Kriging has long been known as the Best Linear Unbiased Estimate, that is, BLUE. That sounded catchy in the early days of geostatistics, but now it just sounds quaint. Also, in the modern era of machine learning (ML), a linear estimate seems antiquated and simplistic. Kriging is also portrayed as model-driven (and not data-driven like ML techniques). I argue that Kriging is highly non-linear and data-driven.
The figure summarizes my argument. Five data are shown as red dots, the mean is shown as the horizontal black line, linear regression is the blue line, and Kriged estimates (using a Gaussian variogram with a range of 8) are shown by the dark red line. The Kriged estimates are highly nonlinear and match the data values exactly. This is contrary to how some people portray Kriging: (1) a linear estimate evokes the blue linear regression, not the complexity of the dark red curve, and (2) a model-driven estimate based on the variogram model does not communicate the strong reliance that Kriging has on the data.
Kriging could be considered a locally linear estimate. There is no doubt that the equation for the Kriging estimate at an unsampled block location is a linear function of the data included within a local search; however, the linear weights change at every unsampled location. The nonlinearity of a Kriged block model becomes strongly apparent in 2-D and 3-D. The spatial response surface of Kriged estimates is nonparametric and highly nonlinear.
Kriging could be considered as partially model driven. Kriging estimates require a variogram model that represents the spatial variability of the variable. The variogram model is itself non-linear and parameterizes data-driven calculated values, analogue geological information including specific input from similar deposits, general knowledge from different deposits, and analytical functions that have proven themselves in thousands of other estimated models. Variogram modeling provides a powerful mechanism to inject supplementary geological knowledge.
There are limitations. Although the use of locally varying anisotropy (LVA) is increasing, the variogram is a two-point measure of spatial variability that does not directly encode nonlinear features. The promise of multiple point statistics has not been fulfilled for continuous variables. The use of secondary variables is increasing, but Kriging estimates largely remain linear functions of the surrounding data; there is no use of nonlinear spatial or variable features.
The application of ML techniques to local estimation is inevitable. Ensemble trees, support vectors, kernels and networks all have unique strengths that are being explored and exploited. Combining ML estimates and Kriging with coKriging or ensemble estimation appears promising. The trend of data science remains toward using varied interdisciplinary aspects of mathematics, statistics, computer science, and any relevant domain specific knowledge. This is expected to accelerate in geostatistics as flexible modeling platforms like RMSP are more widely adopted.
The rush to ML is admirable. Most significant change comes from paradigm shifts and not gradual incrementalism. There is a risk, however, that important functionality is lost if a change is adopted without attention to detail. In the case of resource estimation, we must maintain (1) the ability to integrate analogue geological understanding – there are situations where we have distressingly few drill holes and additional input is required, (2) the ability to calibrate the smoothness of our estimates to anticipate the information effect and the practical scale/support of mining operations – commonly we have access to 1% of the final data when resource estimates are calculated, and (3) data exactitude and conditional unbiasedness for final grade control estimates – the absolute best final estimates are needed when final production decisions are made; sending material to the wrong destination is costly.
Most resource estimates are based on kriging. ML techniques will be introduced with cautious haste to improve those estimates. The paradigm shift to watch for is not embracing ML techniques; the real paradigm shift will be embracing a probabilistic and risk-qualified world view where uncertainty is quantified and managed at all stages of the mining value chain. We continue to incrementally improve estimates while we advocate for correct uncertainty management with simulation. Of course, ML techniques will find a place in simulation.
The Resource Modeling Solutions platform (RMSP) within our more interactive and automatic modeling platform (AMP) have the workflows for both ML and kriging-based estimation and simulation.
