Integration of Soft Data Into Geostatistical Simulation of Categorical Variables

Abstract

Uncertain or indirect "soft" data, such as geologic interpretation, driller's logs, geophysical logs or imaging, offer potential constraints or "soft conditioning" to stochastic models of discrete categorical subsurface variables in hydrogeology such as hydrofacies. Previous bivariate geostatistical simulation algorithms have not fully addressed the impact of data uncertainty in formulation of the (co) kriging equations and the objective function in simulated annealing (or quenching). This paper introduces the geostatistical simulation code tsim-s, which accounts for categorical data uncertainty through a data "hardness" parameter. In generating geostatistical realizations with tsim-s, the uncertainty inherent to soft conditioning is factored into both 1) the data declustering and spatial correlation functions in cokriging and 2) the acceptance probability for change of category in simulated quenching. The degree or sensitivity to which soft data conditions a realization as a function of hardness can be quantified by mapping category probabilities derived from multiple realizations. In addition to point or borehole data, arrays of data (e.g., as derived from a depth-dependency function, probability map, or "prior realization") can be used as soft conditioning. The tsim-s algorithm provides a theoretically sound and general framework for integrating datasets of variable location, resolution, and uncertainty into geostatistical simulation of categorical variables. A practical example shows how tsim-s is capable of generating a large-scale three-dimensional simulation including curvilinear features.

Using sequential conditioning to explore uncertainties in geostatistical characterization and in groundwater transport predictions

“…The sparsity of information with which to develop geostatistical models of heterogeneity structure has motivated efforts to combine 'direct' observations made of mapped lithological properties with 'indirect' data (Refsgaard et al, 2012;Carle & Fogg, 2020) that require another level of interpretation that carries with it uncertainty. Pumping test data (Harp et al, 2008;Harp and Vessilinov, 2010;Harp & Vessilinov, 2012) and geophysical data (Engdahl & Weissmann, 2010;Koch, 2013;He et al, 2014;Zhu et al, 2016) are examples of 'indirect' observational data often used to condition and reduce the uncertainty of geostatistical models.…”

“…However, rejection sampling is used if stochastic inversion risks degrading the representation of important spatially defined geological features, such as a connected flow pathway, as demonstrated by Dorn et al (2012) with observations of cross-borehole connectivity in a fractured rock aquifer. While rejection sampling is too computationally inefficient for most groundwater modelling contexts, this burden is alleviated when using it only in final conditioning steps (Dorn et al, 2012;Linde et al, 2015;Cirpka & Valocchi, 2016;Carle & Fogg, 2020). In this way the different strengths of conditioning methods can be employed where appropriate, while the respective weaknesses of each method are mitigated.…”

Stochastic multi-fidelity joint hydrogeophysical inversion of consistent geological models

“…This approach can be combined with stochastic models to populate the main stratigraphic units with lithologies and represent that level of heterogeneity. Furthermore, one can use geophysical inversion results and borehole data to estimate the probability of occurrence of several lithologies (e.g., clay or sand) and use these probabilities as soft information to generate stochastic realizations that are both constrained by some geological reasoning, borehole, and geophysical data (Jørgensen et al, 2015;Carle and Fogg, 2020). This approach ensures consistency with available knowledge, it reproduces accurately the soft information in terms of probability, but nothing ensures that if the final models were used in a forward geophysical model they would reproduce the field measurements.…”

Assessing the root system of urban trees by geostatistical analysis of GPR data

“…In this regard, geostatistics [18][19][20] is a well-known and well-documented spatial statistical discipline aimed at the analysis of spatial and spatiotemporal data, that has been applied to a wide range of disciplines, such as earth sciences, ecology, biology, soil sciences, and engineering. The adoption of geostatistical approaches for the analysis and modelling of geophysical data has proven to provide promising tools and methodologies in the context of geophysical prospecting and stratigraphic 3D modelling 21,22 . The aim of this research is therefore to propose an analysis that combines GPR and geostatistics.…”