Direct upscaling of semivariograms and cross semivariograms for scale-consistent geomodeling

Transactions of the Society for Mining, Metallurgy, and Exploration , 2013, Vol. 334, No. 1, pp. 544-552

Babak, O.; Cuba, M.A.; Leuangthong, O.

Integration of data from multiple sources and/or multiple scales is a common, yet challenging aspect of geostatistical resource modeling. Common approaches to data integration are based on a cokriging framework that assumes a consistency of scale in the information that leads to a requirement to composite the data prior to model construction. It is well known that the geostatistical model parameters are scale dependent. In particular, the volume variance of integrating these various scales has a direct impact on the semivariogram that should be used in model construction and/or the assessment of volume-variance relations as it affects resource estimation. This paper reviews and compares two different approaches for semivariogram upscaling. The first approach is the scaling laws. It is an empirically derived approach, involving several simplifying assumptions, one of which is a strict assumption of invariance of the semivariogram shape. The second approach is a theoretically derived direct upscaling of the semivariogram. This approach has been around for a number of years, but is seldom implemented due to software limitations. In this paper, we present a numerical integration approach to facilitate direct semivariogram upscaling. The advantage of direct upscaling over the scaling laws approach is shown by considering a synthetic example and a case study. Further, in the case of multiple codependent random variables, the extension of the direct semivariogram upscaling to the cross semivariograms is derived and an upscaled consistent linear model of coregionalization is presented.