Ausgewählten Publikationen


Perspectives of Physics-Based Machine Learning Strategies for Geoscientific Applications Governed by Partial Differential Equations

pic Perspectives of Physics-Based Machine Learning Urheberrecht: © CG3

Denise Degen, Daniel Caviedes Voullième, Susanne Buiter, Harrie-Jan Hendricks Franssen, Harry Vereecken, Ana González-Nicolás, and Florian Wellmann (2023)

An accurate assessment of the physical states of the Earth system is an essential component of many scientific, societal and economical considerations. These assessments are becoming an increasingly challenging computational task since we aim to resolve models with high resolutions in space and time, to consider complex coupled partial differential equations, and to estimate uncertainties, which often requires many realizations. Machine learning methods are becoming a very popular method for the construction of surrogate models to address these computational issues. However, they also face major challenges in producing explainable, scalable, interpretable and robust models. In this manuscript, we evaluate the perspectives of geoscience applications of physics-based machine learning, which combines physics-based and data-driven methods to overcome the limitations of each approach taken alone. Through three designated examples (from the fields of geothermal energy, geodynamics, and hydrology), we show that the non-intrusive reduced basis method as a physics-based machine learning approach is able to produce highly precise surrogate models that are explainable, scalable, interpretable, and robust.

For more information, please visit the GMD website. An open preprint version of the paper is also available.


Estimating uncertainties in 3-D models of complex fold-and-thrust belts: A case study of the Eastern Alps triangle zone

Sofia Brisson, Florian Wellmann, Nils Chudalla, Jan von Harten, Christoph von Hagke (2023)

Geological modeling commonly results in a single 2D prescribed geometric representation of the subsurface without consideration of uncertainties. Implicit modeling using GemPy can be used to create automated 3D model realizations, and multiple models can be assessed within an assigned uncertainty range. In this study, we build an implicit 3D model of a portion of the Eastern Alps Subalpine Molasse, for the first time, within a probabilistic framework that includes a full assessment of uncertainties.
Implicit modeling, however, often yields artifacts that lead to geologically unrealistic scenarios. Geological topology can be used to constrain modeling outputs to eliminate models with artifacts from the final model ensemble. A synthetic experiment was designed to determine what combination of topological information is necessary to reject all non-meaningful models, without compromising the possible range of geological diversity.
Shannon cell entropy can be used to capture areas of lower or higher uncertainty. By using a topological constraint, high entropy due to artifact-prone areas is eliminated and entropy can thus be reliably assessed. Areas of high entropy can be linked to the triangle zone and to areas where there is a lack of data.


PySubdiv 1.0: open-source geological modeling and reconstruction by non-manifold subdivision surfaces

Mohammad Moulaeifard, Simon Bernard, Florian Wellmann (2023)

Sealed geological models are commonly used as an input to process simulations, for example, in hydrogeological or geomechanical studies. Creating these meshes often requires tedious manual work, and it is therefore difficult to adjust a once-created model. In this work, we propose a flexible framework to create and interact with geological models using explicit surface representations. The essence of the work lies in the determination of the control mesh and the definition of semi-sharp-crease values, which, in combination, enable the representation of complex structural settings with a low number of control points. We achieve this flexibility through the adaptation of recent algorithms from the field of computer graphics to the specific requirements of geological modeling, specifically the representation of non-manifold topologies and sharp features. We combine the method with a particle swarm optimization (PSO) approach to enable the automatic optimization of vertex position and crease sharpness values. The result of this work is implemented in an open-source software (PySubdiv) for reconstructing geological structures while resulting in a model which is (1) sealed/watertight, (2) controllable with a control mesh and (3) topologically similar to the input geological structure. Also, the reconstructed model may include a lower number of vertices compared to the input geological structure, which results in reducing the cost of modeling and simulation. In addition to enabling a manual adjustment of sealed geological models, the algorithm also provides a method for the integration of explicit surface representations in inverse frameworks and the consideration of uncertainties.


Uncertainty quantification of geologic model parameters in 3D gravity inversion by Hessian-informed Markov chain Monte Carlo

Zhouji Liang, Florian Wellmann, Omar Ghattas (2022)

Geologic modeling has been widely adopted to investigate underground structures. However, modeling processes inevitably have uncertainties due to scarcity of data, measurement errors, and simplification of the modeling method. Recent developments in geomodeling methods have introduced a Bayesian framework to constrain the model uncertainties by considering the additional geophysical data in the modeling procedure. Markov chain Monte Carlo (MCMC) methods are normally used as tools to solve the Bayesian inference problem. To achieve a more efficient posterior exploration, advances in MCMC methods use derivative information. Hence, we introduce an approach to efficiently evaluate second-order derivatives in geologic modeling and adopt a Hessian-informed MCMC method, the generalized preconditioned Crank-Nicolson (gpCN), as a tool to solve the 3D model-based gravity Bayesian inversion problem. The result is compared with two other widely applied MCMC methods, random-walk Metropolis–Hastings and Hamiltonian Monte Carlo, on a synthetic geologic model and a realistic structural model of the Kevitsa deposit. Our experiment demonstrates that superior performance is achieved by the gpCN compared with the other two state-of-the-art sampling methods. This indicates the potential of the proposed method to be generalized to more complex models.


3-D Strukturgeologische Modelle: Konzepte, Methoden und Unsicherheiten

Strukturgeoologisches Modell Urheberrecht: © G. Caumon

Florian Wellmann and Guillaume Caumon (2018)

Our new review paper on “3-D Structural geological models: Concepts, methods, and uncertainties” has been published in the 2018 volume of “Advances in Geophysics”!

In this review paper, Florian Wellmann and Guillaume Caumon (Universitée de Lorraine, France) provide a detailed overview of the current state of 3-D geological modelling approaches, conceptual aspects of uncertainties in these models, and techniques to quantify them. With more than 100 pages and almost 400 references, we hope to cover many important aspects and describe relevant contributions to this field (without any claim of completeness - for such an extensive topic).

For more information, please visit the sciencedirect website. An open preprint version of the paper is also available.


An Explanation to the Nusselt–Rayleigh Discrepancy in Naturally Convected Porous Media

Convection cell entropy Urheberrecht: © CGRE

Po-Wei Huang and Florian Wellmann (2021)

We model hydrothermal convection using a partial differential equation formed by Darcy velocity and temperature—the velocity formulation. Using the Elder problem as a bench- mark, we found that the velocity formulation is a valid model of hydrothermal convection. By performing simulations with Rayleigh numbers in the non-oscillatory regime, we show that multiple quasi-steady-state solutions can be one of the reasons that caused the Nusselt–Rayleigh discrepancy found in previous experiments. The results reveal more understandings about the nature of uncertainty of convection modes in porous media.

Publication available as open-access on the Transport in Porous Media website.


3D multi-physics uncertainty quantification using physics-based machine learning

Abbildung der nicht intrusiven reduzierten Basismethode für die Fallstudie Groß Schönebeck

Denise Degen, Mauro Cacace and Florian Wellmann (2022)

Quantitative predictions of the physical state of the Earth’s subsurface are routinely based on numerical solutions of complex coupled partial differential equations together with estimates of the uncertainties in the material parameters. The resulting high-dimensional problems are computationally prohibitive even for state-of-the-art solver solutions. In this study, we introduce a hybrid physics-based machine learning technique, the non-intrusive reduced basis method, to construct reliable, scalable, and interpretable surrogate models. Our approach, to combine physical process models with data-driven machine learning techniques, allows us to overcome limitations specific to each individual component, and it enables us to carry out probabilistic analyses, such as global sensitivity studies and uncertainty quantification for real-case non-linearly coupled physical problems. It additionally provides orders of magnitude computational gain, while maintaining an accuracy higher than measurement errors. Although in this study we use a thermo-hydro-mechanical reservoir application to illustrate these features, all the theory described is equally valid and applicable to a wider range of geoscientific applications.

The full text is available at Scientific Reports.


From Google Earth to 3D Geology Problem 2: Seeing Below the Surface of the Digital Earth

Google Earth to Geology Urheberrecht: © CGRE

Florian Wellmann, Alexander Schaaf, Miguel de la Varga, Christoph von Hagke (2019)

The visualization of structural features in a three-dimensional (3D) context is an essential aspect of geological understanding—both implicit in the context of imagination and explicit in the form of generated 3D geometric models. Still, this understanding and intuition does not come naturally to all students, and we aim a lot of effort in teaching structural geology at the development of this understanding. In this project and teaching module, we strive to support this aim with an exercise to generate a 3D geological model directly out of combined satellite images and digital ele- vation models (DEMs) as an aid to answering geological questions. This method is technically possible on the basis of all programs that provide access to, and interaction with, these types of data. We make use here of the excellent pos- sibilities provided by the desktop version of Google Earth to view remote sensing data on DEMs and to select and extract points with relevant geological information.

Paper erhältlich über sciencedirect.


Pattern Extraction of Topsoil and Subsoil Heterogeneity and Soil‐Crop Interaction Using Unsupervised Bayesian Machine Learning

Wang 2019 abstract fig Urheberrecht: © CGRE

Hui Wang, Florian Wellmann, Tianqi Zhang, Alexander Schaaf, Robin Maximilian Kanig, Elizabeth Verweij, Christian von Hebel, and Jan van der Kruk (2019)

The link between remotely sensed surface vegetation performances with the heterogeneity of subsurface physical properties is investigated by means of a Bayesian unsupervised learning approach. This question has considerable relevance and practical implications for precision agriculture as visible spatial differences in crop development and yield are often directly related to horizontal and vertical variations in soil texture caused by, for example, complex deposition/erosion processes. In addition, active and relict geomorphological settings, such as floodplains and buried paleochannels, can cast significant complexity into surface hydrology and crop modeling. This also requires a better approach to detect, quantify, and analyze topsoil and subsoil heterogeneity and soil‐crop interaction. In this work, we introduce a novel unsupervised Bayesian pattern recognition framework to address the extraction of these complex patterns. The proposed approach is first validated using two synthetic data sets and then applied to real‐world data sets of three test fields, which consists of satellite‐derived normalized difference vegetation index (NDVI) time series and proximal soil measurement data acquired by a multireceiver electromagnetic induction geophysical system. We show, for the first time, how the similarity and joint spatial patterns between crop NDVI time series and soil electromagnetic induction information can be extracted in a statistically rigorous means, and the associated heterogeneity and correlation can be analyzed in a quantitative manner. Some preliminary results from this study improve our understanding the link of above surface crop performance with the heterogeneous subsurface. Additional investigations have been planned for further testing the validity and generalization of these findings.

Full text available on the JGR Biogeosciences page.


GemPy 1.0: open-source stochastic geological modeling and inversion

de la Varga 2019 fig Urheberrecht: © CGRE

Miguel de la Varga, Alexander Schaaf, Florian Wellmann (2019)

This paper introduce the new open-sourcelibrary GemPy. GemPy is a tool for generating 3D structural geological models in Python. With GemPy, you can create complex models that include stratigraphical and structural features such as folds, faults and unconformities. It was furthermore designed to enable stochastic modeling to adress parameter and model uncertainties.this paper delves into the mathematical methods behind the software explaining some of its most prominent features.

You can find the paper on Geosci. Model Dev and more details about the software here.


Certified Reduced Basis Method in Geosciences: Addressing the challenge of high dimensional problems

Paper RB Urheberrecht: © CGRE

Denise Degen, Karen Veroy, Florian Wellmann

One of the biggest challenges in Computational Geosciences is finding ways of efficiently simulating high-dimensional problems. In this paper we present the reduced basis method that constructs low-dimensional approximations to (high-dimensional) solutions of partial differential equations. In contrast to other widely used geoscientific reduction techniques, the reduced basis method reduces the Galerkin approximation space and not the physical space and is consequently much less restrictive. Another advantage is that, fort the problems considered in this paper, the method provides a bound to the error in the reduced order approximation, thus permitting an objective evaluation of the approximation quality. Using a geothermal conduction problem we demonstrate that depending on the model we obtain a maximum speed-up of three orders of magnitude with an error in the order of 10\textsuperscript{-8}. This significant reduction of the cost of the forward simulation allows to perform uncertainty quantification, inversions, and parameter studies for larger and more complex models than currently possible.

Paper now accepted for publication in Computational Geosciences , preprint available on EarthArXiv.


Uncertainty estimation for a geological model of the Sandstone greenstone belt, Western Australia – insights from integrated geological and geophysical inversion in a Bayesian inference framework

Greenstone Paper Urheberrecht: © CGRE

Florian Wellmann, Miguel de la Varga, Ruth E. Murdie, Klaus Gessner and Mark Jessell

The spatial relationship between different rock types and relevant structural features is an important aspect in the characterization of ore-forming systems. Our knowledge about this geological architecture is often captured in 3D structural geological models. Multiple methods exist to generate these models, but one important problem remains: structural models often contain significant uncertainties. In recent years, several approaches have been developed to consider uncertainties in geological prior parameters that are used to create these models through the use of stochastic simulation methods. However, a disadvantage of these methods is that there is no guarantee that each simulated model is geologically reasonable – and that it forms a valid representation in the light of additional data (e.g. geophysical measurements). We address these shortcomings here with an approach for the integration of structural geological and geophysical data into a framework that explicitly considers model uncertainties. We combine existing implicit structural modelling methods with novel developments in probabilistic programming in a Bayesian framework. In an application of these concepts to a gold-bearing greenstone belt in Western Australia, we show that we are able to significantly reduce uncertainties in the final model by additional data integration. Although the final question always remains whether a predicted model suite is a suitable representation of accuracy or not, we conclude that our application of a Bayesian framework provides a novel quantitative approach to addressing uncertainty and optimization of model parameters.

Full text on Lyell Collection, Geological Society of London.


    Actors, actions and uncertainties: Optimizing decision making based on 3-D structural geological models

    Paper Actors, Actions and Uncertainties Urheberrecht: © CGRE

    Fabian Antonio Stamm, Miguel de la Varga and Florian Wellmann (2019)

    Uncertainties are common in geological models and have a considerable impact on model interpretations and subsequent decision making. This is of particular significance for high-risk, high-reward sectors, such as hydrocarbon exploration and production. Recent advances allows us to view geological modeling as a statistical problem that we can address with probabilistic methods. Using stochastic simulations and Bayesian inference, uncertainties can be quantified and reduced by incorporating additional geological information. In this work, we propose custom loss functions as a decision-making tool that builds upon such probabilistic approaches.

    Our results show that the optimizing estimators shift according to the characteristics of the underlying value distribution. While overall spread leads to separation, risk-averse and risk-friendly decisions converge in the decision space and decrease in expected loss given narrower distributions. We thus consider the degree of decision convergence to be a measure for the state of knowledge and its inherent uncertainty at the moment of decision making. This decisive uncertainty does not change in alignment with model uncertainty but depends on alterations of critical parameters and respective interdependencies, in particular relating to seal reliability. Additionally, actors are affected differently by one set of information, depending on their risk affinity. It is therefore important to identify the model parameters which are most influential for the final decision in order to optimize the decision-making process.

    Paper available as open-access, published in journal “Solid Earth”