Framework for Estimating Discretization Error and Optimal Expansion Terms in Random Field Discretization

The spatial variability of soil can be characterized using random field theory. A widely adopted method for discretizing random fields is the orthogonal series expansion (OSE), which implements the Karhunen–Loève (KL) expansion. Accurately estimating the discretization error and determining the optimal number of expansion terms ( M 0 ) are very important, as M 0 governs both the accuracy and efficiency of random field discretization and the subsequent reliability analysis of geostructures in spatially variable soils. This paper systematically investigates the influencing factors on the discretization error and develops a framework for determining M 0 that accounts for multiple influencing factors. Representative examples of one- and two-dimensional (1D and 2D) random field discretization using the OSE shows that the discretization error exhibits an inverse relationship with the autocorrelation distance (ACD) and a direct relationship with the domain size of the random field. By defining a dimensionless R ratio as the product of ACDs (along each coordinate direction) divided by the domain size, the combined effects of ACD and domain size can be quantified conveniently in a unified manner. For random field simulations where both the single-exponential (SNX) and square-exponential (SQX) autocorrelation functions are applicable, the latter is preferable as it requires lower M 0 than the former.