Abstract. This paper examines the characteristics of four different methods of estimating the fractal dimension of profiles. The semi-variogram, roughness-length, ...
Mathematical Geology, Vol. 26, No. 4, 1994
E s t i m a t i n g F r a c t a l D i m e n s i o n of Profiles: A C o m p a r i s o n of M e t h o d s John C. Gallant, 1 Ian D. Moore, 1 Michael F. Hutchinson, ! and Paul Gessler 2
This paper examines the characteristics of four different methods of estimating the fractal dimension of profiles, The semi-variogram, roughness-length, and two spectral methods are compared using synthetic 1024-point prt~les generated by three methods, and using two profiles derived from a gridded DEM and two profiles from a laser-scanned soil surface. The analysis concentrates on the Hurst exponent H, which is linearly related to fraetal dimension D, and considers both the accuracT and the variability of the estimates of H. The estimation methods are fi)und to be quite consistent fi~r H near 0.5, but the semivariogram method appears to be biased for H approaching 0 and I, and the roughness-length method for H approaching O. The roughness-length or the ma.rimum entropy spectral methods are recommended as the too'st suitable methods fi~r estimating the fractal dimension of topographic profiles. The fractal model fitted the soil surface data at fine scales but not at broad scales, and did not appear to fit the DEM profiles well at any scale.
KEY WORDS: power spectrum, maximum entropy, semi-variogram, roughness-length, confidence interval. INTRODUCTION The use of fractal geometry and, in particular, a fractal dimension (D) to describe the scaling structure in natural phenomena is now quite common. The theory is based largely on the work of Mandelbrot (Mandelbrot, 1983; Mandelbrot, 1975). Fractal dimensions have been calculated for surfaces such as landscapes, sea floors, and rock fracture surfaces (Bell, 1979; Brown, 1987; Brown and Scholz, 1985; Klinkenberg and Goodchild, 1992; Mark and Aronson, 1984; Polidori et al., 1991; Power and Tullis, 1991; Roy et al., 1987); for spatial patterns such ~Centre for Resource and Environmental Studies, Australian National University, Canberra ACT 0200, Australia. -'Centre for Resource and Environmental Studies, Australian National University, and CSIRO Division of Soils, Canberra ACT, Australia.
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0.045 for SPEC and M E M , respectively (SRA and IFT data only), and 0.040, 0.027 for SV and RL, respectively (SRA, IFT, and W M data). Figure 6 shows the distribution of Hm~.~ for R L and SPEC analysis on SRA data with Hgen = 0, 0.2, 0.4 . . . 1.0. For each value of Hge, the range of the 100/4. .... outcomes was divided into ten equal classes, and the frequency of each class was calculated. This figure highlights the difference in precision of the S P E C and R L methods, and demonstrates the difficulty of accurately estimating H for these data. When only the 20 estimates with highest R 2 are used, the range of an using the same values of H~n is from 0.005 (RL on W M profiles) to 0.087 (SPEC on SRA profiles). The results are clearly less variable, but the improvement is not uniform across all analysis/generation method pairs. For example, R L on
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S R A improves only slightly, while SV on both IFT and W M has a n roughly halved by using only the best estimates. Some of the mean values in Fig. 4 are biased relative to those in Fig. 3. This is particularly noticeable for SPEC on IFT and SRA with Hg~n near 0, and for R L on SRA with Hge, > 0.5. An important point, previously noted by Malinverno (1990), is that there is no apparent relationship between the goodness-of-fit of the least-squares regression and the value of Hme~s: the linearity of the l o g - l o g plot is no guarantee of the accuracy of Hmea.~. This is illustrated in Fig. 7 in two different ways using SV on Brownian motion (H = 0.5). Fig. 7a shows the distribution of R 2 values against the slope of the semi-variogram for the 100 estimates used in the preceding analysis; the expected slope is 1.0. Although the results are clustered around 1.0, there are estimates with high R 2 (over 0.997) with slopes ranging
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from 0.87 to 1.13. Figure 7b shows the semi-variograms of four realizations of Brownian motion generated by summing four sets of independent Gaussian random numbers. Up to a lag of 100, they all show a good fit to a straight line with R 2 of 0.997 to 0.999, but the slopes vary from 0.87 to 1.17. Finally, we consider the correlation between pairs of estimates obtained from the four different analysis methods on I F T data, using 100 realizations for each Hge, = 0.1, 0.3, 0.5, 0.7, and 0.9. Table 2 shows the ranges of R 2 values obtained using linear regression on the estimates o f H for each pair of analysis methods. Figure 8 shows the scatter plots for the two extremes, SPEC vs. SV with Hge, = 0.9 (R 2 = 0.003) and M E M vs. SV with Hge, = 0.9 (R 2 = 0.589).
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M E M and SV e s t i m a t e s are fairly well c o r r e l a t e d with R 2 r a n g i n g f r o m 0 . 0 0 3 to 0 . 0 0 8 i n d i c a t i n g very little c o r r e l a t i o n . T h e v a r i a t i o n s o f / 4 , ..... tbr a g i v e n Hg~, are not c o n s i s t e n t b e t w e e n a n a l y s i s m e t h o d s . T h e effect o f s m o o t h i n g o n the S P E C e s t i m a t e s w a s s t u d i e d by r e - a n a l y z i n g the IFT data. T h e s m o o t h e d s p e c t r a w e r e c o m p u t e d u s i n g the D F T m e t h o d at 1000 l o g a r i t h m i c a l l y s p a c e d f r e q u e n c i e s and a v e r a g i n g g r o u p s o f ten f r e q u e n c i e s to give 100 spectral e s t i m a t e s as for the u n s m o o t h e d c a s e . T h e a v e r a g e o u for 0 _< Hg~. _< 1 w a s r e d u c e d from 0 . 0 7 6 to 0 . 0 6 8 , a n d the m e a n H ..... was o n a v e r a g e slightly c l o s e r to Hee n. T h e c o r r e l a t i o n s with o t h e r e s t i m a t e s w e r e essentially u n c h a n g e d .
Estimating Fracta] Dimension of Profiles
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Analysis of DEM Profiles The analysis results for the two profiles from the Brindabella DEM are shown in Figs. 9 and 10. The most noticeable feature of these results are that the spectral slope ~ of both profiles is greater than 3 for high frequencies. SPEC gives a slope of about - 4 . 0 while MEM gives a slope of about - 4 . 5 . These steep spectral slopes extend down to about 0.003 m - ~ (330 m) except for SPEC on the N-S profile where it continues to 0.0008 -~ (1250 m). At frequencies less than about 0.001 m -~ both MEM and SPEC indicate periodic rather than fractal behavior, with a period of about 2000 m. The minor peak in the MEM spectrum at 0.0025 m-~ (400 m) corresponds to the shorter fluctuations in the profile (Fig. la). SV shows a slope of 2 (H = 1) at short lags for both profiles, indicating extreme smoothness. Straight lines can be fitted in intermediate ranges, giving H = 0.8 from 150 to 1000 m for the N-S profile and H = 0.44 from 600 to 2000 m for the E - W profile. The N-S profile also shows a straight segment from 1000 to 2000 m with H = 0.3, although this could be spurious. The RL results change slopes at similar scales to SV, but show different slopes. The N-S profile gives H = 1.8 for windows shorter than 200 m, H = 1 between 200 and 600 m, and H = 1.45 from 600 to 2000 m. The E - W profile
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Estimating Fractal Dimension of Profiles
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non-fractal behavior of the profiles in these intermediate ranges, the measured values are probably spurious. Some study of the way the estimates of fractal dimension behave when applied to non-fractal data would provide some additional insights into the nature of the inconsistencies experienced with this data. The results of Fig. 3 indicate that the SPEC, MEM, or RL methods would perform better than SV at identifying increased smoothness of DEMs at fine scales because they can estimate H values greater than 1.
Analysis of Soil Surface Profiles The soil surface results are shown in Figs. 11 and 12. The X profile results in Fig. 11 show fairly straight lines with SPEC, MEM, and SV all giving H 0.7 and RL giving an overall slope of H = 0.8. RL shows marked changes in slope with segment slopes ranging from 0.4 to 1.2, and some oscillations at longer window lengths. The Y profile results (Fig. 12) are more variable, with SV giving H = 0.63 and RL H = 0.8. SPEC and MEM give H = 0.7 for frequencies greater than about 0.1 mm-~, and clear indication of periodic behavior with a frequency of about 0.007 mm -~, a period of about 150 mm. SV also shows clear evidence of periodicity above 100 mm. The consistency of the measures of H for fine scale probably indicates that the fractal model is applicable at those scales. The strong periodicity in the Y profile is probably an artifact from the reconstruction of the soil surface.
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CONCLUSIONS The three generating methods used in this paper, the successive random approximation method, the inverse Fourier transform method, and the Weierstrass-Mandelbrot function, produce quite consistent results for H in the range 0 to 1 corresponding to continuous, non-differentiable profiles typified by fractional Brownian motion. The successive random approximation method produced sample profiles with the highest variability of estimated H (and thus fractal dimension D) and the Weierstrass-Mandelbrot function produced the lowest variability. The inverse Fourier transform method should be modified to account for aliasing, particularly for values of H near 0. The four estimation methods, periodogram power spectrum, maximum entropy power spectrum, the semivariogram, and the roughness length method, are all consistent for H near 0.5, but the semivariogram becomes increasingly biased near H = 0 and H = 1 and the roughness length method is biased near H = 0. The roughness length has the least variability of estimated H and the periodogram power spectrum has the greatest variability. The maximum entropy power spectrum method performs significantly better than the periodogram power spectrum method in terms of bias and variability, but is more complex to implement. The roughness length and semivariogram methods are the simplest to implement.
Estimating Fractal Dimension of Profiles
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Based on the results in this paper, the authors recommend the inverse Fourier transform method or the Weierstrass-Mandelbrot function for generating fractal profiles and the roughness length method or maximum entropy power spectrum method for estimating fractal dimension of profiles. The WeierstrassMandelbrot function is unsuitable if the generated profile is to be subsequently analyzed using a spectral technique. The 95% confidence limits on the estimates of H depend on the method and the data, but were between +0.02 and +0.2. The variations of estimated H for a given generating H are not consistent between analysis methods. When applied to datasets which depart significantly from the fractal model, such as the DEM profiles used here, the various estimation methods give widely differing estimates of H. One significant implication of these results is that the low correlation of fractal dimension with traditional morphometric variables observed by Klinkenberg (1992) may reflect the high variability of the estimates of D and not new information extracted from the surface as Klinkenberg assumed. The DEM profiles are dominated by periodic behavior at broad scales and excessive smoothness at fine scales, and estimated H values varied substantially between methods. The fractal model appears to be a poor model for these profiles. The soil surface profiles produced stable estimates of H at fine scales, but were again periodic at broad scales which appears to reflect the method of reconstruction of the soil surface. ACKNOWLEDGMENTS The authors thank Dr. Tony Norton and Dr. Jann Williams for assistance in developing the DEM for the Brindabella Range, and Mr. Jim Taylor and Dr. Chris Moran of CSIRO for the use of the laser-scanned soil data. Mr. Adam Lewis and Mr. Tingbao Xu and the anonymous reviewers provided valuable contributions to the development of this paper. The study was funded in part by Grant 1991/92-ANU3 from the Land and Water Resources Research and Development Corporation and by the Water Research Foundation of Australia.
REFERENCES Ables, J., 1974, Maximum Entropy Spectral Analysis, in Proc. Syrup. on the Collection and Analyses of Astrophysical Data, Vol. 15, Astron. Astrophys. Suppl. Series: p. 383-393. Armstrong, A., 1986, On the Fractal Dimensionsof Some TransientSoil Properties: J. Soil Sci., v. 37, p. 641-652. Bell, T., 1979, Mesoscale Sea Floor Roughness: Deep Sea Res., v. 26A, p. 65-76. Berry, M., and Lewis, Z., 1980, On the Weierstrass-MandelbrotFractal Function:Proc. R. Soc. Lond. A., v. 370, p. 459--484. Brown, S., 1987, A Note on the Description of Surface Roughness Using Fractal Dimension: Geophys. Res. Lett., v. 14, n. 11, p. 1095-1098.
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Brown, S. and Scholz, C., 1985, Broad Bandwidth Study of the Topography of Natural Rock Surfaces: J. Geophys. Res., v. 90(B14), n. 12, p. 12,575-12,582. Burg, J., 1967, Maximum Entropy Spectral Analysis: 37th Meeting Soc. of Exploration Geophysicists, Oklahoma City, Oct. 31. 1967. Burg, J., 1968, A New Analysis Technique for Time Series Data: NATO Advanced Study Institute on Signal Processing, Enschede, Netherlands, Aug. 12-23, 1968. Carr. J., and Benzer, W., 1991, On the Practice of Estimating Fractal Dimension: Math. Geol., v. 23, n. 7, p. 945-958. Denham, C,, 1975, Spectral Analysis of Paleomagnetic Time Series: J. Geophys. Res., v. 80, n. 14, p. 1897-1901. Dicke. M., and Burrough, P., 1988, Using Fractal Dimensions for Characterizing Tortuosity of Animal Trails: Phys. Entomol., v. 13, p. 393-398. Fox, C., 1989, Empirically Derived Relationships Between Fractal Dimension and Power Law Form Frequency Spectra: Pure Appl. Geophys., v. 131, n. 1/2, p. 211-239. Goodchild, M., 1980, Fractals and the Accuracy of Geographical Measures: Math. Geol., v. 12, n. 2, p. 85-98. Hakanson, L., 1978, The Length of Closed Geomorphic Lines: Math. Geol., v. 10, p. 141-167. Hough, S., 1989, On the Use of Spectral Methods for the Determination of Fractal Dimension: Geophys. Res. Lett., v. 16, n. 7, p. 673-676. Huang, C., White, 1., Thwaite, E., and Bendeli, A., 1988, A Noncontact Laser System for Measuring Soil Surface Topography: Soil Sci. Soc. Am. J., v. 52, n. 2, p. 350-355. Hutchinson, M., 1989, A New Procedure for Gridding Elevation and Stream Line Data with Automatic Removal of Spurious Pits: J. Hydrol., v. 106, p. 211-232. Jones, J., Thomas, R., and Eat'wicker, P., 1989, Fractal Properties of Computer-Generated and Natural Geophysical Data: Comput. Geosci., v. 15, n. 2, p. 227-235. Klinkenberg, B., 1992, Fractals and Morphometric Measures: Is There a Relationship?: Geomorphology, v. 5, p. 5-20. Klinkenberg, B., and Goodchild, M., 1992, The Fractal Properties of Topography: A Comparison of Methods: Earth Surf. Proc. Landforms, v. 17, p. 217-234. Lam, N., 1990, Description and Measurement of Landsat TM Images Using Fractals: Photogr. Eng. Rem. Sensing, v. 56, n. 2, p. 187-195. Mandelbrot, B., 1967, How Long is the Coast of Britain: Statistical Self-Similarity and Fractional Dimension: Science, v. 156, p. 636-638. Mandelbrot, B., 1975. Stochastic Models for the Earth's Relief, the Shape and the Fractal Dimension of the Coastlines, and the Number-Area Rule for Islands: Proc. Nat. Acad. Sci. USA, v. 72, n. 10, p. 3825-3828, Mandelbrot, B., 1983, The Fractal Geometr3' of Nature: Freeman, New York, 468 p. Mandelbrot, B., 1985, Self-Affine Fractals and Fractal Dimension: Phys. Scripta, v. 32, p. 257260. Mandelbrot, B., and van Ness, J., 1968, Fractional Brownian Motions, Fractional Noises and Applications: SlAM Rev., v. 10, n. 4, p. 422-437. Mark, D., and Aronson, P., 1984, Scale-Dependent Fractal Dimensions of Topographic Surfaces: An Empirical Investigation, with Applications in Geomorphology and Computer Mapping: Math Geol., v. 16, n. 7, p. 671-683. Matheron, G., 1963, Principles of Geostatistics" Econ. Geol., v. 58, p. 1246-1266. Oliver. M., and Webster, R., 1986, Semi-Variograms for Modelling the Spatial Pattern of Landform and Soil Properties: Earth Surf. Proc. Landforms, v. 11, p. 491-504. Owen, M., and Wyborn, D., 1979, Geology and Geochemistry of the Tantangera and Brindabella Area: Bull. 204, Bureau Mineral Resources., Geol. and Geophys., AGPS, Canberra. Palmer, M., 1988, Fractal Geometry: A Tool for Describing Spatial Patterns of Plant Communities: Vegetatio, v. 75, p. 91-102.
Estimating Fractal Dimension of Profiles
481
Pietgen, H., and Saupe, D. (Eds.), 1988, The Science of Fraetal Images: Springer Verlag, New York. Polidori, L., Chorowicz, J., and Guillande, R., 1991, Description of Terrain as a Fractal Surface, and Application to Digital Elevation Model Quality Assessment: Photogr. Eng. Rem. Sensing, v. 57, n. 10, p. 1329-1332. Power, W., and Tullis, T., 1991, Euclidean and Fractal Models for the Description of Rock Surface Roughness: J. Geophys. Res., v. 96(B1), p. 415-424. Press, W., Flannery, B., Teukolsky, S., and Vetterling, W. T., 1989, Numerical Recipes. The Art of Scientific Computing: Cambridge University Press, Cambridge, 702 p. Pryor, L., and Brewer, R., 1954, The Physical Environment, in H. White (Ed.), Canberra--A Nation's Capital: Angus and Robertson, London. Robert, A. and Roy, A., 1990, On the Fractal Interpretation of the Mainstream Length-Drainage Area Relationship: Water Res. Res., v. 26, n. 5, p. 839-842. Rosso, R., Bacchi, B., and La Barbera, P., 1991, Fractal Relation of Mainstream Length to Catchment Area in River Networks: Water Res. Res., v. 27, n. 3, p. 381-387. Roy, A., Gravel, G., and Gauthier, C., 1987, Measuring the Dimension of Surfaces: A Review and Appraisal of Different Methods, in N. Chrisman (Ed.). Proc. Auto-Carto 8: Baltimore, Maryland, March 29-April 3, 1987, p. 68-77. Saupe, D., 1988, Algorithms for Random Fractals, in H. Pietgen and D. Saupe (Eds.), The Science of Fractal Images: Springer Verlag, New York, p. 71-136. Taylor, J., Moran, C., White, I., Hairsine, P., and Carrigy, N., 1995, Behaviour Characterisation and Calibration of a Soil Surface Laser Scanner: Trans. Am. Soc. Agr. Engrs., in review. Vignes-Adler, M., Page, A. L., and Adler, P., 1991, Fractal Analysis of Fracturing in Two African Regions, from Satellite Imagery to Ground Scale: Tectonophysics, v. 196, p. 69-86. Voss, R., 1985a, Random Fractal Forgeries, in R. A. Eamshaw (Ed.), Fundamental Algorithms .]'or Computer Graphics, Vol. FI7, NATO ASI Series: Springer-Verlag, Berlin, p. 813-829. Voss, R., 1985b. Random Eractals: Characterization and Measurement, in R. Pynn and A. Skjeltorp (Eds.), Scaling Phenomena in Disordered Systems: Plenum Press, New York, p. 1-11. Voss, R., 1988, Fractals in Nature: From Characterization to Simulation, in H. Pietgen and D. Saupe (Eds.), The Science of Fraetal Images:Springer Verlag, New York, p. 21-70. Xu, T., Moore, I., and Gallant, J., 1993, Fractals. Fractal Dimensions and Landscapes: A Review: Geomorpho[ogy, v. 8, p. 245-262.
