Statistical stationarity is a key assumption for the many modeling techniques based on variograms and transiograms used for geostatistical reconstruction of the subsurface. Stationarity expresses the property that the rules of geometry and neighborhood in the model are translation invariant, that is, no directional change in either mean or variance is observed. These criteria are met when the lateral arrangement of lithologic elements into a facies mosaic is isotropic. The balance between isotropy and anisotropy is a defining statistic in the configuration of both real and modeled carbonate landscapes. Even a cursory look at a satellite image of a modern carbonate platform shows that gradients in environment and hydrodynamics cause radical departures from isotropy.
Although reef-forming organisms have changed through time, we do not expect that ancient reef systems behaved any differently than today. Hence, significant anisotropy should also be anticipated in the vertical and lateral arrangements of lithologies in the subsurface. To maintain sufficient geologic realism, it is paramount that process-imitating and pattern-replicating models alike be capable of honoring an expected degree of nonstationarity.
Despite this need, few studies exist that provide quantitative information to the reach and location of zones of geometric isotropy and anisotropy in carbonate systems, let alone methods with which this property can be assessed. In an effort to close this disjoint, we develop a method for evaluating a modern Pacific depositional system, the Saipan Lagoon, for which we have created a geographic information system stack consisting of mapped facies distributions and a seabed topographic model, both at meter-scale resolution. By developing a lagged spatial metric based on the Markov property of facies transitions, we demonstrate that the degree of anisotropy is influenced by water depth; the shallowest areas (<5 m [<16 ft]) of the platform interior tend to be anisotropic whereas areas at greater depth are isotropic. This behavior suggests a possible extension to a genetic rule set that could be imparted to subsurface models based on the environment of deposition. This marks an advance in the understanding and, ultimately, handling of geometric nonstationarity in models of carbonate depositional systems.