Transitive Geostatistics for Stepwise Modeling Across Boundaries between Rock Regions

A single intrinsic stationary random field may not account for transitional heterogeneity and abrupt dissimilarity of geological properties across boundaries between rock type regions. This paper proposes the stepwise construction of transitive covariance models for modeling continuous properties correlated across boundaries of multiple disjoint physical domains such as rock type bodies. Modeling in geology is usually simplified by splitting the geological space into rock type geo-domains (e.g., strata, sedimentary facies, soil series, diagenetic regions and alteration zones). Due to the limitations of simultaneous solutions, a simplification is to model each domain independently at the cost of losing the conditioning of properties across domains. This paper proposes to organize the modeling process in a triangular array which follows events in the geological time domain; for example, the younger formations are at the top of the pyramid and the older formation at the base. The estimation may go from top to base by assuming that younger events have perturbed older formations. Geology shows the scars of events that cumulate in rock formations before they are finally eroded. In some cases, older formations may be parent material for younger formations. The continuous property within each geo-domain has a conditional covariance in the main diagonal of the array which may belong to a specific event in the geological time. This sequence leads to transitive estimation and simulations in the physical space. If a simultaneous solution is sought (i.e., the future and past are correlated both ways), the complex covariance functions can be constructed stepwise from conditional spectra.