Inversion of geological knowledge for fold geometry
Introduction
Building three-dimensional (3D) geological models is a complicated task requiring an assimilation of available datasets, prior geological interpretation, and geological knowledge. Recent developments in 3D modeling algorithms and techniques (Calcagno et al., 2008; Caumon et al., 2013; Cowan et al., 2003; Grose et al., 2017, 2018; Hillier et al., 2013, 2014; Laurent et al., 2016; Massiot and Caumon, 2010) have significantly improved how direct observations can be used to constrain surface geometries. The process of creating 3D geological models to represent subsurface geometries can be framed as an inverse problem where the aim is to infer parameter values for the interpolation algorithm given geological observations (Grose et al., 2018). Geological models, and as a result, geological inversions are usually under constrained. However, there are often geological rules and additional geological knowledge that are not directly incorporated into the interpolation schemes that would further constrain subsurface geometries (Jessell et al., 2014, 2010a).
A priori geological knowledge is usually indirectly incorporated into 3D models using subjective form lines, cross sections and level maps (Jessell et al., 2014, 2010a). Usually these interpretive constraints are defined by the geologist summarising available observations and geological knowledge. The geologist's interpretation is subjective and will typically produce a single non-unique solution to a complex problem (Jones et al., 2004). For example, the geologist may want to refine the fold style obtained by interpolating available observations, e.g., if folds in the field have been observed to be tighter than the modeled geometries. A common way to produce expected geometries is to introduce interpretive cross-sections that locally describes the expected shape. This approach is problematic because the interpolation algorithms used for implicit modeling treat these interpretive constraints in the same way as direct observations, i.e., they would locally define the orientation and position of the interpolated surfaces whereas the intention was only to control the fold style. When used in this way, geological knowledge results in over-constraining the range of possible results. In this paper, we propose to incorporate geological knowledge into an inverse problem framework allowing for the mathematical descriptions of geological knowledge to be used directly without requiring subjective interpretive constraints.
A new framework for modeling folded surfaces has significantly improved the incorporation of structural geology concepts (Grose et al., 2017, 2018; Laurent et al., 2016). Using these developments, structural modeling can be framed as an inverse problem and structural data can be inverted for fold geometries (Grose et al., 2018). This approach uses Bayesian inference to sample from the possible combination of parameters (joint posterior distribution) describing the fold axis geometry and the fold shape for local structural observations (e.g. foliations, bedding, fold axis and intersection lineations). These developments allow for refolded folds with complex non-cylindrical geometries to be modeled.
The description of an inverse problem using Bayesian inference can be easily generalised to incorporate a diverse range of observation types (Malinverno and Parker, 2006). Wood and Curtis (2004) suggest a framework for incorporating Geological Prior Information explicitly for modeling sedimentary structures. They show that the addition of additional geometrical information can significantly improve the modeled sedimentary structures away from data. Wellmann et al. (2017) and de la Varga and Wellmann (2016) have used Bayesian inference to include additional information into implicit modeling schemes, for example fault displacement, layer thickness and fault movement direction. In this contribution, we incorporate geological knowledge into the inversion framework from Grose et al. (2018) using a combination of informative prior distributions and additional geological likelihood functions. This approach allows for a combined inversion of local structural observations and global geological knowledge. The probabilistic framework can also be used to determine where the model is relatively well constrained and where it is not. The combined inversion of local observations and global knowledge allows for feedback in terms of the quality of the observations, for example inconsistent observations and knowledge can be determined by analysing the joint posterior distribution. We present proof-of-concept synthetic examples to demonstrate the application of these new likelihood functions to known examples and a 3D case study from the Davenport Province in the Northern Territory, Australia. Our results show that the addition of geological knowledge significantly reduces the variability in interpolated model geometries where the direct observations are sparse or highly ambiguous. The combined inversion of geological data and knowledge provides a promising framework for real-time geological mapping where geological uncertainties can be propagated allowing for different structural interpretations to be tested, identifying targets for additional data collection.
Section snippets
Geological knowledge and interpretation
Geologists use visual features in an outcrop to define geological observations (Frodeman, 1995; Rudwick, 1976). These characteristics and patterns in the rocks by themselves do not provide all of the possible geological information. It is only by correlating between outcrops and applying geological rules that the observations can be related to geological features such as fault planes, tectonic cleavages or stratigraphic horizons. Frodeman (1995) argues that the process of geological reasoning
Parametrisation of fold geometries
Recent developments in implicit modeling techniques and statistical methods for characterising fold geometries have significantly improved the use of geological observations for modeling folded surfaces (Grose et al., 2017, 2018; Laurent et al., 2016). These methods have incorporated the fundamental concepts from structural geology into implicit modeling schemes. A curvilinear coordinate system is defined to characterise the structural elements of the fold with three coordinates represented by
Adding geological knowledge to the geological inversion
The inversion of geological observations can be easily generalised using Bayesian inference to incorporate a diverse range of observations for solving the inverse problem (Malinverno and Parker, 2006). For example, fault offset and stratigraphic unit thickness can be incorporated into geological modeling using standard implicit schemes (de la Varga and Wellmann, 2016; Wellmann et al., 2017). This approach uses geological data as the parameters for the inverse problem and uses geological
Synthetic proof of concept examples
In Fig. 4, the reference fold is a parasitic fold train that is only sampled in a single limb of the main fold. In this example, only a quarter wavelength of the main fold is captured in the data and the main fold wavelength cannot be identified using the S-Variogram (Fig. 4C). The geological inversion captures the main high frequency features in the S-Plot associated with the parasitic folds, but does not capture the main fold wavelength (Fig. 4E). The interpolated fold geometry (Fig. 4D)
Discussion
3D geological models are constructed using available geological observations that directly constrain the surfaces being modeled. Geological knowledge has usually been incorporated into the modeling process by the geologist first interpreting the resulting geometry and adding in interpretive constraints (Jessell et al., 2014, 2010a; Maxelon et al., 2009). In this contribution, we have used the flexibility of Bayesian inference to combine both geological data and knowledge into a geological
Conclusion
New geological likelihood functions are proposed for integrating additional geological data and knowledge into an inversion scheme for fold geometries. Geological knowledge is incorporated directly into the interpolation scheme using Bayesian inference. This allows for poorly constrained features such as the fold axes to be modeled with minimal hard data and still produce results consistent with the geological interpretation. The incorporation of geological knowledge directly into the
Acknowledgements
This research has been supported by LP170100985: Loop - Enabling Stochastic 3D Geological Modelling is a OneGeology initiative funded by the Australian Research Council and supported by Monash University, University of Western Australia, Geoscience Australia, the Geological Surveys of Western Australia, Northern Territory, South Australia and New South Wales as well as the Research for Integrative Numerical Geology, Universite de Lorraine, RWTH Aachen, Geological Survey of Canada, British
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