On generating conductivity fields with known fractal dimension and nonstationary increments
 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.
 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].
 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  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.  has a high computational cost.
 In the work of O'Malley and Cushman , 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.
2. Fractional Brownian Motion With Nonlinear Spatial Transformations
 FBm can be extended so that the increments are nonstationary by transforming the coordinates. We can represent this mathematically by letting . We then say that is a fBm with nonlinear space (fBm-nls), because the most interesting cases are when is not a linear transformation. From equation (1) it is easy to see that is a Gaussian process with mean zero and covariance given by
Note that when for some constant , is an fBm with an altered scaling factor, . Although the covariance structure and the distribution of increments of fBm-nls differs significantly from ordinary fBm, the fractal dimension is generally unaffected by the transformation . For example, if is bi-Lipschitz, i.e., there is a constant such that , then the dimension of the graph of is the same as the dimension of the graph of . Here the graph of is the set . O'Malley and Cushman  give a proof which covers a broader class of functions, , in the one-dimensional case. This proof can be generalized to higher dimensions and even broader classes of coordinate transformations, but the technicalities render these proofs unsuitable for presentation here.
 Simulating fBm-nls can be approached in two ways. One approach is to attempt to compute an ordinary fBm on a nonregular grid determined by the desired grid for and the coordinate transformation . In particular, if we wish to generate a sample on a grid given by , we would generate on the nonregular grid given by . This would provide all of the values needed for on the desired grid. A second approach is to use the covariance properties for directly. When examining methods for simulating ordinary fBm it is helpful to keep these two approaches in mind.
3. Computational Approaches
3.1. Hybrid Method for Classical fBm
 Brouste et al.  describe a method for generating fractional Gaussian random fields with an arbitrary nonnegative definite covariance function. Their approach combines the Cholesky decomposition method with a generalization of the random midpoint displacement method (RMD). Initially, a coarse field is generated with exact statistics using the Cholesky decomposition method. Once this is obtained, the values at intermediate points are determined by adding a best linear unbiased estimate (BLUE) to a normal random variable with zero mean and appropriate variance. The BLUE may be generated from an arbitrary number of neighboring points. In the case where the covariance function is that of fBm and the BLUE is calculated from the four nearest neighbors, the method reduces to RMD.
 This hybrid method has several advantages for simulating fBm-nls. First, it is able to simulate fields that have a covariance function approximately equal to that of fBm-nls. Second, it is fast, as the computational complexity of the refinement process is . As long as the number of points on the coarse grid is , the entire process remains . Third, it can be easily extended to three or more dimensions. Although the generalized RMD method used for refinement is inexact, the exactness of the coarse field provides more accurate statistics than a standalone RMD method.
3.2. Simulating fBm-nls
 We employ two approaches: The first is based on the hybrid method described previously, and the other is based on the iterated Cholesky decomposition. The former approach allows for the values at intermediate points to be interpolated from an arbitrary number of neighboring points, but we always use the four nearest neighbors for an intermediate point on the interior and the three nearest neighbors for an intermediate point on the boundary. This simplifies the implementation and improves the execution speed by requiring that only a very small linear system be solved for each interpolation. The latter approach to generating approximate Gaussian fields involves repeated use of the Cholesky decomposition. To our knowledge, this approach has not been presented in the literature. It is very flexible, and is capable of generating approximate fBm-nls fields.
3.3. Hybrid Method for fBm-nls
 To compute the fBm-nls on the coarse grid whose points are given by , , we first compute the covariance matrix whose components are given by
Next, we compute , a square root of , so that and a white noise vector , where the components of are independent, normally distributed with zero mean and unit variance. The value of is then given by the i-th component of . The exactness of this method is seen by observing that
where denotes the i-th row of . Note that the properties of the white noise vector are used in the first equality.
 The next step is to repeatedly refine to a finer grid. The value at each new point on the refined grid is the sum of an interpolated value and a normal random variable with appropriate variance. The interpolation is done via ordinary kriging and the appropriate variance is given by the kriging error.
 The refinement is done in three steps. The first step is to determine the values of at the points on the center of each square in the coarse grid using the four corners of the square to interpolate via a BLUE. Next, we determine the values of at the midpoints of the horizontal and vertical line segments connecting neighboring coarse grid points. This interpolation is done using two coarse grid values of and two values determined in the first step of the refinement. Finally, we determine the values of on the boundary using two coarse grid values of and one value from the first step of the refinement process. Note that the boundary points use only three neighbors while the interior points use four neighbors for the interpolation.
 Figure 1a displays an image of standard fBm with H = 0.3. The reader should note that all panels in Figure 1 have the same Hurst exponent, and hence the same fractal dimension. The nonlinear transform of the spatial coordinates provides the distinct character of each subsequent panel within the figure. Figure 1b displays an image of fBm-nls with H = 0.3 where the coordinates have undergone a transformation to polar coordinates. Figures 1c and 1d again display images of fBm-nls with H = 0.3, but now the coordinates have been transformed to produce a layering effect. The images are drawn in the customary fashion for conductivity fields, but could easily be drawn, for example, to represent surface terrain. Of course, in generating surface terrain, a different set of coordinate transformations would be more relevant.
3.4. Iterated Cholesky Decomposition for fBm-nls
 We begin by describing a two-scale version of an iterated Cholesky method. As in the hybrid method, the first step is to generate a field on a coarse grid using Cholesky decomposition. Between each set of four neighboring points (which form a square) on the coarse grid, a fine field is generated using again Cholesky decomposition. This time, however, the covariance matrix is conditioned on the coarse scale field (and also of the neighboring fine-scale fields, if desired).
 The two-scale description given above can be extended to an arbitrary number of scales. For example, in a three-scale scheme, each of the meso-scale fields would be conditioned on the coarse-scale field and (and also of the neighboring meso-scale fields, if desired). Each of the fine-scale fields would be conditioned on the coarse-scale field, the meso-scale field in which the fine-scale field is contained (and also of the neighboring meso-scale and microscale fields, if desired). When the number of points in each of the grids at each scale is , the running time is , where is the number of scales used. Therefore, the efficiency of the algorithm improves as the number of scales increases and converges to . For a small number of scales the hybrid method is significantly more efficient.
 We have implemented a two-scale version of this algorithm where the fine-scale fields are conditioned on the coarse-scale field and the neighboring fine-scale data. Figure 2 is similar to Figure 1 with the main differences being that the Hurst exponent is now 0.7 and the two-scale iterated Cholesky method has been employed instead of the hybrid method.
4.1. Hydraulic Conductivity Fields
 Lu et al.  have used nonstationary stochastic fields such as fBm and truncated Levy motions to model conductivity and log conductivity fields. Their use of these nonstationary fields, in contrast to stationary fields like fractional Gaussian noise, is corroborated by extensive data collection and analysis [Molz and Boman, 1993; Painter and Paterson, 1994; Painter, 1995, 1996a, 1996b, 2001; Liu and Molz, 1997; Lu and Molz 2001; Lu et al., 2002]. Di Federico and Neuman  have observed that the Hurst exponents for log hydraulic conductivity fields fall in the antipersistent range, . Veneziano and Essiam  have studied flow in multifractal log conductivity fields where they make the common assumption of normality of the log conductivity.
 The procedures described here are capable of generating nonstationary log conductivity fields that are normally distributed and have a Hurst exponent in the range . It is desirable to be able to incorporate known information about the conductivity fields, and this is possible with fBm-nls using the conditioning method described in section 3.5. In addition to these properties, it is possible to control the geometry to create, for example, layering or circular patterns. The spatial transformations used in Figures 1c, 1d and 2c, 2d introduce a weak layering effect. By using step (or step-like) functions to transform the coordinates, it is possible to create a stronger layering effect.
 We have presented an extension of fBm with spatial variables nonlinearly transformed, which we label fBm-nls. FBm-nls is a Gaussian process with a flexible covariance structure that can be used to model many geophysical processes. Two methods were employed to generate the fields. A key point for fBm-nls is that the fractal dimension plays only a minor role in determining the underlying geometric structure of the process, relative to the nonlinear transformation of the spatial coordinates. This makes the process radically different from classical fBm and opens up entirely new areas for random field generation.
 J.H.C. acknowledges the NSF for support under contracts CMG-0934806 and EAR-0838224.