New anisotropic covariance models and estimation of anisotropic parameters based on the covariance tensor identity

Abstract

Many heterogeneous media and environmental processes are statistically anisotropic, that is, their moments have directional dependence. The term range anisotropy denotes processes that have variograms characterized by direction-dependent correlation lengths and directionally independent sill. We distinguish between two classes of anisotropic covariance models: Class (A) models are reducible to isotropic after rotation and rescaling operations. Class (B) models are separable and reduce to a product of one- dimensional functions along the principal axes. We present a Class (A) model for multiscale processes and suggest applications in subsurface hydrology. This model is based on a truncated power law with short and long-range cutoffs. We also present a family of Class (B) models generated by superellipsoidal functions that are based on non- Euclidean distance metrics. We propose a new method for determining the orientation of the principal axes and the degree of anisotropy (i.e., the ratios of the correlation lengths). This information reduces the degrees of freedom of anisotropic variograms and thus simplifies the estimation procedure. In particular, Class (A) models are reduced to isotropic, and Class (B) models to one-dimensional functions. Our method is based on an explicit relation between the second-rank slope tensor (SRST), which can be estimated from the data, and the second-rank covariance tensor. The method is conceptually simple and numerically efficient. It is more accurate for regular (on-grid) data distributions, but it can also be used for irregular (off-grid) spatial distributions. We illustrate its implementation with numerical simulations.