Dual Random Fields and their Application to Mineral Potential Mapping

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.