Functional Decomposition Kriging for Embedding Stochastic Anisotropy Simulations

Abstract

Functional analysis of the kriging algorithm is accomplished with consecutive projections of vectors in Hilbert space. The innovation unveils ``functional decomposition kriging{''} (FDK), which can forecast fields on spatially continuous domains without using blocks, cells, or elements. FDK assembles the random field as a summation of field analytic functions, which are sample pivoted and nonstationary. Furthermore, spatially variable uncertain anisotropy is represented as a continuous tensor random field, which is formed from non-orthogonal members. FDK predicts tensor members using physical data collected at sparse sample locations. Particular interest is on structural anisotropy tensor fields representing curvilinear and folded patterns of structural uncertainty. Therefore, spatially variable eigenvector and eigenvalue tensor fields give continuously varying orientation and range of principal stochastic anisotropy of covariances that are used as input to stochastic functionals. FDK enables simulation of anisotropic properties (e.g., permeability, rock stiffness, or structural anisotropy), with stochastic covariance parameter fields. Integration of field analytic functions delivers upscaled multiresolution moments. Since FDK can be stopped, optimized, and updated without repeating computations, it is suitable for inverse, adaptive, and real-time modeling.