Structural geologic modeling as an inference problem: A Bayesian perspective

Abstract

Structural geologic models are widely used to represent the spatial distribution of relevant geologic features. Several techniques exist to construct these models on the basis of different assumptions and different types of geologic observations. However, two problems are prevalent when constructing models: (1) observations and assumptions, and therefore also the constructed model, are subject to uncertainties and (2) additional information is often available, but it cannot be considered directly in the geologic modeling stepalthough it could be used to reduce model uncertainties. The first problem has been addressed in recent work. Here we develop a conceptual approach to consider the second aspect: We combine uncertain prior information with geologically motivated likelihood functions in a Bayesian inference framework. The result is that we not only reduce uncertainties in the ensemble of generated models, but we also gain the potential to learn additional features about the model parameters. We develop an implementation of this concept in a probabilistic programming framework, in which we extend the functionality of a 3D implicit potential-field interpolation method with geologic likelihood functions. With schematic examples, we show how this combination leads to suites of models with reduced uncertainties and how it provides a deeper insight into parameter correlations. Furthermore, the integration into a hierarchical Bayesian model provides an insight into potential extensions of the method, for example, the interpolation functional itself, and other types of information, such as gravity or magnetic potential-field data. These aspects constitute promising paths for future research.

Inversion of Structural Geology Data for Fold Geometry

“…For example, Aydin and Caers () introduce a likelihood function that represents the mismatch between fault observations (seismic interpretations) where the priors are the strike and dip from analog areas. Wellmann et al () and de la Varga and Wellmann () use geology‐based likelihood functions that characterize geological and geophysical observations such as fault geometry, probability of folding, probability of a discontinuity, the probability of an unconformity, or potential field responses where the model parameters (prior knowledge) includes the structural observations. These approaches are suitable for faults where the orientation of the fault plane is a key description of the fault geometry.…”

“…An alternative approach would be to incorporate these additional observations using an additional likelihood function. For example, de la Varga and Wellmann () use multiple likelihood functions to incorporate additional geological knowledge such as fault offset and layer thickness that cannot normally be incorporated into the geological modeling system.…”

“…Geological modeling has previously been considered as an inverse problem (e.g., Aydin & Caers, 2017;de la Varga & Wellmann, 2016;Wellmann et al, 2017;Wood & Curtis, 2004). In these approaches the structural data represent the prior geological knowledge, or model parameters (P( )) and the data (P(D)) are usually not the structural data sets used to create geological models.…”

“…We assume that the structural observations are sampled from an unknown Gaussian distribution, a common assumption in previous work (de la Varga & Wellmann, ; Wellmann & Regenauer‐Lieb, ) and other natural sciences where the underlying probability distributions are unknown. Hence, we use a Gaussian likelihood function: …”

Methods and uncertainty estimations of 3-D structural modelling in crystalline rocks: a case study

“…We therefore apply a computational sampling method based on an adaptive Metropolis MCMC approach (Haario et al, 2001) implemented in the probabilistic programming package PyMC 2 (Patil et al, 2010) and previously successfully used in a geological context (de la Varga and Wellmann, 2016).…”