Bayesian deep learning for spatial interpolation in the presence of auxiliary information

Earth scientists increasingly deal with ‘big data’. For spatial interpolation tasks, variants of kriging have long been regarded as the established geostatistical methods. However, kriging and its variants (such as regression kriging, in which auxiliary variables or derivatives of these are included as covariates) are relatively restrictive models and lack capabilities provided by deep neural networks. Principal among these is feature learning: the ability to learn filters to recognise task-relevant patterns in gridded data such as images. Here, we demonstrate the power of feature learning in a geostatistical context by showing how deep neural networks can automatically learn the complex high-order patterns by which point-sampled target variables relate to gridded auxiliary variables (such as those provided by remote sensing) and in doing so produce detailed maps. In order to cater for the needs of decision makers who require well-calibrated probabilities, we also demonstrate how both aleatoric and epistemic uncertainty can be quantified in our deep learning approach via a Bayesian approximation known as Monte Carlo dropout. In our example, we produce a national-scale probabilistic geochemical map from point-sampled observations with auxiliary data provided by a terrain elevation grid. By combining location information with automatically learned terrain derivatives, our deep learning approach achieves an excellent coefficient of determination (R2=0.74) and near-perfect probabilistic calibration on held-out test data. Our results indicate the suitability of Bayesian deep learning and its feature-learning capabilities for large-scale geostatistical applications where uncertainty matters.