is known as the sensitivityof a model to its parameters. This important stage ofanalysis will be dealt with in more detail below.

1.6.2 Defining the sampling strategy

Like models, measurements are also abstractions ofreality, and the results of a measurement campaign willdepend as much upon the timing, technique, spatialdistribution, scale and density of sampling as on thereality of the data being measured. As in modelling,it is imperative that careful thought is put into theconceptualization and design of a sampling strategyappropriate to the parameter being measured and theobjective of the measurement. This is particularly truewhen the sampled data are to be used to parameterizeor to validate models. If a model under-performs interms of predictive or explanatory power, this can bethe result of inappropriate sampling for parameterizationor validation as much as model performance itself. Itis often assumed implicitly that data represents realitybetter than a model does (or indeed that data is reality).Both are models and it is important to be critical of both.

We can think of the sampling strategy in termsof (a) the variables and parameters to be measuredfor parameterization, calibration and validation; (b) the

direct or indirect techniques to be used in measurementand their inherent scale of representation; (c) the spatialsampling scheme (distributed, semi-distributed, lumped)and its density; and (d) the temporal sampling scheme(duration and temporal resolution). Choosing whichvariables will be measured for parameterization andthe intensity of measurement will depend very muchon the sensitivity of the significant model outputs tothose (see below). Highly sensitive parameters shouldbe high on the agenda of monitoring programmesbut since model sensitivity to a parameter is usuallydependent also on the value of other parameters, thisis not always as straightforward as it might at firstappear. Where variables are insensitive, either theyshould not be in the model in the first place ortheir measurement can be simplified to reflect this.Calibration parameters should be as much as possiblethose without physical meaning so as not to compromisethe physical basis of the model and their measurementwill be necessary for the application of models tonew environments or epochs. Validation parametersand variables should be those which are the criticalmodel outputs in terms of the purpose of the model.A robust validation of the key model output wouldtend to indicate that the model has performed well ina predictive sense. This outcome does not mean thatthe results have been obtained for the correct reasons,in other words, good prediction is no guarantee ofgood explanation. In this way, if one were to validatethe output of a catchment hydrological model usingmeasured discharge data and obtain good agreementbetween model and data, this success can come aboutas the result of many different configurations of thedriving variables for discharge. It is thus importantto validate the output required but also some internalvariable which would indicate whether that outputhas been arrived at for the correct reasons, in thiscase the spatial distribution of soil moisture aroundthe catchment.

The techniques used for measurement will dependupon a number of logistic constraints such as avail-ability, cost, dependence on power supplies, trainingrequired for use and safety, but must also depend uponthe spatial and temporal structure of the model forwhich these techniques will provide data since it isimportant that the model and the data are represent-ing the same thing. A good example is soil mois-ture. Soil is a three-phase medium consisting of thesoil matrix, rock fragments greater than 2 mm in diam-eter and porosity. Soil moisture occupies the porositywhich is usually around half of the soil volume. Inmany soils, rock-fragment content can be in excess of

Modelling and Model Building 53

30% (van Wesemael et al., 2000) and while rock frag-ments sometimes have a small porosity, it is usuallypretty insignificant for the purposes of moisture reten-tion. Volumetric measurement of soil moisture usuallyprovides an output of m3 water per m3 soil fine fractionwhich does not usually contain rock fragments. Thesetend to be avoided in the installation of electronic sen-sors of soil moisture and not accounted for in calibration,and which tend to be avoided or sieved out of gravimet-ric samples. Soil-moisture measurements are usually anaggregate of small sample measurements of the fine soilfraction. However, soil tends to be represented as largeblocks with dimensions of tens to hundreds of metresin hydrological models. This must therefore incorporatea significant loss of available porespace because of thepresence of rock fragments and thus the nature of soilmoisture at this scale is quite different to that at thepoint scale of measurement. The need to balance dataand model attributes is particularly clear where indirectmeasurements, in particular remote sensing, are used formodel parameterization.

Over recent decades there has been a move awayfrom lumped models in which spatial heterogeneity isnot represented towards distributed models in which itis. Advances in computing power and GIS technolo-gies have enabled the development of complex spatialmodels based on the discretization of landscapes intovector polygons, triangular irregular networks, objectsof complex form or simple raster grids. Despite recentadvances in remote sensing, there are still very manyparameters that cannot be measured using electromag-netic radiation and thus remote sensing. The sophistica-tion of spatial models has rapidly outgrown our abilityto parameterize them spatially and they thus remain con-ceptually lumped (Beven, 1992). The appropriate scaleof distribution and the optimum configuration of mea-surements for model parameterization or calibration arethe subject of much debate. For example, Musters andBouten (2000) used their model of root-water uptaketo determine the optimal sampling strategy for thesoil-moisture probes used to parameterize it. Fieldworkis an expensive, labour-intensive, time-consuming andsometimes uncomfortable or even hazardous activity.Traditional random or structured sampling proceduresusually require that a very large number of samples becollected in order to fulfil the assumptions of statisticalinference. In order to reduce the sampling effort, priorknowledge about the system under study may be used toguide convenience or nonrandom sampling which is stillstatistically viable, with the appropriate method depend-ing on the type of prior knowledge available (Modeet al., 2002). Ranked set sampling (Mode et al., 1999)

reduces the cost of sampling by using ‘rough but cheap’quantitative or qualitative information to guide the sam-pling process for the real, more expensive samplingprocess. Chao and Thompson (2001) and others indi-cate the value of optimal adaptive sampling strategiesin which the spatial or temporal sampling evolves overtime according to the values of sites or times alreadysampled. A number of authors indicate how optimalsampling can be achieved by algorithmic approacheswhich maximize entropy in the results obtained (e.g.Bueso et al., 1998; Schaetzen et al., 2000). The lux-ury of optimizing your sampling scheme in this wayis, however, not always available to the modeller, espe-cially within the context of policy models which areapplied using existing datasets generated by governmentagencies, for example, where ‘you get what you aregiven’ and which may not be collected with uniformor standard protocols (e.g. as outlined for soils data inSpain by Barahona and Iriarte, 2001) or where the pro-tocol may evolve over time, affecting the legitimacyof time-series analysis. Usually the spatial samplingscheme chosen is a compromise between that whichbest represents the system under review and the compu-tational resources and data available. This compromiseis most clearly seen in the extensive discussions on theproblem of grid size and subgrid variability in generalcirculation models (GCMs). May and Roeckner (2001),among others, indicate the importance of grid resolutionin affecting the results of GCMs. Smaller grid sizes pro-duce more realistic results, especially in highly moun-tainous areas, but smaller grids also have substantiallyhigher computational and data costs.

Wainwright et al. (1999a) indicated the importance ofthe temporal detail of climate data for accurate hydro-logical modelling. The calculation of evapotranspirationusing the Penman–Monteith formula for hourly dataand then the same data aggregated to a single valuefor each day and then separately for each day and nightindicates that the day–night aggregation produces muchcloser results to the original hourly data than does thedaily aggregation because of the domain change in netradiation values from daylight hours when they are pos-itive to night-time hours when they are negative. Theerror induced by aggregation to daily timestep is of theorder of 100% and varies with the month of the yeartoo. This indicates that one must pay attention to thenatural scales and boundaries of the processes beingmodelled when devising the time (or space) scale forsampling. Similarly, Mulligan (1998) demonstrated theimportance of high temporal resolution rainfall inten-sity data in understanding the partitioning between infil-tration and overland flow. Where soil infiltration rates

54 Mark Mulligan and John Wainwright

fall within the range of measured instantaneous rain-fall intensities (as they often do), it is important tounderstand rainfall intensities as the distribution functionof instantaneous intensities. The greater the timescaleover which these intensities are aggregated, the lowerthe measured intensity would be. Such aggregation canhave major effects on the predicted levels of Hortonianor infiltration-excess overland-flow production – whichis, after all, a threshold process (see Wainwright andParsons, 2002, for spatial implications). Hansen et al.(1996) suggested the importance of data quality in deter-mining streamflow prediction for the lumped IHACRESrainfall-runoff model to conclude that rain-gauge densityand the sampling interval of rainfall are the most crit-ical across a range of catchments. Understanding thesesensitivities is critical to designing an appropriate sam-pling scheme.

1.6.3 What happens when the parametersdon’t work?

It is frequently the case that initial parameter estimateswill produce model outputs that are incompatible withthe known behaviour of the system. There are usuallygood reasons for this outcome, so do not give up!Given that the parameter base for distributed modelsis generally small relative to the detail simulated, it isperhaps not surprising. Similarly, lumped models havea sparse parameter representation relative to naturalvariability in the system. Point measurements or spatialaverages are often poor representations of the parameterinteractions in this case. Evaluating errors from thesesources will be dealt with later.

However, in terms of model parameterization, it maybe impossible to return to the field to carry out moremeasurements, but we still need to obtain results forour model application. Thus, we need to adopt aniterative approach to the evaluation of the correct modelparameters. This procedure is generally known as modelcalibration.

1.6.4 Calibration and its limitations

Kirkby et al. (1992) distinguish between physical param-eters, which define the physical structure of the systemunder study, and process parameters, which define theorder of magnitude of processes. Most models will con-tain both types of parameter. Definition of these processparameters is known as calibration or model tuning.Where they are physically based, this can be achieved bytheir measurement, where not, they are calibrated using a

process of optimization (optimized) against a measure ofthe agreement between model results and a set of obser-vations used for calibration. The calibration dataset mustbe independent from any dataset which is used later tovalidate the model, if the same dataset is used for bothit should be no surprise that the model is a perfect pre-dictor! Split sample approaches, in which the availabledata is separated into a calibration set and a separatevalidation set, are usually the solution to this problem.

Calibration should pay particular attention to the sen-sitivity of parameters with sensitive parameters beingcalibrated carefully against high quality datasets toensure that the resulting model will produce reliable out-comes. The simplest form of optimization is trial anderror whereby model parameters are altered and a mea-sure of goodness of fit between model results and thecalibration dataset is noted. This process is repeated iter-atively to obtain the best possible fit of observed againstpredicted. Of course the calibration will be specific tothe model results calibrated against and will producea model which should forecast this result well at theexpense of other model outputs not involved in thecalibration procedure. The choice of calibration param-eters, measures and techniques will thus depend uponthe purpose to which the model will be put. Moreover,a model calibration by one user with a particular under-standing of its function may be quite different fromthat of another (Botterweg, 1995) and a model cali-brated to a particular objective such as the predictionof peak runoff may then be useless in the predictionof total annual runoff. Some prior knowledge of, forexample, the reasonable ranges of parameter values, willalso be necessary and calibration will usually follow apreliminary sensitivity or uncertainty analysis which isperformed to test the validity of the model. The relation-ship between the range of values for a parameter andthe model agreement is known as the calibration curvefor that parameter. A parameter that shows a signifi-cant change in error with a change in its value (with allother parameters held constant) is known as a sensitiveparameter. If a model has only one parameter, it is usu-ally fairly straightforward to find the optimal value forthat parameter. This procedure becomes only marginallymore difficult for models with more than one param-eter where the parameters are independent. In mostmodels, parameters are highly interdependent and thiswill confound the definition of an optimum parameter-ization. In these cases other – automated – techniquesare used to define the optimal parameter set. Thesetechniques include genetic algorithms and fuzzy logicapproaches as used to calibrate a rainfall-runoff modelto multiple objectives (peak discharge, peak time and

Modelling and Model Building 55

total runoff volume) by Cheng et al. (2002). Calibra-tion is particularly challenging in distributed modelswhich tend to have a large number of parameters.Stochastic or evolutionary genetic algorithms seem tobe the most successful approaches under these circum-stances (Eckhardt and Arnold, 2001) and have beenapplied widely where there are multiple objectives of thecalibration (Madsen, 2000; 2003). In distributed models,there may also be advantages of calibrating differentareas, such as subcatchments, separately and indepen-dently rather than as an integrated whole (e.g. Seibertet al., 2000). Ratto et al. (2001) highlight the utility ofthe global sensitivity analysis (GSA) and generalizedlikelihood uncertainty estimation (GLUE) approaches(see below) in the calibration of over-parameterizedmodels with strong parameter interaction. GSA is amodel-independent approach, which is based on esti-mating the fractional contribution of each input factorto the variance in the model output, accounting also forinteraction terms. GLUE allows model runs to be clas-sified according to the likelihood of their being a goodsimulator of the system, recognizing that many differentcombinations of parameter values can produce accuratemodel results (the issue of equifinality). By applyingGSA to the GLUE likelihoods, the parameter sets driv-ing model simulations with a good fit to observations areidentified along with the basic features of the parameterinteraction structure. Extensive model calibration tendsto remove the physical basis of a model. Part of theobjective in building a physically based model shouldbe to produce a sufficiently good model structure andconceptualization to avoid the need for substantial cali-bration.

1.6.5 Testing models

The terms verification and validation have very specificmeanings in computer science. Verification is used todenote the process of checking that the computer code(program) does exactly what the algorithm is designedto do. As well as a formal confirmation, the procedurealso involves the removal of bugs that have crept intothe program during its writing (due to typing mistakes aswell as mis-conceptualizations). Validation, on the otherhand, refers to the testing of the model output to confirmthe results that should be produced in reality (Fishmanand Kiviat, 1968). One common method of validationis the comparison of a numerical model against theanalytical solution for specific boundary conditions oragainst field measured data for the period and place ofthe model simulation.

However, Oreskes et al. (1994) point out that thedifference between these specific uses and the commonusage of the same terms can often lead to confusion,particularly when model results are being presented tononmodellers. Rykiel (1996) suggests that the terms areessentially synonymous in everyday language, so thedistinction is hard to see to a nonuser. Furthermore, theroots of the words may imply that a model is betterthan was actually intended when the author of a papernoted that the model was verified and validated. Theroot meaning of verify comes from the Latin verus,meaning truth, while the Latin validare means to declareor give legal authority to something. Thus Oreskes et al.(1994) suggest that the nonmodeller may tend to feela verified model presents the truth, and one that isvalidated can have legal authority, or is at least ‘doesnot contain known or detectable flaws and is internallyconsistent’. They suggest that benchmarking is a moreappropriate term for verification and model evaluationshould be used in place of validation. However, it couldbe argued that these are equally value-laden terms. Inreality, most model output actually seems to generatea healthy dose of scepticism in nonmodellers (see thedebate in Aber, 1997, 1998; Dale and Van Winkle,1998; Van Winkle and Dale, 1998, for example). Laneand Richards (2001), on the other hand, suggest thatvalidation is used as a linguistic means of hiding fromsuch criticism.

Much more fundamentally in this debate, the natureof environment systems and scientific practice meansthat whatever term is used for validation/model eval-uation, it will always tend to overstate the case forbelief in the model results. There are six reasons statedby Oreskes et al. (1994) for this problem. First, allenvironmental systems are open. Logically, it is onlypossible to demonstrate the truth of a closed system(although even this proposition is called into ques-tion by Godel’s theorem – see the excellent overviewby Hofstadter, 1979). Second, there are problems dueto the presence of unknown parameters and the scalingof nonadditive parameters (see below and Chapter 19).Third, inferences and embedded assumptions under-lie all stages of the observation and measurement ofvariables – dependent and independent alike. Fourth,most scientific theories are developed by the addi-tion of ‘auxiliary hypotheses’, that is, those not centralto the principal theory, but fundamental in a specificcontext for putting it into action. Thus, it is impos-sible to tell whether the principal or an auxiliaryhypothesis is incorrect should deductive verification fail.Fifth, as we have seen, more than one model formu-lation can provide the same output. This property is

56 Mark Mulligan and John Wainwright

known formally as nonuniqueness or underdetermina-tion (the Duhem–Quine thesis: Harding, 1976). Sixth,errors in auxiliary hypotheses may cancel, causing incor-rect acceptance of the model. Many modellers wouldnow accept that full validation is a logical impossi-bility (e.g. Refsgaard and Storm, 1996; Senarath et al.,2000). Morton and Suarez (2001) suggest that in mostpractical contexts the term ‘model’ can be thought of asbeing synonymous with ‘theory’ or ‘hypothesis’, withthe added implication that they are being confronted andevaluated with data. Often the models represent simplifi-cations of complex, physically based theories, analogiesof other systems, summaries of data, or representationsof the theories themselves. It is this set of approachesthat allows the provisional nature of scientific knowl-edge to be tested. Conversely, it is possible for modelsto continue being used for a range of reasons relatingto the social, economic and political contexts of sci-ence (Oreskes and Bellitz, 2001).

Rykiel (1996) provides an overview of how valida-tion has been employed in modelling, and distinguishes(a) operational or whole-model validation (correspon-dence of model output with real-world observations);(b) conceptual validation (evaluation of the underlyingtheories and assumptions); and (c) data validation (eval-uation of the data used to test the model). He suggeststhat there are at least 13 different sorts of validationprocedure that are commonly employed, explicitly orimplicitly. These procedures are:

(a) face validation – the evaluation of whether modellogic and outputs appear reasonable;

(b) Turing tests – where ‘experts’ are asked to distin-guish between real-world and model output (byanalogy with the test for artificial intelligence);

(c) visualization techniques – often associated with astatement that declares how well the modelledresults match the observed data;

(d) comparison with other models – used for examplein general circulation model evaluations (althoughnote the high likelihood of developing an argumentbased on circular logic here!);

(e) internal validity – e.g. using the same data setrepeatedly in a stochastic model to evaluate whetherthe distribution of outcomes is always reasonable;

(f) event validity – i.e. whether the occurrence andpattern of a specific event are reproduced bythe model;

(g) historical data validation – using split-sample tech-niques to provide a subset of data to build a modeland a second subset against which to test the modelresults (see also Klemes, 1983);

(h) extreme-condition tests – whether the model be-haves ‘reasonably’ under extreme combinationsof inputs;

(i) traces – whether the changes of a variable throughtime in the model are realistic;

(j) sensitivity analyses – to evaluate whether changesin parameter values produce ‘reasonable’ changesin model output (see below);

(k) multistage validation (corresponding to the stagesa, b and c noted above);

(l) predictive validation – comparison of model outputwith actual behaviour of the system in question;

(m) statistical validation – whether the range of modelbehaviour and its error structure match that of theobserved system (but see the discussion on errorpropagation below).

Clearly, all these tests provide some support for theacceptance of a model, although some are more rigorousthan others. The more tests a model can successfullypass, the more confidence we might have in it, althoughthis is still no reason to believe it absolutely forthe reasons discussed above. But in complex models,validation is certainly a nontrivial procedure – Brownand Kulasiri (1996: 132) note, for example, that ‘a modelcan be considered to be successfully validated if allavailable techniques fail to distinguish between field andmodel data’. Any model test will in part be evaluatingthe simplifications upon which the model is based,and in part the reasonableness of its parameterization.However, if a number of parameterizations fail fora specific model, we might seriously reconsider itsconceptual basis. As with other aspects of modelling,evaluation is an iterative process.

1.6.6 Measurements of model goodness-of-fit

Calibration and validation or model evaluation as out-lined above all require some measurement of how wellthe model represents actual measurements. These mea-surements are often known as objective functions, orgoodness-of-fit statistics. It should be recognized thatthere are a number of different goodness-of-fit mea-sures, and they will each be sensitive to different aspectsof model (mis.)behaviour. The choice of an appropriatemeasure is therefore vital in a robust model evaluation.In the following discussion, we use Mi as a sequence ofmodel outputs to be compared to Oi observed systemvalues, M is the mean model output and O the meanobserved value.

We can use the coefficient of determination, r2, torepresent the proportion of variance in the observed

Modelling and Model Building 57

data explained by the model results. The value iscalculated as:

r2 =

n∑i=1

[Oi − O] · [Mi − M]

√√√√ n∑i=1

[Oi − O]2 ·

√√√√ n∑i=1

[Mi − M]2

2

(1.56)

or the ratio of the covariance in the observed andmodelled values to the product of the individual vari-ances. The value of r2 varies from 1, which meansall of the variance in the data is explained by themodel, to 0, where none of the variance is explained.As noted by Legates and McCabe (1999), this mea-sure is insensitive to constant, proportional deviations.In other words, perfect agreement occurs if the modelconsistently underestimates or overestimates by a spe-cific proportion, so that there is a linear relationshipbetween Mi and Oi which does not lie on the 1:1line (compare the use of the coefficient of determina-tion in evaluating regression models). As correlation isnot a robust statistical measure, the value of r2 willbe sensitive to outliers (values that lie a large dis-tance away from the mean), which will tend to increaseits value spuriously if there are extreme events in theobserved dataset.

The Nash and Sutcliffe (1970) model-efficiency mea-sure is commonly used:

NS = 1 −

n∑i=1

(Oi − Mi)2

n∑i=1

(Oi − O)2

(1.57)

which is a measure of the mean square error to theobserved variance. If the error is zero, then NS = 1,and the model represents a perfect fit. If the error is thesame magnitude as the observed variance, then NS = 0and the observed mean value is as good a representa-tion as the model (and thus the time invested in themodel is clearly wasted). As the error continues toincrease, the values of the index become more negative,with a theoretical worst-case scenario of NS = −∞. NSdoes not suffer from proportional effects as in the caseof r2, but it is still sensitive to outliers (Legate andMcCabe, 1999).

A modified ‘index of agreement’ was defined asthe ratio of mean-square error to total potential error

by Willmott (1981):

W = 1 −

n∑i=1

(Oi − Mi)2

n∑i=1

(|Mi − O| + |Oi − O|)2

(1.58)

where 1 again means a perfect model fit, but the worstmodel fit is represented by a value of 0. As with the twoprevious measures, the squared terms make it sensitiveto outliers. Legates and McCabe suggest that it may bebetter to adjust the comparison in the evaluation of NSand W to account for changing conditions, so that ratherthan using the observed mean, a value that is a functionof time (e.g. a running mean) is more appropriate ifthe model covers highly variable situations. It may alsobe desirable to calculate absolute errors such as theroot-mean square (RMSE : see below) and the meanabsolute error:

MAE =

n∑i=1

|Oi − Mi |

n(1.59)

and to use the extent to whichRMSE

MAE> 1 as an indica-

tor of the extent to which outliers are affecting the modelevaluation. In a comparison of potential evapotranspi-ration and runoff models, Legates and McCabe foundthe best way of evaluating the results was to ignorer2, and to use baseline-adjusted NS and W indices,together with an evaluation of RMSE and MAE. Otherauthors have suggested that it is also useful to pro-vide a normalized MAE, by dividing by the observedmean (thus giving a ratio measurement – compare thenormalized RMSE discussed below) (Alewell and Man-derscheid, 1998). The effects of outliers can be min-imized by making the comparison on values below aspecific threshold (e.g. Madsen, 2000) or by using log-transformed values.

The choice of appropriate objective functions andmodel-evaluation procedures is a critical one. UsingBayesian analysis, Kirchner et al. (1996) demonstratehow insufficiently strict tests will cause a reasonablesceptic to continue to disbelieve a model, while themodel developer may continue regardless. Only if amodel has a chance of 90% of being rejected whenit is a poor representation of observed values will thetesting procedure be found convincing. As with allaspects of science, we must endeavour to use the mostrigorous tests available that do not wrongly reject themodel when the data are at fault.

58 Mark Mulligan and John Wainwright

1.7 SENSITIVITY ANALYSIS AND ITS ROLE

Sensitivity analysis is the process of defining modeloutput sensitivity to changes in its input parameters.Sensitivity analysis is usually carried out as soon asmodel coding is complete and at this stage it hastwo benefits: to act as a check on the model logicand the robustness of the simulation and to define theimportance of model parameters and thus the effortwhich must be invested in data acquisition for differentparameters. The measurement of the sensitivity of amodel to a parameter can also be viewed relative tothe uncertainty involved in the measurement of thatparameter in order to understand how important thisuncertainty will be in terms of its impact on the modeloutcomes. If sensitivity analysis at this stage indicatesthat the model has a number of parameters to whichthe model is insensitive, then this may indicate over-parameterization and the need for further simplificationof the model.

Sensitivity analysis is usually also carried out whena model has been fully parameterized and is often usedas a means of learning from the model by understand-ing the impact of parameter forcing, and its cascadethrough model processes to impact upon model out-puts (see, for example, Mulligan, 1996; Burke et al.,1998; Michaelides and Wainwright, 2002). In this waythe behaviour of aggregate processes and the nature oftheir interaction can better be understood. After cal-ibration and validation, sensitivity analysis can alsobe used as a means of model experiment and this isvery common in GCM studies where sensitivity exper-iments of global temperature to greenhouse forcing,to large-scale deforestation or to large-scale deserti-fication are common experiments. Sensitivity analy-sis is also used in this way to examine the impactsof changes to the model structure itself, its bound-ary or initial conditions or the quality or quantity ofdata on its output (for example, May and Roeckner,2001).

The sensitivity of model parameters is determinedby their role in the model structure and if this roleis a reasonable representation of their role in the sys-tem under study, then there should be similaritiesbetween the sensitivity of model output to param-eter change and the sensitivity of the real systemresponse to physical or process manipulation. Never-theless one must beware of attributing the model sen-sitivity to parameter change as the sensitivity of thereal system to similar changes in input (see Baker,2000).

The methods of sensitivity analysis are covered insome detail by Hamby (1994) and, more recently, by

Saltelli et al. (2000) and will not be outlined in detailhere. In most sensitivity analyses a single parameter isvaried incrementally around its normal value, keepingall other parameters unaltered. The model outputs ofinterest are monitored in response to these changes andthe model sensitivity is usually expressed as the propor-tional change in the model output per unit change inthe model input. In Figure 1.26 we show an examplesensitivity analysis of the simple soil-erosion model(Equation 1.1), first in terms of single parameters andthen as a multivariate sensitivity analysis. The for-mer demonstrates the relative importance of vegeta-tion cover, then slope, runoff and finally soil erodi-bility in controlling the amount of erosion accordingto the model. The multivariate analysis suggests thatspatially variable parameters can have significant andsometimes counter-intuitive impacts on the sensitivityof the overall system. A sensitive parameter is one thatchanges the model outputs of interest significantly perunit change in its value and an insensitive parameteris one which has little effect on the model outputs ofinterest (though it may have effects on other aspectsof the model). Model sensitivity to a parameter willalso depend on the value of other model parameters,especially in systems where thresholds operate, evenwhere these remain the same between model runs. Itis important to recognize the propensity for parameterchange in sensitivity analysis, that is a model can behighly sensitive to changes in a particular parameter butif changes of that magnitude are unlikely ever to berealized, then the model sensitivity to them will be oflittle relevance. In this way, some careful judgement isrequired of the modeller to set the appropriate boundsfor parameter variation and the appropriate values ofvarying or nonvarying parameters during the process ofsensitivity analysis.

Sensitivity analysis is a very powerful tool for inter-acting with simple or complex models. Sensitivity anal-ysis is used to do the following:

1. better understand the behaviour of the model, par-ticularly in terms of the ways in which parame-ters interact;

2. verify (in the computer-science sense) multi-compo-nent models;

3. ensure model parsimony by the rejection of parame-ters or processes to which the model is not sensitive;

4. target field parameterization and validation pro-grammes for optimal data collection;

5. provide a means of better understanding parts of orthe whole of the system being modelled.

Modelling and Model Building 59

−100

0

100

200

300

400

500

600

700

800

−100 −75 −50 −25 0 25 50 75 100

% Change in input variable

% C

hang

e in

out

put (

eros

ion

rate

)

QkSV

(a)

−150

−100

−50

0

50

100

150

200

250

300

350

0 25 50 75 100

Variability as % of input variable base value

% C

hang

e in

out

put (

eros

ion

rate

)

(b)

Figure 1.26 Example sensitivity analysis of the simple erosion model given in Equation 1.1: (a) univariate sensitivityanalysis of the simple erosion model given in Equation 1.1. Base values are Q = 100 mm month−1, k = 0.2, S = 0.5,

m = 1.66, n = 2.0, i = 0.07 and V = 30%. The variables Q, k, S and V are varied individually from −100% to +100%of their base values and the output compared. Note that k has a positive linear response; Q a nonlinear response fasterthan k; S a nonlinear response faster than Q (because Q is raised to the power m = 1.66 while S is raised to the powern = 2); and V a negative exponential response. The order of parameter sensitivity is therefore V > S > Q > k.; and(b) Multivariate sensitivity analysis of the same model, where normally distributed variability is randomly added to eachof the parameters as a proportion of the base value. Note the large fluctuations for large amounts of variability, suggestingthat the model is highly sensitive where variability of parameters is >50% of the mean parameter value. No interactionsor autocorrelations between parameter variations have been taken into account

1.8 ERRORS AND UNCERTAINTY

1.8.1 Error

No measurement can be made without error. (If youdoubt this statement, get ten different people to write

down the dimensions in mm of this page, without tellingeach other their measurements, and compare the results.)Although Heisenberg’s uncertainty principle properlydeals with phenomena at the quantum scale, there isalways an element of interference when making an

60 Mark Mulligan and John Wainwright

observation. Thus, the act of observation perturbs whatwe are measuring. Some systems may be particularlysensitive to these perturbations, for example, whenwe introduce devices into a river to measure patternsof turbulence. The very act of placing a flow meterinto the flow causes the local structure of flow tochange. If we were interested in the section dischargerather than the local changes in velocity, our singlemeasuring device would have less significant impacts byperturbation, but the point measurement would be a verybad measurement of the cross-section flow. To counterthis problem, we may repeat the measurement at anumber of positions across the cross-section and providean average discharge (usually weighting by the widthof flow represented by each measurement). But thisaverage will only be as good as the choice of positionstaken to represent the flow. Sampling theory suggeststhat a greater number of measurements will provide abetter representation, with the standard error decreasingwith the square root of the number of measurementsmade. However, a larger number of samples will take alonger time to make, and thus we have possible temporalchanges to contend with in giving added error. Clearly,this approach is impractical when flows are rapidlychanging. If we require continuous measurements, wemay build a structure into the flow, such as a flumeor weir (e.g. French, 1986) which again perturbs thesystem being measured (possibly with major effects ifthe local gradient is modified significantly, or if sedimentis removed from the system). Furthermore, the devicesused to measure flow through these structures will havetheir own uncertainties of measurement, even if theyare state-of-the-art electronic devices. In the flume case,the measurement is usually of a depth, which is thencalibrated to a discharge by use of a rating curve. Therating curve itself will have an uncertainty element,and is usually proportional to depth to a power greaterthan one. Any error in measurement of depth willtherefore be amplified in the estimation of discharge.Such measurements also tend to be costly, and the costand disturbance therefore prevent a large number ofmeasurements being taken in a specific area, which isproblematic if we are interested in spatial variability.

Other environmental modelling questions might re-quire even further perturbation of the system. Forexample, soil depth is a critical control of water flowinto and through the soil and thus has an impact onother systems too, such as vegetation growth or massmovement. In a single setting, we might dig a soilpit. Even if we try to replace the soil as closely aspossible in the order in which we removed it, there willclearly be a major modification to the local conditions

(most notably through bulk-density changes, which oftenmean a mound is left after refilling; a lower bulk den-sity means more pore space for water to flow through,so we have significantly altered what we intended toobserve). Even if we were foolish enough to want touse this approach at the landscape scale, it is clearly notfeasible (or legal!), so we might use a spatial sampleof auger holes. However, the auger might hit a stoneand thus have difficulty penetrating to the soil base – instony soils, we usually only reach into the upper partof the C horizon. We might therefore try a noninva-sive technique such as ground-penetrating radar (GPR).GPR uses the measured reflections of transmitted radiofrequency waves (usually in the 25–1200 MHz range)to provide ‘observations’ of the changes in dielectricconstant, in engineering structures such as building andbridges or in the ground. Dielectric constant is con-trolled by changes in material density and water content.Where transmitted waves encounter a change in dielec-tric constant, some energy passes through the interfaceand some is reflected. The reflected energy from a GPRtransmitter is recorded on a nearby receiver with thetime delay (in nanoseconds) between the transmissionof the pulse and its receipt indicating the distance of thereflecting object from the transmitter-receiver array. Inthis way GPR can image the subsurface and has foundapplications in archaeology (Imai et al., 1987), hydrol-ogy (van Overmeeren et al., 1997), glaciology (Nicollinand Kofman, 1994) and geology (Mellett, 1995). Thedifficulty with electromagnetic noninvasive techniquesis that while dielectric discontinuities can be fairly obvi-ously seen, the techniques provide little information onwhat these discontinuities are (rocks, roots or moisture,for example). Thus, noninvasive techniques are also sub-ject to significant potential error. The implication is thatall measurements should have their associated error citedso that the implications can be considered and due carebe taken in interpreting results. Care is particularly nec-essary in the use of secondary data, where you may havevery little idea about how the data were collected andquality controlled.

Field measurements are often particularly prone toerror, because of the difficulty of collecting data. Wemay choose to use techniques that provide rapid resultsbut which perhaps provide less precise measurements,because of the high costs involved in obtaining fielddata. Note the difference between error and precision(sometimes called the tolerance of a measurement) – thelatter relates only to the exactness with which a measure-ment is taken. A lack of precision may give very specificproblems when measuring features with fractal charac-teristics, or when dealing with systems that are sensitive

Modelling and Model Building 61

to initial conditions. Thus, a consideration of the mod-elling requirements is often important when deciding theprecision of a measurement.

Specification errors can arise when what is being mea-sured does not correspond to the conceptualization ofa parameter in the model. This problem may arise ifa number of processes are incorporated into a singleparameter. For example, if erosion is being considered asa diffusion process in a particular model, diffusion mayoccur by a number of processes, including rainsplash,ploughing, animal activity and soil creep. The first twomight be relatively easy to measure, albeit with theirown inherent problems (e.g. Torri and Poesen, 1988),while the latter may be more difficult to quantify eitherbecause of inherent variability in the case of bioturbationor because of the slow rate of the process in the case ofcreep. Interactions between the different processes maymake the measurement of a compound diffusion param-eter unreliable. It is also possible that different ways ofmeasurement, apparently of the same process, can givevery different results. Wainwright et al. (2000) illustratehow a number of measurements in rainfall-runoff mod-elling can be problematic, including how apparent differ-ences in infiltration rate can be generated in very similarrainfall-simulation experiments. Using pumps to removewater from the surface of the plot led to significantover-estimation of final infiltration because of incom-plete recovery of ponded water, when compared to directmeasurement of runoff from the base of the plot (whichitself incorporates ponding into the amount of infiltrationand thus also over-estimates the real rate). The pump-ing technique also introduces a significant time delayto measurements, so that unsaturated infiltration is verypoorly represented by this method. Differences betweeninfiltration measured using rainfall simulation, cylinderinfiltration and the falling-head technique from soil coresfor the same location can be orders of magnitude (e.g.Wainwright, 1996) because each are representing infil-tration in different ways. Different infiltration modelsmay be better able to use measurements using onetechnique rather than another. Such specification errorscan be very difficult to quantify, and may in fact onlybecome apparent when problems arise during the mod-elling process. It should always be borne in mind thaterrors in model output may be due to incorrect param-eter specification. When errors occur, it is an importantpart of the modelling process to return to the parame-ters and evaluate whether the errors could be caused inthis way.

Environmental models operate in a defined space inthe real world. However, the representation of that spacewill always be some form of simplification. At the

Area or volume being measured

Val

ue o

f mea

sure

d pa

ram

eter

REAor

REV

Figure 1.27 Definition of the representative elementalarea (REA) or volume (REV) concept

extreme case, the system will be completely lumped,with single values for parameters and each input andoutput. Such models can be a useful generalization, forexample in the forecasting of flooding or reservoir fill-ing (e.g. Blackie and Eeles, 1985 – another example isthe population models discussed above, although thesecan be spatialized as shown by Thornes, 1990). How-ever, the definition of each parameter may be non-trivial for all but the simplest of catchments. Woodet al. (1988) used the term representative elemental area(REA) to evaluate the scale at which a model parame-ter might be appropriate (Figure 1.27). At the oppositeend of the spectrum is the fully distributed model, inwhich all parameters are spatialized. There still remainsthe issue of the REA in relation to the grid size used(distributed applications may still have grid sizes ofkilometres or hundreds of kilometres in the case of Gen-eral Circulation Models). However, in addition, thereis the issue of how to estimate parameters spatially.Field measurements are costly so that extensive datacollection may be impossible financially, even if the peo-ple and equipment were available on a sufficient scale.Therefore, it is usual to use some sort of estimationtechnique to relate parameters to some easily measuredproperty. For example, Parsons et al. (1997) used sur-face stone cover to estimate infiltration rates, findingthat the spatial structure provided by this approach gavea better solution than simply assigning spatial valuesbased on a distribution function of measured infiltrationrates, despite the relatively high error in the calibrationbetween stone cover and infiltration rate. Model sensi-tivity to different parameters (see above) may mean thatdifferent techniques of spatialization are appropriate forthe parameters of the model in question. It is impor-tant that distributed models are tested with spatiallydistributed data, otherwise it is possible to arrive at

62 Mark Mulligan and John Wainwright

completely different conclusions about their reliability(Grayson et al., 1992a,b). Similarly, if the scale of theoutput is not related to the scale of the test data, errorsin interpretation can arise. It is for this reason that tech-niques of upscaling or downscaling of model results areimportant (see Chapter 19).

Engeln-Mullges and Uhlig (1996) classify all of theabove problems as input errors. They may be presentas errors in model parameterization, or in data againstwhich the model is tested. It is possible that a modelcould be rejected because of the latter errors, so weshould be careful to specify the magnitude of errors asfar as possible at every stage in our approach. There arethree further types of error associated with the modellingprocess itself. These are procedural, computational andpropagation errors (Engeln–Mullges and Uhlig 1996).

Procedural errors arise from using a numericalapproximation to a problem where there is no knownanalytical solution. These errors are the differencebetween the exact and approximate solutions. Indeed,a useful test for model behaviour is a comparison ofa numerical approximation against a known analyticalsolution. We saw above how the backward-differencesolution to the kinematic wave equation for waterflow (Equations 1.34 to 1.39) had errors associated withO(h) and O(k) – that is of the same order of magnitudeof the timestep and grid size. In terms of the procedu-ral error, a central difference solution is better in that itserrors would be O(h2) and O(k2) and thus much smaller

Binary representation of number Decimalequivalent

Sign Exponent Mantissa

0 1 0 0 0 0 0 1 0 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 30 0 1 1 0 1 0 0 1 1 1 0 1 0 1 1 0 1 0 1 1 1 1 1 1 1 0 0 1 0 1 0 10−7

because of the higher order differences involved. Thus,some balance must be made between the amount of pro-cedural error, the computational resources required andthe potential for side effects such as numerical diffusion.

Computational errors arise during the computer rep-resentation of the algorithm that operationalizes themodel. Accumulated computational error is the sum oflocal computational errors incurred during the carryingout of the algorithm. Local computational errors are typ-ically due to rounding or truncation. To see how thismight be a problem, let us look at how a computer (orcalculator, for that matter) actually stores a number.

All computers store numbers in binary format. Allinteger numbers can be exactly represented in this

format. However, we may encounter two commonproblems when using integer values. First, overflow (orunderflow) may occur if the space (memory) allocatedto storing the number is not sufficient. For a two-byte,signed integer (i.e. 15 binary bits used to store thenumber and one bit to store the sign), the range of valuespossible is ±32 767 (i.e. ±215). Using four bytes, therange increases to ±2 147 483 647 (i.e. ±231 as 31 bitsare used to store the number and one bit for the sign).Second, if we divide one integer by another, there maybe a remainder, which will be discarded if the result isalso stored in integer format.

Floating-point numbers are stored in a three-part for-mat, made up of a single bit representing the sign,eight bits representing the exponent and 23 bits rep-resenting the mantissa. These make up any numberas sign × Mantissa × 2exponent−bias, where the bias ofthe exponent is a fixed integer for any given com-puter (Press et al., 1992). Some numbers cannot be rep-resented exactly in this format – either because they areirrational (e.g. π , e,

√2), or because the binary form is

not precise enough to accommodate them. In the formabove, numbers with more than six or seven decimaldigits cannot be represented. The resulting number mayeither be chopped (truncated) or rounded depending onthe computer, and the associated error is called round-offerror. Press et al. (1992) give the example of calculating3 + 10−7. With a bias of 151, the two values are storedas shown in this table:

To carry out the addition in binary arithmetic, theexponent must be identical. To maintain the identicalnumber, every time the exponent is increased, the binarynumber in the mantissa must be right-shifted (all thedigits are moved one space to the right and a zeroadded in the empty space at the beginning). To convert10−7 (exponent 01101001 binary = 105 decimal) intothe same form as 3 (exponent 10000010 binary = 130decimal), we require 25 right shifts of the mantissa. Asthe length of the mantissa is only 23 bits, it will now befull of zeroes, so the result of 3 + 10−7 would simply be3, and a truncation error of 10−7 would have occurred.

Many programming languages allow the possibilityof storing numbers with 64 bits (called either long real

Modelling and Model Building 63

or double precision). This increase of storage spaceextends the range of values that can be represented fromc. ±1 × 10±38 to c. ±1 × 10±308 (but also increases theamount of memory required to run a program and storethe results). In the example given above, it would bepossible to right-shift the mantissa of 10−7 25 timeswhile still storing all the information present, so 3 +10−7 would be correctly represented. Truncation errorswill still occur, but will be of a lower order of magnitudethan storage at single precision. We should not assumethat truncation errors will cancel. It is possible thatrepeated calculations of this sort will be carried out,so that the total error due to truncation will increasethrough time.

Propagation errors are the errors generated in thesolution of a problem due to input, procedural andcomputational errors. At each stage a calculation is madebased directly or indirectly on an input value that hasan associated error, that error affects the subsequentcalculation, and thus is said to propagate through themodel. As more model iterations are carried out, themagnitude of the propagated errors will tend to increase,and affect more of the solution space (Figure 1.28).Similarly, procedural errors and rounding errors willalso affect subsequent calculations, and thus propagatethrough the model. Analysis of error propagation canbe carried out analytically using the approach outlined

in Engeln-Mullges and Uhlig (1996: 7–9). If x is theactual value and x the value measured with error, thenthe relative error of the measurement is:

εx = |x − x||x| , x = 0 (1.60a)

or

εx = |x − x||x| , x = 0, (1.60b)

so that for basic arithmetic operations, propagationerrors are as follows:

1. A sum is well conditioned (reasonably bounded) ifboth inputs have the same sign. If y = (x1 ± εx1) +(x2 ± εx2) then:

εy ≈ x1

x1 + x2εx1 + x2

x1 + x2εx2 . (1.61)

Because the fractionsx1

x1 + x2+ x2

x1 + x2≡ 1, and

the relative errors are always positive, only a fractionof each of the errors is propagated. This process canbe repeated for the addition of any number, n, of

variables x, using the fractionxi∑n

j=1 xj

to multiply

each error term. If the errors are known as standard

t

t = 0x

Initial values of error

Propagated effects of error after solution through time

t = 1

t = 2

t = 3

t = 4

e0 0 0 00

e e

e e

e

t = 5

t = 6

0 0 0

0

−2e

−4e −4e

−6e −6e

−8e

6e

15e −20e 15e

28e 28e70e−46e −46e

−232e190e 190e−110e −110e 45e

844e−722e −722e455e 455e

0

e

0

0

−8e

45e

−210e −210e

Figure 1.28 Effects of using a forward difference scheme in propagating errors through the solution domain of a model.Note that the errors spread spatially and increase through time

64 Mark Mulligan and John Wainwright

deviations σxi, then the propagated error has a slightly

different form:

εy ≈√√√√ n∑

i=1

σ 2xi

(1.62)

2. However, if the values of x1 and x2 have oppositesigns and are nearly equal, amplification of errorsoccurs, because even though the fractions above stillsum to one, at least one will be < −1 or >1. Thus,the magnitude at least one of the errors will increasethrough propagation. This form of model shouldtherefore be avoided. (But note that if both x1 and x2

are negative, the first rule applies.)3. In the case of a product (y = (x1 ± εx1) · (x2 ± εx2)),

error propagation is well conditioned, and related tothe sum of the individual errors of the inputs:

εy ≈ εx1 + εx2 (1.63)

Again, any number of multiplications can be propa-gated by successive adding of the error terms. Wherethe errors are known as standard deviations, we use:

εy ≈√√√√ n∑

i=1

σ 2xi

xxi

(1.64)

4. For a quotient

(y = x1 ± εx1

x2 ± εx2

, assuming x2 = 0

),

error propagation is well conditioned, and relatedto the difference in the error of the numerator anddenominator in the function:

εy ≈ εx1 − εx2 (1.65)

5. Propagation error of a power (y = (x1 ± εx1)p)

is related to the product of the power and theinput error:

εy ≈ pεx1 (1.66)

Thus roots (powers less than one) are well condi-tioned, and powers (>1) are ill conditioned. As thevalue of the power term increases, so does the errorpropagation. Wainwright and Parsons (1998) note theimplications of this results in soil-erosion modelling,where certain model formulations will be signifi-cantly more susceptible to error than others. For anerror defined as a standard deviation, we use:

εy ≈√

pσ 2x1

xx1

(1.67)

6. Combinations of the above operations can be carriedout by combination of the relevant rules in theappropriate order.

7. Other functions require differentiation to be carriedout, so that if y = f(x), then:

εy ≈ df(x)

dxεx (1.68)

so that if y = sin(x ± εx), εy ≈ (cos x) · εx or εy ≈√[(cos x) · σx]2. For multiple functions y = f(xi),

i = 1, . . . , n, partial differentiation with respect toeach variable xi must be carried out and summed:

εy ≈n∑

i=1

∣∣∣∣∂f(xi)

∂xi

∣∣∣∣ εxi(1.69)

For the example, if y = sin(x1 ± εx) · e(x2±εx2 ), theassociated error will be εy ≈ (cos x1) · εx1 + ex2εx2 orεy ≈ √

[(cos x1) · σx1 ]2 + [ex2σx2 ]2.

An alternative approach to looking at error propa-gation is to use Monte Carlo analysis. This approachuses multiple realizations of a model using distribu-tion functions to generate errors in inputs (e.g. De Rooet al., 1992; Phillips and Marks, 1996; Fisher et al.,1997; Parsons et al., 1997; Nitzsche et al., 2000). Fromthese multiple realizations, error bands can be definedto give the range of possible outputs or even confidenceintervals on outputs (e.g. Hakanson, 1996). The advan-tages of Monte Carlo analysis are that it can deal withthe effects of spatially distributed parameters, whereasthe analytical approach outlined above is more limitedin this case, and that it can deal with relatively com-plex interactions of parameter error, which is impossibleusing the analytical approach. As a disadvantage, reli-able results often require many tens or hundreds of sim-ulations (Heuvelink, 1992), which are extremely costlyin terms of computer resources for large simulations.

Beven and Binley (1992) developed the GeneralizedLikelihood Uncertainty Estimation (GLUE) technique toevaluate error propagation. The technique uses Bayesianestimators to evaluate the likelihood that specific com-binations of model parameters are good predictors ofthe system in question. Monte Carlo simulations areused to provide the various parameter combinations, andthe likelihood measured against a specified objectivefunction (Figure 1.29). The choice of parameter com-binations may be subjective, based on the experienceof the user of the technique, and often fails to accountfor necessary relationships between parameter values.An advantage of the technique is that it requires noassumptions to be made about the distribution functions

Modelling and Model Building 65

Com

bine

d lik

elih

ood

Sand content (%)

80604020

0.137

0.135

0.133

0.131

0.129

0.127

0.1250

Com

bine

d lik

elih

ood

Clay content (%)0 20 40 60

0.137

0.135

0.133

0.131

0.129

0.127

0.125

0.125

0.127

0.129

0.131

0.133

0.135

0.137

20,00,000 40,00,000 60,00,000 80,00,000 1,00,00,000

Ki - Interrill erodibility (kg s m−4)

Com

bine

d lik

elih

ood

0.125

0.127

0.129

0.131

0.133

0.135

0.137

0 0.02 0.04 0.06

Kr - Rill erodibility (s m−1)

Com

bine

d lik

elih

ood

00.125

0.127

0.129

0.131

0.133

0.135

0.137

10 20 30

Kb - Effective hydraulic conductivity (mm hr−1)

Com

bine

d lik

elih

ood

0.125

0.127

0.129

0.131

0.133

0.135

0.137

0.5 1.5 2.5 3.5 4.5

Tau - Critical shear (Pa)

Com

bine

d lik

elih

ood

Com

bine

d lik

elih

ood

Initial rill width (m)

10.80.60.40.2

0.137

0.135

0.133

0.131

0.129

0.127

0.1250

Com

bine

d lik

elih

ood

Canopy cover coefficient

14 1610 1264 82

0.137

0.135

0.133

0.131

0.129

0.127

0.125

Figure 1.29 Example of error analysis using the GLUE approach: the study by Brazier et al. (2000) looking at theWEPP soil-erosion model showing the joint likelihood of goodness-of-fit for different combinations of eight parameters

66 Mark Mulligan and John Wainwright

of the input data (Beven, 1993). Choice of the objec-tive function requires care that the key outputs of themodel simulations are being tested reliably, as notedby Legates and McCabe (1999). Other techniques thatcan be used to estimate model uncertainty include Latinhypercube simulation, Rosenbleuth’s and Harr’s point-estimation methods, all of which can be used to samplethe input-parameter space efficiently. Yu et al. (2001)found that Latin hypercube simulation provided a veryefficient means of replicating more time-intensive MonteCarlo techniques (see Press et al., 1992, for more detailon Latin hypercube sampling).

Hall and Anderson (2002) criticize error-estimationtechniques based on objective measures because thelevel of data and detailed measurements required arelacking in many environmental modelling applications.For example, in the case of an extreme flood, the effectsof the flood might be measurable (water depth fromtrash lines, aerial photographs of flood extent) after theevent, but there may be little in the way of detailedmeasurements during the event. Rainfall fields areparticularly difficult to reconstruct in high resolution inspace and time; even if flow monitoring equipment werepresent, it may have been damaged during the passageof the flood wave (Reid and Frostick, 1996). Techniquestherefore need to be developed that incorporate allinformation appropriate to model evaluation, includingqualitative indications of permissible model behaviour.

1.8.2 Reporting error

We noted how the relative error of a measurement can bemade (Equation 1.60). Different types of measurementsof error also exist (Engeln-Mullges and Uhlig, 1996).Using the same notation as before, the true error of x is:

x = x − x (1.70)

while the absolute error of x is:

|x | = |x − x| (1.71)

and the percentage error of x is:

|x − x||x| . 100 x = 0 (1.72)

If we have a series of values, xi , representing measure-ments with error on true values xi , it is common tocombine them as a root-mean square (RMS) error asfollows:

RMS x =√√√√ n∑

i=1

(xi − xi )2 (1.73)

RMS errors are in the same units as the originalmeasurements, so are easy to interpret on a case-by-case basis. When comparing between cases withvery different values, it may be useful to calculate anormalized RMS error:

NRMS x =

√√√√ n∑i=1

(xi − xi)2

n∑i=1

xi

(1.74)

which gives the proportion of error relative to theoriginal value, and can thus be easily converted intoa percentage error. In other cases, the goodness-of-fitmeasures described above may also be appropriate.

1.8.3 From error to uncertainty

Zimmermann (2000) defines six causes of uncertaintyin the modelling process: lack of information, abun-dance of information, conflicting evidence, ambiguity,measurement uncertainty and belief. A lack of informa-tion requires us to collect more information, but it isimportant to recognize that the quality of the informa-tion also needs to be appropriate. It must be directedtowards the modelling aims and may require the modi-fication of the ways in which parameters are conceivedof and collected. Information abundance relates to thecomplexity of environmental systems and our inabil-ity to perceive large amounts of complex information.Rather than collecting new data, this cause of uncer-tainty requires the simplification of information, perhapsusing statistical and data-mining techniques. Conflict-ing evidence requires the application of quality controlto evaluate whether conflicts are due to errors or arereally present. Conflicts may also point to the fact thatthe model being used is itself wrong, so re-evaluationof the model structure and interaction of componentsmay be necessary. Ambiguity relates to the reportingof information in a way that may provide confusion.Uncertainty can be removed by questioning the originalinformant, although this approach may not be possiblein all cases. Measurement uncertainty may be reducedby invoking more precise techniques, although it mustbe done in an appropriate way. There is often a tendencyto assume that modern gadgetry will allow measurementwith fewer errors. It must be noted that other errors canbe introduced (e.g. misrecording of electronic data ifa data logger gets wet during a storm event) or thatthe new measurement may not be measuring exactly thesame property as before. Beliefs about how data are to be

Modelling and Model Building 67

interpreted can also cause uncertainty because differentoutcomes can result from the same starting point. Over-coming this uncertainty is a matter of iterative testingof the different belief structures. Qualitative assessmentis thus as much an aspect of uncertainty assessment asqualitative analysis.

The quantitative evaluation of uncertainty has beendiscussed above in detail. Error-propagation techniquescan be used in relatively simple models (or their sub-components) to evaluate the impact of an input error(measurement uncertainty) on the outcome. In morecomplex scenarios, Monte Carlo analysis is almostcertainly necessary. If sufficient runs are performed,then probability estimates can be made about the out-puts. Carrying out this approach on a model with alarge number of parameters is a nontrivial exercise,and requires the development of appropriate samplingdesigns (Parysow et al., 2000). The use of sensitiv-ity analysis can also be used to optimize this pro-cess (Klepper, 1997; Barlund and Tattari, 2001). Halland Anderson (2002) note that some applications mayinvolve so much uncertainty that it is better to talk aboutpossible outcomes rather than give specific probabili-ties. Future scenarios of climate change evaluated usingGeneral Circulation Models (see Chapters 2 and 12)are a specific case in point here. Another approachthat can be used to evaluate the uncertainty in out-come as a function of uncertain input data is fuzzyset theory. Torri et al. (1997) applied this approach tothe estimation of soil erodibility in a global datasetand Ozelkan and Duckstein (2001) have applied it torainfall-runoff modelling. Because all measurements areuncertain, the data used for model testing will alsoinclude errors. It is important to beware of rejectingmodels because the evaluation data are not sufficientlystrong to test it. Monte et al. (1996) presented a tech-nique for incorporating such uncertainty into the modelevaluation process.

Distributed models may require sophisticated visu-alization techniques to evaluate the uncertainty of thespatial data used as input (Wingle et al., 1999). Animportant consideration is the development of appropri-ate spatial and spatio-temporal indices for model eval-uation, based on the fact that spatial data and theirassociated errors will have autocorrelation structuresto a greater or lesser extent. Autocorrelation of errorscan introduce significant nonlinear effects on the modeluncertainty (Henebry, 1995).

Certain systems may be much more sensitive to theimpacts of uncertainty. Tainaka (1996) discusses theproblem of spatially distributed predator–prey systems,

where there is a phase transition between the occur-rence of both predator and prey, and the extinction ofthe predator species. Such transitions can occur para-doxically when there is a rapid increase in the numberof the prey species triggered by instability in nutrientavailability, for example. Because the phase transitionrepresents a large (catastrophic) change, the model willbe very sensitive to uncertainty in the local region ofthe parameter space, and it can thus become difficult orimpossible to interpret the cause of the change.

1.8.4 Coming to terms with error

Error is an important part of the modelling process(as with any scientific endeavour). It must thereforebe incorporated within the framework of any approachtaken, and any corresponding uncertainty evaluated asfar as possible. A realistic approach and a healthy scep-ticism to model results are fundamental. It is at best mis-leading to present results without corresponding uncer-tainties. Such uncertainties have significant impacts onmodel applications, particularly the use of models indecision-making. Large uncertainties inevitably lead tothe rejection of modelling as an appropriate techniquein this context (Beck, 1987). Recent debates on possiblefuture climate change reinforce this conclusion (see theexcellent discussion in Rayner and Malone, 1998).

In terms of modelling practice, it is here that wecome full circle. The implication of error and uncertaintyis that we need to improve the basic inputs into ourmodels. As we have seen, this improvement does notnecessarily just mean collecting more data. It maymean that it is better to collect fewer samples, butwith better control. Alternatively, it may be necessaryto collect the same number of samples, but with amore appropriate spatial and/or temporal distribution.Ultimately, the iterations involved in modelling shouldnot just be within computer code, but also between fieldand model application and testing.

1.9 CONCLUSION

Modelling provides a variety of tools with which we canincrease our understanding of environmental systems.In many cases, this understanding is then practicallyapplied to real-world problems. It is thus a powerful toolfor tackling scientific questions and answering (green!)engineering problems. But its practice is also some-thing of an art that requires intuition and imagination toachieve appropriate levels of abstraction from the realworld to our ability to represent it in practical terms. As

68 Mark Mulligan and John Wainwright

a research tool, it provides an important link betweentheory and observation, and provides a means of testingour ideas of how the world works. This link is importantin that environmental scientists generally deal with tem-poral and spatial scales that are well beyond the limitsof observation of the individual. It is important to recog-nize that modelling is not itself a ‘black box’ – it formsa continuum of techniques that may be appropriatelyapplied in different contexts. Part of the art of mod-elling is the recognition of the appropriate context forthe appropriate technique.

Hopefully, we have demonstrated that there is nosingle way of implementing a model in a specificcontext. As we will see in the following chapters,there are many different ways to use models in similarsettings. How the implementation is carried out dependson a range of factors including the aims and objectivesof the study, the spatial and temporal detail required,and the resources available to carry it out. This chapterhas presented the range of currently available techniquesthat may be employed in model building and testing.More details of these techniques are found on thebook’s website and in the numerous references provided.We have also addressed a number of the importantconceptual issues involved in the modelling process,again to suggest that the modelling process is not asuniform as is often assumed, and that debate is oftenfruitful within and between disciplines in order to tacklesome of these issues.

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