Bayesian Deep Learning for Spatial Interpolation in the Presence of Auxiliary Information

Abstract

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 ($$R^{2} = 0.74$$ R 2 = 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. Graphic Abstract

Accuracy Assessment of Kriging, artificial neural network, and a hybrid approach integrating spatial and terrain data in estimating and mapping of soil organic carbon

“…The relationship between soil and environment variables in digital soil mapping can be explained by using linear models with a few standard soil properties and environmental factors. However, the relationship between three or more factors and soil property and environmental variables can only be explained using advanced models such as non-linear machine learning algorithms, ANNs or fuzzy logic due to the non-linear and unstable relationship between soil properties and environmental variables [ 81 – 84 ]. An MLP network was developed to both increase the accuracy of estimation and reveal the relationship between SOC content and environmental factors by using many environmental variables as well as soil properties ( Fig 1 ).…”

Uncertainty Quantification in Geochemical Mapping: A Review and Recommendations

“…Bayesian methods allow for quantifying uncertainty by leveraging both sample data and prior knowledge about model components (Gelman et al., 2014). In the field of geospatial modeling, many Bayesian methods have been developed and applied, including Bayesian hierarchical models, Bayesian‐based spatiotemporal methods, Bayesian mixture model, Bayesian spatial autoregressive models, and Bayesian evidential learning (e.g., Fuentes & Raftery, 2005; Kirkwood et al., 2022; Lawson, 2018; LeSage, 1997; Li & Revesz, 2004; Lindgren & Rue, 2015; Scheidt et al., 2018; Yang et al., 2015; Yin et al., 2020).…”

Landscape domains and information surfaces: Data collection, recording and citation using decimal latitude‐longitude geolocation via the FAIR principles

“…Bayesian and ML techniques are methodologies that rely on digital data (Kirkwood et al, 2022) for mapping in three dimensions. Such ML approaches can also be used to exploit auxiliary variables in geochemical mapping (Kirkwood et al, 2016) and emphasise interpretabil-…”