On generating conductivity fields with known fractal dimension and nonstationary increments

Authors


Abstract

[1] 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.

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, inline image, is a Gaussian process with zero mean and covariance given by

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The subscript inline image 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 inline image is given by inline image [Voss, 1988]. The value inline image corresponds to Brownian motion. When inline image, the increments of inline image are positively correlated. In this case, inline image is called persistent because values greater or less than the mean are likely to persist. When inline image, they are negatively correlated, resulting in a more jagged surface. In this case, inline image 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 inline image 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.

Figure 1.

Four fBm-nls fields with inline image generated via the hybrid method. (b) inline image and inline image. The axis numbering denotes pixel numbers while inline image and inline image vary between 0 and 31.

Figure 2.

Four fBm-nls fields with inline image generated with the iterated Cholesky method. (b) inline image and inline image. The axis numbering denotes pixel numbers while inline image and inline image vary between 1 and 16.

2. Fractional Brownian Motion With Nonlinear Spatial Transformations

[5] FBm can be extended so that the increments are nonstationary by transforming the coordinates. We can represent this mathematically by letting inline image. We then say that inline image is a fBm with nonlinear space (fBm-nls), because the most interesting cases are when inline image is not a linear transformation. From equation (1) it is easy to see that inline image is a Gaussian process with mean zero and covariance given by

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Note that when inline image for some constant inline image, inline image is an fBm with an altered scaling factor, inline image. 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 inline image. For example, if inline image is bi-Lipschitz, i.e., there is a constant inline image such that inline image, then the dimension of the graph of inline image is the same as the dimension of the graph of inline image. Here the graph of inline image is the set inline image. O'Malley and Cushman [2010] give a proof which covers a broader class of functions, inline image, 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.

[6] 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 inline image and the coordinate transformation inline image. In particular, if we wish to generate a sample inline image on a grid given by inline image, we would generate inline image on the nonregular grid given by inline image. This would provide all of the values needed for inline image on the desired grid. A second approach is to use the covariance properties for inline image 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

[7] Brouste et al. [2007] 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.

[8] 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 inline image. As long as the number of points on the coarse grid is inline image, the entire process remains inline image. 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

[9] 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

[10] To compute the fBm-nls on the coarse grid whose points are given by inline image, inline image, we first compute the covariance matrix whose components are given by

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Next, we compute inline image, a square root of inline image, so that inline image and a white noise vector inline image, where the components of inline image are independent, normally distributed with zero mean and unit variance. The value of inline image is then given by the i-th component of inline image. The exactness of this method is seen by observing that

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where inline image denotes the i-th row of inline image. Note that the properties of the white noise vector inline image are used in the first equality.

[11] 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.

[12] The refinement is done in three steps. The first step is to determine the values of inline image 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 inline image 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 inline image and two values determined in the first step of the refinement. Finally, we determine the values of inline image on the boundary using two coarse grid values of inline image 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.

[13] 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

[14] 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 inline image of the neighboring fine-scale fields, if desired).

[15] 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 inline image 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 inline image of the neighboring meso-scale and microscale fields, if desired). When the number of points in each of the grids at each scale is inline image, the running time is inline image, where inline image is the number of scales used. Therefore, the efficiency of the algorithm improves as the number of scales increases and converges to inline image. For a small number of scales the hybrid method is significantly more efficient.

[16] 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.

3.5. Conditioning

[17] Suppose that we have an fBm-nls, inline image, with Hurst exponent inline image and coordinate transformation inline image so that inline image. The covariance of inline image is given by

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If the values of inline image are known for inline image, then inline image for inline image has a multivariate normal distribution with mean vector inline image and covariance matrix inline image, where

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and

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[18] Once the mean vector and covariance matrix are determined, either of the methods described above can be employed to generate realizations. These approaches use the Cholesky decomposition method to generate a coarse field based on the covariance matrix (the Cholesky decomposition takes a square root of the covariance matrix). The only modifications required to the procedures described above are to use the conditioned covariance matrix and to shift each point by the mean, inline image, when generating the coarse field.

4. Applications

4.1. Hydraulic Conductivity Fields

[19] Lu et al. [2003] 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 [1998] have observed that the Hurst exponents for log hydraulic conductivity fields fall in the antipersistent range, inline image. Veneziano and Essiam [2003] have studied flow in multifractal log conductivity fields where they make the common assumption of normality of the log conductivity.

[20] The procedures described here are capable of generating nonstationary log conductivity fields that are normally distributed and have a Hurst exponent in the range inline image. 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.

4.2. Landscape Generation

[21] When generating landscapes, we make the choice of inline image. This is motivated by the earlier work of Mandelbrot [1983], where he employed fBm to generate landscapes that matched the fractal dimension of natural landscapes on some scales of observation. The spatial transformation that we have employed does not change the fractal dimension of the surface. Therefore, the landscape in Figure 3 also matches the fractal dimension of the same scales.

Figure 3.

A conditioned fBm-nls field with a small mountainous area is shown (a) overhead and (b) using the Terragen rendering software.

[22] Depicted in Figure 3a is a conditioned realization of an fBm-nls using inline image and inline image using the hybrid method. Conditioning is used to create two peaks and to fix the upper and lower boundaries. A logistic function was used to transform the inline image coordinate so that there would be high volatility when the inline image coordinate is near the two peaks. Any function with a large value inline image near the peaks would also create this high volatility. In order to create low volatility away from these peaks a small value of inline image is required. The logistic function satisfies both of these requirements and results in a landscape that models a mountain range near the peaks and a relatively flat landscape elsewhere. The data in Figure 3a is being viewed from directly above with elevation displayed in gray scale ranging from highest (white) to lowest (black).

[23] Figure 3b depicts the same realization as Figure 3a, but the Terragen Classic program developed by Planetside Software has been employed to better bring out the nature of the landscape. Water was inserted if the elevation was below a certain threshold and clouds have been inserted to give the figure realism.

5. Discussion

[24] 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.

Acknowledgment

[25] J.H.C. acknowledges the NSF for support under contracts CMG-0934806 and EAR-0838224.

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