Towards Hyper-Dimensional Variography Using the Product-Sum Covariance Model

Modeling hyper-dimensional spatial variability is a complex task from both practical and theoretical standpoints. In this paper we develop a method for modeling hyper-dimensional covariance (variogram) structures using the product-sum covariance model initially developed to model spatio-temporal variability. We show that the product-sum model can be used recursively up to an arbitrarily large number of dimensions while preserving relative modeling simplicity and yielding valid covariance models. The method can be used to model variability in anisotropic conditions with multiple axes of anisotropy or when temporal evolution is involved, and thus is applicable to “full anisotropic 3D+time” situations often encountered in environmental sciences. It requires fewer assumptions than the traditional product-sum modeling approach. The new method also presents an alternative to classical approaches to modeling zonal anisotropy and requires fewer parameters to be estimated from data. We present an example by applying the method in conjunction with ordinary kriging to map photosynthetically-active radiation (PAR) for 2006, in Oklahoma, CA, USA and to explore effects of spatio-temporal variability in PAR on gross primary productivity (GPP).