Review of Methods Fractals 24pp

8/12/2019 Review of Methods Fractals 24pp

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Math ematic al Geology Vol. 26 No. 1 1 994

A R e v i e w o f M e t h o d s U s e d t o D e t e r m i n e t h e F r a c t alD i m e n s i o n o f L i n e a r F e a t u r e s I

Brian Kiinkenberg

An in-depth review o f the more commonly applied methods used in the determination of the fractaldimension of one-dimensional curves is presented. Many often conflicting opinions about the dif-ferent methods have been collected and are contrasted with each other. In addition several littleknown but potentially useful techniques are also reviewed. General recommendations which shouldbe considered whenever applying any method are nmde.K E Y W O R D S : fractal measurement, divider method , box counting, spec tral techniques, vario-gram.

I N T R O D U C T I O NAlthough an increas ing number of papers p rov ide a theore t i ca l bas i s fo r observ-ing f rac ta l behavior in geomorphologica l phenomena, the se lec t ion of a methodwhich can prov id e a cons i s ten t and re l i ab le de termina t ion of the f rac tal d imen-s ion remains unreso lved . Many methods have been developed , bu t mos t havethe i r p rac t i ca l and /or theore t i ca l l imi ta t ions . In thi s paper I wi l l rev iew m any o fthe methods used to de termine the f rac tal d imen s ion of curves or one-d im en-s ional p rof i l es . Theore t i ca l bases fo r the methods descr ibed be low can be foundin a number of t ex t s and papers (e .g . , F eder , 1988; Turco t te , 1992; Pe i tgen ,JiJrgens, and Saupe, 1992).

Determin ing the f rac ta l d imens ion of l inear fea tures - -na tura l fea tures wi tha topolog ica l d imen s ion of on e- - i s a p rocess under taken in a wide var ie ty off i e lds (Table 1 ). L inear fea tures ex i s t in the i r own r igh t (e .g . , coas t l ines , r iverne tworks , fau lt t races ) , and a wide v ar ie ty of appropr ia te methods to de terminethe i r f rac ta l d imens ion have been developed (Table 2 ) . However , cu t s o r s l i cesof g rea ter d imens ional phenomena (e .g . , topographic contours o r p rof i l es ) canal so be analyz ed us ing those same method s , methods which can be eas ier tot Received 2 February 1993; accepted 22 March 1993.- Department of Geography , UBC, Vancouver, British Columbia V6T IZ2.

3

088 2 8121/94/011~0 0023507.00/l Q 1994 International Association for Mathematical eology


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4 KlinkenbergTable 1 . A s e l e c t i o n o f A p p l i c a t io n s o f M e t h o d s U s e d t o D e t e r m i n e th e F r a c t al D i m e n s i o n s

o f L i n e a r F e a t u r e s

M e t h o d A p p l i e d t o C o m m e n t s C i t a t io nA r e a / p e r i m e t e r

B o x c o u n t i n g

D i v i d e r r e l a ti o n

K o c a k ' s l a wP o w e r s p e c t r u m

D i g i t i z e d s h o r e l i n e s a n dc o n t o u r s

D i g i t a l c l o u d i m a g e s

D i g i t a l i m a g e s o f c r a t e rs o nM a r sS i n k h o l e p e r i m e t e rsD i g i t i z e d s h o r e l i n e s a n d

c o n t o u r sP h o t o g r a p h s o f v e g e t a t io nR i v e r n e t w o r k s e x t r a c te d

f r o m a D E MFrac tu re pa t t e rns de t e rmined

f r o m r e m o t e i m a g e r yU s e d p h y s i c a l d iv i d e r s t o

m e a s u r e l a v a f lo w sD i g i t i z e d c o n t o u r s C u l l i n g a n d D a t k o ,

1987D i g i t i z e d s h o r e l in e s a n d D - e l e v a t i o n ( i n c re a s e d ) G o o d c h i l d , 1 9 8 2

c o n t o u r sDig i t i z ed shore l i ne s De f in i t e b reak a t a Ken t and W ong , 1982

cons i s t en t d i s t anceL ine ske l e tons o1 cav e Exh ib i t ed f rac ta l beha v io r Lave r ty . 1987

p a s s a g e s c o n t o u r s D - e l e v a t i o n ( i n c re a s e d ) R o y e t a L 1987D r a i n a g e b a s i n p e r i m e t e r s D - f a i rl y c o n s i s t e n t B r e y e r a n d S n o w .(be tween 1 .06-1 .12) 1992

Dig i t i z ed ca r tograph ic l i ne s D - sca l e o f ma p MOi ler , 1986 , 1987A r e a s o f l a k e s K e n t a n d W o n g , 1 9 82N a t u r a l r o c k s u r f a ce s D s c a l e o f a n a l y s i s B r o w n a n d S c h o l z ,

1985D i g i t a l m o d e l o f s e a f l o o r D s c a l e a n d d i r e c ti o n , F o x a n d H a y e s , 1 9 85

t ~ m I t o 1 00 ? k mN a t u r a l r o c k s u r f a c e s D - s c a l e a n d d i r e c ti o n , P o w e r e t a t . 1987

f r o m 1 0 - s t o I 0~ mDig i t i z ed waces o f f au l t s D sca l e f rom I0 5?? t o Scho lz and Av i l e s ,105 m 1986

D e l e v a t io n ( i n c re a s e d ) G o o d c h i l d , 1 9 82D - c o n s t a n t f ro m 0 . 5 k m K e n t a n d W o n g , 1 9 82

t o 2 0 k mD cons t an t be tween t and Lov e joy , 1982

I 0 0 0 k mW o r o n o w , 1 9 8 1

Large r s inkho le s - f r ac t a l Ream s, 1992D e l e v a t i o n ( i n c re a s e d ) G o o d c h i l d , 1 9 82B r e a k s i n p l o t d e l i m i t e d M o r s e e t a l . 1985

v e g e t a t i o n t y p e sA b o v e a c e r t a in s c a l e t h e T a r b o n t o n e t a L 1988

ne twork i s space - f i l i ng( D 2 )

D m u l t i f ra c t a l f ro m o n e V i g n e s - A d l e r e t a l .a rea ; D - 1 .5 f rom 1991a n o t h e r

Lava shape i s sca l e - Bruno et al.~ 1992inva r i an t

D - e l eva t ion ( i nc rea sed)


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F r a c t a l i m e n s i o n o f L i n e a r F e a t u r e s

T a b l e I. ContinuedMethod Applied to Comments Citation

25

Variogram Various geophysical Burrough, 1981, 1984phenomenaSoil profiles Found very high valuesof Burrough,1984DSoil pH Found very high valuesof Culling,1986DDigitized maps Consistently identified wo Culling nd Datko,fractal regimes 1987DEMs Mixed results; D - direction Klinkenberg ndand the method used to Goodchild, 992create the DEMIce sheet height profiles Clear fractal structure; D Rees, 1992derived from Landsat data ice thickness

implement, and require far less computational power, than those methods whichmust be used when analyzing the greater dimensional phenomena. In addition,reducing the dimensionality of, for example, a surface or volume, means thatthe nature of the fractal can be altered (Mandelbrot, 1989). Thus, although atopographic surface or a cloud are self-affine fractals (see discussion below),horizontal slices or contours through those phenomena exhibit self-similar be-havior, and the fractal dimension of the slice or contour can be directly relatedto that of the surface (Orey, 1970). This dimensionality-reductionprocess doesn'talways alter the fractal nature since profiles of self-affine surfaces remain self-affine.

Determining the fractal dimension of a self-similar feature is generallyeasier than determining the fractal dimension of a self-affine feature, as will bediscussed in the following sections. Determining whether a one-dimensionalfeature is self-affine is also a simpler process. If one can interchange or rotatethe axes of the coordinate system used to map the feature without producingany fundamental changes, then the feature has the minimum requirements forself-similarity. For example, exchanging the (x, y) values which define a contou rline alters nothing essenti al--both sets of axes carry the same information. How-ever, one cannot exchange the (x, y) values of a topographic profile--eventhough both axes may be measured in the same uni ts- -since the vertical axisintegrates different information (i.e ., gravity) than does the horizontal axis. Evenmore obviously, the trace of a particle through time has axes which representdifferent information (time and distance). More formal ly, with self-affine fractalsthe variation in one direction scales differently than the variation in anotherdirection (Mandelbrot, 1985).


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2 6

Name of method

K l i n k e n b e r g

T a b l e 2 . Me t hods U se d t o Es t i ma t e t he F ra c t a l D i me ns i onRelation Estimate of the fractal dimension

Area/perimeter A c~ p , - / o Plot log A against log P , slope isrelation 2 / D

A = a r e aP = perimeter

Box counting n e ~ b - o Plot log n against log b, slope is- D

n = num ber of f i ll ed boxes ,b = box size

Divider relation L ( r ) c ~ r I - o Plot log L ( r ) against log r, slope isI - D

Korcak ' s l awL(r) = length of tra i lz = step sizeN r ( A > a ) c~ a . o / 2 ~

(Empirical relation for N , ( A > a ) = numbe r o f a re a sislands) above size a

Line-scaling X - N V ~ ; Y - N v 'VrH ' = - -VxX. Y = standard deviat ion of x-,

),-coordinates, respectively (seetext for details)

Power spectrum P ( t o ) c~ w ~ - z m

VariogramP ( w ) = the powerw = the frequency( l ( Z p - Zq ) 2 l ) o t (d p , ) ( 4 2 1 J ,

Z e , Z,~ = elevations at points pa nd q

d p q = distance between p and q

Plot log N , ( A > a ) against log a,s l o p e i s - ( D / 2 )

1I f H ' = 1, D = - Votherwise D = 2 - H'

Plot log P ( w ) against log w, slopei s - ( 5 - 2D )

P l o t l o g [ . . . ] ) against log d p q ,slope is (4 - 2D)

= s ta t i s tica l expe c ta t ion. For re fe rences see sou rces l is ted in Table I and throu gho ut the text .

T h e d i s t i n c t i o n b e t w e e n s e l f - a f f i n e a n d s e l f - s i m i l a r p r o f i l e s i s a n e l u s i v eo n e , h o w e v e r ( M a n d e l b r o t , 1 9 8 5 ). C o n s i d e r a f r a c t u r e s u r f a c e . O n e c o u l d a r g u et h a t s i n c e t h e f o r c e s w h i c h c r e a t e d t h e f r a c t u r e a c t e d d i f f e r e n t l y i n d i f f e re n td i r e c t i o n s ( i . e . , t h e s t r e s s w a s a n i s o t r o p i c ) , t h e r e s u l t a n t p r o f i l e i s a n a l o g o u s t oa t o p o g r a p h i c p r o f i l e , a n d s h o u l d t h e r e f o r e b e c o n s i d e r e d a s a s e l f - a ff i n e p r o f i l e( s e e S a k e l l a r i o u e t a l . , 1 9 91 f o r a f u r t h e r d i s c u s s i o n o f t h e s e l f - a f f in i t y o f ro c ks u r f a c e s ) . T h a t t h e m a t e r i a l p r o p e r t ie s t h e m s e l v e s w i ll h a v e a n o b v i o u s e f f e c to n t h e n a t u r e o f t h e p r o f i l e a d d s a f u r t h e r c o m p l i c a t i o n . T h u s , f o r a n a l y t i c a lp u r p o s e s , w h e t h e r a p r o f i l e i s s t u d i e d a s a s e l f - a f f i n e c u r v e o r a s a s e l f - s i m i l a r


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F r a c t a l i m e n s i o n o f L i n e a r F e a t u r e s 27c u r v e m a y u l t i m a t e ly d e p e n d o n th e o b j e c t i v e s o f th e r e se a r c h e . g . , C a r r a n dW a r r i n e r , 1 9 8 7 ) .

T h e n u m b e r o f m e t h o d s w h i c h c a n b e u s e d t o d e t e r m i n e t h e f ra c ta l d i m e n -s i o n o f a l in e a r f e a t u r e i s r e l a t i v e l y l a r g e T a b l e 2 ) , a n d n e w m e t h o d s c a n a n da r e) a l w a y s b e i n g d e r i v e d . T h e t y p e o f l i n e a r f ea t u re a n a l y z e d , w h e t h e r s e l f -s i m i l a r o r s e l f -a f f in e , m u s t b e c a r e f u l l y c o n s i d e r e d w h e n s e l e c t in g a n a p p r o p r i a t em e t h o d . S o m e m e t h o d s c a n w o r k w i t h f e a tu r e s o f e it h e r t y p e , w h i le o t h e rss h o u l d o n l y b e a p p l i e d t o s e l f - s i m i l a r f e a t u r e s o r , i f t h e y a r e a p p l i e d to s e l f -a f fi n e f e a t u r e s , m u s t b e a p p l i e d in a v e r y c o n s i d e r e d m a n n e r . F o r s o m e m e t h o d s ,i f t h e c u r v e i s s e l f - s i m i l a r , t h e n t h e f r a c ta l d i m e n s i o n i s c o m p u t e d o n e w a y . I ft h e c u r v e i s s e l f - a f f i n e , t h e n t h e f r a c t a l d i m e n s i o n i s c o m p u t e d i n a n o t h e r w a y .T h e i n t er p r e ta t io n o f t he d i m e n s i o n s p r o d u c e d b y t h e v a r io u s m e t h o d s a l s or e q u i r e s c a r e f u l c o n s i d e r a t i o n o f s e v e r a l a s p e c t s , s u c h a s w h e t h e r t h e d i f f e r e n c e sb e t w e e n t h e o r e t i c a l ) e x p e c t a t i o n s a r e t h e r e s u l t o f t h e m e t h o d i . e . , th e p r a c t i c a li m p l e m e n t a t i o n ) o r o f t h e f r a c t a l m o d e l B r o w n , 1 9 87 ; D u b u c e t a l . 1989a ;A n d r l e , 1 9 9 2 ) .

T w o m a i n a s p e c t s o f m e t h o d - b a s e d p r o b l e m s c a n b e i d e n t i f ie d . I s t h e m e t h o db e i n g u s e d i n a p r o p e r m a n n e r ? T h i s a s p e c t i s d i s c u s s e d i n d e t a i l in th e f o l l o w i n gs e c t i o n s . T h e s e c o n d m e t h o d - b a s e d p r o b l e m r e l a t e s t o t h e u s u a l l y s t a t i s t i c a lm e a n s b y w h i c h t h e f r a c t al d i m e n s i o n i s c a l c u l a t e d . I f l e a s t - s q u a r e s r e g r e s s io ni s u s e d t o d e te r m i n e t h e s l o p e o f a c u r v e - - a n d f o r m o s t m e t h o d s t h e f r a ct a ld i m e n s i o n i s s o m e f u n c ti o n o f s l o p e - s t a t i s t ic a l c o n s i d e r a t io n s m u s t b e p r o p e r l yo b s e r v e d o r t h e d e r iv e d p a r a m e t e r s m a y b e s u s p e c t. A l t h o u g h t h e sl o p e c an b ed e t e r m i n e d f r o m a h a n d - d r a w n li n e e . g . , M a r k a n d A r o n s o n , 1 9 8 4 ), th e r e s u lt sc a n b e u n r e l i a b l e . F o r t h a t r e a s o n , a n d s i n c e c o n f i d e n c e l i m i t s f o r t h e d e r i v e dp a r a m e t e r s a r e o f t e n a l s o d e s i r e d , a s t a t i st i c a l p r o c e d u r e s u c h a s l e a s t s q u a r e si s t h e p r e f e r r e d m e t h o d M c B r a t n e y a n d W e b s t e r , 1 9 86 ) . T h i s a s p e c t is c o n s i d -e r e d f u r t h e r i n t h e f o l l o w i n g s e c t i o n s .

A l l o f t h e m e t h o d s d e s c r i b e d i n t h is r e v ie w T a b l e 2 ) p r o d u c e a m o n o -f r a c ta l d i m e n s i o n . T h e l i t e r a tu r e o n m u l t i d i m e n s i o n a l f r a c t a l s i s a d e v e l o p i n go n e L o v e j o y a n d S c h e r t z e r , 1 9 8 6 ; M a n d e l b r o t , 1 9 8 8, 1 9 8 9 ; F e d e r , 1 9 88 ). F e wo f t h e m e t h o d s p r e s e n t e d h e r e i n h a v e b e e n o r e v e n c o u l d b e ) e x t e n d e d to m u l t i -f r a c t a l d i m e n s i o n a l a n a l y s e s , a s h a s , f o r e x a m p l e , t h e b o x c o u n t i n g m e t h o d b yL o v e j o y a n d S c h e r t z e r 1 9 8 7 ) . I t i s i m p o r t a n t t o s e p a r a t e t h e o b s e r v a t i o n o fd i s c r e t e f r a c t a l e l e m e n t s c f . A n d r l e a n d A b r a h a m s , 1 9 8 9, 1 9 9 0) f r o m tr u l yc o n t i n u o u s m u l t i d i m e n s i o n a l i t y s i n c e , i n m o s t c a s e s , t h e q u e s t i o n o f w h e t h e r am o n o - o r a m u l t i d i m e n s i o n a l f ra c t a l is t h e m o r e a p p r o p r i a t e m o d e l h a s n o t b e e na n s w e r e d . T h e r e s u lt s o f a n y a n a l y s i s m u s t b e c a r e f u l l y s c ru t i n iz e d f o r e v i d e n c eo f m u l t i d i m e n s i o n a l i t y . O n e w a y t o c h e c k f o r s u c h b e h a v i o r i s t o p l o t t h e r e -s i d u a l s f r o m t h e b e s t - f i t ti n g li n e o f s l o p e a n d c h e c k f o r e v i d e n c e o f n o n l i n e a r i t yA n d r l e , 1 9 9 2 ) .

O n e d r a w b a c k c o m m o n t o e v e r y m e t h o d o f o b t a i n in g t he f r ac t a l d i m e n s i o n


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8 Klinkenbergis that discretization of the phenomenon being investigated will result in a meas-ured fractal dimension that is different from its theoretical fractal dimensioni.e., the dimension which would be obtained if we were able to work in con-

tinuous space with its infinite detail). The coarser the discretization, the greaterthe expected difference. However, the direction that that difference will take isopen to question Gilbert, 1989, Table 1) found instances of both a decreaseand an increase in the fractal dimens ion with greater decimation of the originaldata. Thus, to assume that discretiza tion--achieved by removing finer deta ils- -will always result in a lesser fractal di men sio n-- as some authors have assumede.g., Shelberg, Moellering, and Lam, 1982)--is inappropriate. Since the im-

plementations of some methods are themselves discretizations of continuousexpressions, that, in itself, may affect the derived fractal dimension Dubuc e ta l . 1989a).

A different type of problem, one which afflicts many of the methods dis-cussed below, is that the dimension of the whole may not equal the dimensionof the parts. Consider, for example, the case of horizontal cuts e.g. , contours)of a self-affine surface. For most surfaces, this will produce a suite of self-similar curves Fig. 1). Matsushita, Ouchi, and Honda 1991) prove that thefmctal dimension of the entire suite of curves De.~.) will generally be greaterthan the fractal dimension of a single curve D ~ ) extracted from that suite Dsc.

=I~ 0 -


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F r a c t a l i m e n s i o n o f L i n e a r F e a t u r e s 29< De.s.). The example they present, of curves derived from a fractal surface ofdimension 2.5 , shows D~+~. = 1.50 and, for the single longest curve, D .... =1.32 also see Ouchi and Matsushi ta, 1992).

Matsushita e t a l 1991) present a formula which relates the fractal dimen-sion of a sing le contou r line to the fractal dimension of the surface: D .... =2/1 + H). In comparing the results of that formula to the more often used D= 2 - H we can see why so many studies have reported that contour-baseddimens ions are often lower than those reported for the surface Fig. 2). Thisfact should be taken into consideration whenever a subset of the level set isused.

The divider method is arguably the most popular method of determiningthe fractal dimens ion of linear features Gilbert, 1989; Clarke and Schweizer,1991). It also has the most published variants, and it will be the first methodconsidered in this paper. The second method which will be considered is thebox counting method, another widely used method, particularly in physics andmeteorology. This is, in part, because it can be applied to any dimensional set --a fact which increases its usefulness significantly. Following the review of thebox counting method, there will be a discussion of the power spectrum method.

ooie n t i r e s u i t e

o~:

o . o 0 3 o . i o . ~ o ; . . .. .. . oHFig. 2. The relation between the fractal dimension and the value of H, as determined by Matsushita

eta/ 1991).

t--.O ,.zttJE

t.L


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3 KlinkenbergTh i s m e t h o d o l o g y r e q u ir e s t h e m o s t s o p h i s ti c a t e d d a t a p r e p r o c e s s i n g , a re q u i r e -me n t w h i c h p r o b a b l y l i m i t s i t s a p p l i c a t i o n s . Th e v a r i o g r a m me t h o d a n d s e v e r a la r e a - s c a l i n g me t h o d s w i l t t h e n b e d i s c u s s e d . F i n a l l y , t h e r e l a t i v e n e w l i n e -s c a l i n g m e t h o d o f M a t s u s h i ta a n d O u c h i ( 1 9 8 9 ) w i ll b e p r e s e n t e d , f o l l o w e d b ya b r i e f me n t i o n o f s e v e r a l o t h e r me t h o d s .

F o l l o w i n g t h e r e v i e w s o f th e v a r i o u s m e t h o d s , s o m e g e n e r a l c o n c l u s i o n sw i l l b e ma d e .

T H E D I V ID E R M E T H O DTh e d i v i d e r me t h o d h a s l o n g b e e n u s e d t o d e t e r mi n e t h e le n g t h o f c a r t o -

g r a p h i c l in e s ( M a l i n g , 1 9 9 2 ). R i c h a r d s o n ' s ( 1 9 6 1 ) i n v e s t i g a t io n s i n t o t h e s c a l ed e p e n d e n c i e s o f b o r d e r l e n g t h s , o n e o f t h e k e y b u i ld i n g b l o c k s in th e d e v e l -o p m e n t o f M a n d e l b r o t ' s c o n c e p t o f fr a c ti o n a l d i m e n s i o n s ( M a n d e l b r o t , 1 9 6 7) ,h a s j u s t if i a b ly b e c o m e o n e o f t h e m o r e c i t e d r e f e r e n c e s i n th e d i v i d e r m e t h o dl it e ra t u re . B e c a u s e o f t h e e a s e w i t h w h i c h t h is m e t h o d c a n b e i m p l e m e n t e d - -u s i ng e i th e r p h y s i c a l o r c o m p u t a t i o n a l d i v i d e r s - - a l a rg e n u m b e r o f s tu d ie s h a v eu s e d t h e d i v i d e r me t h o d t o d e t e r m i n e f r a ct a l d i m e n s i o n s o f f e a tu r e s r a n g i n gf r o m p a r t ic l e s h a p e s t o la v a f l o w s ( Ta b l e 1 ). Th i s b r e a d t h o f a p p l i c a t i o n s h a sr e s u lt e d i n a la r g e n u m b e r o f in d e p e n d e n t r e v i e w s a n d a s u i t e o f c o n t r a d i c t i n gr e c o m m e n d a t i o n s ( s e e b e l o w ) . A s a n i l l u st r at io n o f t h e i s o la t e d e f fo r ts w h i c hc a n b e o b s e r v e d , s o me h a v e e v e n r e f e r r e d t o t h i s me t h o d a s b e i n g a " r e l a t i v e l yn e w " t e c h n iq u e ( P o w e r a n d T u ll is , 1 9 91 )

T h e d i v i d e r m e t h o d c a n b e i m p l e m e n t e d i n a n u m b e r o f w a y s , b u t th e b a s ici m p l e m e n t a t i o n i s t o " w a l k " t h e d i v i d e r a l o n g t h e l i n e a n d r e c o r d t h e n u m b e ro f s t ep s r e q u i r ed t o c o v e r th e l in e . B y s y s t e m a t i c a l l y i n c r e a s i n g t h e w i d t h o ft h e d i v i d e r a n d r e p e a t i n g t h e s t e p p i n g p r o c e s s , t h e r e l a t i o n b e t w e e n s t e p s i z ea n d l i n e le n g t h o v e r a r a n g e o f r e s o l u ti o n s c a n b e d e t e r mi n e d . C a l c u l a t i o n o ft h e f r a c t a l d i me n s i o n f o l l o w s ( Ta b l e 2 ) .

T h e m a i n p r o b l e m w i t h t h e d i v i d e r m e t h o d r e l a t e s t o t h e r e m a i n d e r - - t h ef a c t t h a t mo s t o f t e n a n o n - i n t e g e r n u m b e r o f s te p s i s r e q u i re d t o c o v e r th e l i n e.Th e mo s t c o m m o n l y a p p l ie d s o l u t i o n - - t h e a d d i t io n o f t h e fr a c ti o n a l s t ep l e n g t h - -w a s f o u n d b y A v i l e s e t a l . ( 1 9 8 7 ) t o p r o d u c e s l i g h t l y h i g h e r d i me n s i o n s t h a nw h e n t he n u m b e r o f st e ps w a s e it h e r r o u n d e d u p o r d o w n . M a n d e l b r o t ( p e rs o n a lc o m m u n i c a t i o n to a n o n y m o u s r e v i e w e r ) r e c o m m e n d s r e t a in i n g t h e r em a i n d e r .Wh i l e G i l b e r t ( 1 9 8 9 ) c o n c l u d e d t h a t t h e d i v i d e r me t h o d w a s f l a w e d a n d p r o -d u c e d u n r e l ia b l e r e s u lt s , m o s t a u t h o r s h a v e f o u n d t h e m e t h o d t o b e f a i rl y r e li a b lew h e n t e st e d o n c u r v e s o f k n o w n f r a c ta l d i m e n s i o n ( e . g . , A v i l e s e t a l . 1987 ;P e i t g e n , J f ir g e n s , a n d S a u p e , 1 9 9 2 ). T h e r e s u lt s o f p r a c t ic a l a p p l i c a t io n s h a v eb e e n s o m e w h a t m i x e d , h o w e v e r . I n s o m e c a s e s , t h e d i v i d e r d i m e n s i o n s h a v ec l o s e l y c o r r e s p o n d e d t o t h o s e o b t a i n e d b y s p e c t r a l a n a l y s e s ( B r o w n , 1 9 8 7 ) ,v a r i o g r a m a n a l y s e s ( R o y , G r a v e l , a n d G a u t h i e r , 1 9 8 7 ), a n d o t h e r m e t h o d s ( K e n t


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F r a c t a l i m e n s i o n o f L i n e a r F e a t u r e s 31a n d W o n g , 1 9 82 ). H o w e v e r , i n o t h e r c a s e s t he d i v id e r d i m e n s i o n s h a v e n o tc o r r e s p o n d e d a t a ll ( C a r t a n d B e n z e r , 1 9 9 1 ; K l i n k e n b e r g a n d G o o d c h i l d , 1 9 9 2 ).G i v e n t h e r e c e n t f i n d i n g s o f M a t s u s h i t a e t a l (1991) i t i s no t su rp r i s ing tha ts o m e d i f f e r e n c e s h a v e b e e n r e p o r t e d .

S e v e r a l d if f e re n t i m p l e m e n t a t i o n s o f t h e d i v i d e r m e t h o d h a v e b e e n d e v e l -o p e d : S c h w a r z a n d E x n e r ' s ( 1 9 80 ) f a s t a l g o r it h m , B a t ty a n d L o n g l e y ' s ( 1 9 86 )e q u ip a c e d p o l y g o n m e t h o d , C l a r k ' s ( 1 98 6 ) H Y B R I D m e t h o d , a n d K e n n e d y a n dL i n ' s (1 9 8 6 ) F A E N A m e t h o d . T h e s e a l te r n a ti v e i m p l e m e n t a t io n s w e r e d e v e l -o p e d p r im a r i ly to s p e e d u p t he p a c i n g p r o c e s s . G i v e n t h e e v e r i n c re a s in gp r o c e s s i n g p o w e r o f w o r k s t a t i o n s , t h e n e e d t o l o o k f o r c o m p u t a t i o n a l s h o r t c u tsh a s b e e n g r e a t l y r e d u c e d , a n d t h e t r a d i t i o n a l i m p l e m e n t a t i o n s h o u l d b e t h em e t h o d o f c h o i c e ( c f . H a y w a r d , O r f o r d , a n d W h a l l e y , 1 9 89 ). I n p a r ti c ul a r,s i n c e th e s e i m p l e m e n t a t i o n s a r e a l l a p p r o x i m a t i o n s o f t h e o r i g i n a l te c h n i q u e ,t h e y a l l p r o d u c e d i m e n s i o n s d i f f e r e n t f r o m t h o s e p r o d u c e d b y t h e t r a d i t i o n a li m p l e m e n t a t i o n .

T h e d i v i d e r m e t h o d m a y a l s o p r o d u c e d i f f e r e n t d i m e n s i o n s b e c a u s e s o m em e t h o d s - - w h e n d e t e r m i n i n g t h e l e n g th o f t h e c u r v e - - u s e t he f ir st po i n t a l o n gt h e c u r v e in t e r s e c te d b y t h e d i v i d e r a s t h e n e x t w a l k i n g p o i n t ( e . g . , a n y o ft h e f a s t m e t h o d s , a n d t h e w a y t h a t s o m e a u t h o r s a p p e a r t o h a v e i m p l e m e n t e dt h e d i v i d e r m e t h o d - - M u l l e r , 1 9 8 6 is o n e e x a m p l e ) , w h e r e a s o t h e r m e t h o d s u s et h e l a st p o i n t i n t e rs e c t e d b y t h e d i v i d e r a s t h e n e x t w a l k i n g p o i n t ( e , g . , t h et r a d it io n a l a p p r o a c h ) . M a n d e t b r o t ( 1 9 8 6 ) d e m o n s t r a t e s t h a t t h e t w o c h o i c e s p r o -d u c e d if fe r e n t d i m e n s i o n s - - ( 1 / H ) a n d 2 - - H , r e s p e c ti v e ly . T h i s e x p la i n s w h y th ed i v i d e r d i m e n s i o n f o r a n e x a c t m a t h e m a t i c a l f r a c t al s u c h a s a K o c h c u r v e isd i f f e ren t f rom i t s t heore t i ca l d imens ion , s ince tha t d imens ion i s de t e rmined , i ne f fec t , by us ing the f i r s t po in t i n t e r sec t ed .

A d d i t i o n a l p r o b l e m s w i th t h e d i v i d e r m e t h o d i n c l u d e a d e p e n d e n c y o f t h ef r ac t a t d i m e n s i o n o n t h e s t a r ti n g p o s i t i o n w h e n c o u n t i n g , o n t h e s e l e c t io n o f t h em i n i m u m a n d m a x i m u m '~ te p s iz e , a n d t h e s m a l l i s la n d s p r o b l e m ( se e b e l o w ) .A s m a n y a u t h o r s h a v e n o t e d , w h e r e t h e d i v id e r c o m m e n c e s i ts w a l k c a n g r e a t l yi n f lu e n c e t h e r e s u lt s . T o a l l e v i a t e th i s p r o b l e m , a l a r g e n u m b e r o f s t a rt in g p o -s i t i o n s s h o u l d b e s e l e c t e d a n d t h e v a l u e s a v e r a g e d . A n d r l e ( 1 9 9 2 ) s u g g e s t sr a n d o m l y s e l e c t i n g a p o i n t a l o n g t h e l i n e a n d t h e n s t e p p i n g o u t t o e i t h e r e n df r o m t h a t p o i n t . H e f o u n d t h a t a f t e r u s i n g a p p r o x i m a t e l y 5 0 r a n d o m l y d e t e r -m i n e d s t a rt in g p o i n t s , t h e s t a n d a r d d e v i a t i o n o f t h e d i m e n s i o n s w a s m i n i m i z e d .

T he se l ec t ion o f t he sm a l l e s t and l a rges t s t ep s i ze s i s no t wi thou t it s p rob-l ems a l so . Se l ec t ing too sma l l an in i t i a l s t ep s i ze wi l l r e su l t i n l eng th va luest h a t d o n ' t v a r y w i th s t e p s i z e . I f s u c h v a l u e s a r e i n c l u d e d i n th e s l o p e d e t e r -m i n a t i o n , t h e y w i ll t e n d t o d e c r e a s e t h e s l o p e a n d , t h e r e b y , r e d u c e t h e a p p a r e n tf r a c ta l d i m e n s i o n o f t h e f e a t u r e . B e c a u s e o f t h is , v a r i o u s c r it e r ia h a v e b e e ns u g g e s t e d f o r t h e s e l e c t io n o f th e s m a l l e s t s te p s i z e . T h e s m a l l e s t s t e p s iz e s h o u l db e : ( 1 ) t w i c e t h e s h o r t e s t d i s t a n c e b e t w e e n a n y t w o p o i n t s ( A n d r l e , 1 9 9 2 ) , a n d


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3 Klinkenberg

( 2) o n e - h a l f th e a v e r a g e d i s t a n c e b e t w e e n a d j a c e n t p o in t s ( S h e l b e r g , M o e l l e r i n g ,a n d L a m , 1 9 8 2 ). S e l e c t i n g t o o l a rg e a n i n i t ia l s t e p s i z e w i l l r e d u c e th e n u m b e ro f d a t a p o i n t s u s e d i n t h e s l o p e d e t e r m i n a t i o n .

S e l e c t i n g th e l a r g e s t s te p s i z e is a m u c h m o r e s u b j e c t i v e d e c i s i o n . T h el a r g e r t h e s t e p s i z e , t h e g r e a t e r t h e e f f e c t t h a t p a r t i a l s t e p s h a v e a n d , a s w a sd i s c u s s e d a b o v e , p a r t i a l s te p s a f f ec t th e v a l u e a n d v a r i a b i l i t y o f t h e d e r i v e dd i m e n s i o n . A n d r l e ( 1 9 9 2 ) r e c o m m e n d e d t h a t t h e l a r g e s t s t e p s i z e ( s ) n o t b e u s e di n o r d e r to m i n i m i z e t h e e f f ec t s o f p a r t i a l s t e p s . C o n c o m i t a n t l y , t h is r e d u c e s t h el i k e l i h o o d th a t m u l t i p l e f ra c t a l e l e m e n t s o r m u l t i fr a c t a l b e h a v i o r c an b e i d e n t i -f i ed , s i n c e t h e f r a ct a l d i m e n s i o n w o u l d b e c a l c u l a t e d o v e r a n a r r o w e r r a n g e o fs c a l e s .

O t h e r r e a s o n s c a n b e f o u n d f o r s p e c i f y i n g a r e d u c e d m a x i m u m s t e p s i zew h e n u s i n g t h e d i v i d e r m e t h o d o n s e l f - af f i n e c u r v e s . W o n g ( 1 9 8 7 ) a n d B r o w n( 1 9 8 7 ) d e m o n s t r a t e d t h a t th e d e r i v e d d i m e n s i o n s c o n v e r g e t o a v a l u e o f o n e i ft h e s t e p s i z e s a r e g r e a t e r t h a n a v a l u e r e f e r r e d t o a s t h e c r o s s o v e r l e n g t h ( b ) . I ft h e m a x i m u m s t e p si z e r e m a i n s s m a l l e r t h a n th e c r o s s o v e r l e n g t h , t h e d i m e n s i o ns h o u l d b e r e p r e s e n t a t i v e - - w h a t M a n d e l b r o t ( 1 9 8 5 ) r e f e rs t o a s t h e l o c a l f r a ct a ld i m e n s i o n ( a l s o s e e P e i tg e n a n d S a u p e , 1 9 8 8, p . 6 4 f o r f u r th e r e l a b o r a t i o n ) .W h e n w o r k i n g w i t h s e l f - s i m i l a r l i n e a r fe a t u r e s th e c r o s s o v e r l e n g th i s n o t ac o n c e r n .

A g e n e r a l r u l e f o r s e t t i n g t h e m a x i m u m s t e p s i z e w h e n w o r k i n g w i t h s e l f -a f f i n e f e a t u r e s c a n b e d e r i v e d f r o m t h e f o r m u l a p r e s e n t e d i n B r o w n ( 1 9 8 7 ) : t h eg r e a t e r t h e ( e x p e c t e d ) f r a c t a l d i m e n s i o n o f t h e p r o f i le , th e s m a l l e r t h e m a x i m u ms t e p s iz e s h o u l d b e r e l a t iv e to t h e s t a n d a r d d e v i a t i o n i n h e i g h t s . S i n c e t h e c r o s s -o v e r l e n g t h i s d e p e n d e n t o n b o t h t h e f r a c t a l d i m e n s i o n ( w h i c h w o n ' t b e k n o w nu n t il a f t e r t h e m e a s u r e m e n t i s m a d e ) a n d t h e v e r t i c a l r o u g h n e s s ( B r o w n , 1 9 87 ;a l s o s e e M a n d e l b r o t , 1 9 8 5, l a s t s e n t e n c e o f s e c t io n 7 ) s u g g e s t s th a t g r e a t l ye x a g g e r a t i n g t h e v a l u e s o f t h e v e r t i c a l c o m p o n e n t s i g n i f i c a n t ly d e c r e a s e s t h el i k e l i h o o d o f th e s t e p s i z e i n c r e a s i n g p a s t t h e c r o s s o v e r l e n g t h . T h i s i s b e c a u s et h e c r o s s o v e r l e n g t h i n c r e a s e s c o n c o m i t a n t l y . A f t e r a n a p p r o p r i a t e a m o u n t o fm a g n i f i c a t io n o f th e v e r t i c a l c o m p o n e n t v a l u e s , t h e d e r i v e d f r a c ta l d i m e n s i o ns h o u l d b e c o m e s t a b l e ( B r o w n a n d S c h o l z , 1 9 8 5 , F i g . 5 ) .

T h e n o t i o n o f a c r o s s o v e r l e n g t h i s n o t w i t h o u t i ts c r i t ic s , h o w e v e r ( C a r ra n d B e n z e r , 1 9 9 1) . R e s c a l i n g t h e v e r t i c a l c o m p o n e n t i n d e p e n d e n t l y o f t h e h o r -i z o n t al c o m p o n e n t c h a n g e s t h e g e o m e t r i c r e l a ti o n b e t w e e n t h e t w o w h i c h , t h e r e -f o r e , w i l l c h a n g e t h e r e s u l ts o f t h e d i v i d e r - l e n g t h r e l a t i o n - - a n i m p o r t a n t d i s -t i n c t io n i f o n e i s i n t e r e s te d i n a n a l y z i n g t h e a p p a r e n t s e l f - s i m i l a r i t y o f a p r o f i l e .W h e n s c a l i n g a s e lf - a ff i n e c u r v e , i f t h e h o r i z o n t a l c o o r d i n a t e s a r e m u l t i p l i e d b ya f a c t o r r , t h e v e r t i c a l c o o r d i n a t e s s h o u l d b e m u l t i p l i e d b y a f a c t o r o f r 2 D( M a n d e l b r o t , 1 9 82 ; M a l i n v e r n o , 1 9 8 9) . I n li g h t o f t h i s w e m u s t r a t io n a l i z eB r o w n ' s ( 1 9 8 7) r e s c a l i n g f i x . I n t e r e s t i n g l y , O u c h i a n d M a t s u s h i t a ( 1 9 9 2 )r e p o r t e d t h a t v e r t i c a l e x a g g e r a t i o n o f t h e s u r f a c e s u s e d i n th e i r a n a l y s e s d i d n o t


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F r a c t a l D i m e n s io n o f L i n e a r F e a t u r e s

s i g n i fi c a n t ly a l t e r t h e f r a c ta l d i m e n s i o n s d e r i v e d f r o m t h o s e s u r f a c e s . O f c o u r s e ,t h e q u e s t i o n o f w h e t h e r t h e d i m e n s i o n o b t a i n e d u s i n g t h e d i v i d e r m e t h o d ist h e o r e t i c a l l y m e a n i n g f u l m u s t a l w a y s b e c o n s i d e r e d s i n c e i t s d e t e r m i n a t i o n i s am e c h a n i c a l p r o c e s s w h i c h w i l l a l w a y s r e t u r n a n u m e r i c a l v a l u e .

C l a r k e a n d S c h w e i z e r ( 1 9 9 1 ) e x a g g e r a t e d t h e v e rt ic a l c o m p o n e n t o f s o m ep r o f il e s a n d f o u n d t h a t d o i n g s o c h a n g e d l it tl e o f th e i r re s u lt s . H o w e v e r , t h el a rg e s t e x a g g e r a t io n f a c t o r t h e y u s e d w a s o n l y a te n t im e s m a g n i f i c a t i o n - - B r o w n( 1 9 8 7 ) u s e d f a c t o r s o f u p t o 10 4 , a n d o n l y a f t e r th e v a l u e s h a d b e e n e x a g g e r a t e db y a m i n i m u m o f 102 w e r e a n y s i g n if ic a n t c h a n g e s n o t e d . F u r t h e r m o r e , C l a r k ea n d S c h w e i z e r ( 1 9 9 1 ) s e l e c t e d s te p s i z e s w h i c h s p a n n e d t h e c r o s s o v e r l e n g t h ,e n s u r i n g t h a t t h e d e r i v e d d i m e n s i o n s w o u l d a l w a y s b e c l o s e t o o n e - - t h e a p -p r o x i m a t e v a l u e o f th e i r r e p o r t e d d i m e n s i o n s . A l t h o u g h t h e y f e lt th a t th e i r r e su l tss e r v e d t o c a st d o u b t s o n B r o w n s w o r k , t h e r es u lt s a c t u a ll y l e nd s o m e c r e d e n c et o t h e w o r k . T h e d i m e n s i o n s r e p o r t e d f r o m t h e t e n t i m e s m a g n i f i c a t i o n w e r ec l o s e r to th e d i m e n s i o n s o b t a i n e d f r o m a v a r i o g r a m a n a l y s i s o f t h e s u r f a c e th a nw e r e t h e u n m a g n i f i e d d i m e n s i o n s .

O t h e r p r o b l e m s w i t h t h e d i v i d e r m e t h o d i n c l u d e d e c i d i n g h o w m a n y s t e pi n c r e m e n t s a r e n e c e s s a r y t o p r o d u c e r e l i a b l e r e s u l t s , a n d d e c i d i n g t h e i n t e r v a lb e t w e e n d i v i d e r w i d t h s . S i n c e th e f r a c ta l d i m e n s i o n is d e r i v e d f r o m t h e s l o p eo f th e l in e o b t a i n e d f r o m t h e l o g - l o g p l o t o f fe a t u r e l e n g t h v s . s t e p s i z e , t h e o -r e t ic a l ly a m i n i m u m o f t w o p o i n t s a r e a ll th a t a r e r e q u i re d . H o w e v e r , f e w p e o p l ew o u l d c o n s i d e r t h e r e s u l ts o f s u c h a n a n a l y s i s r e l ia b l e o r r e p r e s e n t a t iv e , a n d iti s u s u a l ly s u g g e s t e d t h a t b e t w e e n f iv e a n d e i g h t d e t e r m i n a t i o n s b e m a d e ( i . e . ,t he s t ep s i ze be inc remented f ive t o e igh t t imes) . S t a t i s t i c a l r ea son ing sugges t st h a t e v e n l y s p a c e d v a l u e s o n t h e i n d e p e n d e n t a x i s w i l l i m p r o v e t h e r e l i a b i l i t yo f th e e s t im a t e s d e r i v e d f r o m l e a s t - s q u a r e s r e g r e s s i o n . T h i s r e q u ir e s t h a t th ed i v i d e r w i d t h s d o u b l e b e t w e e n e a c h s e t o f li ne d e t e rm i n a t i o n s . H o w e v e r , t h es et w o c o n c e r n s w o r k a g a i n s t e a c h o t h e r , s i n c e d o u b l i n g t h e s t e p s i z e m e a n s t h a tf e w e r l e n g t h d e t e r m i n a t i o n s c a n b e m a d e . T h i s a l s o m e a n s t h a t s m a l l f e a t u r e s(e spec i a l ly sma l l i s l ands) wi l l t end to be exc luded f rom the ana lys i s , poss ib lyp r o d u c i n g a n e f f e c t o n t h e d e r i v e d d i m e n s i o n s i m i l a r t o t h a t r e p o r t e d a b o v e b yM a t s u s h i t a et al ( 1 9 9 1 ) . T h i s i s o n e a r g u m e n t a g a i n s t u s i n g g e o m e t r i c a l l yinc reas ing s t ep s i ze s .

T h e t r a d i t i o n a l d i v i d e r m e t h o d s u f f e r s f r o m b o t h p r a c t i c a l a n d t h e o r e t i c a ll im i t a t io n s . B e c a u s e o f th i s , t h e s u r f ic i a l d i v i d e r m e t h o d is a p o te n t i a ll y m o r er o b u s t r e p l a c e m e n t f o r a p p l i c a t i o n s w h i c h u t i li z e i s o li n e s d e r i v e d f r o m D E M s .T h i s m e t h o d i s c o n c e p t u a l l y e q u i v a l e n t t o m e a s u r i n g l i n e a r f e a t u r e s f r o m m a p so f va r iou s sca l e s us ing a cons t an t w id th d iv ide r . S t a r t ing w i th t he i so l ine s de -r ived f rom the h ighe s t r e so lu t ion da t a , t he i r t o t a l l eng th i s de t e rm ined . Th en ,t h e r e s o l u t io n o f t h e d a t a is r e d u c e d b y s e l e c t i n g , f o r e x a m p l e , e v e r y s e c o n dc e l l , a n d t h e i s o l i n e l e n g t h s a r e r e d e t e r m i n e d . T h e d a t a s u b s a m p l i n g p r o c e s s i sr epea t ed un t il t he dec im a t ion is such tha t t he na tu re o f t he o r ig ina l su r face is


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4 linkenbergl o s t . Th e to t a l l eng th o f t he i so l ine s a t e ach d e t e rm ina t ion i s t hen used in t hec a l c u la t io n o f t h e f ra c ta l d i m e n s i o n . T h e s t e p s i z e f o r a g i v e n d e t e rm i n a t i o ni s o b t a i n e d b y d i v i d i n g t h e t o t a l c o n t o u r l i n e l e n g t h b y t h e t o t a l n u m b e r o fc o o r d i n a t e p a i r s ( m i n u s o n e ) . T h e s l o p e d e r i v e d f r o m t h e l o g - l o g p l o t o f t h et o ta l c o n t o u r l in e l e n g t h a g a i n s t t h e s t e p s i z e ( a v e r a g e c h o r d l e n g t h ) i s t h e n u s e dt o d e t e r m i n e t h e f ra c t a l d i m e n s i o n - - a s i n t h e tr a d i ti o n a l m e t h o d ( T a b l e 2 ) . T h i s

s u r f ic i a l d i v i d e r s m e t h o d w a s d e s c r i b e d in P a u m g a r t n e r e t a l . ( 1 9 8 1 ) a s t h ev a r i a b l e m a g n i f i c a t i o n m e t h o d .

A f t e r e a c h r e s o l u ti o n c h a n g e , t h e p o s s i b l e n u m b e r o f d a ta s u b s a m p l e s i si n c r e a s e d b y a f a c t o r o f fo u r . I f all p o s s i b l e s a m p l e s a r e u s e d i n t h e a n a l y s i s ,t h is s h o u l d i n c r e a s e t h e m e t h o d ' s r e l i a b il it y . T h i s m e t h o d d o e s n o t s u f fe r f r o mt h e s m a l l i s la n d s p r o b l e m , t h e s i n g l e l in e b e h a v i o r o u tl i n e d b y M a t s u s h i t a e t a l .( 1 9 9 1 ) , o r f r o m th e c r o s s o v e r l e n g th p r o b l e m . K l i n k e n b e r g a n d G o o d c h i l d ( 1 9 9 2 )f o u n d t h a t t h i s m e t h o d p r o d u c e d d i m e n s i o n s s i m i l a r t o t h o s e p r o d u c e d b y t h ev a r i o g r a m m e t h o d w h e n a p p l i e d t o th e s u r f a c e , w h i le t h e m o r e tr a d i ti o n a l d i v i d e rm e t h o d s ( i n c l u d i n g s e v e ra l o f th e f a s t e r a l t e rn a t i v e s ) p r o d u c e d m u c h l o w e r d i-m e n s i o n s . W h e n t h e i r d a t a w a s r e w o r k e d u s i n g t h e s i n g l e c o n t o u r l i n e f o r m u l ao f M a t s u s h i t a e t a l . ( 1 9 9 1 ) th e m a g n i t u d e o f t h e d i ff e re n c e s w a s r e d u c e d b u tn o t e l i m i n a t e d .

T H E B O X C O U N T I N G M E T H O DT h e b o x c o u n t i n g m e t h o d i s w i d e l y u s e d t o d e t e r m i n e t h e fr a ct a l d im e n s i o n

o f m a n y d i f f e r e n t p h e n o m e n a ( T a b l e 1 ). P r i o r t o its a p p l i c a t io n s i n f ra c t al r e -s e a r c h , b o x c o u n t i n g w a s m a i n l y u s e d t o q u i c k l y d e t e r m i n e t h e a r e a o f i r r e g u l a rc a r t o g r a p h i c f e a t u r e s ( e . g . , G i e r h a r t, 1 9 5 4 ; M a l i n g , 1 9 6 8 ). S i n c e it c a n b eapp l i ed wi th equa l e f fec t iveness t o po in t se t s , l i nea r f ea tu re s , a rea s , and vo l -u m e s , t he b o x c o u n t i n g m e t h o d is a w i d e l y u s e d m e a n s o f d e t e r m i n i n g f r a c ta ld i m e n s i o n s . T h i s m e t h o d i s a l s o k n o w n a s t h e g r i d o r r e t i c u l a r c e l l c o u n t i n gm e t h o d ( G a g n e p a i n a n d R o q u e s - C a r m e s , 1 9 8 6 ; P e i t g e n a n d S a u p e , 1 9 8 8) , a ndh a s b e e n s h o w n to b e e q u i v a le n t to t he M i n k o w s k i - B o u l i g o n d ( o r s a u s a g e )d i m e n s i o n ( D u b u c e t a l . 1989b) .

T h e b a s i c i m p l e m e n t a t i o n , u s i n g t h e d e t e r m i n a t i o n o f a l i n e a r f e a t u re a st h e e x a m p l e , i s a s f o l l o w s :

1 . C o v e r t h e f e a tu r e w i t h a s i ng l e b o x .2 . D i v i d e t h e b o x i n t o f o u r q u a d r a n t s , a n d c o u n t t h e n u m b e r o f c e l ls o c -

c u p i e d .3 . D i v i d e e a c h s u b s e q u e n t q u a d r a n t i n t o f o u r s u b q u a d r a n t s , a n d c o n t i n u ed o i n g s o u n ti l th e m i n i m u m b o x s i z e i s e q u a l t o t h e r e s o l u ti o n o f t h e

d a t a , k e e p i n g t r a c k o f th e n u m b e r o f q u a d r a n t s o r c e ll s o c c u p i e d a t e a c hstep .

T h e f r a c t a l d i m e n s i o n i s e a s i l y o b t a i n e d f r o m t h e l o g - l o g p l o t ( T a b l e 2 ) .


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F r a c t a i i m e n s i o n o f L i n e a r F e a t u r e s 35T h e b o x c o u n t i n g m e t h o d , w h e n a p p l i e d t o l i n e a r f e a t u r e s , i s u s u a l l y a p -

p l i e d to c u t s o f a s u r f a c e ( e . g . , c o n t o u r s ) w h e r e t h e b o x e s o v e r l a y t h e c u t l i n e s.T h e m e t h o d c a n b e a p p l i e d to p r o f il e s , h o w e v e r , a s d e s c r i b e d b e l o w .

D i v i d e t h e p r o f i l e l e n g t h w i s e i n t o , s a y , f o u r e q u a l p a r t s a n d c o u n t t h en u m b e r o f i n t e r s e c t i o n s o f a h o r i z o n t a l l i n e a t s o m e s p e c i f i e d v e r ti c a l v a l u e( u s i n g o n l y t h o s e v a l u e s w h i c h o c c u r a t t h e s e c t io n d i v i d e s t o d e t e r m i n e i f a ni n t e r s e c ti o n o c c u r s ) . T h e n , i n c r e a s e t h e n u m b e r o f d i v i s i o n s a n d r e d e t e r m i n et h e n u m b e r o f i n t e rs e c ti o n s . C o n t i n u e ( g e o m e t r i c a l l y ) in c r e a si n g t he n u m b e r o fd i v i s i o n s a n d r e - d e t e r m i n i n g t h e n u m b e r o f i n t e r s e c t i o n s u n t i l t h e m i n i m u mr e s o lu t io n o f t h e d a t a is r e a c h e d . U s i n g a l o g - l o g p l o t o f b o x s i z e a g a i n s tt h e n u m b e r o f i n t e r s e c t i o n s , t h e f r a c t a l d i m e n s i o n i s e q u a l t o t h e s l o p e t i m e sm i n u s o n e . S h e l b e r g , M o e l l e r i n g , a n d L a m ( 1 98 3 ) d e s c r i b e d a s i m i l a r p r o c e d u r ef o r d e t e r m i n i n g t h e f r a c t a l d i m e n s i o n o f a s u r f a c e , f r o m w h i c h t h i s p r o c e d u r ew a s a b s t r a c t e d .

S e v e r a l p r o b l e m s h a v e b e e n i d e n t i f i e d w i t h t h e s t a n d a r d i m p l e m e n t a t i o n o ft h e b o x c o u n t i n g m e t h o d ( L i e b o v i t c h a n d T o t h , 1 9 8 9) . T h e m e t h o d r e q u i r e s as i g ni f ic a n t a m o u n t o f c o m p u t e r m e m o r y a n d c o m p u t a t i o n a l ti m e s in c e a v e ryl a rg e n u m b e r o f c e ll s h a v e t o b e s t o re d . B e c a u s e o f th i s p r o b l e m , L i e b o v i t c ha n d T o t h ( 1 9 8 9 ) i n t r o d u c e d a f a s t a l g o r i t h m w h i c h , u s i n g a s t a t i s t ic a l l y - b a s e ds a m p l i n g a p p r o a c h , d o e s n o t r e q u i r e a c o m p l e t e e n u m e r a t io n o f e v e r y c e ll att h e h i g h e r r e s o l u t i o n s . T h e i r m e t h o d i s b e s t a p p l i e d o n l y t o d a t a s e t s o f l o wf r ac t al d i m e n s i o n s , h o w e v e r , S a r r a i ll e ( 1 9 91 ) h a s p r o d u c e d a n i m p l e m e n t a t i o no f t h e i r m e t h o d .

B o x c o u n t i n g a l s o r e q u i r e s a l a r g e n u m b e r o f d a t a p o i n t s in o r d e r to p r o d u c ec o r r e c t d i m e n s i o n - - D u b u c e t a l ( 1 9 8 9 a ) r e p o r t e d i n s t a b i l it i e s i n t h e m e t h o dw h e n t h e n u m b e r o f d a t a p o in t s u s e d w a s s m a l l . T h e y a l s o fo u n d t h a t th e m e t h o dw a s s e n s i t i v e to t h e l e v e l o f q u a n t i z a t io n o f th e d a t a . T h i s s e n s i t i v i t y m a y a p p l yt o o t h e r m e t h o d s , a n d s h o u l d b e i n v e s t i g a t e d .

T h e q u e s t i o n o f d e f i n in g th e m i n i m u m a n d m a x i m u m b o x s iz e w a s a d -d r e s s e d b y L i e b o v i t c h a n d T o t h ( 1 9 8 9 ) . T h e f ir st t w o b o x c o u n t s ( i . e . , w h e no n e a n d f o u r b o x e s a r e u s e d ) s h o u l d n o t b e u s e d i n t h e s l o p e d e t e r m i n a t i o n , n o rs h o u l d t h e b o x c o u n t s w h i c h o c c u r w h e n t h e c e l l s i z e a p p r o a c h e s t h e r e s o l u t i o no f t h e d a t a ( i . e . , w h e n e a c h d a t a p o i n t f a l l s i n a s i n g l e b o x ) . D u b u c e t a l( 1 9 8 9 a ) i d e n t i f i e d p r o b l e m s w i t h t h e s t a b i l i t y o f t h e s l o p e w h e n t h e l a r g e r b o xs i z e s w e r e i n c l u d e d i n t h e a n a l y s i s . A s w i t h e v e r y o t h e r m e t h o d w h i c h d e t e r -m i n e s t h e s l o p e i n l o g - l o g s p a c e , t h e c e l l s i z e s s h o u l d c h a n g e a s a f u n c ti o n o fa p o w e r o f tw o s o t h a t t h e y w i l l b e e v e n l y s p a c e d i n t h e l o g s p a c e . D u b u c e ta l ( 1 9 8 9 a ) f o u n d t h a t d o i n g s o t o o r a p i d l y c o v e r e d t h e d a t a , r e s u l t in g i n t o of e w p o i n t s in t h e l o g - l o g p l o t. T h e r e f o r e , t h e y d i d n o t u se d y a d i c b o x e s .

B o x c o u n t i n g c a n a l s o s u ff e r f r o m a r e m a i n d e r p r o b l e m . I f t h e b o x e sc a n n o t c o v e r t h e d a t a e v e n l y ( e . g . , i f t h e r a w d a t a c o n s i s t s o f a 2 17 b y 2 1 7a r m y ) , t h e n s o m e c e l l s w i l l b e m i s s e d i f t h e b o x s i z e s i n c r e a s e g e o m e t r i c a l l y .


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6 Klinkenberg

B e c a u s e o f t h is p o t e n t ia l p r o b l e m, s o m e a u t h o r s s u g g e s t f ir st m a p p i n g t h e r a wd a t a o n t o a s q u a r e u n it . H o w e v e r , i f t h e r a w d a t a w a s c o l l e c t e d o n a n o n - s q u a r ef r a m e ( e . g . , a 1 4 9 b y 2 1 7 a r m y ) , t h e n t h a t o p t i o n w i l l n o t b e a v a i l a b l e . V e r yl i t t l e ha s been wr i t t en abou t t h i s p rob lem in t he box coun t ing l i t e ra tu re .

Lo v e j o y , S c h e r t z e r , a n d Ts o n i s ( 1 9 8 7 ) p r e s e n t a d e t a i le d d i s c u s s i o n o f t h eb o x c o u n t i n g me t h o d a n d n o t e t h a t t h e b o x e s n e e d n o t b e s q u a r e . B y u s i n gr e c t a n g u l a r b o x e s , f o r e x a m p l e , n o n - i s o t r o p i c , m u l t i d ime n s i o n a l f r a c ta l s c a n b es t u d ie d . H o w e v e r , d i s c u s s i o n o f s u c h e x t e n s i o n s t o t h e b a si c me t h o d a r e o u t s i d eo f th e s c o p e o f th i s p a p e r , h o w e v e r .

S P E C T R A L M E T H O D SU s i n g s p e c t ra l m e t h o d s t o o b t a i n t h e f r a c ta l d i me n s i o n s o f f e a t u re s is a n -

o t h e r w i d e l y u s e d me t h o d ( Ta b l e 1 ). A l t h o u g h a r i g o r o u s me t h o d , i t i s c o m -pu ta t iona l ly d i f f i cu l t and compute r i n t ens ive (Ba r t l e t t , 1991) . Th i s me thod re -q u i re s m u c h m o r e d a ta p r e p r o c e s s i n g t h a n a n y o f t h e o th e r m e t h o d s , a ll o f w h i c hw o r k w i t h t h e d a t a a s i s . S p e c t r a l m e t h o d s r e q u i re t h e r a w d a t a t o b e d e -t r e n d e d a n d t a p e r e d , a n d n o t d o i n g s o p r o p e r l y c a n g r e a t l y a l t e r t h e r e s u l t s ( F o xa n d H a y e s , 1 9 8 5 ; C a r t a n d B e n z e r , 1 9 9 1 ) . Th e y s h o u l d o n l y b e a p p l i e d t o s e l f -a f f ine cu rv es ( i . e , , p ro f i le s ) s ince the me thod wi l l a lw ays r e tu rn a f r ac t a l d i -me n s i o n o f o n e f o r s e l f - s im i l a r c u r v e s ( P e i tg e n a n d S a u p e , 1 9 8 8 ), D e s c r i p t i o n so f t h e s te p s r e q u i r e d t o p e r f o r m a s p e c t r a l a n a l y s i s f o r f r a ct a l p u r p o s e s c a n b ef o u n d in H u a n g a n d Tu r c o t t e ( 1 9 8 9 , 1 9 9 2 ), P e n t l a n d ( 1 9 8 4 ) , P e i t g e n a n d S a u p e( 1 9 8 8 ) , a n d Tu r c o t t e ( 1 9 9 2 ) .

A n e a r l y a p p l i c at io n o f th e p o w e r s p e c t r u m m e t h o d t o th e s t u d y o f s u r fa c et o p o g r a p h y w a s r e p o rt e d i n S a y l e s a n d T h o m a s ( 1 9 7 8 ) . B e r ry a n d H a n n y ( 1 9 7 8 )subsequen t ly p l aced those r e su l t s i n to a f r ac t a l f r amework . I t was no t un t i l Be r rya n d Le w i s ( 1 9 8 0 ) t h a t a f o r ma l l i n k b e t w e e n t h e f r a c t a l d i me n s i o n a n d t h e p o w e rs p e c t r u m w a s p r o v i d e d , h o w e v e r . F i n a l l y , M a n d e l b r o t e t a l . 1 9 8 4 ) c l e a r e d u ps o m e o f t h e p ra c t ic a l i s su e s r e l a t e d t o th e m e t h o d w h i c h h a d a r i se n i n t h el i te ra ture .

P o w e r a n d Tu l l is ( 1 9 9 1 ) me n t i o n s e v e r a l p r o b l e m s w i t h a p p l ic a t i o n s o fs p e c t r a l me t h o d s . O n e p r o b l e m i s t h a t t h e t y p i c a l f r e q u e n c y p r o g r e s s i o n u s e d i nF F T a l g o r i t h ms i s a r i t h me t i c . Th i s r e s u l t s i n t h e h i g h e r f r e q u e n c i e s b e i n g o v e r -r e p r e s e n t e d - - r e la t i v e t o t h e l o w e r f r e q u e n c i e s - - i n t h e l o g - l o g p l o t s u s e d to d e -t e r mi n e t h e f r a c t a l d i me n s i o n ( Ta b l e 2 ) . O t h e r me t h o d s a r e e a s i l y a d j u s t e d t op r o d u c e e q u a l l y - s p a c e d d a t a p o i n t s o n t h e i n d e p e n d e n t a x i s , w h i c h r e s u l t s i nm o r e s t a b le l e a s t - s q u a r e s p a r a m e t e r s . A m o r e s i g n i fi c a n t p r o b l e m is t h a t se v e r a la u t h o r s h a v e m a d e i n c o r r e c t a s s u m p t i o n s a b o u t th e a p p r o p r i a t e s l o p e o f t h ep o w e r s p e c t r u m p l o t, p e r h a p s b e c a u s e t h e r e a r e a v a r i e t y o f w a y s o f e x p r e s s i n gt h e p o w e r o r a m p l i tu d e s p e c t r a ( P o w e r a n d Tu l l i s, 1 9 9 1 ; a ls o s e e C a r r a n dB e n z e r , 1 9 9 1 ) . D u b u c e t a l . ( 1 9 8 9 a ) r e p o r te d l o w e r p r e c i s i o n s f r o m p o w e r


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Fractal imen s ion of Linear Featu res 37spec t rum methods when compared to the i r o ther resu l t s . They found tha t thepoints in the log-log plot rarely formed a s t raight l ine, increasing the ins tabi l i tyof the l eas t - squares der ived va lues .

Th e co mp uta t iona l com plex i ty o f spec t ra l m ethods i s wel l i llus tra ted by theser i es o f papers by Hu ang and Turco t t e 1989 , 1990) and G off 1990). In the i rf irst pape r , Hua ng and Tu rcot te 1989) reported that the fractal d im ension s ofone-d im ens iona l p ro f i les were s ign i f ican t ly low er 0 .5 ) than the f rac ta l d im en-s ions o f the su r faces minus 1 ) as de te rm ined us ing two -d imens iona l FFT . Sub-sequen t ly , an e r ro r in the i r fo rmu la t ion was no ted Goff , 1990). Hu ang andTurc o t t e 1990) re -exa m ined the i r o r ig ina l wo rk and repor ted tha t the d imen s ionsp lus one) p roduced by the one-d imens iona l FFT were wi th in s t a t i s t i ca l uncer -

t a in ty o f the two-d im ens iona l FF T va lues . Thu s , it appea rs tha t th is m ethodproduces cons i s t en t f rac ta l d imens ions when app l i ed to su r faces wi th d i f fe r ingtopo log y d ime ns ions . Non ethe less , the inheren t com plex i ty o f spec tra l me thodsrequ ires tha t they be carefu l ly im ple m en ted - -P ow er and Tul l i s 1991) foundsevera l examples where subsequen t re - in te rp re ta t ion o f p rev ious ly pub l i shedspec t ra l ana lyses have d ramat ica l ly a l t e red the conclus ions .

T H E V A R I O G R A M M E T H O DTh e va r iogram m ethod i s wide ly used in the de te rmina t ion o f the f rac ta l

d imen s ion o f su r faces Burrough , 1981 , 1984) and appears to have p rop er t i es - -in par t i cu la r i t s ease o f use- -which make i t a p refer red method over spec t ra lana lys i s Ca r t and Ben zer , 1991 ; Kl inken berg and Go odch i ld , 1992L Al thoughit has been m uch t ess com m only app l i ed to s tr i ct ly l inear phen om ena , the methodi s very easy to implement when ana lyz ing se l f -a f f ine p rof i l es . By sampl ing ala rge num ber o f pa i r s o f po in t s o f d i f fe ring spac ings ) a long the p rof i le andcom put ing the d i f fe rences in the i r ver t ica l va lues e .g . , z va lues ) the f rac ta ld imen s ion i s eas i ly der ived f rom the log- log p lo t o f expec ted d i ffe rences in z 2vs . d is t ance be tween the po in t pa i r s Tab le 2 ).

Th e cho ice o f the m ax im um p o in t -pa i r d i s t ances used in the ana lyses re -qu i res some thought . The maximum d i s tance be tween po in t pa i r s i s usua l lytaken to be one-h a l f o f the abso lu te m ax im um d i s tance be tween po in t s, a l thoughs o me au t h o r s h av e s u g g es t ed co n si d e ri n g m u ch s h o r t e r m ax i m u m s , s u ch a s o n eq u a r t e r t he max i m u m d is t an ce Ro y , G rav e l , an d G au t hi e r, 1 98 7) . H o w ev e r ,th is ru le m ay be m ore res t r i c tive than necessary . Very l inear re la tions ex tend ingo u t to ab s o lu t e ma x i m u m d i s tan ces h av e b een n ot ed K l i n k en b erg an d G o o d -ch i ld , 1992) . More o f t en , however , the d imens ions assoc ia ted wi th d i s t ancepa i rs tha t a re g rea te r than the sugges ted cu to f f a re l a rger than those assoc ia tedwi th the d i s t ance pa i rs tha t a re l ess than the sugges ted cu to f f M ark and A ronson ,1984 ; K l inkenberg and Go odch i ld , 1992). The shor tes t po in t -pa i r d i s tance usedin the analys es wi l l dep end on the resolut ion o f the data .


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38 KlinkenbergF o r s ta t is t ic a l p u r p o s e s , i t i s i mp o r t a n t t h a t th e s a mp l e o f p o i n t p a i rs u s e d

i n th e a n a l y s e s u n i f o r m l y s p a n t h e d i s ta n c e r a n g e . I n o r d e r t o o b t a in s t a t i st ic a l lyv a l i d a v e r a g e s f o r t h e z d i f f e r e n c e s , t h e p o i n t - p a i r d is t a n c e s a r e u s u a l l y p l a c e di n a n u m b e r o f d e f i n e d c l a s s e s . T h i s m e a n s t h a t , a s w i t h t h e o t h e r m e t h o d sw h i c h u s e l i n e a r r e g r e s s io n i n t h e d e t e r mi n a t i o n o f th e s l o p e , t h e d i st a n c e c l a s s e ss h o u l d b e s e l e c t e d s o t h a t t h e y a r e e v e n l y s p a c e d i n t h e l o g s p a c e u s e d i n t h el e a s t- s q u a r e s r e g r e s s i o n . S c a t t e r o f t h e d a t a p o i n t s i n t h e l o g - l o g p l o t c a n c a u s esome ins t ab i l i t i e s i n t he I ea s t - squa re s de r ived pa rame te r s .

T h i s m e t h o d d o e s a p p e a r t o p r o d u c e c o n s i s t e n t f r a c t a l d i m e n s i o n s w h e na p p l i e d t o f e a t u r e s w i t h d i f f e r i n g t o p o l o g i c a l d i me n s i o n s . K l i n k e n b e r g a n dG o o d c h i l d ( 1 9 9 2 ) f o u n d t h a t th e v a r i o g r a m - d e r i v e d f ra c ta l d i m e n s i o n s o f p ro f il e sw e r e e q u a l t o th e f r a c ta l d i me n s i o n o f t h e s u r fa c e s . Lo v e j o y a n d S c h e r t z e r ( 1 9 8 7 )c r i t i c i z e d t h e v a r i o g r a m me t h o d , s t a t i n g t h a t t h e me t h o d o n l y e x p l o r e s t h e s c a l -i n g n a t u re o f t h e v e r t ic a l f l u c t u a t io n s , a n d d o e s n o t i n c o r p o r a t e t h e s c a l i n g o fh o r i z o n t a l s t ru c t u r e s . A l t h o u g h t h e v a r i o g r a m is a f u n c t io n o f d i f fe r e n c e s i n ,u s u a l l y , e l e v a t i o n , a n d h o r i z o n t a l d i s t a n c e , o n e c o u l d i ma g i n e a f l i p p e d v a r -i o g r a m, w h e r e i n t h e v e r t i c a l d i f f e r e n c e s b e c o me t h e i n d e p e n d e n t v a r i a b l e a n dt h e a v e r a g e ( h o r i z o n t a l d i f f e r e n c e s ) 2 a s s o c i a t e d w i t h e a c h v e r ti c a l d i f f e r e n c e t h ed e p e n d e n t v a r i a b l e . F r o m t h i s p e r s p e c t i v e t h e i r c r i t i c i s m d o e s w a r r a n t s o mec o n s i d e r a t i o n .

A R E A B A S E D M E T H O D ST w o a r e a - b a s e d m e t h o d s c a n b e u s e d t o d e te r m i n e t h e f ra c ta l d i m e n s i o n

o f li n e a r f e a t u r e s i f t h o s e f e a t u r e s f o r m c l o s e d l o o p s ( M a n d e l b r o t , 1 9 7 5 ) . Th ea r e a - p e r i m e t e r r e l a ti o n a n d K o r c a k ' s e m p i r i c a l r e l a t io n f o r is l a n d s b o th h a v eb e e n u s e d t o d e t e r mi n e t h e f r a c t a l d i m e n s i o n o f la k e s , c o n t o u r lo o p s , a n d i s la n d s(Tab le 1 ). I f t he da t a is app ropr i a t e , t he se a re r e l a t i ve ly s imple m e tho ds to usea n d t h e i r i mp l e m e n t a t i o n i s fa i r ly s i mp l e , r e q u i r in g f e w o f t h e d e c i s io n s m o s to f t h e o t h e r me t h o d s d e m a n d ( Ta b l e 2 ) . W h e n u s i n g t h e a r e a - p e r i m e t e r r e l at io n ,h o w e v e r , o n e mu s t n o t c o n f o u n d t h e r e l a t i o n b y u s i n g p e r i me t e r s d e r i v e d f r o ms o u r c e s o f d i f fe r e n t sc a l e s ( i . e . , t h e a r e a o f a g i v e n f e a t u r e w i ll n o t c h a n g e m u c hw h e n m e a s u r e d f r o m s o u r c e s o f d i ff e re n t s c a le s , b u t t h e p e r i m e t e r m e a s u r e m e n t sc a n c h a n g e s i g n i f i c a n t l y ) .

Th e r e q u i r e me n t t h a t t h e f e a t u r e s f o r m c l o s e d l o o p s r e s t r i c t s t h e u s a g e o fa r e a - b a s e d m e t h o d s . F o r e x a m p l e , a l th o u g h a l l c o n t o u r s m u s t e v e n t u a ll y l o o pb a c k u p o n t h e ms e l v e s , t h e y ma y n o t d o s o w i t h i n t h e p a r t i c u l a r s t u d y a r e a .T h u s , f e w o b je c t s m a y b e a v a i la b l e f o r a n a l y si s . H o w e v e r , t h es e m e t h o d s s h o u l db e f a i r l y r o b u s t , s i n c e t h e y w i l l n o t s u f f e r f r o m t h e p r o b l e ms i d e n t i f i e d a b o v ew i t h r e s p e c t to t h e w h o l e v s . p a r t s d i me n s i o n , o r f r o m t h e s ma l l i s la n d s p r o b l e m .Sake l l a r iou e t a l ( 1 9 9 1 ) f o u n d t h e a r e a - p e r i me t e r me t h o d t o b e t h e mo s t s t a b l ea n d l ea s t a m b i g u o u s m e t h o d o f t h e f o u r t h e y c o n s i d e r e d ( w h i c h i n c l u d e d v a r -


17/24

F r a c t a l D i m e n s i o n o f L i n e a r F e a t u r e s 9

i o g r a m a n d s p e c tr a l m e t h o d s ) . M a n d e l b r o t et al ( 1 9 8 4 ) f o u n d t h e d i m e n s i o no b t a i n e d f r o m a n a r e a - p e r i m e t e r a n a l y s i s to a g r e e w i t h t h e d i m e t , s io n o b t a i n e df r o m a s p e c t r a a n a l y s i s .

T H E L I N E S C A L I N G M E T H O DO n e e s s e n t i a l a s p e c t o f f r a c t a l f e a t u r e s i s t h e i r s c a l i n g n a t u r e . T h a t m e a n s

t h a t t h e m o m e n t s t a t is t i c s o f a f r a c t a l f e a t u r e a r e ( t h e o r e t i c a l l y ) u n s t a b l e ( M a n -d e l b r o t , 1 9 8 2) . F o r e x a m p l e , t h e lo n g e r t h e t o p o g r a p h i c p r o f i le , t h e g r e a t e r t h eo b s e r v e d v a r i a b i l i t y i n e l e v a t i o n s ( A h n e r t , 1 9 8 4 ). B u i l d i n g o n t h a t c o n c e p t ,M a t s u s h i t a a n d O u c h i ( 1 9 8 9 ) d e v e l o p e d a m e t h o d f o r t h e d e t e r m i n a t i o n o f t h ef r a c t a l d i m e n s i o n o f a l i n e a r f e a t u r e w h i c h u s e d t h e o b s e r v e d r e l a t i o n s h i p b e -t w e e n s a m p l e s i z e s a n d s a m p l e v a r i a n c e s ( T a b l e 2 ) .

T h e m e t h o d i s c o n c e p t u a l l y v e r y s i m p l e t o i m p l e m e n t a n d w o r k s o n b o t hs e l f - s i m i l a r a n d s e l f- a f f in e c u r v e s . S a m p l e s o f v a r i o u s l e n g t h s ( s a y N ) a r e t a k e nf r o m t h e c u r v e a n d t h e s t a n d a r d d e v i a t i o n s o f t h e c o o r d i n a t e s a l o n g e a c h a x i sa r e d e t e r m i n e d ( s a y X a n d Y ) . F r o m t h e tw o l o g - l o g p l o t s o f s ta n d a r d d e v i at i o n( X a n d Y ) v s . s a m p l e s i z e ( N ) t h e r e s p e c t i v e s l o p e s a re d e t e r m i n e d ( s a y V x a n dVr) . I f V x a n d V a r e e q u a l o r n e a r l y s o , t h a t i s a g o o d i n d i c a t i o n t h a t t h e c u r v ei s s e l f - s i m i l a r . F o r th e s e c u r v e s th e f r a c ta l d i m e n s i o n i s c o m p u t e d a s D = 1 /V x = 1 / V r I f V x a n d V r a r e u n e q u a l - - a g o o d i n d i c a t i o n t h a t th e c u r v e i s s e l f -a f fi n e s i n c e t h e c o o r d i n a t e s s c a l e d i f f e r e n t l y - - t h e t w o v a r i a n c e s a r e r e l a t e d t oe a c h o t h e r v i a t h e H u r s t s c a li n g p a r a m e t e r ( H ) a s H = Vr /V x M a t s u s h it a a n dO u c h i ( 1 9 8 9 ) t e s t e d t h e i r m e t h o d u s i n g l i n e a r f e a t u r e s o f k n o w n f r a c t a l d i m e n -s i o n s ( b o t h s e l f - s i m i l a r a n d s e l f - a tt i n e ) a n d f o u n d t h e r e s u l ts t o b e v e r y r e a s o n -a b l e .

T h i s m e t h o d h a s y e t t o b e t e s te d e x t e n s i v e l y , b u t i t w o u l d a p p e a r to h a v ec h a r a c t e r i s t i c s w h i c h c o u l d m a k e i t a u s e f u l t o o l i n f r a c t a l a n a l y s i s g i v e n t h a t i tw o r k s w i t h b o t h s e l f - s i m i l a r a n d s e l f - a f fi n e f e a t u r es . F u r t h e r m o r e , O u c h i a n dM a t s u s h i t a ( 1 9 9 2 ) h a v e e x t e n d e d t h i s m e t h o d t o h a n d l e s u r f a c e s a n d r e p o r t e dg o o d r e s u l t s .

M I S C E L L A N E O U S M E T H O D ST h e r e a r e s e v e r a l a d d i t i o n a l m e t h o d s w h i c h r e q u ir e s o m e d i s c u s s i o n . T h e

i n t e r s e g m e n t a n g l e s m e t h o d i n t r o d u c e d b y E a s t m a n ( 1 9 8 5 ) i s b a s e d o n r e l a ti n gl o c a l m e a s u r e s o f a c u r v e s s i n u o s i t y to i t s f r a c ta l d i m e n s i o n . E a s t m a n a n a l y z e ds e v e r a l w e l l k n o w n n a t u r a l c u r v e s ( e . g . , t h e w e s t c o a s t o f B r i t a in ) a n d o b t a i n e dr e s u l t s t h a t m a t c h e d t h o s e r e p o r t e d i n t h e l i t e r a t u r e . T h e m e t h o d w a s c o n c e i v e db y o b s e r v i n g t h e r e l a t i o n b e t w e e n t h e c h a r a c t e r i s t i c g e n e r a t o r s o f s y s t e m a t i cf r a c t a l c u r v e s , t h e i r f r a c t a l d i m e n s i o n , a n d t h e i r s i n u o s i t y . E x t e n d i n g t h e i d e a


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4 Klinkenberg

o f a g e n e r a t o r t o r a n d o m f r a c ta l s, E a s t m a n ( 1 98 5 ) d e v e l o p e d t h e f o r m u l a b e l o w .I t d i r e c t l y r e l a te d t h e s i n u o s i t y o f a n a t u r al c u r v e t o i t s fr a c t a l d i m e n s i o n .

l o g ( 2 )D = , o ga , b = s e g m e n t l e n g t h s a b o u t p o i n t j ( i . e . , a a n d b w i l l s p a n l e n g t h s f r o mp t ( j ) t o p t ( j ___ k ) , w h e r e k = 1 t o 5 ) , a n d c = d i s t a n c e f r o m p t ( j - k ) t op t ( j + k ) .

T h e v a r i at io n m e t h o d , r e c en t ly d e v e l o p e d b y D u b u c e t a l . (19 89a ) , i s av a r i a t io n o f t h e M i n k o w s k i m e t h o d . I n s t e a d o f u s in g f i x e d d i sk s ( o r b o x e s ) t oc o v e r th e f e a tu r e , t h e y u s e an a d a p t i v e c o v e r i n g p r o c e d u r e - - b o x e s w i t h d i m e n -s i o n s th a t a r e a f u n c t io n o f t h e c u r v e w i t h i n a l o c a l n e i g h b o r h o o d . T h e o p t i m a ls i z e o f t h e lo c a l n e i g h b o r h o o d i s c h o s e n b y o b s e rv i n g t h e r e l a t io n b e t w e e n t h ed a t a p o i n t s c a t t e r a n d t h e n e i g h b o r h o o d s i z e . D e t a i l s o n h o w t o i m p l e m e n t t h ev a r i a t i o n m e t h o d a r e p r e s e n t e d i n D u b u c e t a l . ( 1 9 8 9 a ) . T h e m e t h o d c a n a l s ob e a p p l i e d t o s u r f a c e s ( D u b u c e t a t . , 1987 , 1989b) .

T e s t s o f t h e m e t h o d - - u s i n g m a t h e m a t i c a l l y d e f i n e d ra n d o m f r ac t a l c u r v e s - -p r o d u c e d d i m e n s i o n s a l m o s t i d e n t i c a l t o t h e c u r v e s ' t h e o r e t i c a l d i m e n s i o n s .H o w e v e r , o t h e r m e t h o d s s h o w n t o p r o d u c e a c c u r a t e d i m e n s i o n s w h e n w o r k i n gw i t h w e l l - b e h a v e d f r a c ta l c u r v e s h a v e n o t a l w a y s b e h a v e d a s w e l l w h e n a p p l i e dt o r e a l w o r l d f r a c ta l s . I t r e m a i n s t o b e s e e n h o w t h e v a r i a t i o n m e t h o d w o r k sw h e n a p p l i e d t o n a tu r a l d a t a s e t s . A n o t h e r v a r ia n t o f t h e b o x c o u n t i n g m e t h o dw a s p r e s e n t e d i n R i g a u t ( 1 9 9 1 ) . I t t o o r e q u i r e s f u r t h e r t e s t i n g .

I n th e s t u d y o f d i f f u s i o n - l i m i t e d a g g r e g a t i o n ( D L A ) , t h e m a s s f r a c t a l d i -m e n s i o n i s c o m m o n l y d e t e r m i n e d . A c o m p r e h e n s i v e r e v i e w o f D L A a n d d e te r -m i n a t i o n o f th e m a s s d i m e n s i o n w a s p r e s e n te d b y M e a k i n ( 1 9 9 1 ) . T h e m e t h -o d o l o g y i s e a s i l y a d a p t e d t o l i n e a r p h e n o m e n o n : p l o t t h e ( l o g o f t he ) a c c u m u l a t e dl e n g t h L R ) o f t h e f e a t u r e a s a f u n c t i o n o f t h e ( l o g o f th e ) d i s t a n c e ( R ) f r o m ac e n t r a l p o i n t ( D = t h e s l o p e ) . I t h a s b e e n l i t t l e u s e d i n a r e a s o t h e r t h a n D L A ,b u t t h e m e t h o d w i l l p r o b a b l y p l a y a m u c h l a r g e r r o l e i n t h e f u t u r e a s f r a c t a la n a l y s e s in t h e g e o s c i e n c e s b e c o m e m o r e s o p h i s t ic a t e d ( e . g . , B a r to l i e t a l . ,1 9 9 1) . B a t t y a n d L o n g l e y ( 1 9 8 7 ) a n d B e n q u i q u i a n d D a o u d ( 1 9 9 1 ) p r o v i d ea t y p i c a l e x a m p l e s o f t h e a p p l i c a t io n o f t h i s m e t h o d .

D I S C U S S I O N A N D C O N C L U S I O N SS o m e g e n e r a l o b s e r v a ti o n s ca n b e m a d e a b o u t th e m e t h o d s d e s c r i b e d a b o v e .

W h i l e e v e r y m e t h o d a p p e a r s to h a v e i ts s p e ci a l c o n c e r n s - - a s p e c t s o f t h e ir p a r-t i c u l a r a p p l i c a t i o n w h i c h m u s t b e c a r e f u l l y c o n s i d e r e d b e f o r e r e s u l t s a r e r e l i a b l yp r o d u c e d - - m a n y s h a r e c o m m o n c o n c e r n s . U s i n g l e a st - s q u a re s a n a l y s i s t o d e -


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F r a c t a i i m e n s i o n o f L i n e a r F e a t u r e s 41t e r m i n e t h e s l o p e o f th e l o g - l o g c u r v e m e a n s t h a t th e s e l e c t io n ( i . e . , t h e r a n g ea n d s p a c i n g ) o f th e i n d e p e n d e n t v a r i a b le v a l u e s c a n g r e a t l y i n f l u e n c e th e r e s u lt s .T h e a u t o c o r r e l a t e d n a t u r e o f th e d a t a s h o u l d a l s o b e c o n s i d e r e d , b u t r a r e ly i s( R e e v e , 1 9 9 2 ). A n o t h e r a s p e c t w h i c h h a s n t b e e n m e n t i o n e d in t h e f ra c t a l li t-e r a tu r e is th e l a c k o f c o r r e s p o n d e n c e b e t w e e n p a r a m e t e r s o b t a i n e d f r o m n o n l i n -e a r r e g r e s s i o n a n d t h o s e o b t a i n e d f r o m l i n e a r i z e d l o g - l o g r e g r e s s i o n . T h a t t h ev a l u e s p r o d u c e d b y t h e t w o f o r m s o f re g r e s s i o n c a n b e v e r y d if f e re n t h a s l o n gb e e n k n o w n ( e .g . , Z a r , 1 9 6 8 ). I n i s la n d b i o g e o g r a p h y , a n o t h e r f ie ld w h i c hs i m i l a r l y u s e s l o g - l o g r e g r e s s i o n t o d e t e r m i n e m e a n i n g f u l p a r a m e t e r s , i t h a sb e e n s h o w n t h a t d i f fe r e n c e s i n t h e i n t e rp r e t a ti o n o f s u p p o s e d l y m e a n i n g f u l p a -r a m e t e r s c a n b e t r a c e d b a c k t o th e l i n e a r i z a ti o n o f t h e u n d e r l y i n g p o w e r r e l a ti o n( K l i n k e n b e r g , 1 9 8 8 ). T h e e f f ec t o f th i s o n t h e in t e r p re t a t io n s o f f r ac t a l d im e n -s i o n s r e q u i r e s e x p l o r a t i o n s i n c e t h e e f f e c t s o f l i n e a r i z a t i o n m a y n o t b e c o n s t a n tf r o m m e t h o d t o m e t h o d .

F o r m o s t m e t h o d s t h e c h o i c e o f t h e s m a l l e s t ( a n d l a r g e s t) s a m p l e e l e m e n t sr e q u i r e s c a r e f u l c o n s i d e r a t i o n . R e d u c i n g t h e b a n d w i d t h w i t h i n w h i c h t h e p a r -t i c u l a r m e t h o d i s e m p l o y e d w i l l n e c e s s a r i l y r e d u c e t h e l i k e l i h o o d t h a t f r a c t a le l em en t s be i den t i f i ed , i f p re sen t , o r t ha t mul t i f r ac t a l be ha v io r be i den t i fi ed .F u r t h e r m o r e , i m p o s i n g a d y a d i c s a m p l i n g s t r a te g y f o r r e g re s s io n p u r p o s e s c a nre su l t in s ign i f i can t r educ t ions o f t he sam ple s i ze .

T h e r e h a s b e e n a r a n g e o f c o n t r a s t i n g o p i n i o n s r e g a r d i n g t h e c h o i c e o f o n em e t h o d o v e r a n o th e r . H u a n g a n d T u r c o t t e ( 1 9 8 9 ) p r e f e r s p e c tr a l m e t h o d s o v e rm e t h o d s s u c h a s t h e v a r i o g r a m m e t h o d b e c a u s e t h e i n t e r c e p t s e r v e s a s a q u a n -t it a ti v e m e a s u r e o f te x t u r e . H o w e v e r , t h is i s n o t r e a l ly a r e a s o n f o r u s i n g th em e t h o d b e c a u s e t h e v a r i o g r a m i n t e r c e p t c a n b e u s e d i n a s i m i l a r f a s h i o n ( K l i n -k e n b e r g a n d G o o d c h i l d , 1 9 9 2 ) . S e v e r a l a u t h o r s h a v e c o m m e n t e d t h a t t h e y f e e lt h a t t h e v a r i o g r a m m e t h o d i s m o r e r o b u s t / a c c u r a t e t h a n t h e p o w e r s p e c t r u mm e t h o d , a n d i s b e t te r a b l e to d e t e c t th e p r e s e n c e o f fr a c t a l e l e m e n t s ( e . g . ,M a l i n v e r n o , 1 9 8 9; K l i n k e n b e r g a nd C l a rk e , 1 9 92 ). H o w e v e r , v a r i o g r a m m e t h -o d s , s i n c e t h e y a r e b a s e d o n s e c o n d - o r d e r s t a t i s t i c s , m a y n o t a l w a y s b e a b l e t od e t e c t b r e a k d o w n s in t he s c a le i n v a ri a n c e o f p h e n o m e n o n - - s p e c t r a l m e t h o d sm a y b e b e t t e r a b l e t o d e t e c t s u c h b r e a k d o w n s ( J o n e s , T h o m a s , a n d E a r w i c k e r ,1 9 8 9 ) . Wi t h a p p r o p r i a t e d a t a , S a k e l l a r i o u et al. 1991) f o u n d t h e a r e a - p e r i m e t e rm e t h o d t h e m o s t p r e f e r a b l e .

B r o w n ( 1 9 8 7 ) f o u n d t h a t t h e s p e c t r a l a n d d i v i d e r m e t h o d s y i e l d e d s i m i l a rd i m e n s i o n s f o r s e l f - a f f i n e p r o f i l e s p r o v i d e d t h a t t h e d i v i d e r m e t h o d w a s u s e dp r o p e r l y ( i . e . , t h e c r o s s o v e r l e n g t h w a s r e s p e c t e d ) . O n t h e o t h e r h a n d , C a r r a n dB e n z e r ( 1 9 9 1 ) f e e l t h a t th e r e i s n o p r e re q u i s i te t h a t th e t w o m e t h o d s y i e l d s im i l a rf r a c t a l d i m e n s i o n s s i n c e t h e d i v i d e r m e t h o d i s b a s e d o n g e o m e t r i c r e l a t i o n s ,w h i l e t h e s p e c t r a l m e t h o d ( a s w i t h m o s t o t h e r m e t h o d s ) i s b a s e d o n s t o c h a s t i cr e la t io n s . F u r t h e r m o r e , s i n c e s p e c t ra l m e t h o d s a r e a p p l i e d t o p r o f il e s , w h i led i v i d e r s a r e m o s t o f t e n a p p l i e d t o h o r i z o n t a l c u t s , t h e r e a r e m a n y r e a s o n s w h y


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4 Klinkenbergthe two methods produce different dimensions. Much of the controversy in thefractal literature derives from the observation that different methods often yielddifferent and sometimes ambiguous dimensions. It should be apparent that, insome cases, the differences may be the result of inappropriate ly applied methods.On the other hand, some differences should be expected, since the methods arenot all measuring the same fractal quantity.

No one method appears to be the method with which to determine thefractal dimension of linear features. Some of the methods have only recentlybeen introduced (e.g., the line scaling and the variation methods) and they needto be applied to a variety of natural features before any conclusion is reachedabout their overall reliability and consistency. Some methods can be applied toboth self-s imilar and self-affine curves , while others should only be applied toone type or the other. The particular choice of a method should depend on anumber of factors, some of which are outlined below.

I. Whether the feature is self-similar or self-affine.2. The format in which the data will be provided, which can make certain

methods much easier to implement than others.3. Whether features from a range of topological dimensions will be inves-

tigated, in which case one of the extensible methods --such as box count-ing, variogram, or spectral methods--may be the preferred choice.4. Which method fits most comfortably within the current methodological

framework of the researcher.5. Whether multidimensiona lity is known to exist, in which case the box

counting method as extended by Lovejoy e t a l (1987) should be em-ployed.

The main conclusion to be reached is that while there are many ways of deter-mining the fractal dimension of linear features, the application of every methodrequires careful consideration of a range of methodological concerns. Withoutadequate consideration of the potential problems the results from any analysiswill not truly reflect the fractal nature of the feature.

AC KNOW L E DGM E NT SThe comments of an anonymous reviewer are gratefully acknowledged.

This research is supported in part by grants from NSERC.REFERENCES

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