EARTH SURFACE PROCESSES AND LANDFORMS, VOL. 12,475480 (1987)

FRACTALS IN KARST MARTIN LAVERTY

23, Brunswick Street, Canton, Cardin; CF5 ILH

Received 26 November 1986 Revised 6 February 1987

ABSTRACT

The analysis of cave survey data is used to establish that cave length exhibits fractal (self-similar) behaviour, with dimensions in the range 1-1.5, over a range of measurement resolutions from 1 to 100 m. It is suggested that fractal dimensions, together with their range of applicability, will prove to be useful parameters for the study of caves.

KEY WORDS Karst Cave Fractals

INTRODUCTION

The ‘natural feel’ of geomorphological features is largely a function of the seemingly random repetition of similar features at different scales, in contrast with the smoothness and well-defined scale at which most models operate. The mathematical work of Mandelbrot (1967,1977,1982) has stimulated the growth of novel means of describing, modelling, and, perhaps, explaining some aspects of such features through the fractal concept. The ideas studied and brought together by Mandelbrot range from art to zoology, encompassing brain morphology, coastlines, drainage networks, economics . . . galaxies . . . linguistics . . . turbulence, and much else (Mandelbrot, 1977, 1982; Batty, 1985). Of particular note to geomorphologists are the computer- generated landscapes which have been produced through the use of models based, if sometimes rather loosely, on fractals. They are often thought to be surprisingly realistic and, as such, figure strongly in popular accounts (Batty, 1985; Dewdney, 1986).

Fractals are closely related to the concept of self-similarity whereby a part can be viewed at different levels of detail without changing the overall form: like a Spanish onion, or a Russian doll, peeling off more layers reveals nothing essentially new, although an element of randomness may give rise to ‘inessential’, albeit notable, differences. Fractals are also notable for the way in which very complex features arise from elegantly simple seeds. Mandelbrot (1967) has noted that ‘the concept of dimension is elusive and very complex’. However, the familiar idea of a line being one dimensional and an area two dimensional, are clear enough. A fractal, with a non-integer dimension, blurs the distinction: a line can writhe and kink itself to such an extent that it is more easily seen as a blurred area. Mandelbrot shows that an equation of the form:

m = a . r 1-D

relates the (estimated) measured length, rn, of such a continuous but kinked line to the length, r, of the straight line unit yardstick used to measure it. The D in the exponent is known as the fractal dimension, and the a is a constant. As expected, a straight (one dimensional) line has a well-defined length which is not dependent on r, but a curve with D greater than one increases in length as it is more assiduously inspected: its length is only meaningful if the details of the measurement are given. The observation is by no means new, but the theory erected around it is.

0197-9337/87/05047546$05.00 0 1987 by John Wiley & Sons, Ltd.

476 M. LAVERTY

The above equation is easily recast into the familiar doubly logarithmic linear relationship often called Zipf’s law, or the Pareto distribution. Many empirical relationships thus come within its ambit.

METHODOLOGY

This study is based on the use of cave survey data stored for processing and retrieval by a microcomputer. This is the most detailed data pertaining to the line skeleton of a cave passage which is usually available in numeric form, and is normally used to provide spatial coordinates plus figures for vertical range, and for plan and total survey traverse lengths. The line thus obtained will often be close to the centre line of the passage, especially in narrow passages, but often tends to zig-zag in larger passages. It tends to approximate to a line parallel to and 1-1.5 m above the floor, except where practical considerations militate against it. The measurements used here conform to the standards for a BCRA grade 5 survey (angles to the nearest degree, distances to the nearest 10 cm; Ellis in Ford and Cullingford, 1976).

The details of the data stored, its storage, and its processing will not be discussed here beyond saying that the essential elements used are:

1. The distance between successive stations 2. The forward bearing from one station to the next 3. The forward inclination from one station to the next

and that the measuring algorithm used is designed to cater for contiguous legs without branches, each such unit forming a section of the whole body of survey data. For each section, the algorithm simulates the walking of the survey skeleton with a pair of compasses opened to a fixed yardstick length, using trigonometrical formulae to calculate the intersection points, and to combine legs shorter than the yardstick until the resultant does intersect. Figure 1 illustrates how the use of this technique with progressively larger yardsticks removes detail from the survey line, and consequently reduces the measured length.

The calculations are applied in two ways;giving either a plan length estimate or a total survey length estimate, depending on whether the yardstick is confined to the horizontal plane or allowed to follow the cave in three dimensions.

Several other related techniques are described by Mandelbrot, and have been implemented in many fields. A technique relating area to perimeter has been applied to the geomorphology of Martian craters ( Woronow, 1981); such a technique, or even an extension to utilize volumetric data, would be interesting to apply to caves: the scatter introduced by the lack of control on the position of the survey line, as used here, would be reduced, at the expense of requiring more rigorous data collection. A very different methodology for estimating fractal dimensions, using statistical data fitted to a function known to behave as a fractal, is described by Burrough (198 1).

DATA SOURCES AND RESULTS

The technique has been tested using data collected from two caves in Sarawak, and two in Spain. The Sarawak caves are in the Bau-Serian karst to the south of K u c h g . Tang Dorog Penyok (Local dialect for Penyok mountain cave-Figure 1) is a short vadose cave, rather atypical of Bornean caves in being an active sink cave. It provides a conveniently small data set for the illustrations here. Tang Baan is a much larger cave system with about 5.8 km of surveyed passages, although only two of the sections of survey data are considered here. It has the typically large phreatic passages arranged in horizontal levels linked by steeply dipping passages as found in many tropical caves. The Spanish caves are located in the alpine karst of the Picos de Europa. Pozo Jorcada Blanca is one of two shaft systems, each almost 600 m deep, which enter a horizontal level. Pozo la Cistra is another shaft system, this time entering the 1139 m deep Pozu del Xitu system (OUCC, 1984, 1986).

Representative examples of the relationships between measured length (in metres) against various yardstick lengths are shown in Figure 2. Both plan and total length estimates are shown. The slope of the graphs is numerically equal to 1 - D, so inspection shows that D values vary from close to 1 to about 1.5 (Figure 2a illustrates both extremes). Table I gives values for D obtained by linear regression, the corresponding product

FRACTALS IN KARST 477

I

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Figure 1. Tang Dorog Penyok: Plan, skeleton line survey, and line surveys for yardsticks of $10, and 20 m. The measured length decreases as corneware cut by longer yardsticks

moment correlation coefficient, and the standard reported survey statistics of plan and total surveyed lengths, and vertical range (depth). The fractal D values apply to the steeper, central, part of the graphs in Figure 2, which appear to be generally divisible into three linear segments. (The calculations use some additional data omitted from Figure 2 for clarity.)

DISCUSSION

It is clear that cave length exhibits fractal properties in that the length estimate drops as yardstick length increases. The plan length is inevitably less than the total survey length, the difference being more marked the greater the vertical component of the cave. However, the fractal dimension of each measure can differ from the

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Figure 2. Doubly logarithmic relationships between length measured, m, and yardstick length, r, (both in metres) for: (a) T. Baan (two passages) (b) T. D. Penyok (c) P. Cistra (d) P. Jorcada Blanca. In each case, total length is shown by filled, and plan length by open, circles

other in either direction, depending on the particular cave morphology. For example, Cistra has slightly lower D in the plan case, whereas the other examples show a lower D for the total length. The fractal relationship is usually very good, although the tendency to maximize the length, thus minimizing the number, of survey legs results in a dimension close to 1 at yardsticks of befow average leg length: at large yardstick lengths the dimension becomes one as the length becomes constant at the end to end straight line distance.

Although further work is clearly needed, especially in analysing further datasets, and in testing alternative measuring techniques, it seems reasonable to suggest that the combination of characteristic lengths and fractal dimensions will be of use as indices for comparison of different cave passages both with each other and with experimental and theoretical models for passage development.

Theoretical models yield values for the fractal dimension: a Brownian motion gives a value of 3/2, whereas a self-avoiding random walk gives the lower value of 4/3 if confined in two dimensions, or 5/3 in 3 - D. In practice, it has been found that rivers (Hack, 1957), and coastlines and frontiers (in plan) often have values around 1.2 (Mandelbrot, 1982).

Curl (1966) studied the cumulative frequency distribution of cave lengths in a region and established that the best fit relationship has what can now be described as a fractal form. He has now combined this observation

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Table I. Values for D, the corresponding product moment correlation coefficient, and the standard reported survey statistics of plan and total surveyed lengths

No of survey Baan 1 Baan 2 Penyok Cistra J. Blanca stations 26 23 23 207 245

yardstick range no of yardsticks Plan reported length fractal D correlation Total vertical range reported length fractal D correlation

25-55 7

564 1.457

- 0.758

15 572

1.546 -0815

15-40 4

312 1.402

- 0.975

55 354

1.2 18 - 0.884

5-50 7

103 1.20 1

- 0.763

36 119

1.206 - 0,965

4-50 8

838 1.074

- 0.963

593 1346 1.102

- 0.998

4-40 7

1039 1.366

- 0.994

589 1559 1.185

- 0‘994

All measurements are in metres.

with the assumption that the type of relationship demonstrated for the length of an individual, enterable, and explorable cave applies to the total length of all caves in a region (Curl, 1986). From this he develops regional area and volume equations and presents a new measuring algorithm for explorable caves from which a value for the fractal dimension in his regional length equation can be obtained (provided that height, width, and length measurements are available for each survey station). It is notable that Curl’s theory applies not only to a whole region, but also includes unenterable and unexplorable subterranean spaces; these are of particular importance to biologists and hydrologists. Thus the two fractal dimensions proposed by Curl to characterize karst on a regional basis complement those presented in this paper which characterize individual caves.

CONCLUSION

The long and the short of it is: both the plan length and the total surveyed length of a cave behave as fractals. The actual dimension values differ between, and within, caves but values in the range 1-1.5 are found in the data analysed here. This is similar to the range reported for coastlines and includes the theoretical value for a planar random walk.

The fractal concept illustrated in this paper has implications for all who utilize dimensional data (lengths, areas etc) derived from indirect measurements, and highlights the importance of specifying the methods and scales used.

ACKNOWLEDGEMENTS

The data used here were collected by Oxford University Cave Club members, who have also published further details of the caves. The flourishing of this club, and of this author, owes not a little to Marjorie Sweeting’s assistance and encouragement over many years.

REFERENCES

Batty, M. 1985. ‘Fractals-geometry between dimensions’, New Scientist, 4 April, 31-35. Burrough, P. A. 1981. ‘Fractal dimensions of landscapes and other environmental data’, Nature, 294, 24CL242. Curl, R. L. 1966. ‘Caves as a measure of karst’, J . Geology, 74, 798-830. Curl, R. L. 1986. ‘Fractal dimensions and geometries of caves’, Mathematical Geology, 18, 765-783. Dewdney, A. K. 1986. ‘Of fractal mountains, graftal plants and other computer graphics at Pixar’, Sci. Am., Dec, 1418. Ellis, B. M. 1976. ‘Cave surveys’, in Ford and Cullingford.

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Ford, T. D. and Cullingford, C. H. D. 1976. The Science of Speleology, Academic Press, London. Hack, J. T. 1957. ‘Studies of longitudinal streams in Virginia and Maryland‘, USGS Prof. Pap., 294B. Mandelbrot, B. B. 1967. ‘How long is the coast of Britain? Statistical self-similarity and fractional dimension’, Science, 156, 636638. Mandelbrot, B. B. 1977. Fractals: Form, Chance and Dimension, Freeman, San Francisco. Mandelbrot, B. B. 1982. The Fractal Geometry of Nature, Freeman, San Francisco. Oxford University Cave Club 1984. Proceedings, 11, 8-12 and 51-63. Oxford University Cave Club 1986. Proceedings, 12. Woronow, A. 1981. ‘Morphometric consistency with the Hausdorff-Besicovich Dimension’, Math. Geol., 13, 201-216.