Interpreting Deepkriging for spatial interpolation in geostatistics

Abstract

In the current era marked by an unprecedented abundance of data, the usage of conventional methods such as kriging persists in some applications of geostatistics, despite their limitations in adequately capturing the intricate relationships found in contemporary, multivariate datasets. Although deep neural networks (DNNs) have demonstrated remarkable efficacy in capturing complex nonlinear feature relationships across various domains, their success in geostatistical applications has been limited. This can be partly attributed to two significant challenges. Firstly, the opaque nature of these black box models raises concerns about the dependability of their outputs for critical decision-making, as the inner workings of the model remain less interpretable. Secondly, DNNs do not explicitly capture spatial dependencies within data. To address these shortcomings, we employ a methodology to interpret the recently proposed spatial DNNs known as Deepkriging, and we apply it to dry bulk rock density estimation, an often-overlooked aspect in mineral resource estimation. Through our adaptation of Shapley values—Batched Shapley—we overcome significant computational challenges to quantify feature importance for Deepkriging. This approach takes into account feature interactions, which is crucial for DNNs, as they rely on high-order interactions, especially in a complex application like mineral resource estimation. Additionally, we demonstrate in the 3D case that Deepkriging outperforms ordinary kriging and regression kriging in terms of mean squared errors, in both the purely spatial case and in the presence of auxiliary variables. Our study produces the first methodology to interpret Deepkriging, which is suitable for any model with a large number of features; it reaffirms the efficacy of Deepkriging through several comparisons in a 3D application, and most importantly; it underscores the adaptability and broader potential of DNNs to cater to various challenges in geostatistics.