Assessment of spectral representation and Karhunen–Loève expansion methods for the simulation of Gaussian stochastic fields

Abstract

From the wide variety of methods developed for the simulation of Gaussian stochastic processes and fields, two are most often used in applications: the spectral representation method and the Karhunen–Loève (K–L) expansion. In this paper, an in-depth assessment on the capabilities of the two methods is presented. The spectral representation method expands the stochastic field as a sum of trigonometric functions with random phase angles and/or amplitudes. The version having only random phase angles is used in this work. A wavelet-Galerkin scheme is adopted for the efficient numerical solution of the Fredholm integral equation appearing in the K–L expansion. A one-dimensional homogeneous Gaussian random field with two types of autocovariance function, exponential and square exponential, is used as the benchmark test. The accuracy achieved and the computational effort required by the K–L expansion and the spectral representation for the simulation of the stochastic field are investigated. The accuracy obtained by the two approaches is examined by comparing their ability to produce sample functions that match the target correlation structure and the Gaussian probability distribution or, alternatively, its low order statistical moments (mean, variance and skewness).