Modeling three-dimensional spatial variability of soil property using non-parametric Gaussian process regression with limited standard penetration test data

Standard penetration test (SPT) is a widely used in-situ test in geotechnical site investigation because it is robust and applicable to a broad range of soils, provides soil samples for visual inspection, and benefits from long-established empirical correlations with key geotechnical design parameters. In practice, however, SPTs are usually performed along the depth in a limited number of boreholes, with relatively large horizontal distances between boreholes. It is therefore challenging to characterize three-dimensional (3D), statistically anisotropic, and non-stationary spatial variability of geotechnical properties, or develop a reliable 3D subsurface geotechnical model, from such limited SPT data. This study proposes a novel Gaussian process regression (GPR) to directly construct 3D subsurface geotechnical model with quantified uncertainty using limited SPT measurements. Building on the recently developed 1D sparse spectrum representation of GPR, the proposed method projects the 3D complex spatial variability of geotechnical properties into an orthonormal space of properly selected basis functions and sparsely represents the 3D spatial variation using limited basis functions. The proposed GPR is non-parametric and does not require a separation of the spatial variation of geotechnical properties into a non-stationary trend and stationary residuals, nor estimating the associated parameters and hyperparameters, as required in the traditional semi-parametric GPR. Illustrative examples show that the proposed GPR effectively constructs 3D subsurface geotechnical model from limited SPT measurements while explicitly quantifying associated uncertainty. It outperforms the semi-parametric GPR, i.e., improving prediction accuracy and enhancing robustness, when only limited site-specific data are available.