17
("l/l/]qlill'f~ I.~ (]t't,~lll'lltt'~ Wol 15. "40 2. pp 167-183. 1q~39 (~)98-31)t)4 $9 $1 IX1 + 018) Printed m Great Brllaln -MI nght,, re,,er',ed Copyright q' 1989 Pergamon Pre,,s pit: FRACTAL MEASUREMENT AND LINE GENERALIZATION PAUL A. LONGLEY and MICHAEL BATTY Department ofTov, n Planning. University of Wales at Cardiff. P.O. Box 906, Cardiff CFI 3YN. U.K. (Receired 30 Juh' 1987: accepted 4 Juh' 1988) Abstract Thts paper describes a standard linear regression formulation through which the fractal dtmen- sion of cartographic lines may be ascertained. Four different algorithms for measuring the fractal character of lines are introduced: these are termed the structured walk. equipaeed polygon, hybrid walk. and cell-count methods. The relative merits of these four methods are assessed using a digitized database which depicts the tov, n of Cardilrs (U.K.) urban growth during the period 1886-1949. A concluding section asse,~ses the tnethodological and substantive rationales for this measurement task. Kcr llord~: Fraclal measurement,, Line generalization. Structured walk. Equipaced polygon, tlybrid walk, Cell count. I. INTRODUCTION Fractal concepts now are being used in many areas of scientific enquiry and tv,'o complcn]cntary approaches can he identilicd. First. the description of irregularity in a I'orntal fralncv, ork has led to the application of fractals in luany traditional areas of scientific mcas- ltrctm'nt which are being inlbrmcd by these ideas. Second. ctm~plementing description and measure- mcnt is tile notion that fractals can be used to generate or simulate irregularity by building computer models of various natural and ilrtilicial forms. Indeed, it has bccn argued that tile use of computer graphics in visuali/inlg fnact:d COllCepts hits played a central part ill popularizing these Meas, and this is clearly the part it] popularizing these ideas, and this is clearly the situation ill nlitlilenlittics where abstract notions of space have been visualized in entirely new ways through the convergence of fractals and computer graphics. Tile pov, er of these concepts has been illu- stral,cd further through their commercial exploitation in the compul,cr-graphics industry. In the spatial sciences, parl,ict, lar attention has been focused on the way in ~shich fractals nlay bc used to generate visually realistic simulal,ions of a variety of natural and man- made phenomena such :Is c:trtographic features en- countered in geographical enquiry (Batty, 1985; Goodchild and Mark. 1987). This paper takes a step backwards from the practice of generating or simulat- ing clirl,ographic lines as fr:tctal forms and reviews a number olwa~'s in ~'hich the fractal dimension ("frac- tality") of such lines may IX measured. A three-fold rationale for this exercise may l'x: outlined. First. in an era in which map management (e~l,her as an end in itself or as a component of geo- graphic:,l inlbrmal,ion systems) increasingly involves sl,oragc o1 digitized data in compul,cr-readable form. the quest,ion arises as to what is the most appropriate number and spacing of points for efficient storage and effective display. An emerging consensus (e.g. Muller, 1987: Butl,cnficld, 19~4) suggests that fractals provide more visually acceptable models of total line charac- ter than other more established methods based upon prespecified point weeding intervals or band-width toler:mcing. Second. there also is increasing support Ibr the notion that fractals offer a panacea I-or carl,o- graphic line simulation in a vltriety of conl,cxl,s. Al- though plausible levels of visual realisn] have been produced in some existing simulations (Batty and Longlcy. 1986), the increased sophisl.ical,ion of such compt, tcr graphic simulations (and of the devices on which they are displayed) is likely to require the speci- lication of empirical fractal dintensions about which statistical fractals are to be measured and associated simt, lations gencr:tted. The measurement task should necessarily precede large-scale simuktl,ion in such in- stances. The third, and in the present example most controversial, ration:de ['or fracl,al measurement sur- rounds its role in enhancing our understanding of the various processes which conspire to produce it linal cartographic delineation on a map. Fracl,als have been used in various natural scientilic dora:tins (c.g Clark, 1986; Dcarnley, 1985; Kayc, Leblanc, and Abbot, 1985; Schertzer and Lovcjoy, 1984) to infer the effects of a single process (or a small numixr of different processes). In moving from object,ire meas- urement of a distinct physical boundary to subjective cartographic representation of ahogel,hcr different, geographical phenomena, inference from form to process inevitably Ixcomes incalculably more dilficult and controversial. Insofar as these processes may be more manifold, multifitcel,ed, complex, and rapidly changing, fractal measurement may offer at best only a "black box" synthesis of line character: nevertheless substantive interpretations in the literal,urc (e.g. Longley and Batty, 1989), remain sufficiently plaus- ible to suggest that this approach should not be dis- missed out-of-hand. For present purposes, however, fractal measure- ment may Ix viewed as an end in itself. That is to slty. the measurement procedure allows us to do two things: first, to diagnose whether a particular curve is 167

Fractal measurement and line generalization

Embed Size (px)

Citation preview

("l/l/]qlill'f~ I.~ ( ] t ' t ,~ l l l ' l l t t '~ Wol 15. "40 2. pp 167-183. 1q~39 (~)98-31)t)4 $9 $1 IX1 + 018) Printed m Great Brllaln -MI nght,, re,,er',ed Copyright q' 1989 Pergamon Pre,,s pit:

FRACTAL MEASUREMENT AND LINE GENERALIZATION

PAUL A . LONGLEY a n d MICHAEL BATTY

Department ofTov, n Planning. University of Wales at Cardiff. P.O. Box 906, Cardiff CFI 3YN. U.K.

(Receired 30 Juh' 1987: accepted 4 Juh' 1988)

Abstract Thts paper describes a standard linear regression formulation through which the fractal dtmen- sion of cartographic lines may be ascertained. Four different algorithms for measuring the fractal character of lines are introduced: these are termed the structured walk. equipaeed polygon, hybrid walk. and cell-count methods. The relative merits of these four methods are assessed using a digitized database which depicts the tov, n of Cardilrs (U.K.) urban growth during the period 1886-1949. A concluding section asse,~ses the tnethodological and substantive rationales for this measurement task.

Kcr llord~: Fraclal measurement,, Line generalization. Structured walk. Equipaced polygon, tlybrid walk, Cell count.

I. INTRODUCTION

Fractal concepts now are being used in many areas of scientific enquiry and tv,'o complcn]cntary approaches can he identilicd. First. the description of irregularity in a I'orntal fralncv, ork has led to the application of fractals in luany traditional areas of scientific mcas- ltrctm'nt which are being inlbrmcd by these ideas. Second. ctm~plementing description and measure- mcnt is tile notion that fractals can be used to generate or simulate irregularity by building computer models of various natural and ilrtilicial forms. Indeed, it has bccn argued that tile use of computer graphics in visuali/inlg fnact:d COllCepts hits played a central part ill popularizing these Meas, and this is clearly the part it] popularizing these ideas, and this is clearly the situation ill nlitlilenlittics where abstract notions of space have been visualized in entirely new ways through the convergence of fractals and computer graphics. Tile pov, er of these concepts has been illu- stral,cd fu r ther through their commercial exploitation in the compul,cr-graphics industry. In the spatial sciences, parl,ict, lar attention has been focused on the way in ~shich fractals nlay bc used to generate visually realistic simulal, ions of a variety of natural and man- made phenomena such :Is c:trtographic features en- countered in geographical enquiry (Batty, 1985; Goodchild and Mark. 1987). This paper takes a step backwards from the practice of generating or simulat- ing clirl,ographic lines as fr:tctal forms and reviews a number olwa~'s in ~'hich the fractal dimension ("frac- tality") of such lines may IX measured.

A three-fold rationale for this exercise may l'x: outlined. First. in an era in which map management (e~l,her as an end in itself or as a component of geo- graphic:,l inlbrmal,ion systems) increasingly involves sl,oragc o1 digitized data in compul,cr-readable fo rm.

the quest,ion arises as to what is the most appropriate number and spacing of points for efficient storage and effective display. An emerging consensus (e.g. Muller, 1987: Butl,cnficld, 19~4) suggests that fractals provide

more visually acceptable models of total line charac- ter than other more established methods based upon prespecified point weeding intervals or band-width toler:mcing. Second. there also is increasing support Ibr the notion that fractals offer a panacea I-or carl,o- graphic line simulation in a vltriety of conl,cxl,s. Al- though plausible levels of visual realisn] have been produced in some existing simulations (Batty and Longlcy. 1986), the increased sophisl.ical,ion of such compt, tcr graphic simulations (and of the devices on which they are displayed) is likely to require the speci- lication of empirical fractal dintensions about which statistical f ractals are to be measured and associated simt, lations gencr:tted. The measurement task should necessarily precede large-scale simuktl,ion in such in- stances. The third, and in the present example most controversial, ration:de ['or fracl,al measurement sur- rounds its role in enhancing our understanding of the various processes which conspire to produce it linal cartographic delineation on a map. Fracl,als have been used in various natural scientilic dora:tins (c .g Clark, 1986; Dcarnley, 1985; Kayc, Leblanc, and Abbot, 1985; Schertzer and Lovcjoy, 1984) to infer the effects of a single process (or a small numixr of

different processes). In moving from object,ire meas- urement of a distinct physical boundary to subjective cartographic representation of ahogel,hcr different, geographical phenomena, inference from form to process inevitably Ixcomes incalculably more dilficult and controversial. Insofar as these processes may be more manifold, multifitcel,ed, complex, and rapidly changing, fractal measurement may offer at best only a "black box" synthesis of line character: nevertheless substantive interpretations in the literal,urc (e.g. Longley and Batty, 1989), remain sufficiently plaus- ible to suggest that this approach should not be dis- missed out-of-hand.

For present purposes, however, fractal measure- ment may Ix viewed as an end in itself. That is to slty. the measurement procedure allows us to do two things: first, to diagnose whether a particular curve is

167

P.A. L()', .(. iLr ' , and ~,1 B.,,rT',

ffactal and to identif.~, the range of scales through which the fractal property o1" self-similarity holds; and. secondly, to assess ~,hether the sell-similarity proper,) is exact statistically through the entire range of scales for ~hich the fractal property has been iden- tiffed. If sell-similarity does not seem to be exact across the entire scale range, we may seek to accom- modate this in our recorded measurements of fractal dimensions.

The remainder of this paper is structured as I'ol- lows. Section 2 defines what a fractal line is and shows how fractal dimensionality can be identified as a par- ameter in a regression model; Section 3 addressed the empirical measurement problem: Section 4 introduces four methods of obtaining measurements and illustra- tes each method using a number of cartographic dis- plays of an empirical case study; and Section 5 dis- cusses the measurenlents obtained I'rom that study. assesses the applicability o1 the techniques, and draws some conclusions its 1o future rescarcff.

Z. EMPIRICAL DI,~G%O,~IS OF FR ~tCF.~L FORM

The central concepts underl~ing fr:,ctals research are the properties ofindeterminaLe scale measure such as length, scale-dependence and self-dependence, and self-~,imil:trity (~hich ma'. be c,~act or statistical) (Goodchild and .Mark. 198"71. Earl.', research on Irac- tal measurement (e.g. Richardson. 1961)detected the property of increasing cur',e length at liner and finer levels of resolution by setting a pair of dp, iders at a succession oF span'~ and I\~r each span. "',aalking" the dividers along the cur~,e, and recording the number o1" steps that were needed to tra,,erse its entirety'. This is a neat method of illustrating the property of ~cule- dependent length, for adjusting the db, iders to ,mlaller :rod smaller span'~ is equivalent to I'ocusing at liner and finer levels o f resolution.

Figure I illustrates ho~ this resuhs in approxima- lions ~vhich are flirted uncreasingly finely to the path of a base curve. In the top diagram, n, = 5 complete

• i / " ~

,,.2

r 2 - I

Figure I. Measuring line length al three adjacent scales.

Fractal measurement and line gencrahzutlon 169

chords arc needed to approximate the curve at divider width Iscale of resolution) r,. Changing the divider ~idth t o r ~ in the middle diagram, such thatr_~ = ! (r 1_ w e determine that n,. L = 13 chords are required. Thus

= 2. but n''~ > 2 . ( I ) /'p - I n ,

H a h m g the d iv ider w id th once again in the Iov, er d iagram requires 31 complete chords to tr;l~erse the cur'.e, and we see that the same property holds. A property of a fractal line is that halving the step intcrxal (or doubling the scale of resolution) in this wa', ~ill dictate that ntorc than twice as man,, steps w-ill be needed to approximate the line. This is because increasing amounts of self-sinfilar detail ',~,ill be detec- ted. Denoting the perimeters of the three ct, rves P;( = n,r,). P,+ I( = lit, i t / , I}" and P; ~2( = n,, _.r,, 2). it also is clear from Equation (I) that P,, : > P,~I > P,.

For a curve ~h ich is ei ther a pure or statisticul fractal ~),e can gcneralize this relationship by assunfing that the ratio of the number of chord sizes at any two scales is in constant relation to the ratio of the lengths of the chords. Thus

r; I)

", , i = . I <~ D <~ 2, ( 2 ) III

~here D is the fractaI dimension common to the tWO llle:iMirenlcnl scales, l:t)r the prescilt, it is assunled

thai the Iractal thmcnsion D is inv:Irmnl to scale

changes, although this asstunption will bc recon-

sidered in tile cntpirical discussion. The lilnits speci- fied in l 'quation (2) deline the special situations of a liuclidcan straightlinc (D = I. i.e. h:,b, ing tile scale yields exactly twice the number of chords) and a I~vo-duncnsitm:d plane (D = 2. i.e. halving the scale yields four times the number of chords :rod the line ench.,es tile .',p:tce). Thus each ol ' thc curves which will bc analyzed in Section 3 can be thought of as lilling more space than a lone-dimension:d) straightlinc, but less space I l lan a (t,.vo-dimension:d) surface.

Rearranging Equation (2) in a form suitable fllr stati.,,tical prediction yields:

r D ",+t = (,,fl,)) r , / i = a ,, L. (3)

where n,r? acts its tt hast constant a against which the number of chords n,. ) from any interval size r,. L may be predicted. Thus Eqt, ation (3) can bc generalized easily as

n = . r i) ( 4 )

With t~o scales o f resolution, D can bc derived by rearranging Equation (2) :is

D = Iog(,,,i , 'n,):.log(r,,r;,l). (5)

Ilm~ever. con l ] r lna t ion thal a COlllmon f rac la l d in lcn- sion D holds across a ~ ide r range o f scales requires

t ha t a ran,,e= of r a n d corresponding+ n values be con- sidcred. Taking logarithms of Equation (4) ',raids

I o g n = Iogu - D l o g r . (6)

The range of r and corresponding n values can be substituted into Equation (6) and the value of D e,,aluated using a simple linear regression procedure. A related equation is based upon the perimeter length P. From Equation (4):

P = l i t = ( i r l l - D) (7)

and. b) taking logarithms of (7):

log P = I o g a + (I - D) logr . (8)

Obviot, sl) D also can be obtained from Equation (8) using simple regression. Equation (8) will be used in the Iollo~,ing empirical analysis, because it allows the most appropriate ssay o f checking tile range of scales tiscd.

3. I-~MPIRICxl. %II:.A.%URI'~MI'~NT OF I:R.~CrAI. CURVES

This section out l ines tbur di f ferent methods o f ascerluinhlg the fraclal d imension o f a curve. [ :our corrcspondhlg algorithms were derived in order Lo tlbl.ain values of perimeter and step lengths which MIhMiIt|Ic intto [ !quat ion (8); mod i l i ca l ions Lo these a lgor i thms and harnessing t l tcm to a graphicul display device also u]lm~s Pedagogic i l lust rat ion o f the scale- dcf~ent lcn t n l ¢ : i s t i r e l l l e l l t p r o c e s s ,

Each of these :dgorithms was applied to the di- gitized urban boundaries of ( 'ardiff, Wales in 1886, 1901, 1922, and 194'1 (Fig. 2). These boundaries were digitized using the in-house MicroPLOT software suite (Bracken. 19X5). The sLihst;.tnlive .'lilt| metho- dologica l rationale for tile selection of this ex:unple might be Sulnrnarwetl its Iblhv.+s. The per iod spanned by these t ime shoes bourlds the pe,-iod of Cardi lFs illOM rapM growth. ~ltich v+a,, conscqtlcnl upon the opening up of the South Wales coallields through r:tilroad construction and the attendant focusing of coal exports upon the new dock facilities of Cardilr. In fact. tile faihire to develop the extensive industrial muhiplicrs characteristic o f some other mining areas led to the rel.',tivc decline of tile city in the post World W:tr I period. At the intraurban scale, the develop- ment of the Cardil1 tramwa.~, system during the kite Igth and early 20th centuries promoted suburban expansion upon nearly opened up tracts of kind (see Daunton. 1977 for aft , II disct, ssion). The period from 1861) to World War I thus represents the period of CardilFs fastest population grmvth and most rapid industrial change, unsurpassed pcrhups even by the de,.elopntent of Card i l l - : i s the Welsh capital in the po',t-World War II period.

Cartographic line representation is an inherently subjective process, anti the delineation of a (fast-

[ - I I

1 8 8 6

%; \ /

1 (

P.A. LONGLEY and M. Ba~r'~

1 9 0 1

/ .~,t! ,e i7> 7 7 ' , ;,"

/' t ¢

1 9 2 2

t ,~. . ,

\

-" ~:?~- I ?

'\\

1 9 4 9

S

p:~--.Z

t N

Figure 2. Digitized urban boundaries of

changing and cphcmcr:d) feature of historical urban growth inevitably tests cartographic codification con- ,.cntions to the limits• In the situation of Cardiff's urban edge, decisions about the representation of "'urban" allotments, the classification of disused rail- gay sidings and the assimilation of "inliers" of rural kind-use within the urban fabric poscd particularly daunting problems. Fractal measurement in natural science has been applied to edges whose form are the direct consequence of a single, or only a very few, ph.*,sical processes_ In moving into the humam do- ma*m. at similar presupposition does not hold. for an urba*n edge is clearly the spatial expression of a my- riad o1" different social and economic factors which in

• i - ~ z

. t ~ - J ~_~ '

i - - r C" ,.

)

, i ~" ' ; /

Cardiffin IN~6. 1901. ~'~'~ and I ~._, 1949.

many instances evade precise quantification Never- thclcss, this is not to say that we c:mnot envisage urban phenomena as being fractal in struclure: neigh- borhoods, shopping centers, the spatial distribution of industry and public open space, and the jux- taposition of suburbs themselves exhibit self-similar- it)' at a variety of scales, and it is intuitively plausible to suggest that this should result in spatial expression in fractal mcasurcnlenls.

The central questions are. first, whether the com- bined effects of different urban processes find spatial expression in a fractal form, given also that delinea- lion of this Ibrm mainly is the result o1 a ht,m~,n judgernental process; and. secondly. ~ helher the sell'-

Fracta! measurement and line gcnera|izatmn 171

similanD consequent upon the organization of a wide range of urban phenomena manifests itself as a single unchanging fractal index. As such. this paper im.es- tigates prex ious findings that curses exhibit statistical fractal properties only o~er a hmned scale range. :rod extends them b_,, countenancing ft, nct,on:tl changes in the nature of fractal shapes o,,er the ~.idest possible scale range_

In empirical terms, statistical I'ractals can be diag- nosed by plotting the log of scale of resolution (step length) against the log of the associated perimeter length at this scale. This standard display is known as a Richardson Plot {alter Richardson. 1961 )_ Fitting a best-fit regression line through this scatter of points .yields the fractal dimension D in accordance with Equation (8). As.~ssmcnt of whether the curve su=tis- tically is fractal requires judgement as to how closely the log-linear functional form established through regression fits the scatter of points. Values of R: will be expected to be high. Muller 11987) uses a threshold of 0 05, in view of the generally low number of fitted points attd the high precision of the measurement estimates: it should be stressed (following Shelberg, Modlcr ing. and Lain, 1982) that in this context R: is used in a descriptive rather than inferential sense.

It shouM be noted that ordinary least-squares rc- grcs,,ion estimates ~i l l be most relktble when the given number of observations is scaled at more-or-less equal intcr,.als along the axes of the scatterplots, because the observations then approxim:ltely are weighted cquall). Previous empirical studies have invoked rules-of-thumb in order to specify a range through which scale changes arc likely to detect meaningful changes in corresponding curve perimeters. At the Io,,,,c," end of the scale range spectrum, the finest scaJe de;My will be dependent upon the level of resolution' at ~,hich the b:,se-level curve was digitized. Previous research (Batty and Longley, 1987) suggests an appro- priate lower limit to be approximately equal to the mean length of the chords which comprise the curve, whereas an upper limit is a scale which approximates the curve with no less than eight chords. In the empiri- cal measurement exercise outlined here, about 30 changes of se:tle are interpolated within these limits in accordance with a geometric criterion. In fact, meas- urements at or close to the upper limit frequently depend upon the starting point used to begin measur- ing the curve, because on the one hand this can deter- mine whether near-unique large-scale features arc de- tected at all and, on the other, rounding errors caused by closing the meast, rcd curve with an incomplete chord {or chords) will be far greater at such scales (Kaye. 1978). To minimize this problem, some writers (e.g Kayc and Clark. 19N5) have taken coarse so:de measurements from a small number of different start- ing points, and then have averaged them to get a more stable fin:d value. In the I'olh)~ing empirical measure- ment exercises the cxch,sive dedication of a Mi- croVAX II computer It) the project allov.cd this prac- tice to he taken to its logical extreme, and each mc:|s-

urement at e~,erv scale change is the average of the lengths recorded by commencing measurement from e~.er~, successive point on the digitized curse.

4. XlEASL REXlE~,I" MEI"tlODS: SI 'Rt (FI I RED ~ i l K , EQUIP~CED POIA GON. I-I~ BRID ~AI.K,

AND (.ELL COt:~,T

4./. The structured uulk nwlhnd This is the original method used by Richardson

(1961) to measure the lengths of national boundaries and coastlines. In the original application. Richard- son manually' walked a pair of dividers along a map- pod bound~,ry, and obtained sc:,lc-dcpendent meas- urentents b.v adjusting the divider span. Shelberg. Moellering. :md Lain 11982) first automated this procedure with an algorithm designed to approximate a digitized curve using a prespecified range of chord lengths_ In the application of a simiktr :dgorilhm to the CardilTdata series, the initial (base) scale level Ibr each curve was computed by l i rstcalculal ing the dis- tances between each adjacent pair of (x,y) coordinates i :md ( i+ I) using:

,1,.,,, = [ ( x , - x , , i )= + 0 ' , - .r , ~ i )'1~ 2.

i = I . . . . . N - I. (t})

The I~rimeter of each base curve is given hv

'x' P = d,,,, i. 1101

The initial scale level for each chord was taken to be the average chord length, tha! is

r, = P / ( N - I). (11)

In practice the upper limit of a minimum of eight or so chords was met by basing the scale changes on the progression r~ = 2 ~ r , , k = 1,2 . . . . 31. This gave 30 changes in scale which conformed to the criteria iden- tiffed. In fact. this sequence of scale adjustments was used to gauge the cffcct of scale change using each of the four different measurement methods, ahhough strictly speaking the use o f the mean digitizing inter- val is of less direct relevance in the initiation of the cell-count progression.

The walk at any given scale begins by calculating the distance dr,., from a starting point (x,,. y,.,) to the second coordinate p:tir (x,, y,) using Equation 19). If this distance is less than the chord length, r. the next coordinate pair Ix,. ~, r , . ~) is selected, the distance d r , . , , ~ is computed and the test against chord length r is made again. This process continues until the dis- tance d , . , , > r and when this is :,thieved. a new point [ b .~ . rr.~] is int,:rpolated onto the line segment ,.~hich joins points [x,,~ i . . r , . , i] and [~ . . . r , , , l The ~alk then recommences from this interpolated point and proceeds through painstaking u',e of tr igonometry to span the curse ~,ilh chords ol-exacll', length r. As the end of the curse ix approached, the distance between the last interpolated point and the

i 7.

|'i,~

ure

] ]l[

u,,tr

,iHL~

n ,o

l M

.rUCl

Lzrc

d ~

d[k

proc

edur

e,

u_,ln

~_ il

l[er

J~'L

l~ C

~'p~I

]pLL

tL*r ~

rlLp[

llC~

di~p

ld~

i:! P e.,

i.

r: [-

!/. :-2

[ - 4 P.'%. Lt "~ ( , tF~ a q d %! B,,TT'~

end poin t Jh'qc-,[ v~,tt ~, tn'+arijhl ' , bc lc+s than r: m th:~

m-t_,_ncc. ,,. t-rJ.,.".:on o f the chord length r is compu ted

hl, ,+rJcr to ci~+.c 'he in te rpo la ted cur~e. N lea ,urcd

pcr~mctcr t J->i ;cn:.:th,, at an ; -,ca!c (r) ,:,in be ob ta ined [I,',111 an ' , .;ta<,',.mg p,+mt ,m the digi t ized ba`4e cur ' ,c: n

the x~. ,ilk hc.~in, o ther than at ei ther o f the cad po in t , . the intcrpolat~,m proccud-, a long the curxc in bo th

&rcct ion, , and thc fiqal recordud length c o m p r h e s the .,tim. or thc`4c computatt ,m~,. A_, s ta ted previously, the empir ical mca- t i rcmcnt> recorded in the Cardiff" c \ c r -

ci ,c compri`4c the a, .cragc o f the Icrlgths measured troln cxcr,, po>_',lblc s tar t ing po in t cm the cur,.c, re- [ ; c a l l e d . o t Ct ' , t i rnc . for exer t scale change .

Phc rudiment- , ~,1-tin-, p rocedu re arc gl ' .cn ~i<ual c \ p r c s s i t m iri Figttrc 3. These di,,pluys tq- tile 1')4'* ( "<lr d l r l ill b a i l c d ~c ',', e r e p r o d u l : o . I u', ing an interact l ' , c

,. c t - . m m t + l - t h e , t r uc tu rcd x~ :ilk alg( , r i thm, in which the ,q~cr',ilor specific`4 the ,nilial and lhml cho rd lengths (the I'ornlcr a`4 ;I pcrccnt,t<..:c o f mean chord length on Ihc ba~;c ¢t]l,,c. the la t ter a'4 an abso lu te ~,aluc). the

,,tJ+tNlg pt)ml , m t l i c b a s e curxe, and the n u m b e r or gci tc i ' , ih/athm., i l cxc ls ) th , t t arc ttl hc p roduced using

the ~+alk algt+l i lhnl . ~CICClI Cll l l lOlt l l l im for each lc~cl

rcc~,rtb, ( l ' l ,Ut l b tq ton i h+ h+pi Ihc chord Icn e.th to be"

in tc r : ' tqa tcd ontn the hu:,c cur ' , , : , the :":c:l-,urcd

~ ' r m ' c t c r t,l-~hc cur'+c at thL', .,ca!c. <t;'d t}:c tYr.H+lhCr ,'1' I . -ompictc o~ p-trt~a!" tr:.~,--+r-c ~, t!'.,,l'. ,!%_" ..'q.c,..~'--:lr', t.~

C!0-,2 t k c cu,'- ' ,c t m I t ' . . :~ ' , . ! .D , ; ' , ~ ' , . 1"!':C -CqJCP,,.:C [q"

,.]l-pl,t:, ', a!lou+., the opc: 'a:- ,r to - : t in .~ ' .hu,l ] , lpprccLi -

rIOt1 t ' , l the m a n n e r ip "+,,h~cl+ I I ]Cd%IrL 'd pcFi r ' r :c [ , : r

lengths l a r d i l U l l l b c r o [ C ! l o ; d . ) dccr . :d . , . " J., .,c;t!,."

incrca`4c',, and cou ld aF, o a>,l:,t a dcc l , l ,m ~+l] thc Ic,,cl

ot" fractal detail n],~+~-,l appr , ,pr l . t tc it, the -.lo'.agc and

di.,pl<t', o f <l g h c n digi t iz. :d d<tta ,.._'t The rc',ult~ o f the rui] j l ) ~c;<i]c-ch.tnkec mlorpol<t-

t t on~ o ' , ¢ r t he f o u r t i m e pc r i ud . J r¢ pic~,mnt,2d ',1,

R i c h d r d , , o n P lo t , , i n l : l ~ t l r c 4 x,t. 'hdt bCCOll"¢-, l l l l t l l -

c d i a l c l ) , i p p , l r c n t i'~ t h a t t he 4c i t t l c i ' o l r , ~ + I N t , doe , , tit+!

t \ + i l , ~ t h e " , t r<t !ThLl i [ IC t r e n d ~t~Jc't] ~t+ci lct h , l t c 111

dlc '<t tcd t h d t tilL" I o + - I l l l c a r io i2 rc - , , , Ion t+l [ : q t i . i l i + q l IX )

' ~ t ]mr r i c i r i l cd a d L ' q u a h ' l ) . I h# t rend in tl.c c]d ld R , i l h c t .

'401117 o t h e r r u n ¢ t i o n < i l (o1111 xCCI l l , [ o bc rCL lU i l cd

P r c x i O t l <, rL",Cal~;h ha ' , p O l n t c t l o t l l I h A I I l a c l , l l t r i t l -

c t 'p t , , t+ll l~ ¢;.1ll hi._, c l p p h c d u , ,¢ lu l l ~ CtLl~-~ .t i ' c ' , l i iC lCd

ra l l~C O f _-,¢alc,, ;,,hlcl] tlt+L " u t l tc l t lC h+ ~1 I 'L I r l lL ' t l l , l i

[ ' q l L ' r l tm ] c l l tm u n d e r , , iu t l ~, ( ) t h o r r 'c, ,c<i ich hLi'. } n ~ h -

hghicd ,lpl+drL'rll d]nctml inUi l lC <, in the I. l ich,l ld.. tul

Phq,, tu" 7. N d k L I I I O . [ t ) ~ 1-, ~ c h c r l l C l , I nd I o;L' l t+~..

11 -a '1

13.

g ..J

lO

1886

I I ~i i

I I I I

QI i

11 J!

o.

o = ..J

Io

1901

g l g l i l m l B

I I I

I I O

O I I I I

O I I

4 4 6 L o g ='. x Log : . x

8

A l l

.) v

o .

-I Io

1 9 2 2

' J I I I I t I l l l O

I I . i

Jl

O.

,,J Io

+ i i

1949

i

. . . . . r "

I I t O i i i i I l l i

I Q 9 I Q

Q I Q i I Q I

I D I i

4 4 +3 8 Log ~x Log . :x

] i g u r c 4 l,P. nu'll,tr-tb.,m Plot ' , ,q" pcrnrnclt+'r-',t:Ll]c rclJtnor~'4 f o r f o u r ttnP, c 'sht. c',, ",trtKlxJ+<d ',",J[k n t c t h , , d ILt~c Lur',L" L~,tt,[ 'ACIC dJ~.[ih/cd tl,.l11~ hlnlt+rtc,tl 111,1['~, ol d l l ]u ' rcn l .,ca]¢', ,llT.d tht£', "~tlhni,il:tt',c.- Jltl,Cti~T.'t,111t+ll ,+I t hc ,c rc,u]t- , c.t l l CI~I]kUT[1 (+it['+, the Conl1111111%..2dig F~IT1EC ob,,L'F ~, ,lilt+It -, x?~ !IICH Jl'u" ['Cpl c , . r l h :d I+,. ~t+[Itl ~.'lfL'Jk'~ II1 I 1.~LIrL'n 4 ( I ~' ,111~[ L+ I [ [ i)II~]L'_~ . l [ ld HLIII),. 1~,",9) Irl '.1¢', +- t , f prml,Lr l ] : , nlClht-.!tdt>_'lL,i i ItlL'tl,, t t l [[1In p, tpUl. Ihl,, d l~ l l l IC l l tq ] hi:Iv+,_-ell ~:tl[ll l l iOH ' . ta lc FLIII~C ~lll'~] t qhq r t th~cl-xAtl t t l ln 1~ iq,,i[ Ill,talL" I ,q ptlF[+OxCh ol [L'!-~lU~liq]

<tr;,t] , , .c. t~h tuh []+i+tti'~ ~, • l )en,+tc- , oh-cr~.,tth,t++ f, l l h n - ,, ' .H,~m .c . t le r.H:: 'c u~mul ; ,m It, a/[ h::." 4~Lc-,

Fracfa[ measurement and line generalization

Table I. Structured v,alk method: computational costs and statistical performances of competing functional forms

175

CPU u s a g e

D a t e D a y s : h o u r s : minutes

1886 15:23

1901 19:11

1922 2:07:10

1949 7:49

Log-linear form [8)

Log a D R 2

11.080 1.239 0.914

10.886 1.184 0.927

11.393 1.186 0.907

12.150 1.267 0.975

Transient dimension model (12)

Log a d * b2x10 -5 R2

10.719 1.141 5.865 0.982

10.622 1.117 3.947 0.985

11.114 1.109 3.901 0.984

11.883 1.211 1.202 0.991

* d = l + b 1 ( w h e n r = O)

1984) and has applied separate regressions to the observations lying to the left and to the right of these points. An interesting feature of Figure 4, which per- haps arises from the use of a large number ofobserva- lions in this study, is that the Richardson Plots each imply a single continuous curvilinear functional form, rather than an amalgam of juxtaposed log-linear forms. In effect it is not possible to identify clear breaks in the slopes of the plots, and thus the fitting of any number of log-linear functions to the data woukl remain arbitrary.

Experimentation with a n t , mbcr of different re- grcssion lines led to the selection of

log P = Ioga - h l l o g r - h, r l o g r (12)

as the best statistical fit. The goodness-of-fit (R'-) of this form compared to that specified in Equation (8) is presented in Table I_ The interpretation of Equa- tion (12) is that the fraetal dimension itself is a func- tion of scale and that the scaling coelticient (I - D) is given by

(1 - D ) = b~ + b , r . (t3)

As the scale r ~ 0 in Equation (12), so D --, I - b,. As such, the term h,r log r in Equation (12) acts as a dispersion factor which increases the fractal dimen- sion at increased scales [Eq. (12) redtices to Eq. (8) when h: = 01. This form is referred to as the transient dimension model.

Slructurod Walk Equ~lced Polygon

301~-

• 2 0 0 0 - 5

! g

I 1000

~1ooo •

2000-

°t o 2~ ~oo ~oo

NO. Of P o m ~

I ! I n 1 i 1000 2000 3 0 0 0 4 0 0 0 5 0 0 0

Hyl~rld WelL. Cal~l Court!

3 0 0 0 -

2 0 0 0 -

1 0 0 0 -

i IOO0

3 0 0 0 -

2 0 0 0 -

, i | 1 0 , 2 0 0 0 3 0 0 0 4 0 0 0 5 0 0 0 1 0 0 0

Figure 5. CPU usage of four measurement methods.

J 31O0O ¢ 0 0 0

CAGEO 1 5 : 2 - D

170 P.A. LO".OLE'~ and M. B . r r~

The R e valucs presented in Table I suggest that the transient dimension model Equation (12) produces a consistently better statistical fit than the standard log-linear form for e,+ery' one of the four time slices under analysis. The Richardson Plots sho,a n in Figure 4 i l lustrate that the structured ~,alk method produced est,mates v+ hicfi correspond closely to this functional t\~rm, with the clearest continuous trend being dis- cernible for the smaller step lengths. Although the positioning of the points does become slightly' more erratic Ibr the largest step lengths, there is no evidence of any sudden "'flattening" of the curve, v+hich would have indicated that the scales '.,,ere too coarse to pick up further fractal detail. This cohesion of the larger so+de points about the best fitting functional Ibrm is the rcsuh of the averaging of each scale observation through measurements from e',ery single po.ssiblc starting point. Finally Figure 5 shows that the struc- tured walk method is associated consistently with the highest CPU usage of the four methods. This is a consequence of the precise trigonometric interpola- tion of points upon the base curve. It might be conjec- tured that this precision ohvi:ttes the need to average out the measured perimeter lengths by using every conceivuhle starting point, although the decay ill the trend in the points ;.it larger scale steps ~tlggCsls this is possibly n o l the situ:ilion,

4.2. TIw cqttil~uccd pol.r.t,o, method This ~+as first suggested by Kaye (197,~) and ell,-

borated in Kayc and (' lark (Ig,RSI as a measttremcn[ method in which there is no need to compute new base-level points. The lirsl, perimeter length I\~r the sequence of SO:LIe changes is cornptlted by sunlnl ing the distance between :,djacent coordinates; the second perimeter length represents the summed distance bet- ween every second coordinate; the third sums the distance between every fourth coordm:ttc; and st) the progression continues v.ceding out all but every 8lh. 16th, 32nd . . . . ctc. point. This geometric point-weed- ing series is contrived so as to give observations a more equal spacing in the Richardson Plots, and hence a more equal weighting in the regression analy- sis. In terms of the Richardson Plots and regression analysis, the chord length r in Equation (3) which is paired x+ith an associated measured perimeter length is given by the average chord length spanning thc points at the corresponding level of the point-v, ccding sequence.

Formally then, a direction is established Irom :t

given starting point on the ba~,cle'.el curve (v,. r , ) : m d a chord is constructed to :t digitized point (.v,, A..r,, ~ ) which is k steps away from {.v,. y,); k is thus an index of scale. The dislanec d,. ,~ then is computed using Equation (9). and then the reset chord involving the point (%, .'A • .r,+ .+A ) is constructed from <.v,, ~..r, , ~ ). Eventually the endpoint of the baselcvel curve is ap- proached, and the level k cur,,e is closed on this endpoint when the remaining number of base points is less than step length k {this is equivalent to the

" 'remainder" as described lbr the structured walk). Computat ions in both directions from the starting point are added to determine the perimeter and mean chord lengths.

The Richardson Plots associated ~ith this method are sho~n in Figure 6 and the resuhs of regression anal,,sis in Table 2. At an intuiti',e le',el, one might anticipate that this method might yield rest, Its of a slightly more arbitrary nature, because exact per, meter lengths ~ill be dependent upon the evenness with '.~hich the base cur'+e has been digitized. For example, points are unlikely' to ha~e been "Torced" on long straight sections, so these sections arc unlikely to contain chord cndpoints; moreover, the entire shape of a measured cur',e is likely to change ff the base cur~e contains major irregularities or fissures which ,,viii be detected abruptly at a shift bet~een tv, o scale changes. In fact. the Richardson Plots show that any such effect is removed by the averaging process and the points actually follow a clearer trend than the structured v,alk plots. The regression results compare directly with the structured v,.alk results, both m terms of rnea.sured fractal dimensions and the statistical fits of the tV, o competing functional form specific:ttions. The biggest :tpparent difference bet',~ccn the two I11ethods SeelllS Io he ( ' I ' U usage i l:ig. 5 and Table 2). il1 that the eqtilpaced polygtm ntclhod used <_ 5 " , ) o f the resources rcquu-ed by the structured walk for it ft,llv ave, aged run. I lowcver, intermediate polygon plots are IIlore erratic th:ln structured ,.~ alk ones when

full aver.'lging does nc, t take place.

4.3. "lTw krhrid walk mellmd This v, as suggested by (' lark (19,~6) ,is a method

v,.hich I'Clanls sonic favorahle chau':tctct'istics of both the structured walk and the equipaccd polygon meth- ods. h is based directly upon the sltme prcspccilied geometric chord length series as the structured walk, which makes it less vulnerable than the equipaced polygon method to the spacing of point:, on tile base curve. However, it is similar tO the equipaced polygon method in that no new points are interpolated onto the base curve; rather, each chord is either extended or contracted to coincide with the nearest digitized point, which then is used its the origin from which the next chord is sought. Removal of the time-consuming trigonometric interpolations thus ser~,es to speed up the computations. It is based on the same lowest level of resolution (r,) its the pre,.rous two methods and cnt:tils similar tre:ttment of Ihe " 'remainder" distance as the end of the curve is approached.

f:ormally, the method proceeds in the slmle wa.v its the street,,red v, alk. except that when it point (.v,~. j . , , , ) is reached ,.+.here d,,,,~ > r. no new point is interpolated using the Shelberg. Moellcring. and Lain algorithm. Ifldv.,,~ - rl ~< Ide.,+~ t - r[then point (.x,,~..v,,a) is selected: if not. then thc point (v,,~ ~, )',,~ L) is selected, because this point is the closest to the point at v, hich chord length r intersects the base curve.

Fr:,clal measurement and hn¢ gcncrah/:ttlon 177

| !

o = ..I

1886

1o

g

D I g l I I ~ I l l l O I

I Io I

D O i j

1 O

1901

I ! A

N

,3 o.

..4

10 I I I I I g i l i ~ o o i Q i I l I

I I I 0 I I

I I

' +' ' i , ' '+ -' L o g ~ x L o g ~ + z

I I A ml

o.

- I I o

1922 1949

11 i I

t t~ I I I ° O l l l o I i I i I i i i o i ,3 o,%.

o. ".° e l % ~ "'+%

°u ,..I IO

, , v ,

L o g Az L o g ,'%x

l"igurc 0 l',P, ichard:-,orl Plot+ or l+critnclcr-scalu ' r,..'l~iti+,111",, l'or l'ol.lr lilllC slice';: Cqtlipaccd p o l y g o l t mdht)d • l ) L ' n o t c s oh ' . , , . ' l "+ ' ; lho l l l~lllLrl[~ within ,,.'ah: r i l l l ~k " L2OIIIIIIOII It) all lllll'2 slice,

The Rich,lrdson Plots assock, ted with this method (Fig. 7) show a similar pattern to those or the struc- tured walk method (l:ig. 4). The analytical results (Table 3) :,lso arc ct)nlparahle with the first two meth- ods, although the nlelhod is unahle to discrmdn:Lt¢ between the log-l inear l'ornls or the 1901 and 1922 series_ The graphs or CPU usage (Fig. 5) show that only conlparat i ;cly nlotle~,t s:lvhlg~, :ire nlad¢ toni- pared to lhe strl.lCtUled walk method, and the equi- paced polygon method remains the least demanding by Iilr in this respect.

4.4. TIw c('l/-('mml mr lh.d This method is more akin to, a r:tsleri/ed conccp-

l ion ol ' th¢ digi l i /¢d bane Cl.ll'l. C ;.lIId hil~ becI1 Sllg~¢sled by a numhcr or authors (I)¢arnley, 19X5; ( ;oodchi ld, 19,~0; Morse .'knd others, 19,".;_'5). In elt'ccL, the computer :algorithm imposes a square lattice for a range o f d i l l t rent spacing,~ across the base curve. The spacings ol ' t l le ddl'cren[ lattices introduce the sequence or scale changes over which the irregularity or lhc hast curve is Lo be measured. At each scale (grid spacing), the cell-count .'l l~oridml simpl~ enumerates all or ihc cells

r~,hl~ 2. [!qmpaccd polygon method: ¢ompuhltion:ll costs Lind ~tall~tical pcrl'ormancC~, or competing I'unctional I\)rms

CPU usage Bate Days: Log-linear form (8) Transient dimension model (12}

hours: minutes Log a D R 2 Log a d* b2xlO -5 R 2

1886

1901

1922

949

0:36

0:45

2:09

0:20

11.176 1.236 0.875

10.923 1.178 0.917

11.420 1.172 0.902

12.342 1.293 0.992

10.589 1.086 11.200 0.99~

10.594 1.094 5.920 U.99~

11.078 1.085 5.187 0.99i

12.132 1.250 1.211 0.991

* d = 1 + b l ( w h e n r = 0 )

17~ P A. LONGLEY and M. Bal-rY

I 1

M

K

- I

1886

3 1 1 D D O I

m e B

l l •

L o g ~ x

11

g "J I 0

1 9 0 1

0 0 ~ O'JO O 0 9 , 1 1 1 1

I Q I i i 0 I I i I

I

I I i i g I I

i , [ 4 6

L o g ~ x

I I

le .d

L

~' ~o - I

1 9 2 2

~ O ~ O O 0 0 0 0 O i

1 9 4 9

I I

, J

I I I l e l l l I

m I •

I g

I I O I I I I I I

1 1 0 m O I g

I I

I I • " , 'a

9

L o g A i L o g A x

I-'igtnrc 7. Richardson Plots of perimeter-scale relations for I-.ur time slices: hybrid walk method. • -- Denotes observation falling within scale range comm~n to all ttmc shoes.

th.'tt the basclcvcl curve passes through. Counts ;are made at each so:de change for grids originating at each point on the base curve: these are averaged to produce the final observations for each scale ch.'mge in the now-familiar way. Strictly speaking, each grid scale should be defined with respect to the start and endpoints on the baselevel curve, although for reasons of convenience and comparability the empirical re- sults reported and depicted below use the same 31 scales used for the structured and hybrid walk methods.

Formally, from a given starting point (x,. yp) with

a selecLed cell size r and d i rec t ion o f Iraverse. the next

c o o r d i n a t c (x, . y , ) on the base curve is a l ighted t, pon.

A lesl is made to see i f this po in t lies w i th in the s:lmc

square by cons ider ing whether Ix, - xpl >/ r or lY, - Ypl >/ r. If either of these conditions hold, a new point is established where the coordin:tte in question is updated in the direction of greatest increase, Thus if

Ix, - x , [ ~< Lr, - y p [ , x p ~ , = .xp + r and )';,, i = Yp whilst if the converse holds. ?"e,, = xe and r r, ~ = .r e + r. If the increase along both the x and y axes is less than the grid size r, then a new coordinate point (x,, ,. .v,~) is selected and the tests are made once again.

Table 3, t l , , b r i d walk method: computational costs and st:ntistical performances of competing functionvl Forms

CPU usage Date Days: Log-linear form (8) Transient dimension model (12)

hours: minutes Log a O R 2 Log a d* b2x10-5 R 2

886

1901

1922

1949

12:34

15:52

1:21:56

6:52

11.119 1.248 0.913

10.895 1.190 0.929

11.412 1.190 0.906

12.416 1.308 0.989

10.715 I .137 7.256 0.987

10.633 1.117 4.560 0.990

11.111 1.106 4.567 0.989

12.197 I .262 I .001 0.996

* d = 1+b I ( w h e n r = O)

['-'r ac IA ', rP,¢a:,urcrl:¢ <q!

Each time the direction is upd. , ted, a ce[l ha , N.-cn crossed and thu-, i~, counted. Unlike the prc',lou:, n-cih,,d:,_ v~ hen the cP.dpoint ol- the cur', e i- appr,_~acl-- cd. the cell a p p r o \ m l a u o n simply iinibhc~ ~ h c n the cclI in ~hict~ the endpoint c\i.,ts has been Ldentlficd.

The ",~a: in ~h ich this procedure ~o rks is illu- ,,Ir,ltcd in I-:l~turc ~, v. hich shov,,, that the intr tca 'e

fOFm of the line L, lost at an early stage in the cell- count process. It i'~ I \ ,r this rea:,on that the method has bccn ad', ocated as a computat~onal ly ine,~penq,.e firq . ipproximafion to iiqeastlrcmenl. Figure 5 ,~hox~s that the cell-count method is closest to the equipaccd pol_'.- ,'tin method in itg me:leer CPU requirements. Al- though the Richard,,on Plots I lrrig. 9) exhibit generally

,,nlooth trends, there is sonic c ~, idencc o l -" 'bot toming otil '" ol-the cur', e'. tit the Co.lrscq ,,tales. This indicates that the method does not detect fractal detail tit these ,,talcs, dc<,pitc the a ' .eraging that has taken place. l a rge ly because ti[' t i l ls tile fractal d inlcnsion <, and • ,tati,.tical fits sho~,.n in Table 4 bear lens thrccl COnl- p. l r~on ~,<ith the other nlethods lhan has bccn the ' q [L l , l t IO I1 In t h e prcxious b, CCt lOI IS

5. ( ( ) \ ( L I ,"41{i\14

|"racial nlcil'-,tlrenlellt p r e ~ e n t s a po,,~,crltl[ forlll of c.nhLL'raphic line generalization ~hich is uot dcpen- dcnl upon a priori definition o1 tile conslilucnl lea- lures ~luch nlakc up the Iolality tll-line ch:tr:icler. In inelhodohlgical lCln/~, such nleasurenlcnls inighl bc used Io nl',inalte di~ii i /ed d:ll.ibases Ihlotl,,h sdcctive "-~,cedlng" o1' thow poinl~ ~,luch con l r lbu l¢ least IO o~erall line char',lClCr; :llic[-natively. these nleasurc- [ilenl,, inighi pro~ide tile basis to daliibas¢ enhance- niClll IJirotlgh "'relidering'" o1 nnnili lai digitized pohll inl \ui l iahon (Wise. 1987) or coarsely raslerl/¢d tl:lla.

Thcsc and tither molivali,ms ha~e provided the iiIIpCtus to tile e% altiallOn of I~.~tll" fracta[ ntcasurcnlcnl algorltlun~ which lia~e been assessed LIMIIg ilUlllCrous exphmlior_,, displa)~ i l:ig~. _I and 8) l ind repealcdly <i~,cia~ed n l c : l s t l r c m c n l s (l:ig~,. 4, 6, 7. and 9). Re- peated c~pcrnncnlatlon with the display :dgorilhirls h@flightcd a number of characteristics of the I\mr incthodx: these include r:idic:il shifts hi hne ap- pro%imallt ins :it adjacent order o1" nl:lgnJltlde scale chaiigcs using the cqLlip:iccd plll,,gon method, and c~msiderabl¢ ~arkiliims in coarse scale shape ap- pl't l~imalit i l iS con',eqtlenl UpOFI using diffc.rent Marl- iltg pninls in Ihe ccll-count :tlgorithm. It scents plaiis- ~blc Ihal such oh,into occurrences account for the l t loderi i le scattering o1" points : lbout lilt: regression Irc~ld~ Ihal II:l~c beeil identilicd in some previous pcipcrs Such chance ~ariation can be climinaled by a~eragm,g llleBMIr~:nlcnlk ~hich ha%e coillillenced at e~er.', sillgle successi',e di~ltl/cd coordlllal¢, lind II is ihe~e nle,i,,tlrcnlcnl,, ,a hich ha~e bccn used to produce Ihe re~ulls slio\~n in 1-ahle~ I 4. These a,Qnlplolic i-c-~ulD, and Iheir an.,oClalcd Richardson Phll~ sl l ,~ Illal conni.,IcnI lrends are detected ushtg an'~ of Ihe t\KIr nle{istlrenlcn{ n lc ihods, allhough Ihere is some

a[ld '.w.t: gcncr_~]:;a.'_!~,~ i %1

¢'.idcnce from Tables I - 4 to suggest that the cell count and h', hnd walk method.; arc !tcs<~ able to detect

finc t rends in the data (i ¢ lhc result,, shov, a more ambpgu~,u> trend bet,.,,ecn the 1901 and 1'-)22 data

serics than is indicated by the othc, r tv~,o rncthod~l Tim, ma ; in part rellcc~ the " b o t t o m i n g out" of the Rlchard,~on Plots for coarse cell-count a p p r o \ i m a - tions, indicat ing that the cell count in parllct, lar is the least reliable of the four methods at the coarsest scales. The major difference anlongst the four meth- otis concern,, central proctes,a~r usage. ~ i th Figure 5 shinning a measured difference bct,,,.een the modest

rcqt i rements of the cell count and cquipaccd polygon methods on the one hand. and the s t ructured walk and h' ,hrid v, alk methods on the other. In practical i l least l rcnlCll[ sHtl',lLions the c h o i c e w o u l d seeill all lit.'

belx,,een a few passes through the data using a robust but t ime-consuming ~'.alk method, or greater use el" a~eraging (perhaps a l so across :1 narro'.,,er range el- scale c h a n g c q using the equipaccd polygon or cell- cour t [ nlclhl~t ls .

A second m~portarH isstie also :irises from thc<,c empmca l re.,tllts, a n d COllo2rn~ tile I-tinctional I-~)rm of the regression hne which ~s lilted to the points on Ihc Richardson Plots. With Ihe possible exccptmn of the 194q data writ<; (which '.,,ere digit i /ed at a diffcrcnt ItS, C] (]l" b a s e CUl'}e resolution; sec Longle', :rod Ball.,,. 19,R9, I'or a dc t , l l l ed c x p l a n a l i m l , the n n p l i e d f u n c -

l ional form is clearly cur ' , i lmcar. Al though so[tic pl'CVil.ltl',, I l leasureBlcnl %,tvrOr'k h~ts c r e a t e d i.ibscl-~,a[ionb,

'aluch do nol foiler, a clear s t ra ighl-hne trend (e.g Nakano. 1984; Schcrtzcr lind I_ovqov, 1984) thin i:lit~- blcm usually hip, been rcnt)l,,ed by' the insertion oF at step function rote the Iog-hnear regressions Tile in- scr tmn and placing t)f this step is c l ea rh a st, bjecti',¢ Judgcnle,lt ~ hich also is handicapped by chance scat- tering of Ihe restricted nunlber of unaveragcd ob- serxations abou t the mo_,,t plausible trend. It thus is a ntoot point ~,,.hcther extensive use of a,,craging of these data xeric_.,. COUlqcd ,,s'ilh closer spacing of ob- servations, might reveal the sort o f Ct)lllilltlOUs CUF- viline:lr funct ional form thai has been identiIied here. This st, ggcsts that more careful anal.,,sis in reqnircd in fult,rt,, appl icat ions attd that nlanv cxiq ing applica- lions should bc rc,,vorked.

If this is indeed the si tuat ion and the base cur,,en are " 'multufractal", Ihen the) art,. not fractal at all insofar as a single ~,.ell-defined fractal d imension does not occur across the entire, scale range. IIov,-e~.er. Mand lebro t (1984) acknowledges a generic statistical use of the term " 'fractal" ~'.hich. tirstly, allows dis- crct ion to bc exercised ~,hh respect to the range o f scalcs which are used to klentify a single fractal di- IllCnMl.ln a n d . s e c o n d l y , r e c o g i l i / ¢ s th[ll n lan} prob- lems nccessanb, ~.HI in, .oh ¢ more l h a n a ,,melt fraclal sel. It Is this second quahficati(m that is e l greatest i n l p o r l a [ l c e Ill the pl-csenl coTl le \ t , a n d o n e o f the

reason,, '.,.hx Mandlebro t inm~ducen tt c,,nccrn~ thee o c c u r r e n c e o f f r a c t a l s ~,.hich a rc c m b e d d c t [ in spa_tees

(and b,. cxien,qon o f thin a r g u m e [ l l . (lille SlICeS}

J

J ~ L ~ rl~'~'~':~i~'~,~'!~.! ~ ! ' ~ i : ~ ; ~ : - ~ ~, '~!~k~ ~ I~

L~

i t

i , ] P..-~, L()',GLI-'~ .t,qd \ l B x r r ;

• 1886

E t ..I

10

I

9

I I I I I I I I i I

e

i e

4 6 8 L o g ~ x

: 1901 i

E

- I tO

° , . .

4 6 L o g : ~ x

8

1922

~ 1 1 ~ , i <]

.J l O

' , - , ° A t . t t

I t ~ t t r ¢ ()

J e . 1 0

I

1 9 4 9

I I I l I I i I

I I I I I I I i I i I

I I I I i i I

I

, ~ 7 9 L _ _ - r - - i 4 6 0 2 4 6

L ° 9 + z L o g A x

I , t t c h , . ' d . , . n Pk+l'~ o f p c r m l e l e r - s c a l c tel.tilt)H,+ lt." ft)t£r inane +.Ince~ ce l l t'tltlll[ rnctht, t l • l)¢nt)tc', ,~ )h ' , c r ' , ; t l l (m falling within s+.:;llc r : [ n g c Ct)l|llll()ll ILl all tittle ~Jl~-'c'~

enthused ~lth ddl-crcnt notions o1 disl:mcc (sec tl;tr- ~¢~, 19h9, chap 4, p 191-229 I'or an explichly geo- ~lal~hical t rca lmcnl ) . In tlle absence oF map projcc- Don+-, x ~. hlch pel lml ~tll UI)C())ILCnIICq.IS mtcrprctaUon of tll,,t,mce, there ,,+ould scent to he an ine,, i lablc nccd to c~msidcr compound I racla l dimensions

This point brings the argunlcnt around Full circle ,,,,d rcqtnrc:, us to make an explicit statement as It) the rel.nionqfip bct~ccn measurement and theory. In sh,)rl, i~ lhc notion or measurement usel'u] beyond d~c ~impI) descriptive il" we arc only to obtain complex

"'black box" dcscriplms or Ca[h~urapluc phcnomcna • ~,Ifich arc the COIII[~OIII)i[ '~piIllilI cxprcs'qon or inCOlll-

plclcly enumerated or inL.mi~IctcI~ undcrsto(~d processes? On this point MandlebruI (19X4) is un- cquivoc-d in hi.', bchcr m the "'ahsolui¢ primacy or

g¢omclr, , over analytic rc lh)cment" ; H is the role t)r subsequent theoret ical de~elopmcnl to disentangle Illc con.,.tltUCnt "'fractaI uigcndimcnsit)ns'" lhal may be o1" interest. Wi th in Ihc geo~raphic:t l l i tcr: t lure, lhi,, scn l imcnl is o r con t inu ing rclcvam.'e in the context ol- Harvey 's (1969, p. 227)ear l ie r concern that " the nel-

Table 4 (-ell COLlnt method: comput~itional co.,;ts and staU'~lical pcrrormancc'~ t~l'~.'(~UlIl'~Cl- Brig I'unclional forrn,~

Date

1886

901

92Z

1949

CPU usage Days: hours: minutes

3:04

3:51

1 1 : 3 0

1:37

Lag-linear form (8) Translent dimension model (IL)

Log a D R 2 Log a d* b2~10-5 R 2

11.326 1.267 0.953

11.079 1.200 0.967

11.617 1.209 0.95Y

12.288 1.274 0.985

11 . 1 0 9 1 . 2 0 7 3 . 5 2 5 0 . 9 Z _ ~

1 0 . 9 1 9 1 . 1 5 6 2 . 5 v 2 o . g z

1 1 . 4 2 6 1 . 1 5 o 2 . 6 ~ 6 0 . 9 S ~

1 2 . 1 4 " 1 .2 c, ,, 0 . 6 4 6 0 . 9 u :

* d = 1+bl (when r = 0)

Fractal measurement and line generalization 183

work for the in terp lay o f fo rmal m a t h e m a t i c s and

empir ica l p r o b l e m s is poor ly c o n n e c t e d " . T h e pros-

pcct now may exist tha t t h r o u g h the wider involve-

men t o f c o m p u t e r c a r t o g r a p h e r s and g raph ics analysts . ~ e ~ill ident ify the d o m a i n wi th in which

theoret ica l d e v e l o p m e n t can be i n fo rmed mos t use- full~ by rou t in i za t ion o f the m e a s u r e m e n t task.

REFERENCES

Batty, M , 1985. Fractals - - geometry betv,een dimensions: New Scientist, v. 11)5. no. L450, p 31-35.

Batty, M., and Longlcy. P.. 1986. The fractal simulation of urban structure. En'.tronment and Planning A. v. 18, no. 9. p. 1143-1179.

B~LtI.',. M., and longlc,,. P,A.. 1987. Fractal-based descrip- tion of urban lotto: Enxironment and Planning B. v. 14. no. 2. p, 123-134.

Bracken, !,. 19~5. Computer-aided cartography v,'ith micro- computers: a practical guide Ii.) [~11CROPLOT: Working Paper 911, Dept. Tomn Planning. UWIST, Cardiff, U,K., 78p.

Buttcnlield. B., It}84, Line structure in graphic and carto- graphic space UlllV. Wa,,hington :rod University bh- crolihus lnternattonal. Ann Arbor. Michigan, 320 p,

( 'lark. N.N.. [t)S6. "l-firec techniques for implementing di- gital fractal analysis of particle shape: Powder Tcchnul- ogy, v. 46. m~. I, p. 45-52.

l)auulon. M J., 1977. Coal Metropolis: Cardiff 1870-1914: [,eiccslcr tTn~. Press, I,eicc'dcr. U.K., 261)p.

I)carnlc.,,, R., I~$5, lillccts ol'rcsolution tin the mcasurentcnt of grain "siec": Mineralogical Maga~'inc, v. 41}, no. 3. p. 539 546.

(~oodchild, M F.. I~}SI), Fract:|ls anti the :Lccuracy of geo- graphic:tl luc:l~urcs: Jour. Math. (icology, v. 12. no. I, p. 85 t,J.~.

( ;oodchlh.I, M ,:Hlt[ M:mrk, I).M., 1987, The fract;.d nature of gcognq'~hic phenozucna: Ann. Assoc, AIn, Geograpfiers, v. 77. no. 2. p, 2~'~5-278.

I I:Lr'~cy, I ) . I~)(,t), |!xpl;otalion ill gcograpfiy: Edward Ar- nt~ltI, [,ondt,n, 521 p.

Kaye. B.H.. 1978, Specification of the ruggedness and or texture of a fine particle profile by its fractal dimenston: Po'.,,'der Technology, v. 21, no. I, p. 1-16.

Kaye, B.H.. and Clark, G.G.. 1985. Fractal dimension of extraterrestrial fineparticles: Dept. Physics. Laurentian Univ.. Sudbury, Ontario. Canada. 16 p.

Kaye. B.H., Leblanc. J .E . and Abbot. P., 1985, Fractal description of the structure of Fresh and eroded alumi- nium shot fineparticles: Particle Char:tcter]sation. v. 2, no. 1, p. 56-61.

Longley, P.A., and Batty. M.. 1989. Measuring and simulat- ing the structure and form of cartographic lines, m Hauer. J., Timmermans. H.JP. . and Wrigley, N., eds., Contemporary developments in quantitative geography: Reidel, Dordrecht, Holland. in press.

Mandlebrot, B.B.. 1984, Each fractal set has a unique fractal dimension, m Tatsuml. T., ed . Turbulence and chaotic phenomena in fluids: Else,,ier. Amsterdam. p 203-206.

Morse. D.R., Lawton. J.H.. Dodson, M.M.. and Wilham- son, M.H.. 1985. Fractal dimenston of vegetation and the distribution ofar thropod body lengths: Nature, v. 314, p. 731-733,

Muller. J.-C.. 1987, Fractal and automated line gencrahsa- tion: The Cartographic Jour.. v. 2-1, no. I, p. 27-34.

Nakano, T.. 1984, A systcmatics of "'transient fr~lctals" of rias coastline: an example of rias coast from K anta~shi to Sht,'ugawa. northeastern Japan: Ann. Rcpt., Inst. (ico- sci., Umv. Tsukuba, no. 10. p. 66-bX.

Richardson. L.F,. 1961. The problem ot'contlgu]ty" an ~lpp- cndix to statistics of deatlly quarrels: General Systems Ycarbook, v. b, no. I, p. 139 187.

Scficrtzer. I)., and Lovejoy. S,. 1984. On the dimension of atmospheric UtOliOns, i~l Tatsunll. T., ed . q'urbulcnce and chaotic pficnonlena in Ililltl~.: lilscvicr, Aiustcrdanl. p. 505-512.

Shelbcrg, M.C., Mocllcring. I1., and L:im. N., 1982, Measur- ing the fracial dimensions of empirical c,trtographic or.r- yes: Auto-( 'arto, v. 5. p 481 4tJ().

Wise. S., L9~7. The use of fr.'ictals in thematic mappillgs: Working P:ipcr, SWURCC, Univ. Bath. I|,lth. U.K., 12p.