Estimating the Fractal Dimension of Synthetic and Real Topographic Surfaces

Nicholas J. Tate
School of Geosciences, Queens University of Belfast, Belfast BT7 INN, United Kingdom

This paper reports on analysis concerned with the comparison of several statistical methods for the calculation of the self-affine fractal dimension D of both computer generated synthetic surfaces, and real topography in the form of 1:24,000 scale USGS Digital Elevation Models (DEMs). Results suggest that the presence of method produced variation limits the utility of D as an index of discrimination between such surfaces.

The development of fractal geometry has provided appropriate tools for the characterisation of some of the irregularities and complexities inherent in the real world, and has been widely used in those spatially oriented disciplines such as geography and ecology which are interested in the characterisation of spatial patterns. Concern in these disciplines is often with spatial data that is heterogeneous and multiscale in a nested or hierarchical structure (Burrough, 1987). This presents practical difficulties for both (1) the suitable characterisation of spatial heterogeneity and (2) appropriate integration/aggregation of this information across scales (e.g. King, 1991). These problems are extant within geography/ geoscience in a variety of guises: as a general scale/scaling problem (e.g. Jeffers, 1988), the specific detection of scale dependent processes, the modifiable areal unit problem (Openshaw, 1984) and difficulties with the integration of remote sensing and GIS data (Davis et al., 1986) often in the form of upscaling or downscaling (e.g. Urban et al., 1986). Fractal geometry has been widely lauded as a method for the provision of useful information to address both (1) and (2). In a more specific context fractals have been widely detected in both subaerial and submarine topography, however there is little consensus on the expected magnitude of observed fractal dimension D (Evans & McClean, 1995), appropriateness of the fractal model (Herzfeld et al., 1995) and relevance in terms of a possible chaotic process regime (Phillips, 1994). A contributory factor to this confusion is that the variable definitions of fractal dimension and resulting variety of estimation techniques which makes strict comparisons impossible.

A fundamental premise in the measurement of D for topography is that the chosen statistical method of estimation is appropriate, robust, and generates an accurate dimension measure. This cannot be determined without comparison of the performance of both (a) different methods, and (b) different estimation algorithms for each method, in the context of a variety of different data. Unfortunately, there is relatively little such comparative analysis to guide the selection of the optimal method or algorithm for the calculation of D.

Computer simulated data in the form of surfaces of fractional Brownian motion (fBm) are a useful tool for such analysis, given that the fractal parameters of the data are mathematically well defined and can be compared with the fractal parameters derived from the estimation method applied to the surface. A set of simulated surfaces with different fractal dimensions such that D (2,3) has been generated using the method of iterated multidimensional interpolation from Voss (1985). Results are presented for the analysis of this data using three estimators of the ID semivariance function g(h), and an autoregressive maximum entropy based estimator of the 1D power spectral density S(f) to derive D. The analysis is then repeated for a set of real surfaces of digital topography in the form of United States Geological Survey 1:24,000 Digital Elevation Models.

The results indicate that the measure of fractal dimension D for both simulated and real surfaces is sensitive to:

  1. The choice of estimation method and algorithm used. Both influence the magnitude and variability of D measured locally within surfaces and between surfaces.
  2. The details of the analysis, in particular the selection of critical parameters such as the maximum distance lag for semivariance estimation, spectral auoregressive prediction error filter size, and the effective frequency range via data filtering/ detrending prior to analysis.
  3. The critical value of the coefficient of determination used to determine statistical goodness of fit of the fractal model.

The paper concludes with a consideration of the potential of texture based methods for more realistic synthetic surface generation and the use of geostatistical methods of crossvalidation for model testing.

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