Simulation of random field samples directly from sparse measurements using Bayesian compressive sampling and Karhunen-Loève expansion

Geotechnical materials (e.g., soils and rocks) are natural materials, and they are affected by many spatially varying factors during the geological process, such as properties of their parent materials, weathering and erosion processes, transportation agents, and sedimentation conditions. Geotechnical data therefore exhibit spatial variability, and to some extent, are unique in every site. In recent years, random field has been increasingly used to model spatial variability of geotechnical data. In conventional frequentist approach, measurement data at a specific site are used to estimate random field parameters, such as mean and standard deviation, as well as parameters (e.g., correlation length) of a pre-determined parametric form of correlation function (e.g., an exponential correlation function). Estimation of these random field parameters, particularly the correlation length, and selection of the suitable parametric form of correlation function generally require extensive measurements from a specific site, which are generally not available in geotechnical engineering practice. This paper presents a random field generator that is able to simulate random field samples directly from sparse measurements, bypassing the difficulty in the estimation of correlation function and its parameters. The proposed generator is based on Bayesian compressive sensing/sampling and Karhunen-Loève expansion. The proposed method is illustrated and validated using simulated geotechnical data. It is also compared with the conventional random field models. The results show that the proposed generator can rationally simulate the geotechnical spatial variability at a specific site from sparse measurements.