Simulation of Random Fields with Trend from Sparse Measurements without Detrending

Abstract

Although spatially varying quantities in real life (e.g., mechanical properties of soils) often contain a linear or nonlinear trend, stationary random fields with zero trend are often used to model these quantities due to mathematical convenience. To model a random field with a linear or nonlinear trend through a stationary random field, removal of the trend, known as detrending, is often performed first on the available measurement data points to separate the random field into a deterministic trend component and a stationary random field, followed by characterization and simulation of the stationary random field. Detrending is a tricky process. Because the form of the trend function (e.g., linear or nonlinear) is often unknown in application, it is difficult to select the most appropriate form of trend function given the available measurements. Using different forms of trend function results not only in different deterministic trends, but also different parameters for the stationary random field (i.e., different random fields). The situation becomes even more challenging when the measurements are sparse and the difficulty in estimating parameters (e.g., correlation length) of the stationary random field from sparse data becomes significant. This paper proposes an innovative method to generate samples of random fields with a linear or nonlinear trend directly from sparse measurements without detrending. The proposed method is based on Bayesian compressive sampling and the Karhunen-Loève expansion. Because no detrending is needed in the proposed method, the difficulties associated with detrending in the simulation of random fields with a linear or nonlinear trend from sparse measurements are bypassed. The proposed method is illustrated and validated using numerical examples in this paper.