Analysis and simulation of scale-dependent fractal surfaces with application to seafloor morphology

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Common theories on fractal surfaces as observed in geology assume a universality law, in most instances the simplest type of universality, which is self-similarity or self-affinity; in the case of multifractals, another well-known special type of fractals, a more complex form of scale invariance is described using one generating process. The assumption of a scale-invariant universality law, however, implies that a geological object was created by a single underlying process, which is clearly in contradiction to geological knowledge and measurable observations. The processes of crust generation, seafloor spreading, sediment deposition, and erosion work at different specific homogeneity ranges of scale, and such scale dependency is observed in many data sets collected for topographic surfaces. This necessitates the design of methods and algorithms for analysis and simulation of fractal surfaces with scale-dependent spatial characteristica. A suite of algorithms and programs for this purpose is compiled and presented in this paper. Numerical algorithms build on geostatistics, Fourier theory, and some “fractal” methods. The approach presented here uses a dimension parameter for characterization of roughness and an anisotropy factor, given with respect to a principal direction, to capture anisotropic properties. Analytical methods are an isarithm method, a variogram method, a Fourier method, and an isarithm-type Fourier method for estimation of a dimension parameter. In applications to bathymetric data from the western flank of the mid-Atlantic ridge, the variogram method is found most accurate and produces results consistent with geological observations. Interpolation, unconditional simulation and conditional simulation algorithms based on Fourier methods and Fractional Brownian Surfaces localized in scale are combined to construct and merge grids of different scales with specific roughness and anisotropy characteristics, resultant in surfaces with scale-dependent properties which almost exactly reproduce those observed from geophysical data in the seafloor case studies. The scale-dependent simulation methods serve to (1) extrapolate in scale beyond the observed resolution, if roughness and anisotropic properties are known from another area with similar characteristics, and thus provide information on subscale properties for surveys with instrumentation of lower resolution, and (2) extrapolate and simulate in space, if an area has only partly been covered by a survey.