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. In this paper we focus on range anisotropy, that is, stochastic processes with variograms that have direction dependent correlation lengths and direction 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 can be separated into a product of one-dimensional functions oriented along the principal axes. We propose a new Class (A) model with multiscale properties that has applications in subsurface hydrology. We also present a family of Class (B) models based on non-Euclidean distance metrics that are generated by superellipsoidal functions. Next, we propose a new method for determining the orientation of the principal axes and the degree of anisotropy, i.e., the ratio(s) 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 covariance tensor. The procedure is conceptually simple and numerically efficient. It is more accurate for regular (on-grid) data distributions, but it can also be used for sparse (off-grid) spatial distributions. In the case of non-differentiable random fields the method can be extended using generalized derivatives. We illustrate its implementation with numerical simulations.