Sparse Spectrum Representation of Gaussian Process Regression for Modeling Nonstationary Geo-Data from Limited Measurements

Abstract Although geo-data are spatially variable, they are often measured sparsely at a limited number of locations owing to constraints in resources, time, or technology in various geoscience disciplines. Modeling of spatially varying geo-data from limited measurements is therefore necessary using methods such as Gaussian process regression (GPR). Since geo-data often exhibit nonstationary characteristics, such as complex spatial trends, GPR may require a composite kernel or covariance function that combines a nonstationary dot-product kernel and a stationary kernel to effectively model the complex spatial structure of geo-data. In geoengineering practice, however, the spatial structure of spatially varying geo-data is often unknown, posing a significant challenge in the development of a suitable composite kernel function, especially when measurement data points are limited. Additionally, estimating the hyperparameters of a composite kernel from limited measurements—particularly the length scale of the stationary kernel—is challenging. To address these challenges, this study develops a sparse spectrum representation of GPR that requires neither the construction of a composite kernel nor estimation of associated hyperparameters, enabling direct modeling of spatially varying geo-data from limited measurements. The proposed method is illustrated and validated using both simulated and real-world data. The results show that the proposed GPR improves prediction accuracy and reduces prediction uncertainty when predicting nonstationary geo-data from limited measurements.