Benchmarking of Gaussian Process Regression with Multiple Random Fields for Spatial Variability Estimation

Benchmarking is very valuable for evaluating and comparing methodologies. Here, Gaussian process regression using multiple Gaussian random fields (GPR-MR) is applied to benchmarking data for spatial variability problems. The benchmarking data used were from the literature and included four types of virtual ground models (VG1 to VG4) and one real ground measurement data set. The spatial variability of geological properties is often divided into a trend component and a random component. In GPR-MR, the trend component is expressed by a random field with a large scale of fluctuation (SOF), leading to a smooth (slow) variability, whereas the random component is expressed by one with a small SOF, leading to a rapidly changing variability. The SOF and the standard deviation of random fields were estimated using the maximum likelihood method based on the measured data provided in the benchmarking data. GPR-MR was used to estimate the spatial variabilities of all cases, and its performance was evaluated. For the real ground measured data, model selection was also performed with respect to the autocorrelation function of the random component in terms of information criteria, whereas the Markovian autocorrelation function was used for the virtual ground data without the model selection. Based on the results, the Whittle-Matérn (WM) model was selected for the random component. GPR-MR was used to estimate the spatial variability, and its performance with the WM model was evaluated.