Statistical interpretation of spatially varying 2D geo-data from sparse measurements using Bayesian compressive sampling

Spatially varying two dimensional (2D) geo-data are commonly encountered in earth sciences and engineering, such as soil or rock properties. Only sparse measurements are usually taken on the spatially varying quantity of interest due to resource or time constraints, although high resolution data are generally preferred in an analysis. Interpretation of spatially varying geo-data from sparse measurements is frequently carried out using interpolation techniques, such as kriging in geostatistics. However, when the number of measurement data is too small, the geostatistical methods are not applicable. This hence poses a great challenge for interpretation of the spatially varying geo-data when only sparse measurements are available. In addition, there exists an important question of how to quantify properly the statistical uncertainty in the interpreted results from the sparse measurements. The statistical uncertainty may greatly influence subsequent engineering design or analysis. To address the two challenges mentioned above, a 2D Bayesian compressive sampling (2D BCS) approach is developed in this study. The 2D BCS approach not only provides a best estimate of the spatially varying 2D surface but also explicitly quantifies the statistical uncertainty associated with the interpretation. The quantified statistical uncertainty may be used as an index to evaluate the reliability and quality of the 2D BCS interpretation results. Both simulated and real-life data are used to illustrate and validate the proposed approach. The proposed 2D BCS approach is shown to properly interpret spatially varying 2D data from sparsely measured data and quantify explicitly the statistical uncertainty in the interpretation results.