Comparison of Variogram Upscaling Approaches for Consistent Scale Modeling of Mineral Deposits

Resource models are generally constructed using data of different types and/or from different sources. A common issue is the consistency of scale in the information that often leads to a requirement to composite the data prior to model construction. For the purposes of mine planning, development and/or resource assessment, this model may then be upscaled to a relevant selective mining unit. It is well established that the geostatistical parameters for resource model construction is scale dependent. In particular, the volume variance of integrating these various scales has a direct impact on the variogram that should be used in model construction and/or the assessment of volume variance relations as it affects resource estimation. The scaling laws are a set of guidelines that detail how the variogram should change as a result of a change in the volume; these laws are empirically derived and involve some simplifying assumptions, not the least of which is an assumption of shape invariance in the variogram structure. An alternative approach was proposed to directly upscale the variogram via numerical integration, which demonstrates that the assumption of shape invariance is not required and we can directly access the new variogram shape given any volume of interest. This paper compares these two different approaches to variogram upscaling by considering two applications: the first corresponds to a 2D synthetic case and the second looks at a 3D case with data from a porphyry copper deposit. The simulation studies in these cases show that the direct upscaling approach performs better than the scaling laws in fitting the average experimental variogram values at any block size. This is based on a mean absolute error criterion. Moreover, it is also shown that with an increase in the block size, direct variogram upscaling significantly outperforms the scaling laws approach.