Spartan random fields: Smoothness properties of gaussian densities and definition of certain non-gaussian models

Abstract

Spartan spatial random fields (SSRFs) were introduced in [10]. Certain mathematical properties of SSRFs were presented, inference of the model parameters from synthetic samples was investigated [10], and methods for the unconditional simulation of SSRFs were developed [11]. This research has focused on the fluctuation component of the spatial variability, which is assumed to be statistically homogeneous (stationary) and normally distributed. The probability density function (pdf) of Spartan fields is determined from an energy functional H[X(s)], according to the familiar in statistical physics expression for the Gibbs distribution fx[X(s)] = Z exp {H[X(s)]}. (1) The constant Z (called partition function) is the pdf normalization factor obtained by integrating exp (H) over all degrees of freedom (i.e. states of the SSRF). The subscript I denotes the fluctuation resolution scale. The energy functional determines the spatial variability by means of interactions between neighboring locations. One can express the multivariate Gaussian pdf, typically used in classical geostatistics, in terms of the following energy functional H[X(s)]= 1 2 ds ds X(s)c 1 X (s, s)X(s), (2) where cX(s, s) is the centered covariance function; the latter needs to be determined from the data for all pairs of points s and s, or (assuming statistical homogeneity) for all distance vectors s s. In contrast, the energy functional in Spartan models is determined from physically motivated interactions between neighbors. The name Spartan emphasizes that the number Np of model parameters to be determined from the data is small. For example, in the fluctuation gradient curvature (FGC) model, the pdf involves three main parameters: the scale factor 0, the covariance shape parameter 1, and the correlation length. Another factor that adds flexibility to the model is the coarse-graining kernel that determines the fluctuation resolution [10]. As we show below, the resolution is directly related to smoothness properties of the SSRF. In previous work [10, 11], we have used a kernel with a boxcar spectral density that imposes a sharp cutoff in frequency (wavevector) space at kc 1 . We have treated the cutoff frequency as a constant, but it is also possible to consider it as an additional model parameter, in which case Np = 4. A practical implication of an interaction-based energy functional is that the parameters of the model follow from simple sample constraints that do not require the full calculation of two-point functions (e.g., correlation function, variogram). This feature permits fast computation of the model parameters. In addition, for general spatial distributions (e.g., irregular distribution of sampling points, anisotropic spatial dependence with unknown a priori principal directions), the parameter inference does not require various empirical assumptions such as choice of lag classes, number of pairs per class, lag and angle tolerance, etc. [7] used in the calculation of two-point functions. In the case of SSRFs that model data distributed on irregular supports, the definition of the interaction between near neighbors is not uniquely defined. Determining the neighbor structure for irregular supports increases the computational effort [10], but the model inference process is still quite fast. Methods for the non-constrained simulation of SSRFs with Gaussian probability densities on the square lattice (by filtering Gaussian random variables in Fourier space and reconstructing the state in real space with the inverse FFT) and for irregular supports (based on a random phase superposition of cosine modes with frequency distribution modeled on the covariance spectral density), have been presented in [11]. © 2009 Springer Berlin Heidelberg.