Interpolation of non-stationary geo-data using Kriging with sparse representation of covariance function

Abstract

Interpolation of spatially varying measurements, e.g., spatial variation of a geotechnical property along depths, is often needed in many disciplines, including geotechnical engineering. Among various interpolation methods, geostatistical methods, such as Kriging, are popular since they are able to not only provide the best estimates but also quantify the interpolated uncertainty. However, Kriging relies on an assumption of data stationarity, and a transformation from non-stationary data to stationary data by, e.g., detrending, is required when using Kriging to interpolate non-stationary data. Detrending, however, is a tricky and long-lasting challenge in geo-data analyses. To tackle this challenge, this study develops a novel Kriging method for direct interpolation of spatially non-stationary data without detrending. A sparse representation of covariance function is proposed for Kriging interpolation of spatially non-stationary data, bypassing the tricky detrending process. The proposed Kriging method is particularly appealing for non-stationary geotechnical site investigation data often encountered in geotechnical practice (e.g., spatial variation of soil properties over different soil layers along depth). Equations are derived for the proposed Kriging method, and it is illustrated and validated using numerical examples. Results show that the proposed Kriging method with sparse representation of covariance function is directly applicable to spatially non-stationary data and has the advantages of improved accuracy and reduced interpolation uncertainty when interpolating spatially varying, non-stationary data.