Dual Random Fields and their Application to Mineral Potential Mapping

TL;DR

The paper tackles spatial uncertainty in mineral potential mapping by treating the response model itself as a regionalized process through the dual random field (dRF) framework. Local SVM-based response boundaries are constructed and geometrically pooled on the oriented sphere via Grassmannian concepts, enabling spatial inference of both model parameters and predictions. It develops variography, (non-)conditional sphere simulations, and distribution-mapping techniques to handle orientation data, ensuring uncertainty is propagated in a geostatistical fashion. A real case study in the Yilgarn Craton demonstrates the approach by producing mineral-potential maps with quantified uncertainty and by revealing location-dependent input-feature significance to guide exploration.

Abstract

In various geosciences branches, including mineral exploration, geometallurgical characterization on established mining operations, and remote sensing, the regionalized input variables are spatially well-sampled across the domain of interest, limiting the scope of spatial uncertainty quantification procedures. In turn, response outcomes such as the mineral potential in a given region, mining throughput, metallurgical recovery, or in-situ estimations from remote satellite imagery, are usually modeled from a much-restricted subset of testing samples, collected at certain locations due to accessibility restrictions and the high acquisition costs. Our limited understanding of these functions, in terms of the multi-dimensional complexity of causalities and unnoticed dependencies on inaccessible inputs, may lead to observing changes in such functions based on their geographical location. Pooling together different response functions across the domain is critical to correctly predict outcome responses, the uncertainty associated with these inferred values, and the significance of inputs in such predictions at unexplored areas. This paper introduces the notion of a dual random field (dRF), where the response function itself is considered a regionalized variable. In this way, different established response models across the geographic domain can be considered as observations of a dRF realization, enabling the spatial inference and uncertainty assessment of both response models and their predictions. We explain how dRFs inherit all the properties from classical random fields, allowing the use of standard Gaussian simulation procedures to simulate them. These models are combined to obtain a mineral potential response, providing an example of how to rigorously integrate machine learning approaches with geostatistics.

Figures (14)

• Figure 1: General setting in mineral prospectivity mapping illustration. Relations among input features (gravity and magnetic intensity) and mineralized response (colored bullet in red and blue) may change at different locations. Simple local relations between inputs and the response will be overlooked due to the overlapping of datasets if a global model is considered in the domain. How do we extract the information from calibrated models at A, B, and C in a coherent fashion in order to make inferences at the unexplored location?
• Figure 2: Decision tree modeling of mineralization at A, B, and C. Although each model is highly interpretative by itself, the graph-structure representation of the response model conditions the spatial problem in the sense of allowing the inference of a best-estimated model at unexplored locations.
• Figure 3: At the left, a comparison of classifiers: input data, nearest neighbors, support vector machine (using a radial basis function), decision tree, and neural net. This example is meant to demonstrate how different classifiers establish their decision boundaries. The diagrams display the training data in solid hues and the testing data in a translucent form. Classification accuracy on the test set is displayed in the lower right corner scikit-learn. Right: illustrative summary of the boundary situation when comparing these classifiers (after domingos2012fewdomingos2012few).
• Figure 4: The hyper-sphere $\mathbb S^{p-1}$ and a scheme of the most relevant operations on it.
• Figure 5: Realizations of a three-dimensional orientation process (or $\mathbb S^{2}$-RF) drawn over a two-dimensional grid with size 100 $\times$ 100. (a) A non-conditional simulation; (b) the conditioning data points, and two conditional simulations (c, d). Bottom, sample covariance for a set of 100 non-conditional realizations; their average and the theoretical model.