Combination of Machine Learning and Kriging for Spatial Estimation of Geological Attributes

A growing number of studies in the spatial estimation of geological features use machine learning (ML) models, as these models promise to provide efficient solutions for estimation especially in non-Gaussian, non-stationary and complex cases. However, these models have two major limitations: (1) the data are considered to be independent and identically distributed (or spatially uncorrelated), and (2) the data are not reproduced at their locations. Kriging, on the other hand, has a long history of generating unbiased estimates with minimum error variance at unsampled locations. Kriging assumes stationarity and linearity. This study proposes a methodology that combines kriging and ML models to mitigate the disadvantages of each method and obtain more accurate estimates. In the proposed methodology, a stacked ensemble model, which is also referred to as the super learner (SL) model, is applied for ML modeling. We have shown how the estimates generated by the SL model and estimates obtained from kriging can be combined through a weighting function based on a kriging variance. The weights are optimized using the sequential quadratic programming. The proposed methodology is demonstrated in two synthetic case studies containing data with non-stationarity and non-Gaussian features; a real case study using a dataset from an oilsands deposit is also presented. The performance of the combined model is compared with the SL model and kriging using the coefficient of determination (R-squared), root mean squared error, and mean absolute error criteria. The combined model appears to yield more accurate estimates than the ones generated by SL model and kriging in all cases.