Interpolation of Sparse Measurements and Quantification of Measurement Error Using Enhanced Kriging with Sparse Representation of Covariance Function

Abstract

Spatially varying properties of geomaterials (e.g., soils and rocks) are often measured sparsely, and the measurements inevitably contain noises or error. Therefore, interpolation of sparse geotechnical measurements with noises and quantification of measurement error are frequently performed using, for example, geostatistical methods such as kriging. Kriging relies on a semivariogram to account for spatial autocorrelation structure of geodata. A parameter called the nugget effect has been incorporated into semivariograms to consider measurement error, but the development of a semivariogram in kriging generally requires extensive measurements. When the number of measurement data points is limited, the resulting semivariogram might become a pure nugget effect model and is unable to correctly capture the spatial autocorrelation structure of geodata, leading to unreliable kriging interpolations. This poses a long-lasting challenge to existing kriging methods for properly interpolating sparse measurements with noises, a situation often encountered in geoengineering practice. To tackle this challenge, a novel kriging method is developed in this study for simultaneously quantifying the measurement error and interpolating the sparse, noisy measurements, with quantified interpolation uncertainty. The proposed kriging method is extended from a recently developed kriging method with sparse representation of covariance function. The proposed method is illustrated and validated using simulated and real data. Results show that the proposed method accurately estimates the measurement error, properly provides the best prediction, and rationally quantifies the associated interpolation uncertainty from sparse measurements with noises.