Non-parametric machine learning methods for interpolation of spatially varying non-stationary and non-Gaussian geotechnical properties

Spatial interpolation has been frequently encountered in earth sciences and engineering. A reasonable appraisal of subsurface heterogeneity plays a significant role in planning, risk assessment and decision making for geotechnical practice. Geostatistics is commonly used to interpolate spatially varying properties at un-sampled locations from scatter measurements. However, successful application of classic geostatistical models requires prior characterization of spatial auto-correlation structures, which poses a great challenge for unexperienced engineers, particularly when only limited measurements are available. Data-driven machine learning methods, such as radial basis function network (RBFN), require minimal human intervention and provide effective alternatives for spatial interpolation of non-stationary and non-Gaussian data, particularly when measurements are sparse. Conventional RBFN, however, is direction independent (i.e. isotropic) and cannot quantify prediction uncertainty in spatial interpolation. In this study, an ensemble RBFN method is proposed that not only allows geotechnical anisotropy to be properly incorporated, but also quantifies uncertainty in spatial interpolation. The proposed method is illustrated using numerical examples of cone penetration test (CPT) data, which involve interpolation of a 2D CPT cross-section from limited continuous 1D CPT soundings in the vertical direction. In addition, a comparative study is performed to benchmark the proposed ensemble RBFN with two other non-parametric data-driven approaches, namely, Multiple Point Statistics (MPS) and Bayesian Compressive Sensing (BCS). The results reveal that the proposed ensemble RBFN provides a better estimation of spatial patterns and associated prediction uncertainty at un-sampled locations when a reasonable amount of data is available as input. Moreover, the prediction accuracy of all the three methods improves as the number of measurements increases, and vice versa. It is also found that BCS prediction is less sensitive to the number of measurement data and outperforms RBFN and MPS when only limited point observations are available.