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.

翻译：在包括矿产勘探、已建矿山的地质冶金特性表征以及遥感在内的多个地球科学分支中，区域化输入变量在整个感兴趣域内空间采样充分，这限制了空间不确定性量化程序的应用范围。相应地，诸如特定区域的矿产潜力、采矿处理量、冶金回收率或遥感卫星影像原位估算等响应结果，通常仅能基于因可及性限制和高昂采集成本而在某些地点收集的、极为有限的测试样本子集进行建模。由于我们对这些函数在多维因果关系复杂性及对不可及输入的未察觉依赖性方面认知有限，可能导致观察到此类函数随地理位置的变化。在整个研究域内汇集不同的响应函数，对于正确预测结果响应、评估这些推断值的不确定性以及分析未勘探区域预测中输入变量的显著性至关重要。本文引入了双随机场（dRF）的概念，其中响应函数本身被视为一种区域化变量。通过这种方式，地理域内已建立的不同响应模型可被视为dRF实现的一次观测，从而实现对响应模型及其预测的空间推断和不确定性评估。我们阐释了dRF如何继承经典随机场的所有特性，允许使用标准高斯模拟程序对其进行模拟。这些模型被组合以获取矿产潜力响应，为如何将机器学习方法与地质统计学进行严格整合提供了范例。