Block Predictions of Compositional Data with high order geostatistics

Block kriging of compositional data is a challenge even in the case of classical linear statistics. Compositional kriging is not unbiased with respect to the quantity used additively in block integration. Classical block kriging can leave the compositional simplex. Compositional data cannot directly be integrated in a block, when the material density varies within the block. For full compositions the density might be a function of the composition. For subcomposition it is typically still correlated. In a previous publication we developed a geostatistical block prediction of compositional data relying on the additive lognormal property of the compositional random field. This assumption is however not always met. Especially in case of multiple facies, we see bimodal distributions leading to substantially different distribution of block values. Due to the non lognormality of the residuals typical lognormal kriging type bias corrections are not appropriate.
For this situation we propose to apply conditional distribution based type of multiple point geostatistics. Based on a training image or a training model, we generate joint dataset of the observations of the block integral value incorporating all relevant corrections (like density correction). The conditional distribution is described by a very general version of a generalized linear model for the conditional distribution. The parameters of this regression model are estimated from the training dataset. Compositional, Euclidean expected values can be computed from the resulting prediction.
The method has the following properties: The Expected difference of true value and prediction is 0. The method provides the full conditional distribution. Not only the mean, but more selected nonlinear functionals can be predict on average correct from the conditional distribution. Due to its underlying Baysian nature the method can in principle outperform averages over conditional simulations with mps or ordinary kriging.