Means and covariance functions for geostatistical compositional data: an axiomatic approach

This work focuses on the characterization of the central tendency of a sample of compositional data. It provides new results about theoretical properties of means and covariance functions for compositional data, with an axiomatic perspective. Original results that shed new light on the geostatistical modeling of compositional data are presented. As a first result, it is shown that the weighted arithmetic mean is the only central tendency characteristic satisfying a small set of axioms, namely continuity, reflexivity and marginal stability. Moreover, this set of axioms also implies that the weights must be identical for all parts of the composition. This result has deep consequences on the spatial multivariate covariance modeling of compositional data. In a geostatistical setting, it is shown as a second result that the proportional model of covariance functions (i.e., the product of a covariance matrix and a single correlation function) is the only model that provides identical kriging weights for all components of the compositional data. As a consequence of these two results, the proportional model of covariance function is the only covariance model compatible with reflexivity and marginal stability

Lithofacies uncertainty modeling in a siliciclastic reservoir setting by incorporating geological contacts and seismic information

Deterministic modeling lonely provides a unique boundary layout, depending on the geological interpretation or interpolation from the hard available data. Changing the interpreter’s attitude or interpolation parameters leads to displacing the location of these borders. In contrary, probabilistic modeling of geological domains such as lithofacies is a critical aspect to providing information to take proper decision in the case of evaluation of oil reservoirs parameters, that is, applicable for quantification of uncertainty along the boundaries. These stochastic modeling manifests itself dramatically beyond this occasion. Conventional approaches of probabilistic modeling (object and pixel-based) mostly suffers from consideration of contact knowledge on the simulated domains. Plurigaussian simulation algorithm, in contrast, allows reproducing the complex transitions among the lithofacies domains and has found wide acceptance for modeling petroleum reservoirs. Stationary assumption for this framework has implications on the homogeneous characterization of the lithofacies. In this case, the proportion is assumed constant and the covariance function as a typical feature of spatial continuity depends only on the Euclidean distances between two points. But, whenever there exists a heterogeneity phenomenon in the region, this assumption does not urge model to generate the desired variability of the underlying proportion of facies over the domain. Geophysical attributes as a secondary variable in this place, plays an important role for generation of the realistic contact relationship between the simulated categories. In this paper, a hierarchical plurigaussian simulation approach is used to construct multiple realizations of lithofacies by incorporating the acoustic impedance as soft data through an oil reservoir in Iran.This research was funded by the National Elites Foundation of Iran in collaboration with research Institute Petroleum of Industry in Iran under the project number of 9265005

Non-stationary covariance function modelling in 2D least-squares collocation

Standard least-squares collocation (LSC) assumes 2D stationarity and 3D isotropy, and relies on a covariance function to account for spatial dependence in the ob-served data. However, the assumption that the spatial dependence is constant through-out the region of interest may sometimes be violated. Assuming a stationary covariance structure can result in over-smoothing of, e.g., the gravity field in mountains and under-smoothing in great plains. We introduce the kernel convolution method from spatial statistics for non-stationary covariance structures, and demonstrate its advantage fordealing with non-stationarity in geodetic data. We then compared stationary and non-stationary covariance functions in 2D LSC to the empirical example of gravity anomaly interpolation near the Darling Fault, Western Australia, where the field is anisotropic and non-stationary. The results with non-stationary covariance functions are better than standard LSC in terms of formal errors and cross-validation against data not used in the interpolation, demonstrating that the use of non-stationary covariance functions can improve upon standard (stationary) LSC