Weierstrass-Mandelbrot

It is difficult to describe the complex forms and status of natural fracture surfaces with ordinary functions. The fractal theory can be used to describe extremely irregular geometric patterns, and many researches have indicated that it is reasonable to simulate rough fracture surfaces with fractals theory. At present, the self-affine fractal model is regarded as the best fractal model to simulate fracture surface. In the following, random Brown function method (Weierstrass- Mandelbrot function) is used to simulate a fractal fracture surface. 

Over the years, many examples of continuous, nowhere differentiable functions have been published see Edgar [11 (pp. 7, 341)]. One of them, the so-called Weierstrass-Mandelbrot function, assumes a particular significance in environmental science because it constitutes the theoretical basis of the first article that used fractal geometry in connection with soils data. Burrough [19] used the Weierstrass-Mandelbrot function to describe the often erratic-looking spatial variation of soil properties along transects. 

Multivariate Weierstrass-Mandelbrot functions have, however, been defined in the last decade [e.g. 21],... 

Another dimension, occasionally used in environmental science, is referred to as the variogram dimension. In Section 2.2.4, a relationship was mentioned between the variance of increments V (t) and the parameter D appearing in the formulation of the Weierstrass-Mandelbrot function wft). In the case of spatial functions, it is more common to compute the semi-variance (half the variance of increments), also termed the variogram. By analogy with the behavior of V (t) at the origin, for the variogram, one may postulate a power-law dependency on the spatial increment h (or lag ) as 0 and define a variogram dimension [e.g. 19, 39]. 

Ausloos, M. and Berman, D.H. (1985). A multivariate Weierstrass-Mandelbrot function. 

Berry, M.V. and Lewis, Z.V. (1980) On the Weierstrass-Mandelbrot fractal function. Proceedings of Royal Society, London, A 370 459-84. 

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