Geological models are commonly used to represent geological structures in 3D space. A wide range of methods exists to create these models, with much scientific work focusing recently on implicit representation methods, which perform an interpolation of a three-dimensional field where the relevant boundaries are then isosurfaces in this field. However, this method has well-known problems with inhomogeneous data distributions: if regions with densely sampled data points exist, modeling artifacts are common. We present here an approach to overcome this deficiency through a combination of an implicit interpolation algorithm with a local smoothing approach. The approach is based on the concepts of nugget effect and filtered kriging known from conventional geostatistics. It reduces the impact of regularly occurring modeling artifacts that result from data uncertainty and data configuration and additionally aims to improve model robustness for scale-dependent fit-for-purpose modeling. Local smoothing can either be manually adjusted, inferred from quantified uncertainties associated with input data or derived automatically from data configuration. The application for different datasets with varying configuration and noise is presented for a low complexity geologic model. The results show that the approach enables a reduction of artifacts, but may require a careful choice of parameter settings for very inhomogeneous data sets.

Structural geological models are widely used to represent relevant geological interfaces and property distributions in the subsurface. Considering the inherent uncertainty of these models, the non-uniqueness of geophysical inverse problems, and the growing availability of data, there is a need for methods that integrate different types of data consistently and consider the uncertainties quantitatively. Probabilistic inference provides a suitable tool for this purpose. Using a Bayesian framework, geological modeling can be considered as an integral part of the inversion and thereby naturally constrain geophysical inversion procedures. This integration prevents geologically unrealistic results and provides the opportunity to include geological and geophysical information in the inversion. This information can be from different sources and is added to the framework through likelihood functions. We applied this methodology to the structurally complex Kevitsa deposit in Finland. We started with an interpretation-based 3D geological model and defined the uncertainties in our geological model through probability density functions. Airborne magnetic data and geological interpretations of borehole data were used to define geophysical and geological likelihoods, respectively. The geophysical data were linked to the uncertain structural parameters through the rock properties. The result of the inverse problem was an ensemble of realized models. These structural models and their uncertainties are visualized using information entropy, which allows for quantitative analysis. Our results show that with our methodology, we can use well-defined likelihood functions to add meaningful information to our initial model without requiring a computationally-heavy full grid inversion, discrepancies between model and data are spotted more easily, and the complementary strength of different types of data can be integrated into one framework.

Quantitative uncertainty analysis, 2-D and 3-D modeling of the subsurface, as well as their visualization form the basis for decision making in exploration, nuclear waste storage and seismic hazard assessment. Methods such as cross-section balancing are well established and yield plausible kinematic scenarios. However, they are based on geological data with errors and subject to human biases. Additionally, kinematic models do not provide a quantitative measure of the uncertainty of structures at depth. New 3-D modeling approaches have emerged that use computational interpolation, which are less dependent on human biases. Probabilistic extensions enable the quantification of uncertainties for the modeled structures. However, these approaches do not provide information on the time evolution of structures. Here, we compare classical cross-section balancing (2-D, kinematic modeling) with 3-D computational modeling to pave the way towards a solution that can bridge between these approaches. We show the strengths and weaknesses of both approaches, highlighting areas where probabilistic modeling can possibly add quantitative structural uncertainty information to improve section balancing. On the other hand, we show where probabilistic modeling still falls short of being able to cover the observed geometric complexities. We ultimately discuss how a workflow that iteratively combines results of the approaches can improve structural and kinematic constraints. As an example, we use the fold-and-thrust belt of the northern Alpine Foreland, the so-called Subalpine Molasse, focusing on the Hausham Syncline (Bavaria) and adjacent areas. We take advantage of the fact that here the stratigraphy as well as the tectonic history are well constrained. We show that shortening within the syncline progressively increases from west to east, independent from structural uncertainties. Two equally viable models can explain this. First, strain in the west is accommodated underneath the syncline in a triangle zone that progressively tapers out, or second, the strain difference is accommodated in more internal units. This highlights the importance of introducing uncertainty modeling also in kinematic restorations, as it enables identifying key regions, where different hypotheses can be tested.

Gravity is one of the most widely used geophysical data types in subsurface exploration. In the recent developments of stochastic geological modeling, gravity data serves as an additional constraint to the modeling construction and can be included in the modeling process as the likelihood function in a Bayesian workflow. A fast but also precise forward gravity simulation is key to the success of the geological modeling inverse problem. In this study, we present a gravity kernel method, which is based on the widely adopted analytical solution on a discretized grid. As opposed to a globally refined regular mesh, we construct local tensor grids for each sensor, respecting the gravimeter locations and the local sensitivities. The kernel method is efficient in terms of both computing and memory use for meshless implicit geological modeling approaches. This design makes the method well suited for many-query applications like Bayesian machine learning using gradient information calculated from Automatic Differentiation (AD). Optimal grid design without knowing the underlying geometry is not straightforward before evaluating the model. Therefore, we further provide a novel perspective on a refinement strategy for the kernel method based on the sensitivity of the cell to the corresponding receiver. Synthetic results are presented and show superior performance compared to the traditional spatial convolution method.