Karhunen–Loève expansion of Spartan spatial random fields

Abstract

Random fields (RFs) are important tools for modeling space–time processes and data. The Karhunen–Loève (K–L) expansion provides optimal bases which reduce the dimensionality of random field representations. However, explicit expressions for K–L expansions only exist for a few, one-dimensional, two-parameter covariance functions. In this paper we derive the K–L expansion of the so-called Spartan spatial random fields (SSRFs). SSRF covariance functions involve three parameters including a rigidity coefficient η1, a scale coefficient, and a characteristic length. SSRF covariances include both monotonically decaying and damped oscillatory functions; the latter are obtained for negative values of η1. We obtain the eigenvalues and eigenfunctions of the SSRF K–L expansion by solving the associated homogeneous Fredholm equation of the second kind which leads to a fourth order linear ordinary differential equation. We investigate the properties of the solutions, we use the derived K–L base to simulate SSRF realizations, and we calculate approximation errors due to truncation of the K–L series.