Probabilistic estimation of cross-variogram based on Bayesian inference

Site characterization based on measurements is essential for geological and geotechnical engineering. However, measurements are usually limited and sparse because of many limitations, which can hardly be utilized to perform a well site characterization. Therefore, some data fusion methods are commonly utilized to integrate correlated data to improve the performance of site characterization. Among data fusion methods, cokriging is widely utilized to improve the performance of site characterization by integrating measurements of correlated variables. The correlation between correlated variables is expressed by a cross-variogram, which can only be calculated using co-located measurements between correlated variables. However, the measurements in geological and geotechnical engineering are commonly obtained by destructive sampling, which are usually not co-located and cannot be utilized to calculate the cross-variogram. In this study, a Bayesian inference method is developed to tackle this difficulty. The proposed method is illustrated and validated by two real datasets. The results show that the proposed method can estimate a well cross-variogram model, no matter whether the measurements of correlated variables are co-located or not. Moreover, the uncertainty of variogram models and cokriging estimation can be quantified by the proposed method. The proposed method can improve the wide utilization of the cokriging method, which can help characterize geology conditions of geological and geotechnical engineering. •A Bayesian inference method is developed to estimate the cross-variogram model for non-co-located variables.•The proposed method can ensure a positive semi-definite covariance matrix, which can be used in cokriging.•The uncertainty of variogram models and cokriging predictions can be quantified by the proposed method.•The uncertainty of cross-variogram models and cokriging variance reduces as the sampling density increases.