Compositional kriging analysis: A spatial interpolation method for distributions

Abstract

Effective development of subsurface petroleum resources relies on the estimation of spatially distributed parameters at undrilled locations. Established geostatistical algorithms based on kriging exist for the estimation of scalar spatial variables such as porosity and permeability. This may not be suitable for the estimation of distributions of properties, such as grain size, whose whole distribution varies spatially. The current conventional approach is to fit a normal or log-normal distribution to the available data, and then to estimate the parameters of the distribution, the mean and standard deviation, spatially using kriging, taking care to consider any dependence between the mean and standard deviation. The assumption that all the grain size distributions can be approximated by a single distribution type is unsatisfactory, since datasets have very different-looking distributions, with variations in skewness, kurtosis, and modality. This paper presents an alternative approach that can handle significant variability in the distribution shape by separating the distribution into bins, like a histogram, and treating these bins as percentages in a composition. Compositional data needs to be mapped onto a simplex to overcome spurious correlations between those components, in addition, spatial estimation methods for compositional data have already been developed. However, the contribution to this field is the mapping of continuous data from distributions into a composition that enables the compositional kriging method to predict distributions at new locations. Moreover, the results showed the prediction distributions in the presence of varying modality, skewness, and kurtosis. Therefore, the grain size datasets in this paper have been working with the confidentiality restrictions so it explains the technique with a dataset of population ages from the US census 2010 for the state of Texas, which shows similar variability in distribution.