On generating conductivity fields with known fractal dimension and nonstationary increments

Abstract

Fractional Brownian motion (fBm) is a stochastic process that has stationary increments with long‐range correlations and known fractal dimension. We study a multiple‐dimensional extension of fBm with nonstationary increments that allows for trends in the statistical structure while maintaining the Gaussian nature and fractal dimension of fBm. Two methods for simulating this extension are employed and described in detail. One approach combines Cholesky decomposition with a generalization of random midpoint displacement. The other makes repeated use of the Cholesky decomposition. The resulting fields can be employed in various geophysical settings, e.g., as log conductivity fields in hydrology and topographic elevation in geomorphology.