Steven F. Carle

TL;DR: In this paper, a transition probability model for indicator variables is proposed, which is more interpretable compared with the indicator (cross-) variogram or indicator covariance model, and the transition probability elucidates order relation conditions and readily formulates the indicator kriging equations.

Abstract: The continuous-lag Markov chain provides a conceptually simple, mathematically compact, and theoretically powerful model of spatial variability for categorical variables. Markov chains have a long-standing record of applicability to one-dimensional (1-D) geologic data, but 2- and 3-D applications are rare. Theoretically, a multidimensional Markov chain may assume that 1-D Markov chains characterize spatial variability in all directions. Given that a 1-D continuous Markov chain can be described concisely by a transition rate matrix, this paper develops 3-D continuous-lag Markov chain models by interpolating transition rate matrices established for three principal directions, say strike, dip, and vertical. The transition rate matrix for each principal direction can be developed directly from data or indirectly by conceptual approaches. Application of Sylvester's theorem facilitates establishment of the transition rate matrix, as well as calculation of transition probabilities. The resulting 3-D continuous-lag Markov chain models then can be applied to geo-statistical estimation and simulation techniques, such as indicator cokriging, disjunctive kriging, sequential indicator simulation, and simulated annealing.

TL;DR: In this paper, a Markov chain model of vertical-direction transitions based on well data yields a three-dimensional model of sediment variability which includes cross correlation between sediment types and representation of asymmetry (e.g., fining upward tendencies).

TL;DR: In this article, the authors examined the representation of geologic heterogeneity in two types of geostatistical models under the same mean and spatial covariance structure, and subsequently its effect on the hydraulic response to a pumping test based on 3D high-resolution numerical simulation and field data.

TL;DR: In this paper, the authors use a Markov chain model of transition probability, representing spatial correlation within and among the facies, capturing the relevant geologic features while highlighting a new approach for statistical characterization of hydrofacies spatial variability, and demonstrate the suitability of conventional geostatistical approaches based on variograms or covariances for modeling geologic heterogeneity.