Efficient Variography with Partition Variograms

Directional variograms were introduced in geostatistics as a tool for revealing major directions of correlation in spatial data. However, their estimation presents some practical challenges, particularly in the case of large irregularly-sampled data sets where efficient spectral-based estimation methods are not applicable. In this work, we propose a generalization of directional variograms to general partitions of spatial data, and introduce a parallel estimation algorithm that can efficiently handle large data sets with more than 105 points. This partition variogram generalization is motivated by a five-spot point pattern in the petroleum industry, which we named more generally as the isolated-lines arrangement. In such an arrangement, traditional estimators of directional variograms such as r-tube estimators very often fail to incorporate measurements from adjacent lines (e.g. vertical wells) without also incorporating measurements from other planes (e.g. horizontal layers). We provide illustrations of this new concept, and assess the approximation error of the proposed estimators with bootstrap methods and synthetic Gaussian process data.