## 1. Introduction

[2] Fractional Brownian motion (fBm) is a nonstationary process with stationary increments that depend on the entire history of the motion. The process was popularized in one-dimension by Mandelbrot [*Mandelbrot and Van Ness*, 1968], but fBm in higher dimensions is now commonplace. An fBm, , is a Gaussian process with zero mean and covariance given by

The subscript denotes the Hurst exponent and is a measure of the roughness of the surface. This is illustrated by the fact that the fractal dimension for an fBm whose domain has the Euclidian dimension is given by [*Voss*, 1988]. The value corresponds to Brownian motion. When , the increments of are positively correlated. In this case, is called persistent because values greater or less than the mean are likely to persist. When , they are negatively correlated, resulting in a more jagged surface. In this case, is called antipersistent because values greater or less than the mean are likely to be followed by values that are less or greater than the mean, respectively. The coefficient is a scaling factor. FBm can be used in a geophysical setting to generate, for example, terrain or coastlines [*Mandelbrot*, 1983] and conductivity fields [*Cushman*, 1997, pp. 417–434].

[3] Many methods for computing fBm fields in multiple dimensions sacrifice accurate statistics for execution speed. Such methods include fast Fourier transform (FFT) methods, random midpoint displacements (RMD), and successive random additions (SRA). *Voss* [1988] presents an introduction to several common approaches for the simulation of fBm. A method with exact statistics based on the Cholesky decomposition (an algorithm for computing a square root of a matrix) used by *Hoefer et al.* [1992] has a high computational cost.

[4] In the work of *O'Malley and Cushman* [2010], the authors developed a nonlinear extension of classical fBm, which is fBm run with a nonlinear clock. Here we extend that idea to nonlinear spatial (fBm-nls) dimensions and apply the resultant to generate fractal fields for geophysical application. A key point is that like fBm, with fBm-nls the Hurst exponent determines the fractal dimension, but unlike in fBm, the fractal dimension plays a far more limited role in determining the geometric structure of the process which is largely controlled by the nonlinear spatial transformation used with fBm-nls. The implication of this point is that a modeler has more control over the shape of fBm-nls fields than is possible with fBm. For instance, it is possible to introduce layering circular patterns into fBm-nls fields by choosing an appropriate spatial transformation (see Figures 1 and 2). These effects do not arise naturally in fBm.