The characterization of the spatial continuity of categorical variables, such as geological units, is a longstanding subject in geostatistics. Indicator covariances and variograms are used to measure spatial relationships of categorical data between pairs of points. Alternatively, transition probabilities, or transiograms, have been proposed to measure the probability of transition from one category to another as a function of distance. Recently, high-order moments and cumulants built from them have been proposed as measures of complex non-linear spatial relationships for arrangements of multiple points in 3D space. This paper extends the spatial high order statistics, originally conceived for continuous data, to the analysis of categorical spatial datasets. In addition the concept of two-point conditional transition probabilities is expanded to multiple point conditioned probabilities. The algorithm for high-order statistics, HOSC, has been updated to allow for the proposed high-order indicator spatial statistics. A third extension developed is the inference of indicator cross-cumulants and transiograms for two different categories. The experimental spatial indicator cumulants and transition probabilities for different scattered datasets are compared with those obtained from corresponding exhaustive datasets and training images. These comparisons show that significant information about the high-order and multiple point spatial continuity of categorical variables can be extracted from scattered samples. These results open an avenue for the development of simulation algorithms that rely more on data and less on training images.
Published September 2013 , 19 pages