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Spatially distributed natural phenomena represent complex non-linear and non-Gaussian systems. Currently, their spatial distributions are typically studied using second-order spatial statistical models, which are limiting considering the spatial complexity of natural phenomena such as geological applications. High-order geostatistics is a new area of research based on higher-order spatial connectivity measures, especially spatial cumulants as suitable for non-Gaussian and non-linear phenomena. This paper presents HOSC or High-order spatial cumulants, an algorithm for calculating spatial cumulants, including anisotropic experimental cumulants based on spatial templates. High-order cumulants are calculated on two- and three-dimensional synthetic training images so as to elaborate on their characteristics. Spatial cumulants up to and including the fifth-order are found to be efficient in characterizing patterns on both binary and continuous images. The behaviour of spatial cumulants is shown to characterize well the behaviour of the spatial architecture of geological data, including the degree of homogeneity and connectivity. The high-order cumulants are found to be relatively insensitive to the number of data used, and relatively small data sets are sufficient to provide cumulant maps. HOSC has been coded in FORTAN 90 and is easily integrated to the S-GeMS open source platform.