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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 step — although 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.
Read More: http://library.seg.org/doi/abs/10.1190/INT-2015-0188.1
... The consideration of uncertainties in structural geological models has been getting increasing attention [7,[14][15][16][17] and applications to mineral systems have shown relevance in both analysis and visualization [18][19][20]. There are promising results for the integration of additional information to the geological model through the implementation of a Bayesian modeling framework from synthetic studies [17,21], as well as application to a mineral setting [22]. In this paper, we present a similar Bayesian modeling framework for geological modeling, where we combine geophysical and geological data. ...
... The mathematical models relating the magnetic data and the geological observations to the model parameters are the interpolator functions that form the basis of our geological modeling step and the forward magnetic simulator. Essentially, this means that we consider structural geologic modeling as a forward modeling step within the Bayesian inference framework (a deeper explanation of these types of probabilistic models can be found in de la Varga and Wellmann [21] and Wellmann et al. [22]). Such an implementation requires an interpolation method that enables full automation of the geological modeling step so that a model can be updated when relevant input parameters are changed [22]. ...
... Considering geological information in combination with geophysical measurements in an inverse framework is not new [14,[53][54][55][56][57], nor is the idea of explicitly considering geological models as uncertain [14][15][16]58]. Our method of including geological modeling as part of the Bayesian inference framework [21,22], and implementing geophysical information to the geological model in the form of likelihood functions has also been tested before [22]. The novelty here is that we used the most efficient MCMC sampler (NUTS) currently available by computing the gradient of our coupled geological-geophysical model. ...
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.
... Understanding fault uncertainty is, therefore, essential in many geoscience studies but faces challenges as human-based interpretations generally aim at producing one or a few interpretation(s) deemed most likely. Computer-based models have a potential, at the expense of some simplifications, to explore a larger set of acceptable scenarios which can then be scrutinized by experts or used as prior model space for inverse methods (de la Varga & Wellmann, 2016;Tarantola, 2006). This paper describes such a computer-based sampling method to interpret fault structures from sparse data. ...
... Stochastic structural modeling has already been proposed to generate several scenarios while taking account of seismic image quality and faults below seismic resolution (Aydin & Caers, 2017;Hollund et al., 2002;Holden et al., 2003;Irving et al., 2010;Julio et al., 2015aJulio et al., , 2015bLecour et al., 2001); uncertainty related to reflection seismic acquisition and processing (Osypov et al., 2013;Thore et al., 2002); geological field measurement uncertainty Lindsay et al., 2012;Pakyuz-Charrier et al., 2019;Wellmann et al., 2014); structural parameters for folding (Grose et al., 2018(Grose et al., , 2019; and observation gaps (Aydin & Caers, 2017;Cherpeau & Caumon, 2015;Cherpeau et al., 2010b;Holden et al., 2003). Considering several structural interpretations has also proved useful to propagate uncertainties to subsurface flow problems (Julio et al., 2015b), to rank structural models against physical data and ultimately to falsify some of the interpretations using a Bayesian approach (Cherpeau et al., 2012;de la Varga & Wellmann, 2016;Irakarama et al., 2019;Seiler et al., 2010;Suzuki et al., 2008;Wellmann et al., 2014). ...
... Then, the geological formations affected by the fault network should be modeled. Generating such geometries would also be useful to assess the impact of fault network uncertainty on resource assessment (Richards et al., 2015), to incorporate this source of uncertainty in geophysical inverse problems Ragon et al., 2018), or to consider the geological likelihood in a Bayesian inference problem (Caumon, 2010;de la Varga & Wellmann, 2016). • Second, the graph formalism at this stage only considers pairwise associations but does not use the likelihood of associating several pieces of evidence at once. ...
The characterization of geological faults from geological and geophysical data is often subject to uncertainties, owing to data ambiguity and incomplete spatial coverage. We propose a stochastic sampling algorithm which generates fault network scenarios compatible with sparse fault evidence while honoring some geological concepts. This process is useful for reducing interpretation bias, formalizing interpretation concepts, and assessing first‐order structural uncertainties. Each scenario is represented by an undirected association graph, where a fault corresponds to an isolated clique, which associates pieces of fault evidence represented as graph nodes. The simulation algorithm samples this association graph from the set of edges linking the pieces of fault evidence that may be interpreted as part of the same fault. Each edge carries a likelihood that the endpoints belong to the same fault surface, expressing some general and regional geological interpretation concepts. The algorithm is illustrated on several incomplete data sets made of three to six two‐dimensional seismic lines extracted from a three‐dimensional seismic image located in the Santos Basin, offshore Brazil. In all cases, the simulation method generates a large number of plausible fault networks, even when using restrictive interpretation rules. The case study experimentally confirms that retrieving the reference association is difficult due to the problem combinatorics. Restrictive and consistent rules increase the likelihood to recover the reference interpretation and reduce the diversity of the obtained realizations. We discuss how the proposed method fits in the quest to rigorously (1) address epistemic uncertainty during structural studies and (2) quantify subsurface uncertainty while preserving structural consistency.
... Whether models are created to be restorable or through static interpolation, they are non-unique solutions and are subject to uncertainty (Bond, 2015;Wellmann and Caumon, 2018;Cardozo and Oakley, 2019). Data inversion methods, such as Markov chain Monte Carlo (MCMC), can be used to find an optimal solution and to estimate uncertainty in model parameters (de la Varga and Wellmann, 2016;Cardozo and Oakley, 2019). However, as the complexity of the model increases, such as by moving from two to three dimensions, allowing parameters such as fault displacement to vary spatially, and including multiple faults, the number of parameters that an inversion must fit for becomes very large. ...
... In three-dimensional geological modeling, as in fold kinematics, there is also increasing interest in quantifying uncertainty rather than producing a single model (Røe et al., 2014;Cherpeau and Caumon, 2015;Wellmann and Caumon, 2018), and stochastic data inversion methods have been applied in this field as well. Cherpeau et al. (2012), de la Varga and Wellmann (2016), Aydin and Caers (2017), Grose et al. (2018), and Grose et al. (2019) have all applied Markov chain Monte Carlo methods to the problem of building three-dimensional geological models. In these studies, the goal is to build a static model, possibly incorporating geological knowledge in the process (Grose et al., 2019), but not to produce one that is kinematically restorable. ...
... Implicit modeling [1][2][3] of ore bodies consists of two basic processes: the solution of an unknown implicit function and surface reconstruction of the solved implicit function. One of the most significant advantages of implicit modeling is that the topology of the reconstructed surface (the arrangement for how implicit function fields share geometry) is honored automatically when samples are densely distributed. ...
... Due to the sparsity of geological sampling data, there is a large uncertainty [6], especially in the topology between drillholes (the arrangement of how sample segments and non-sample segments share geometry) [2], for the interpolation of regions lacking data support. Therefore, it is necessary to append manual constraints according to the geologist's insight. ...
In this paper, we introduce combination constraints for modeling ore bodies based on multiple implicit fields interpolation. The basic idea of the method is to define a multi-labeled implicit function that combines different sub-implicit fields by the combination operations, including intersection, union and difference operators. The contribution of this paper resides in the application of combination of more general implicit fields with combination rules for the implicit modeling of ore bodies, such that the geologist can construct constraints honoring geological relationships more flexibly. To improve the efficiency of implicit surface reconstruction, a pruning strategy is used to avoid unnecessary calculations based on the hierarchical bounding box of the operation tree. Different RBF-based methods are utilized to study the implicit modeling cases of ore bodies. The experimental results of several datasets show that the combination constraints are useful to reconstruct implicit surfaces for ore bodies with mineralization rules involving multiple fields.
... Several previous efforts have been made to develop methods to estimate and incorporate geometric uncertainties of geological structures into modeling (e.g., de la Varga & Wellmann, 2016;Schneeberger et al., 2017;Wellmann et al., 2010Wellmann et al., , 2018. It has been shown that uncertainty modeling of geometric models is also possible for areas of complex structure, such as the Alpine fold-thrust belt and the Subalpine Molasse (Brisson et al., 2023;Frings et al., 2023). ...
To understand the exhumation history of orogens and their fold‐thrust belts, it is important to accurately reconstruct their time‐temperature evolution. This is often done by employing thermokinematic models. One problem of current approaches is that they are limited in prescribing geometric constraints only as far as they affect transient thermal conditions. This often results in 2‐D plane strain assumptions, and a simple treatment of structural and kinematic uncertainties. In this work, we combine 3‐D kinematic forward modeling with a random sampling approach to automatically generate an ensemble of kinematic models in the range of assigned geometric uncertainties. Using Markov Chain Monte Carlo, each randomly generated model is assessed regarding how well it fits the available paleo‐depth data taken from low‐temperature thermochronology. The resulting, more robust model can then be used to re‐interpret the thermal resetting data. We first apply this method to synthetic experiments with variable structural complexity and sample uncertainties, and later to the Alpine fold‐thrust belt, the Subalpine Molasse. Results show that it is possible to use thermochronological data to make predictions about exhumation, which can be translated into likelihood functions to obtain the range of 3‐D kinematic forward models explaining the data. Though the method performs well for the synthetic models, additional thermochronological parameters may need to be considered to improve the inversion results for structurally complex settings. The method is useful, however, to study alternative mechanisms of exhumation for the thermochronological samples that are not respected by the modeling, even when uncertainty is considered.
... Markov Chain Monte Carlo (MCMC) method, which estimates uncertainty by feeding model generators with probabilistic input data (De La Varga and Wellmann, 2016;Pakyuz-Charrier et al., 2019). ...
This study analyzes the directional properties of geological faults using triangulations to model displaced horizons. We investigate two scenarios: one without elevation uncertainties and one with such uncertainties. Through formal mathematical proofs and computational experiments, we explore how triangular surface data can reveal geometric characteristics of faults. Our formal analysis introduces four propositions of increasing generality, demonstrating that in the absence of elevation errors, duplicate elevation values lead to identical dip directions. For the scenario with elevation uncertainties, we find that the expected dip direction remains consistent with the error-free case. These findings are further supported by computational experiments using a combinatorial algorithm that generates all possible three-element subsets from a given set of points. The results offer insights into predicting fault geometry in data-sparse environments and provide a framework for analyzing directional data in topographic grids with imprecise elevation data. This work has significant implications for improving fault modeling in geological studies, particularly when dealing with limited or uncertain data.
... Multiple surfaces can be represented as the isovalues of a single or several scalar fields (Lajaunie et al., 1997;Wellmann and Caumon, 2018). Implicit surfaces can be created using meshless methods such as radial basis function (RBF) interpolation (Carr et al., 2001;Cowan et al., 2003; or dual kriging (Lajaunie et al., 1997;de la Varga and Wellmann, 2016) and mesh-based methods (Moyen et al., 2004;Caumon and Collon-Drouaillet, 2014). These techniques permit the building of geologic models in areas with a high degree of complexity (Wellmann et al., 2017), making them reproducible and less biased (Grose et al., 2019). ...
To support economic decisions and exploration targeting, as well as to understand processes controlling the mineralization, three-dimensional structural and lithological boundary models of the Kiruna mining district have been built using surface (outcrop observations and measurements) and subsurface (drill hole data and mine wall mapping) data. Rule-based hybrid implicit-explicit modeling techniques were used to create district-scale models of areas with high disproportion in data resolution characterized by dense, clustered, and distant data spacing. Densely sampled areas were integrated with established conceptual studies using geologic conditions and the addition of synthetic data, leading to variably constrained surfaces that facilitate the visualization, interpretation, and further integration of the geologic models. This modeling approach proved to be efficient in integrating local, frequently sampled areas with district-scale, sparsely sampled regions. Dominantly S-plunging lineation on N-S–trending fracture planes, characteristic fracture mineral-fill, and weak rock mass at the ore contact indicated by poor core orientation quality and rock quality description suggest that ore-parallel fractures in the Kiirunavaara area were more commonly reactivated. Slight variation in the angular relationship of fracture sets situated in different fault-bound blocks suggests that strain accommodation across the orebodies was uneven. The location of brittle faults identified in drill core, deposit-scale structural analysis, and aeromagnetic geophysical maps indicate a close relationship between fault locations and the iron oxide-apatite mineralization, suggesting that uneven stress accommodation and proximity of conjugate fault sets played an important role in juxtaposing blocks from different crustal depths and controls the location of the iron oxide-apatite orebodies.
... Interpreting geological structures serves as fundamental data for three-dimensional geological modeling, allowing for the characterization of reservoir distribution trends within study area. [10][11][12][13] In the context of coalbed methane reservoirs, the interpretation of top and bottom structures holds particular significance. Three-dimensional structural models are primarily based on the interpretation of three-dimensional seismic data. ...
China's proven coalbed methane reserves exceeding 30 trillion cubic meters, and two major coalbed methane industrial bases have been established in the Qinshui and Ordos Basin. Among them, the eastern margin of the Ordos Basin is one of the main blocks for shallow coalbed methane development, holding immense exploration and development potential. Nowadays, the reservoir characteristics of the eastern margin of the Ordos Basin remain unclear, necessitating further research to study reservoir features through fine reservoir description studies. Threedimensional geological modeling can effectively characterize reservoir heterogeneity, especially for predicting geological features between wells and in undrilled gas reservoirs. This study primarily focuses on conducting fine characterization of gas reservoirs and establishing a fine threedimensional geological model through techniques such as multi-attribute fusion in the eastern margin of the Ordos Basin, providing guidance for making and adjustment of development schemes.
... However, model structural errors always exist when a real-world system is represented by a model (de la Varga & Wellmann, 2016;Kavetski et al., 2018;Sikorska et al., 2012). If the chosen model does not flawlessly describe the data-generating process, or model errors are not appropriately handled (e.g., via parametric statistical error models), all inferred probability distributions (for parameters, states, predictions) will again converge to complet. ...
Hydrogeological models require reliable uncertainty intervals that honestly reflect the total uncertainties of model predictions. The operation of a conventional Bayesian framework only produces realistic (interpretable in the context of the natural system) inference results if the model structure matches the data‐generating process, that is, applying Bayes' theorem implicitly assumes the underlying model to be true. With an imperfect model, we may obtain a too‐narrow‐for‐its‐bias uncertainty interval when conditioning on a long time‐series of calibration data, because the assumption of a quasi‐true model becomes too strict. To overcome the problem of overconfident posteriors, we propose a non‐parametric Bayesian method, called Tau‐averaging method: it applies Bayesian analysis on sliding time windows along the data time series for calibration. Thus, it obtains so‐called transitional posteriors per time window. Then, we average these into a wider predictive posterior. With the proposed routine, we explicitly capture the time‐varying impact of model error on prediction uncertainty. The length of the calibration window is optimized to maximize goal‐oriented statistical skill scores for predictive coverage. Our method loosens the perfect‐model‐assumption by conditioning only on small windows of the data set at a time, that is, it assumes that “the model is sufficient to follow the system dynamics for a smaller duration.” We test our method on two cases of soil moisture modeling and show how it improves predictive coverage as compared to the conventional Bayesian approach. Our findings demonstrate that the proposed method convincingly overcomes the overconfidence drawback of Bayesian inference under model misspecification and long calibration time‐series.
... One important focus of these developments has also been on methods to estimate uncertainties of the geological structures (e.g., Wellmann et al., 2010;Jessell et al., 2010;Wellmann et al., 2014;de la Varga and Wellmann, 2016;Schneeberger et al., 2017;Ailleres et al., 2019;Brisson et al., 2023). These models provide an estimate of the plausibility of different geometric steady state models of the subsurface. ...
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.
... As a result, it is not always feasible to apply these methods to settings that require a realtime response. Overcoming this limitation has a potential impact on many applications, e.g., medical imaging [39,40], structural health monitoring [41,42], geology [43]. The goal of our paper is to address this drawback. ...
Inverse problems, i.e., estimating parameters of physical models from experimental data, are ubiquitous in science and engineering. The Bayesian formulation is the gold standard because it alleviates ill-posedness issues and quantifies epistemic uncertainty. Since analytical posteriors are not typically available, one resorts to Markov chain Monte Carlo sampling or approximate variational inference. However, inference needs to be rerun from scratch for each new set of data. This drawback limits the applicability of the Bayesian formulation to real-time settings, e.g., health monitoring of engineered systems, and medical diagnosis. The objective of this paper is to develop a methodology that enables real-time inference by learning the Bayesian inverse map, i.e., the map from data to posteriors. Our approach is as follows. We represent the posterior distribution using a parameterization based on deep neural networks. Next, we learn the network parameters by amortized variational inference method which involves maximizing the expectation of evidence lower bound over all possible datasets compatible with the model. We demonstrate our approach by solving examples a set of benchmark problems from science and engineering. Our results show that the posterior estimates of our approach are in agreement with the corresponding ground truth obtained by Markov chain Monte Carlo. Once trained, our approach provides the posterior parameters of observation just at the cost of a forward pass of the neural network.
... Additional data or concepts could also be integrated in the method, as an ancillary geological likelihood to evaluate the results (e.g., de la Varga and Wellmann, 2016). Other posterior analysis on the generated stratigraphic model may be achieved to evaluate the physical likelihood of the correlation outcomes (layer connectivities) like hydraulic or tracer simulation tests (e.g., Lallier et al., 2012). ...
Subsurface modeling is a way to predict the structure and the connectivity of stratigraphic units by honoring subsurface observations. These observations are commonly be sampled along wells at a large and sparse horizontal scale (kilometer-scale) but at a fine vertical scale (meter-scale). There are two types of well data: (1) well logs, corresponding to quasi-continuous (regular sampling) geophysical measurements along the well path (e.g., gamma ray, sonic, neutron porosity), and (2) regions, corresponding to categorical reservoir properties and defined by their top and bottom depths along the well path (e.g., biozones, structural zones, sedimentary facies). Markers are interpreted along the well path and can be associated in order to generate a consistent set of marker associations called well correlations. These well correlations may be generated manually (deterministic approach) by experts, but this may be prone to biases and does not ensure reproducibility. Well correlations may also be generated automatically (deterministic or probabilistic approach) by computing with an algorithm a large number of consistent well correlations and by ranking these realizations according to their likelihood. The likelihood of these computer-assisted well correlations are directly linked to the principle of correlation used to associate markers. This work introduces two principles of correlation, which tend to reproduce the chronostratigraphy and the depositional processes at the parasequence scale: (1) "a marker (described by facies and distality taken at the center of an interval having a constant facies and a constant distality) cannot be associated with another marker described by a depositionally deeper facies at a more proximal position, or a depositionally shallower facies at a more distal position", and (2) "the lower the difference between a chronostratigraphic interpolation (in between markers) and a conceptual depositional profile, the higher the likelihood of the marker association". These two principles of correlation are first benchmarked with analytical solutions and applied on synthetic cases. They have then been used (1) to predict the connectivity of stratigraphic units from well data without strong knowledge on depositional environments by inferring the correlation parameters, or (2) to evaluate the likelihood of a hypothetical depositional environment by generating stochastic realizations and assessing the uncertainties. The methods are applied on a siliciclastic coastal deltaic system targeting a Middle Jurassic reservoir in the South Viking Graben in the North Sea.This work enables (1) to define two specific principles of correlation defined by a few parameters that can be used to generate stochastically well correlations within coastal deltaic systems, and (2) to open the path towards a simple combination of specific principles of correlation to obtain a better characterization of coastal deltaic systems by assessing the uncertainties.
... Other stochastic geomodeling methods have also been successfully used for spatial modeling of geo-domains (Wellmann and Caumon, 2018;Jessell et al., 2018;Grose et al., 2018;Jessell et al., 2014;Lindsay et al., 2013;Caumon, 2010;Jessell et al., 2010). In particular, stochastic implicit modeling approaches have gained some interest and are an active research area (Manchuk and Deutsch, 2019;de la Varga et al., 2019;Clausolles et al., 2019;Martin and Boisvert, 2017;de la Varga and Wellmann, 2016;Cherpeau and Caumon, 2015;Rongier et al., 2014;Lindsay et al., 2012;Henrion et al., 2010;Cherpeau et al., 2010;Wellmann et al., 2010;Tertois et al., 2010;Frank et al., 2007;Aug et al., 2005). ...
The spatial modeling of geo-domains has become ubiquitous in many geoscientific fields. However, geo-domains’ spatial modeling poses real challenges, including the uncertainty assessment of geo-domain boundaries. Geo-domain models are subject to uncertainties due mainly to the inherent lack of knowledge in areas with little or no data. Because they are often used for impactful decision-making, they must accurately estimate the geo-domain boundaries’ uncertainty. This paper presents a geostatistical implicit modeling method to assess the uncertainty of 3D geo-domain boundaries. The basic concept of the method is to represent the underlying implicit function associated with each geo-domain as a sum of a random implicit trend function and a residual random function. The conditional simulation of geo-domains is performed through a step-by-step approach. First, implicit trend function realizations and optimal covariance parameters associated with the residual random function are generated through the probability perturbation method. Then, residual function realizations are generated through classical geostatistical unconditional simulation methods and added to implicit trend function realizations to obtain unconditional implicit function realizations. Next, the conditioning of unconditional implicit function realizations to hard data is performed via principal component analysis and randomized quadratic programming. Finally, conditional implicit function simulations are transformed to conditional geo-domain simulations by applying a truncation rule. The proposed method is constructed to honor hard data and stated rules of how geo-domains interact spatially. It is applied to a lithological dataset from a porphyry copper deposit. A comparison with the classical sequential indicator simulation (SIS) method is carried out. The results indicate that the proposed approach can provide a more reliable and realistic uncertainty assessment of 3D geo-domain boundaries than the traditional sequential indicator simulation (SIS) approach.
... 75 stratigraphic contact location) and secondary (e.g. local formation thickness) geometric information, as well as fault and stratigraphic topological relationships, we are able to export a complete input file for two Open Source geomodelling packages (GemPy de la Varga et al., 2016;LoopStructural, Grose et al. this volume). In principle this workflow could be extended to work with other implicit modelling platforms such as EarthVision (Mayoraz et al., 1992), Gocad-SKUA (Mallet, 2004) and ...
We present two Python libraries (map2loop and map2model) which combine the observations available in digital geological maps with conceptual information, including assumptions regarding the subsurface extent of faults and plutons to provide sufficient constraints to build a reasonable 3D geological model. At a regional scale, the best predictor for the 3D geology of the near-subsurface is often the information contained in a geological map. This remains true even after recognising that a map is also a model, with all the potential for hidden biases that this model status implies. One challenge we face is the difficulty in reproducibly preparing input data for 3D geological models. The information stored in a map falls into three categories of geometric data: positional data such as the position of faults, intrusive and stratigraphic contacts; gradient data, such as the dips of contacts or faults and topological data, such as the age relationships of faults and stratigraphic units, or their adjacency relationships. This work is being conducted within the Loop Consortium, in which algorithms are being developed that allow automatic deconstruction of a geological map to recover the necessary positional, gradient and topological data as inputs to different 3D geological modelling codes. This automation provides significant advantages: it reduces the time to first prototype models; it clearly separates the primary data from subsets produced from filtering via data reduction and conceptual constraints; and provides a homogenous pathway to sensitivity analysis, uncertainty quantification and Value of Information studies. We use the example of the re-folded and faulted Hamersley Basin in Western Australia to demonstrate a complete workflow from data extraction to 3D modelling using two different Open Source 3D modelling engines: GemPy and LoopStructural.
... ML methods are being applied to a whole range of geological and geophysical problems, in particular geophysical inversion (Laloy et al., 2019;Liu and Grana, 2019;Pan et al., 2020;Pirot et al., 2016), earth surface prediction (Carranza & Laborte, 2015;Liu et al., 2015), well log analysis (Bressan et al., 2020;Dunham et al., 2020;Qi and Carr, 2006;Smith et al., 2019) and geochemical prediction and analysis (Dornan et al., 2020;Johnson et al., 2018). Machine learning has often been applied to the study of 3D geological modeling, such as geological modeling based on a support vector machine (Smirnoff et al., 2008;Wang et al., 2014), implicit modeling of a geological structure using a potential field method based on the Kriging model (Calcagno et al., 2008;Gonçalves et al., 2017), and estimating uncertainty for geological models using Bayesian methods (de la Varga and Wellmann, 2016;Wang, 2020;Wellmann et al., 2018), ensemble-based methods (Luo et al., 2015) and Monte Carlo simulations (Pakyuz-Charrier et al., 2018a, 2018b, 2019. ...
Using geophysical inversion for three-dimensional (3D) geological modeling is an effective way to model underground geological structures. In this study, we propose and investigate a 3D geological structure inversion method using convolutional neural networks (CNNs). This method can quickly predict the parameters of a geological structure for constructing a 3D model. First, we sample the geological model space by generating millions of 3D geological models and their corresponding magnetic images. The dataset we use to train CNN classification and regression models includes faults, folds, tilts, tilt-faults and fold-faults. The classification model is used to judge the classification of geological structures. The regression model is used to predict the attitudes of geological structures. The method is applied to synthetic data and real survey data, and the results show that geological structures can be recovered effectively. The classification accuracy is approximately 100%, and the regression accuracy of different structures is mostly between 80% and 97%.
... In addition to the methods mentioned above, geostatistical methods can also be combined with stochastic simulation to assess the comprehensive uncertainty (Wellmann et al. 2010;Wellmann and Regenauer-Lieb 2012;Røe et al. 2014;Schweizer et al. 2017;Hou et al. 2017;Soares et al. 2017;Pakyuz-Charrier et al. 2019). Although cognitive uncertainty is difficult to evaluate (Bárdossy and Fodor 2001), some researchers have presented uncertainty assessment methods that consider a geologist's cognition (Wellmann and Regenauer-Lieb 2012;Wellmann et al. 2017;de la Varga and Wellmann 2016;Demyanov et al. 2019). To assess the impact of spatial variation and cognitive bias, Tacher et al. (2006) assumed that a geological model is the best guess based on modeler's cognition, and used a Gaussian random field around the expected model to evaluate the comprehensive uncertainty. ...
A 3D geological structural model is an approximation of an actual geological phenomenon. Various uncertainty factors in modeling reduce the accuracy of the model; hence, it is necessary to assess the uncertainty of the model. To ensure the credibility of an uncertainty assessment, the comprehensive impacts of multi-source uncertainties should be considered. We propose a method to assess the comprehensive uncertainty of a 3D geological model affected by data errors, spatial variations and cognition bias. Based on Bayesian inference, the proposed method utilizes the established model and geostatistics algorithm to construct a likelihood function of modeler’s empirical knowledge. The uncertainties of data error and spatial variation are integrated into the probability distribution of geological interface with Bayesian Maximum Entropy (BME) method and updated with the likelihood function. According to the contact relationships of the strata, the comprehensive uncertainty of the geological structural model is calculated using the probability distribution of each geological interface. Using this approach, we analyze the comprehensive uncertainty of a 3D geological model of the Huangtupo slope in Badong, Hubei, China. The change in the uncertainty of the model during the integration process and the structure of the spatial distribution of the uncertainty in the geological model are visualized. The application shows the ability of this approach to assess the comprehensive uncertainty of 3D geological models.
... As a result, the fault displacements and overprinting relationships are internally consistent. In comparison, where faults are represented using step functions (Calcagno et al., 2008b;de la Varga and Wellmann, 2016) the fault displacements are added into the interpolation of the faulted surfaces 595 https://doi.org/10.5194/gmd-2020-336 Preprint. ...
In this contribution we introduce LoopStructural, a new open source 3D geological modelling python package ( https://www.github.com/Loop3d/LoopStructural). LoopStructural provides a generic API for 3D geological modelling applications harnessing the core python scientific libraries pandas, numpy and scipy. Six different interpolation algorithms, including 3 discrete interpolators and 3 polynomial trend interpolators, can be used from the same model design. This means that different interpolation algorithms can be mixed and matched within a geological model allowing for different geological objects e.g. different conformable foliations, fault surfaces, unconformities to be modelled using different algorithms. Geological features are incorporated into the model using a time-aware approach, where the most recent features are modelled first and used to constrain the geometries of the older features. For example, we use a fault frame for characterising the geometry of the fault surface and apply each fault sequentially to the faulted surfaces. In this contribution we use LoopStructural to produce synthetic proof of concepts models and a 86 x 52 km model of the Flinders ranges in South Australia using map2loop.
... Neither of these two interpolation methods uses the structural risk minimization principle. Recently, a geological surface uncertainty modeling method combining prior information and a Bayesian reasoning framework has emerged [12], [13]. This approach integrates additional information into the modeling steps and effectively reduces the uncertainty. ...
Reconstruction of complex geological surface is widely used in oil and gas exploration, geological modeling, geological structure analysis, and other fields. It is an important basis for data visualization and visual analysis in these fields. The complexity of geological structures, the inaccuracy and sparsity of seismic interpretation data, and the lack of tectonic morphological information can lead to uncertainty in geological surface reconstruction. The existing geological surface uncertainty characterization and uncertain reconstruction methods have a shortcoming in balancing the interpolation error of high-confidence samples and model structure risk. Based on support vector regression (SVR), a method with confidence constraints for uncertainty characterization and the modeling of geological surfaces is proposed in this article. The proposed method minimizes the structural risk by adding a regularization term representing the model complexity, integrates high-confidence samples, such as drilling data, based on confidence constraints, and utilizes well path points by assigning appropriate inequality constraints to the corresponding prediction points. The results based on a real-world fault data set show that the uncertainty envelopes and fault realizations generated by the proposed method are constrained by well observations and well paths, effectively reducing the uncertainty.
... Inversion has the advantage of providing statistical feedback on solution quality. Specifically, within a Bayesian framework, the objective is to determine the posterior distribution of a set of parameters given prior distributions and likelihood functions that describe how the data relate to those unknown parameters (Tarantola 2005;Aster et al. 2013;De La Varga & Wellmann 2016;Wellmann et al. 2018). The Bayesian approach is particularly useful for geophysical inverse problems, which are in principle ill-posed because they are inherently non-unique. ...
We have developed a linear 3-D gravity inversion method capable of modelling complex geological regions such as subduction margins. Our procedure inverts satellite gravity to determine the best-fitting differential densities of spatially discretized subsurface prisms in a least-squares sense. We use a Bayesian approach to incorporate both data error and prior constraints based on seismic reflection and refraction data. Based on these data, Gaussian priors are applied to the appropriate model parameters as absolute equality constraints. To stabilize the inversion and provide relative equality constraints on the parameters, we utilize a combination of first and second order Tikhonov regularization, which enforces smoothness in the horizontal direction between seismically constrained regions, while allowing for sharper contacts in the vertical. We apply this method to the nascent Puysegur Trench, south of New Zealand, where oceanic lithosphere of the Australian Plate has underthrust Puysegur Ridge and Solander Basin on the Pacific Plate since the Miocene. These models provide insight into the density contrasts, Moho depth, and crustal thickness in the region. The final model has a mean standard deviation on the model parameters of about 17 kg m–3, and a mean absolute error on the predicted gravity of about 3.9 mGal, demonstrating the success of this method for even complex density distributions like those present at subduction zones. The posterior density distribution versus seismic velocity is diagnostic of compositional and structural changes and shows a thin sliver of oceanic crust emplaced between the nascent thrust and the strike slip Puysegur Fault. However, the northern end of the Puysegur Ridge, at the Snares Zone, is predominantly buoyant continental crust, despite its subsidence with respect to the rest of the ridge. These features highlight the mechanical changes unfolding during subduction initiation.
Three-dimensional geological modelling algorithms can generate multiple models that fit various mathematical and geometrical constraints. The results, however, are often meaningless to geological experts if the models do not respect accepted geological principles. This is problematic as use of the models is expected for various downstream purposes, such as hazard risk assessment, flow characterization, reservoir estimation, geological storage, or mineral and energy exploration. Verification of the geological reasonableness of such models is therefore important. If implausible models can be identified and eliminated, it will save countless hours and computational and human resources.
To begin assessing geological reasonableness, we develop a framework for checking model consistency with geological knowledge and test it with a proof-of-concept tool. The framework consists of a space of consistent and inconsistent geological situations that can hold between a pair of geological objects, and the tool assesses a model's geological relations against the space to identify (in)consistent situations. The tool is successfully applied to several case studies as a first promising step toward automated assessment of geological reasonableness.
The stratigraphic sequence of an archaeological site constitutes its biography, which details what was deposited, when it was deposited, and how the deposition occurred. However, many current methods of recording archaeological stratigraphy do not allow for a full reconstruction of a site's biography. This paper introduces a theoretical framework for analysing a site's formation and deformation processes, emphasizing the integration of different temporal, spatial, geological, and archaeological information elements. This approach offers a way to study change and dynamics by deciphering successive episodes, depositional events, their temporal and causal ordering, and, when possible, the speed and rhythm of such depositions. The integration of micromorphological data and absolute dating can increase the chronological resolution of a site's biography by formally defining the temporal boundaries of depositional events. Additionally, the use of multidimensional geometric analysis of occupational floors and sedimentary volumes enhances the understanding of the complex relationships between stratigraphic depth and temporality. These tools enable archaeologists to create multidimensional visualizations, make inferences, and enhance interpretations of archaeological sites.
The coastal plain of Rio Grande do Sul state, in Brazil, is highly vulnerable to expected changes in sea level, while having an increasing population and consequently increasing water demands. Adequate management is essential to restrain contamination, depletion and salinization of the region’s aquifers considering current and future challenges, but geologic knowledge is essential to guide groundwater sustainable practices. To contribute to this discussion, this work integrated existing geological data from the northern coast of Rio Grande do Sul state to create a three-dimensional representation of the main hydrostratigraphical units of the region and its relation to the basement rocks, expanding the current knowledge of the coastal aquifer system. A review of existing data was carried out, consisting of 307 borehole logs from 13 municipalities inside the area of interest, as well as 19 vertical electrical soundings and 37 logs from oil and coal exploratory drillings, that resulted in 315 input points for the model. This work builds up on the conceptual model previously developed for the area, that defined four hydrostratigraphical units for the region, and was able to constrain the geometries of the main aquifers (unit 1 and 3) and aquitards (unit 2 and 4) and their relation to the basement rocks, showing them to be more heterogeneous in thicknesses and extent than previously thought. In addition, this work was able to model what could be a fifth hydrostratigraphical unit, that strongly differs from the other four and could be an indication of the alluvial fans previously described in the literature.
To understand the exhumation history of the Alpine foreland, it is important to accurately reconstruct its time-temperature evolution. This is often done employing thermokinematic models. One problem of many current approaches is that they are limited to 2-D and do not consider structural or kinematic uncertainties. In this work, we combine 3-D kinematic forward modeling with a systematic random sampling approach to automatically generate an ensemble of kinematic models in the range of assigned geometric uncertainties. Using Markov chain Monte Carlo, each randomly generated model will be assessed in regards to how well they fit the available thermochronology data. This is done to obtain an updated set of modeling parameters with reduced uncertainty. The resulting, more robust model can then be used to re-interpret the thermochronological data and find alternative drivers of cooling for certain samples.We apply this approach to a simple synthetic model to test the methodology, and then to the Eastern Alps triangle zone in the Bavarian Subalpine Molasse. Results show that it is possible to translate low-temperature thermochronology data into a likelihood function to obtain a 3-D kinematic model with updated, more probable parameters. The thermochronological data by itself, however, may not be informative enough to reduce the parameter uncertainty. The method is useful, however, to study alternative mechanisms of exhumation for the thermochronological samples that are not respected by the modeling, even when uncertainty is considered.
Ore sorting is a preconcentration technology and can dramatically reduce energy and water usage to
improve the sustainability and profitability of a mining operation. In porphyry Cu deposits, Cu is the primary
target, with ores usually containing secondary ’pay’ metals such as Au, Mo and gangue elements
such as Fe and As. Due to sensing technology limitations, secondary and deleterious materials vary in correlation
type and strength with Cu but cannot be detected simultaneously via magnetic resonance (MR)
ore sorting. Inferring the relationships between Cu and other elemental abundances is particularly critical
for mineral processing.
The variations in metal grade relationships occur due to the transition into different geological
domains. This raises two questions - how to define these geological domains and how the metal grade
relationship is influenced by these geological domains. In this paper, linear relationship is assumed
between Cu grade and other metal grades. We applies a Bayesian hierarchical (partial-pooling) model
to quantify the linear relationships between Cu, Au, and Fe grades from geochemical bore core data.
The hierarchical model was compared with two other models - ’complete-pooling’ model and ’nopooling’
model. Mining blocks were split based on spatial domain to construct hierarchical model.
Geochemical bore core data records metal grades measured from laboratory assay with spatial coordinates
of sample location. Two case studies from different porphyry Cu deposits were used to evaluate
the performance of the hierarchical model. Markov chain Monte Carlo (MCMC) was used to sample the
posterior parameters. Our results show that the Bayesian hierarchical model dramatically reduced the
posterior predictive variance for metal grades regression compared to the no-pooling model. In addition,
the posterior inference in the hierarchical model is insensitive to the choice of prior. The data is wellrepresented
in the posterior which indicates a robust model. The results show that the spatial domain
can be successfully utilised for metal grade regression. Uncertainty in estimating the relationship
between pay metals and both secondary and gangue elements is quantified and shown to be reduced
with partial-pooling. Thus, the proposed Bayesian hierarchical model can offer a reliable and stable
way to monitor the relationship between metal grades for ore sorting and other mineral processing
options.
Implicit neural representation (INR) networks are emerging as a powerful framework for learning three-dimensional shape representations of complex objects. These networks can be used effectively to model three-dimensional geological structures from scattered point data, sampling geological interfaces, units, and structural orientations. The flexibility and scalability of these networks provide a potential framework for integrating many forms of geological data and knowledge that classical implicit methods cannot easily incorporate. We present an implicit three-dimensional geological modelling approach using an efficient INR network architecture, called GeoINR, consisting of multilayer perceptrons (MLPs). The approach expands on the modelling capabilities of existing methods using these networks by (1) including unconformities into the modelling; (2) introducing constraints on stratigraphic relations and global smoothness, as well as associated loss functions; and (3) improving training dynamics through the geometrical initialization of learnable network variables. These three enhancements enable the modelling of more complex geology, improved data fitting characteristics, and reduction of modelling artifacts in these settings, as compared to an existing INR approach to structural geological modelling. Two diverse case studies also are presented, including a sedimentary basin modelled using well data and a deformed metamorphic setting modelled using outcrop data. Modelling results demonstrate the method's capacity to fit noisy datasets, use outcrop data, represent unconformities, and efficiently model large geographic areas with relatively large datasets, confirming the benefits of the GeoINR approach.
The coastal plain of Rio Grande do Sul state is highly vulnerable to expected changes in sea level, while having an increasing population and consequently increasing water demands. Adequate management is essential to restrain contamination, depletion and salinization of the region’s aquifers considering current and future challenges, but geologic knowledge is essential to guide groundwater sustainable practices. To contribute to this discussion this work integrated existing geological data from the northern coast of Rio Grande do Sul state to create a three dimensional representation of the main hydrostratigraphical units of the region and its relation to the basement rocks, expanding the current knowledge of the coastal aquifer system. A review of existing data was carried out, consisting of 307 borehole logs from 13 municipalities inside the area of interest, as well as 19 vertical electrical soundings and 37 logs from oil and coal exploratory drillings, that resulted in 315 input points for the model. This work builds up on the conceptual model previously developed for the area, that defined four hydrostratigraphical units for the region, and was able to constrain the geometries of the main aquifers (unit 1 and 3) and aquitards (unit 2 and 4) and their relation to the basement rocks, showing them to be more heterogeneous in thicknesses and extent than previously thought. In addition, this work was able to model what could be a fifth hydrostratigraphical unit, that strongly differs from the other four and could be an indication of the alluvial fans previously described in the literature.
The fractured reservoir is one of the significant petroleum reservoir types in China, representing over one-third of total reserves. The Kuqa Depression in the Tarim Basin is dominated by fractured low-porosity sandstone gas reservoirs with characteristic tight matrix, developed fractures, and edge and bottom water. However, the continued development of these reservoirs has led to various problems, including strong reservoir heterogeneity, low well control, complex gas-water relationships, and early water invasion. Addressing these issues requires a detailed understanding of the reservoir’s geological characteristics. One method for achieving a fine reservoir description is through the use of 3D geological modeling. This high-level, comprehensive characterization technique is widely used throughout the entire life cycle of oil and gas field development. A 3D geological model can accurately predict the actual underground reservoir characteristics and provide a geological basis for later numerical simulation work. Based on a study of the geological characteristics of the Kuqa Depression in the Tarim Basin, a 3D geological modeling technique was developed, which includes structural modeling, facies modeling, petrophysical modeling, and fracture modeling. This technology has been successfully applied to many deep gas reservoirs in the Kuqa Depression of the Tarim Basin, leading to enhanced gas recovery.
Boreholes are one of the main tools for high-precision urban geology exploration and large-scale geological investigations. At present, machine learning based 3D geological modelling methods for borehole data have difficulty building a finer and more complex model and analysing the modelling results with uncertainty. In this paper, a semisupervised learning algorithm using pseudolabels for 3D geological modelling from borehole data is proposed. We establish a 3D geological model using borehole data from a complex real urban local survey area in Shenyang, and the modelling results are compared with implicit surface modelling and traditional machine learning modelling methods. Finally, an uncertainty analysis of the model is made. The results show that the method effectively expands the sample space, the modelling results perform well in terms of spatial morphology and geological semantics, and the proposed modelling method can achieve good modelling results for more complex geological regions.
A Bayesian Belief Network (BN) has been developed to predict fractures in the subsurface during the early stages of oil and gas exploration. The probability of fractures provides a first-order proxy for spatial variations in fracture intensity at a regional scale. Nodes in the BN, representing geologic variables, were linked in a directed acyclic graph to capture key parameters influencing fracture generation over geologic time. The states of the nodes were defined by expert judgment and conditioned by available datasets. Using regional maps with public data from the Horn River Basin in British Columbia, Canada, predictions for spatial variations in the probability of fractures were generated for the Devonian Muskwa shale. The resulting BN analysis was linked to map-based predictions via a geographic information system. The automated process captures human reasoning and improves this through conditional probability calculations for a complex array of geologic influences. A comparison between inferred high fracture intensities and the locations of wells with high production rates suggests a close correspondence. While several factors could account for variations in production rates from the Muskwa shale, higher fracture densities are a likely influence. The process of constructing and cross-validating the BN supports a consistent approach to predict fracture intensities early in exploration and to prioritize data needed to improve the prediction. As such, BNs provide a mechanism to support alignment within exploration groups. As exploration proceeds, the BN can be used to rapidly update predictions. While the BN does not currently represent time-dependent processes and cannot be applied without adjustment to other regions, it offers a fast and flexible approach for fracture prediction in situations characterized by sparse data.
A variational Gaussian process (VGP) model for implicit geological modeling is proposed, capable of modeling linear and non-linear
trends. A new likelihood capable of honoring geological boundaries is introduced, and its performance is compared with the radial basis function (RBF) model. The model is tested on two case studies in mining, one of which relies only on photogrammetric data. The new approach generates more realistic surfaces on qualitative terms and provides an uncertainty measure. It is available as an open-source Python package.
Implicit neural representation (INR) networks are emerging as a powerful framework for learning three-dimensional shape representations of complex objects. These networks can be used effectively to implicitly model three-dimensional geological structures from scattered point data, sampling geological interfaces, units, and orientations of structural features, provided appropriate loss functions associated with data and model constraints are employed during training. The flexibility and scalability of these networks provide a potential framework for integrating new forms of related geological data and knowledge that classical implicit methods cannot easily incorporate. We present a methodology using an efficient INR network architecture, called GeoINR, consisting of multilayer perceptrons (MLP) that advance existing implicit methods for structural geological modelling. The developed methodology expands on the modelling capabilities of existing methods using these networks by: (1) including unconformities into the modelling, (2) introducing constraints on stratigraphic relations as well as global smoothness along with their associated loss functions, and (3) improving training dynamics through the geometrical initialization of learnable network variables. These three enhancements enable the modelling of more complex geology, improved data fitting characteristics, and reduction of modelling artifacts in these settings, as compared to existing INR frameworks for structural geological modelling. A provincial scale case study for the Lower Paleozoic portion of the Western Canadian Sedimentary Basin (WCSB) in Saskatchewan, Canada is presented to demonstrate the modelling capacity of the MLP architecture using the developed methodology. Modelling results illustrate the method’s capacity to fit noisy datasets, represent unconformities, and implicitly model large regional scale three-dimensional geological structures.
The world-class Jiaodong gold province, which is located in the Jiaodong Peninsula, eastern North China Craton, hosts >5,000t of gold resources. The major gold mineralization in this area is mainly controlled by NE-NNE faults, such as the Zhaoping, Jiaojia, and Sanshandao faults. While intensive efforts have been devoted to the investigation of the architecture of the three detachment faults, the fault geometry at depth remains highly uncertain and hotly controversial due to the scarcity of observations, non-uniqueness of the geophysical inversion, and underutilization of the geological knowledge. Along these lines, the deep three-dimensional (3D) geometry of detachment faults from a Bayesian inference perspective is reconstructed in this work. In the Bayesian framework, the multi-source information and the prior geological knowledge are combined into an integrated probabilistic model and the posterior probability distribution of detachment faults is inferred. The resulting posterior distribution permits an estimation of the uncertainty (information entropy) of the 3D fault geometry and the expected shape undulations compared with a prior reference model. More specifically, it demonstrates that specific and detailed 3D models of faults could be inferred by tracking surfaces with maximum information entropy of the geological units. The reconstructed models exhibit detailed 3D geometry of the fault surfaces and their reliability is validated by a recent drill hole at ∼2,300m depth. In terms of the reconstructed 3D models, the dip variations are analyzed and the possible stress directions at deep zones of the three faults are discussed. The shape features of the reconstructed faults are also quantified, which indicates that the fault sections with a steep-to-gentle transition in dip angle are closely associated with the gold mineralization. The extracted 3D gold prospectivity mapping with the spatial features as predicted indicators, highlights seven deep prospecting targets, which are generally consistent with the gently dipping zones and beneath the known orebodies.
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 model construction. The gravity data can be included in the modeling process as the likelihood function in a probabilistic joint inversion framework and allows the quantification of uncertainty in geological modeling directly. A fast but also precise forward gravity simulation is essential to the success of the probabilistic inversion. Hence, 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 individual gravity receivers, respecting the gravimeter locations and the local sensitivities. The kernel method is efficient in terms of both computing and memory use for mesh-free 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. Numerical results are presented and show superior performance compared to the conventional spatial convolution method.
Geological modeling has been widely adopted to investigate underground structures. However, modelingprocesses inevitably have uncertainties due to scarcity of data, measurement errors andsimplification of modeling method. Recent developments in geomodeling methods have introduced aBayesian framework to constrain the model uncertainties by considering additional geophysical datainto the modeling procedure. Markov chain Monte Carlo (MCMC) methods are normally used as tools tosolve the Bayesian inference problem. To achieve a more efficient posterior exploration, advances inMCMC methods utilize derivative information. Hence, we introduce an approach to efficiently evaluatesecond-order derivatives in geological modeling and adopt a Hessian-informed MCMC method, thegeneralized preconditioned Crank-Nicolson (gpCN), as a tool to solve the 3D model-based gravityBayesian inversion problem. The result is compared with two other widely applied MCMC methods,random walk Metropolis-Hasting and Hamiltonian Monte Carlo, on a synthetic geological model and arealistic structural model of the Kevitsa deposit. Our experiment demonstrates that superior performanceis achieved by the gpCN compared to the other two state-of-the-art sampling methods. This shows thepotential of the proposed method to be generalized to more complex models.
Structural geomodeling is a key technology for the visualization and quantification of subsurface systems. Given the limited data and the resulting necessity for geological interpretation to construct these geomodels, uncertainty is pervasive and traditionally unquantified. Probabilistic geomodeling allows for the simulation of uncertainties by automatically constructing geomodel ensembles from perturbed input data sampled from probability distributions. But random sampling of input parameters can lead to construction of geomodels that are unrealistic, either due to modeling artifacts or by not matching known information about the regional geology of the modeled system. We present a method to incorporate geological information in the form of known geomodel topology into stochastic simulations to constrain resulting probabilistic geomodel ensembles using the open-source geomodeling software GemPy. Simulated geomodel realizations are checked against topology information using an approximate Bayesian computation approach to avoid the specification of a likelihood function. We demonstrate how we can infer the posterior distributions of the model parameters using topology information in two experiments: (1) a synthetic geomodel using a rejection sampling scheme (ABC-REJ) to demonstrate the approach and (2) a geomodel of a subset of the Gullfaks field in the North Sea comparing both rejection sampling and a sequential Monte Carlo sampler (ABC-SMC). Possible improvements to processing speed of up to 10.1 times are discussed, focusing on the use of more advanced sampling techniques to avoid the simulation of unfeasible geomodels in the first place. Results demonstrate the feasibility of using topology graphs as a summary statistic to restrict the generation of geomodel ensembles with known geological information and to obtain improved ensembles of probable geomodels which respect the known topology information and exhibit reduced uncertainty using stochastic simulation methods.
Limited by the survey data and current interpretation methods, the modelling processes of fault networks are fraught with
uncertainties. In hydraulic geological engineering, the location uncertainty of faults plays a vital role in decision-making
and engineering safety. However, traditional uncertainty modelling methods have difficulty obtaining accurate uncertainty
quantification and topology representation. To this end, we proposed a novel solution for uncertainty analysis and threedimensional
modelling for faults via a deep learning approach. A spatial uncertainty perception (SUP) method is first presented
based on a modified deep mixture density network (MDN), which can be used to learn the spatial distributions of fault
zones, calculate the probability of fault models, and simulate stochastic models with certain confidence degrees. After that,
a graph representation (GRep) method is developed to express the topological form and geological ages of fault networks.
The GRep makes it possible to automatically simulate the spatial distributions of fault belts, thus providing an effective
way for the uncertainty modelling and assessment of fault networks. The two methods are then performed in the geological
engineering of a practical hydraulic project. The results show that this solution can conduct accurate uncertainty evaluations
and visualizations on fault networks, thus providing suggestions for subsequent geological investigations.
Earth scientists increasingly deal with ‘big data’. For spatial interpolation tasks, variants of kriging have long been regarded as the established geostatistical methods. However, kriging and its variants (such as regression kriging, in which auxiliary variables or derivatives of these are included as covariates) are relatively restrictive models and lack capabilities provided by deep neural networks. Principal among these is feature learning: the ability to learn filters to recognise task-relevant patterns in gridded data such as images. Here, we demonstrate the power of feature learning in a geostatistical context by showing how deep neural networks can automatically learn the complex high-order patterns by which point-sampled target variables relate to gridded auxiliary variables (such as those provided by remote sensing) and in doing so produce detailed maps. In order to cater for the needs of decision makers who require well-calibrated probabilities, we also demonstrate how both aleatoric and epistemic uncertainty can be quantified in our deep learning approach via a Bayesian approximation known as Monte Carlo dropout. In our example, we produce a national-scale probabilistic geochemical map from point-sampled observations with auxiliary data provided by a terrain elevation grid. By combining location information with automatically learned terrain derivatives, our deep learning approach achieves an excellent coefficient of determination ( R 2 = 0.74 ) and near-perfect probabilistic calibration on held-out test data. Our results indicate the suitability of Bayesian deep learning and its feature-learning capabilities for large-scale geostatistical applications where uncertainty matters.
Graphic Abstract
Without properly accounting for both fault kinematics and observations of a faulted surface, it is challenging to create 3D geological models of faulted geological units. Geometries where multiple faults interact, where the faulted surface geometry significantly deviate from a flat plane and where the geological interfaces are poorly characterised by sparse datasets are particular challenges. There are two existing approaches for incorporating faults into geological surface modelling. One approach incorporates the fault displacement into the surface description but does not incorporate fault kinematics and in most cases will produce geologically unexpected results such as shrinking intrusions, fold hinges without offset and layer thickness growth in flat oblique faults. The second approach builds a continuous surface without faulting and then applies a kinematic fault operator to the continuous surface to create the displacement. Both approaches have their strengths; however, neither approach can capture the interaction of faults within complicated fault networks, e.g. fault duplexes, flower structures and listric faults because they either (1) impose an incorrect (not defined by data) fault slip direction or (2) require an over-sampled dataset that describes the faulted surface location. In this study, we integrate the fault kinematics into the implicit surface, by using the fault kinematics to restore observations, and the model domain prior to interpolating the faulted surface. This new approach can build models that are consistent with observations of the faulted surface and fault kinematics. Integrating fault kinematics directly into the implicit surface description allows for complexly faulted stratigraphy and fault–fault interactions to be modelled. Our approach shows significant improvement in capturing faulted surface geometries, especially where the intersection angle between the faulted surface and the fault surface varies (e.g. intrusions, fold series) and when modelling interacting faults (fault duplex).
Computer-assisted stratigraphic correlation can help to produce several scenarios reflecting interpretation uncertainties. In this work, we propose a method which translates sedimentary concepts into a correlation cost for each possible stratigaphic correlation. All these correlation costs are used to populate a cost matrix in order to apply the Dynamic Time Warping algorithm and to compute the n-best correlation sets having the n-least cumulative costs. The proposed cost function reflects prior knowledge about sediment transport direction, and it is tested on two wells penetrating a Middle Jurassic reservoir in the North Sea. Well markers are described by two parameters: (1) the sedimentary facies corresponding to a depositional environment, and (2) the relative distality of the well computed from its position along the sediment transport direction. The main principle of correlation used in this article assumes that a well marker (described by a facies and a distality) cannot be correlated with another well marker described by a depositionally deeper facies at a more proximal position, or a depositionally shallower facies at a more distal position. This approach produces consistent stratigraphic well correlations, and highlights the sensitivity of the solution to the facies zonation and to the relative well distality. Therefore, the proposed rule offers a way to coherently consider chronostratigraphic correlation and the associated uncertainties at the parasequence scale, i.e., at a smaller scale than generally considered in deterministic correlation.
At a regional scale, the best predictor for the 3D geology of the near-subsurface is often the information contained in a geological map. One challenge we face is the difficulty in reproducibly preparing input data for 3D geological models. We present two libraries (map2loop and map2model) that automatically combine the information available in digital geological maps with conceptual information, including assumptions regarding the subsurface extent of faults and plutons to provide sufficient constraints to build a prototype 3D geological model. The information stored in a map falls into three categories of geometric data: positional data, such as the position of faults, intrusive, and stratigraphic contacts; gradient data, such as the dips of contacts or faults; and topological data, such as the age relationships of faults and stratigraphic units or their spatial adjacency relationships. This automation provides significant advantages: it reduces the time to first prototype models; it clearly separates the data, concepts, and interpretations; and provides a homogenous pathway to sensitivity analysis, uncertainty quantification, and value of information studies that require stochastic simulations, and thus the automation of the 3D modelling workflow from data extraction through to model construction. We use the example of the folded and faulted Hamersley Basin in Western Australia to demonstrate a complete workflow from data extraction to 3D modelling using two different open-source 3D modelling engines: GemPy and LoopStructural.
When too few field measurements are available for the geological modeling of complex folded structures, the results of implicit methods typically exhibit an unsatisfactory bubbly aspect. However, in such cases, anisotropy data are often readily available but not fully exploited. Among them, fold axis data are a straightforward indicator of this local anisotropy direction. Focusing on the so-called potential field method, this work aims to evaluate the effect of the incorporation of such data into the modeling process. Given locally sampled fold axis data, this paper proposes to use the second-order derivatives of the scalar field in addition to the existing first-order ones. The mathematical foundation of the approach is developed, and the respective efficiencies of both kinds of constraints are tested. Their integration and impact are discussed based on a synthetic case study, thereby providing practical guidelines to geomodeling tool users on the parsimonious use of data for the geological modeling of complex folded structures.
Three-dimensional structural geomodels are increasingly being used for a wide variety of scientific and societal purposes. Most advanced methods for generating these models are implicit approaches, but they suffer limitations in the types of interpolation constraints permitted, which can lead to poor modeling in structurally complex settings. A geometric deep learning approach, using graph neural networks, is presented in this paper as an alternative to classical implicit interpolation that is driven by a learning through training paradigm. The graph neural network approach consists of a developed architecture utilizing unstructured meshes as graphs on which coupled implicit and discrete geological unit modeling is performed, with the latter treated as a classification problem. The architecture generates three-dimensional structural models constrained by scattered point data, sampling geological units and interfaces as well as planar and linear orientations. The modeling capacity of the architecture for representing geological structures is demonstrated from its application on two diverse case studies. The benefits of the approach are (1) its ability to provide an expressive framework for incorporating interpolation constraints using loss functions and (2) its capacity to deal with both continuous and discrete properties simultaneously. Furthermore, a framework is established for future research for which additional geological constraints can be integrated into the modeling process.
In this contribution we introduce LoopStructural, a new open-source 3D geological modelling Python package (http://www.github.com/Loop3d/LoopStructural, last access: 15 June 2021). LoopStructural provides a generic API for 3D geological modelling applications harnessing the core Python scientific libraries pandas, numpy and scipy. Six different interpolation algorithms, including three discrete interpolators and 3 polynomial trend interpolators, can be used from the same model design. This means that different interpolation algorithms can be mixed and matched within a geological model allowing for different geological objects, e.g. different conformable foliations, fault surfaces and unconformities to be modelled using different algorithms. Geological features are incorporated into the model using a time-aware approach, where the most recent features are modelled first and used to constrain the geometries of the older features. For example, we use a fault frame for characterising the geometry of the fault surface and apply each fault sequentially to the faulted surfaces. In this contribution we use LoopStructural to produce synthetic proof of concepts models and a 86 km × 52 km model of the Flinders Ranges in South Australia using map2loop.
Quantifying the uncertainty in model parameters and output is a critical component in model-driven decision support systems for groundwater management. This paper presents a novel algorithmic approach which fuses Markov Chain Monte Carlo (MCMC) and Machine Learning methods to accelerate uncertainty quantification for groundwater flow models. We formulate the governing mathematical model as a Bayesian inverse problem, considering model parameters as a random process with an underlying probability distribution. MCMC allows us to sample from this distribution, but it comes with some limitations: it can be prohibitively expensive when dealing with costly likelihood functions, subsequent samples are often highly correlated, and the standard Metropolis–Hastings algorithm suffers from the curse of dimensionality. This paper designs a Metropolis–Hastings proposal which exploits a deep neural network (DNN) approximation of a groundwater flow model, to significantly accelerate MCMC sampling. We modify a delayed acceptance (DA) model hierarchy, whereby proposals are generated by running short subchains using an inexpensive DNN approximation, resulting in a decorrelation of subsequent fine model proposals. Using a simple adaptive error model, we estimate and correct the bias of the DNN approximation with respect to the posterior distribution on-the-fly. The approach is tested on two synthetic examples; a isotropic two-dimensional problem, and an anisotropic three-dimensional problem. The results show that the cost of uncertainty quantification can be reduced by up to 50% compared to single-level MCMC, depending on the precomputation cost and accuracy of the employed DNN.
Structural geomodeling is a key technology for the visualization and quantification of subsurface systems. Given the limited data and the resulting necessity for geological interpretation to construct these geomodels, uncertainty is pervasive and traditionally unquantified. Probabilistic geomodeling allows for the simulation of uncertainties by automatically constructing geomodels from perturbed input data sampled from probability distributions. But random sampling of input parameters can lead to construction of geomodels that are unrealistic, either due to modeling artefacts or by not matching known information about the regional geology of the modeled system. We present here a method to incorporate geological information in the form of geomodel topology into stochastic simulations to constrain resulting probabilistic geomodel ensembles. Simulated geomodel realisations are checked against topology information using a likelihood-free Approximate Bayesian Computation approach. We demonstrate how we can learn our input data parameter (prior) distributions on topology information in two experiments: (1) A synthetic geomodel using a rejection sampling scheme (ABC-REJ) to demonstrate the approach; (2) A geomodel of a subset of the Gullfaks field in the North Sea, comparing both rejection sampling and a Sequential Monte Carlo sampler (ABC-SMC). We also discuss possible speed-ups of using more advanced sampling techniques to avoid simulation of unfeasible geomodels in the first place. Results demonstrate the feasibility to use topology as a summary statistic, to restrict the generation of model ensembles with additional geological information and to obtain improved ensembles of probable geomodels using stochastic simulation methods.
In this paper we address the problem of the prohibitively
large computational cost of existing Markov chain Monte
Carlo methods for large--scale applications with high
dimensional parameter spaces, e.g. in uncertainty quantification
in porous media flow. We propose a new multilevel Metropolis-Hastings
algorithm, and give an abstract, problem dependent theorem on
the cost of the new multilevel estimator based on a set of simple,
verifiable assumptions. For a typical model problem in subsurface
flow, we then provide a detailed analysis of these assumptions
and show significant gains over the standard Metropolis-Hastings
estimator. Numerical experiments confirm the analysis and
demonstrate the effectiveness of the method with consistent
reductions of more than an order of magnitude
in the cost of the multilevel estimator over the standard
Metropolis-Hastings algorithm for tolerances .
Existing three-dimensional (3-D) geologic systems are well adapted to high data-density environments, such as at the mine scale where abundant drill core exists, or in basins where 3-D seismic provides stratigraphic con-straints but are poorly adapted to regional geologic problems. There are three areas where improvements in the 3-D workflow need to be made: (1) the handling of uncertainty, (2) the model-building algorithms themselves, and (3) the interface with geophysical inversion. All 3-D models are underconstrained, and at the regional scale this is especially critical for choosing modeling strategies. The practice of only producing a single model ignores the huge uncertainties that underlie model-building processes, and underpins the difficulty in providing meaningful information to end-users about the inherent risk involved in applying the model to solve geologic problems. Future studies need to recognize this and focus on the characterization of model uncertainty, spatially and in terms of geologic features, and produce plausible model suites, rather than single models with unknown validity. The most promising systems for understanding uncertainty use implicit algorithms because they allow the inclusion of some geologic knowledge, for example, age relationships of faults and onlap-offlap relationships. Unfortunately, existing implicit algorithms belie their origins as basin or mine modeling systems because they lack inclusion of normal structural criteria, such as cleavages, lineations, and recognition of polydeformation, all of which are primary tools for the field geologist that is making geologic maps in structurally complex areas. One area of future research will be to establish generalized structural geologic rules that can be built into the modeling process. Finally, and this probably represents the biggest challenge, there is the need for geologic meaning to be maintained during the model-building processes. Current data flows consist of the construction of complex 3-D geologic models that incorporate geologic and geophysical data as well as the prior experience of the modeler, via their interpretation choices. These inputs are used to create a geometric model, which is then transformed into a petrophysical model prior to geophysical inversion. All of the underlying geologic rules are then ignored during the geophysical inversion process. Examples exist that demonstrate that the loss of geologic meaning between geologic and geophysical modeling can be at least partially overcome by increased use of uncertainty characteristics in the workflow.
ScienceDirect 1876-6102 © 2014 The Authors. Published by Elsevier Ltd. This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/3.0/). Peer-review under responsibility of the Austrian Academy of Sciences Abstract Several stochastic approaches have recently been developed to evaluate uncertainties in 3-D geological models based on imprecise geological data. Multiple models are generated through random changes of input parameters, according to defined probability distributions. This leads to a new problem: randomly generated realizations might be topologically different or they might be in contrast to expected geological constraints or settings. We present here a method to enable automatic checks of models based on reliability filters encapsulating aspects of geological knowledge. We apply those filters to a model of a graben structure and obtain an ensemble of acceptable realizations, representing the range of uncertainty, but preserving expected geological features.
A generalized interpolation framework using radial basis functions (RBF)
is presented that implicitly models three-dimensional continuous geological surfaces
from scattered multivariate structural data. Generalized interpolants can use multiple
types of independent geological constraints by deriving for each, linearly independent
functionals. This framework does not suffer from the limitations of previous RBF
approaches developed for geological surface modelling that requires additional offset
points to ensure uniqueness of the interpolant. A particularly useful application
of generalized interpolants is that they allow augmenting on-contact constraints with
gradient constraints as defined by strike-dip data with assigned polarity. This interpolation
problem yields a linear system that is analogous in form to the previously
developed potential field implicit interpolation method based on co-kriging of contact
increments using parametric isotropic covariance functions. The general form of the
mathematical framework presented herein allows us to further expand on solutions by:
(1) including stratigraphic data from above and below the target surface as inequality
constraints (2) modelling anisotropy by data-driven eigen analysis of gradient constraints
and (3) incorporating additional constraints by adding linear functionals to the
system, such as fold axis constraints. Case studies are presented that demonstrate the
advantages and general performance of the surface modelling method in sparse data
environments where the contacts that constrain geological surfaces are rarely exposed
but structural and off-contact stratigraphic data can be plentiful.
In this paper we address the problem of the prohibitively large computational cost of ex-isting Markov chain Monte Carlo methods for large–scale applications with high dimensional parameter spaces, e.g. in uncertainty quantification in porous media flow. We propose a new multilevel Metropolis-Hastings algorithm, and give an abstract, problem dependent theorem on the cost of the new multilevel estimator based on a set of simple, verifiable assumptions. For a typical model problem in subsurface flow, we then provide a detailed analysis of these assumptions and show significant gains over the standard Metropolis-Hastings estimator. Nu-merical experiments confirm the analysis and demonstrate the effectiveness of the method with consistent reductions of a factor of O(10–50) in the ε-cost of the multilevel estimator over the standard Metropolis-Hastings algorithm for tolerances ε around 10 −3 .
Continuous data are often measured or used in binned or rounded form. In this paper we follow up on Hall's work analyzing the effect of using equally-spaced binned data in a kernel density estimator. It is shown that a surprisingly large amount of binning does not adversely affect the integrated mean squared error of a kernel estimate.
The analysis of multiple data sets is required to select a realistic 3D geological model among an infinite number of possibilities. An inverse method that aims at describing the 3D geometry of geological objects is presented. The method takes into account the geology and the physical properties of rocks, while respecting the topological properties of an a priori model. The a priori model is built from the geological data set and its geometry is largely dependent upon assumptions about inaccessible geology at depth. This method, referred to as “total litho-inversion” is a generalised 3D inversion that results in quantifying the lithology and the distribution of rock property in a probabilistic way. Its application is demonstrated through (i) a simple synthetic case and (ii) the relative distribution characterization of granites and diorites in an orogenic domain.
Probabilistic formulation of inverse problems leads to the definition of a probability distribution in the model space. This probability distribution combines a priori information with new information obtained by measuring some observable parameters (data). As, in the general case, the theory linking data with model parameters is nonlinear, the a posteriori probability in the model space may not be east to describe (it may be multimodal, some moments may not be defined, etc.). When analyzing an inverse problem, obtaining a maximum likelihood model is usually not sufficient, as we normally also wish to have information on the resolution power of the data. In the general case we may have a large number of model parameters, and an inspection of the marginal probability densities of interest may be impractical, or even useless. But it is possible to pseudorandomly generate a large collection of models according to the posterior probability distribution and to analyze and display the models in such a way that information on the relative likelihoods of model properties is conveyed to the spectator. This can be accomplished by means of an efficient Monte Carlo method, even in cases where no explicit formula for the a priori distribution is available. The most well known importance sampling method, the Metropolis algorithm, can be generalized, and this gives a method that allows analysis of (possible highly nonlinear) inverse problems with complex a priori information and data with an arbitrary noise distribution.
Most 3D geological modelling tools were designed for the needs of the oil industry and are not suited to the variety of situations encountered in other application domains. Moreover, the usual modelling tools are not able to quantify the uncertainty of the geometric models generated. The potential-field method was designed to build 3D geological models from data available in geology and mineral exploration, namely the geological map and a DTM, structural data, borehole data and interpretations of the geologist. This method considers a geological interface as a particular isosurface of a scalar field defined in the 3D space, called a potential field. The interpolation of that field, based on universal cokriging, provides surfaces that honour all the data. Due to the difficulty of inferring the covariance of the potential field, the first implementation of the method used an a priori covariance given by the user. New developments allow this covariance to be identified from the structural data. This makes it possible to associate sensible cokriging standard deviations to the potential-field estimates and to express the uncertainty of the geometric model. Practical implementation issues for producing 3D geological models are presented: how to handle faults, how to honour borehole ends, how to take relationships between several interfaces into account, how to integrate gravimetric and magnetic data. An application to the geological modelling of the Broken Hill district, Australia, is briefly presented.
The quantification and analysis of uncertainties is important in all cases where maps and models of uncertain properties are the basis for further decisions. Once these uncertainties are identified, the logical next step is to determine how they can be reduced. Information theory provides a framework for the analysis of spatial uncertainties when different subregions are considered as random variables. In the work presented here, joint entropy, conditional entropy, and mutual information are applied for a detailed analysis of spatial uncertainty correlations. The aim is to determine (i) which areas in a spatial analysis share information, and (ii) where, and by how much, additional information would reduce uncertainties. As an illustration, a typical geological example is evaluated: the case of a subsurface layer with uncertain depth, shape and thickness. Mutual information and multivariate conditional entropies are determined based on multiple simulated model realisations. Even for this simple case, the measures not only provide a clear picture of uncertainties and their correlations but also give detailed insights into the potential reduction of uncertainties at each position, given additional information at a different location. The methods are directly applicable to other types of spatial uncertainty evaluations, especially where multiple realisations of a model simulation are analysed. In summary, the application of information theoretic measures opens up the path to a better understanding of spatial uncertainties, and their relationship to information and prior knowledge, for cases where uncertain property distributions are spatially analysed and visualised in maps and models.
Geological three-dimensional (3D) models are constructed to reliably represent a given geological target. The reliability of a model is heavily dependent on the input data and is sensitive to uncertainty. This study examines the uncertainty introduced by geological orientation data by producing a suite of implicit 3d models generated from orientation measurements subjected to uncertainty simulations. The resulting uncertainty associated with different regions of the geological model can be located, quantified and visualised, providing a useful method to assess model reliability. The method is tested on a natural geological setting in the Gippsland Basin, southeastern Australia, where modelled geological surfaces are assessed for uncertainty. The concept of stratigraphic variability is introduced and analysis of the input data is performed using two uncertainty visualisation methods. Uncertainty visualisation through stratigraphic variability is designed to convey the complex concept of 3D model uncertainty to the geoscientist in an effective manner.Uncertainty analysis determined that additional seismic information provides an effective means of constraining modelled geology and reducing uncertainty in regions proximal to the seismic sections. Improvements to the reliability of high uncertainty regions achieved using information gathered from uncertainty visualisations are quantified in a comparative case study. Uncertainty in specific model locations is identified and attributed to possible disagreements between seismic and isopach data. Further improvements to and additional sources of data for the model are proposed based on this information. Finally, a method of introducing stratigraphic variability values as geological constraints for geophysical inversion is presented. © 2012 Elsevier B.V..
Monte Carlo inversion techniques were first used by Earthscientists more than 30 years ago. Since that time they havebeen applied to a wide range of problems, from the inversion of free oscillation data for whole Earth seismic structure to studies at the meter-scale lengths encountered in explorationseismology. This paper traces the development and application of Monte Carlo methods for inverse problems in the Earth sciences and in particular geophysics. The major developments in theory and application are traced from the earliest work of the Russian school and the pioneering studies in the west by Press [1968] to modern importance sampling and ensemble inference methods. The paper is divided into two parts.The first is a literature review, and the second is a summary of Monte Carlo techniques that are currently popular in geophysics.These include simulated annealing, genetic algorithms, and other importance sampling approaches. The objective is to act as both an introduction for newcomers to the field and a comprehensive reference source for researchers already familiar with Monte Carlo inversion. It is our hope that the paper will serve as a timely summary of an expanding and versatile methodology and also encourage applications to new areas of the Earth sciences.
We consider the numerical solution of elliptic partial differential equations with random coefficients. Such problems arise, for example, in uncertainty quantification for groundwater flow. We describe a novel variance reduction technique for the standard Monte Carlo method, called the multilevel Monte Carlo method. The main result is that in certain circumstances the asymptotic cost of solving the stochastic problem is a constant (but moderately large) multiple of the cost of solving the deterministic problem. Numerical calculations demonstrating the effectiveness of the method for one-and two-dimensional model problems arising in groundwater flow are presented.
Geological information can be used to solve many practical and theoretical problems, both within and outside of the discipline of geology. These include analysis of ground stability, predicting subsurface water or hydrocarbon reserves, assessment of risk due to natural hazards, and many others. In many cases, geological information is provided as an a priori component of the solution (i.e. information that existed before the solution was formed and which is incorporated into the solution). Such information is termed ‘geological prior information’.
The key to the successful solution of such problems is to use only reliable geological information. In turn this requires that: (1) multiple geological experts are consulted and any conflicting views reconciled, (2) all prior information includes measures of confidence or uncertainty (without which its reliability and worth is unknown), and (3) as much information as possible is quantitative, and qualitative information or assumptions are clearly defined so that uncertainty or risk in the final result can be evaluated. This paper discusses each of these components and proposes a probabilistic framework for the use and understanding of prior information.
We demonstrate the methodology implicit within this framework with an example: this shows how prior information about typical stacking patterns of sedimentary sequences allows aspects of 2-D platform architecture to be derived from 1-D geological data alone, such as that obtained from an outcrop section or vertical well. This example establishes how the extraction of quantitative, multi-dimensional, geological interpretations is possible using lower dimensional data. The final probabilistic description of the multi-dimensional architecture could then be used as prior information sequentially for a subsequent problem using exactly the same method.
Geological modelling from field data and geological knowledge, Part II -Modelling validation using gravity and magnetic data inversion, Physics of the Earth and Planetary Interiors (2007), doi:10.1016/j.pepi.2008.06.014 This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.
The modeling of subsurface geometry and properties is a key element to understand Earth processes and manage natural hazards
and resources. In this paper, we suggest this field should evolve beyond pure data fitting approaches by integrating geological
concepts to constrain interpretations or test their consistency. This process necessarily calls for adding the time dimension
to 3D modeling, both at the geological and human time scales. Also, instead of striving for one single best model, it is appropriate
to generate several possible subsurface models in order to convey a quantitative sense of uncertainty. Depending on the modeling
objective (e.g., quantification of natural resources, production forecast), this population of models can be ranked. Inverse
theory then provides a framework to validate (or rather invalidate) models which are not compatible with certain types of
observations. We review recent methods to better achieve both stochastic and time-varying geomodeling and advocate that the
application of inversion should rely not only on random field models, but also on geological concepts and parameters.
KeywordsGeomodeling-Uncertainty-Structural restoration-Inverse methods-Geostatistics
Building a 3D geological model from field and subsurface data is a typical task in geological studies involving natural resource
evaluation and hazard assessment. However, there is quite often a gap between research papers presenting case studies or specific
innovations in 3D modeling and the objectives of a typical class in 3D structural modeling, as more and more is implemented
at universities. In this paper, we present general procedures and guidelines to effectively build a structural model made
of faults and horizons from typical sparse data. Then we describe a typical 3D structural modeling workflow based on triangulated
surfaces. Our goal is not to replace software user guides, but to provide key concepts, principles, and procedures to be applied
during geomodeling tasks, with a specific focus on quality control.
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 step — although 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.
A procedure to predict potential Cu areas in the Polish Fore-Sudetic region using structural surface-restoration and logistic regression (LR) analysis is investigated. The predictor variables are deduced from the restored horizon that contains the ore-series. Curvature attribute are calculated for each restored time after applying flexural slip to unfold/unfault the 3D model of the Fore-Sudetic Monocline. It is shown that the curvature represents one of the main structural features related to the Cu mineralization. The maximum curvature exposes internal deformation in the restored layers, evidencing faulting and damaged areas in the 3D model. Thus, the curvature may highlight the fault-fracture systems that drove the fluid circulation from the basement and host the early mineralization stages. In the Cu potential modeling, the curvature, the distance to the Fore-Sudetic Block and the depth of restored Zechstein at Cretaceous time correspond to predictor variables and well known Cu-potential areas are targets. Then, logistic regression (LR) analysis from the free software R, is used for calculating a discriminant function between mineralized and non-mineralized locations. The LR models show that the predicted probability to find Cu-potential locations is higher when the curvature in the surface that represents the mineralized layer has strong values at each restoration step. This last is more remarkable for the restoration steps of Late Paleozoic and Late Triassic. The methodology above explained can be applied in similar fault-fracture dependent sediment-hosted ore systems.
Publisher's Summary:
From first principles to current computer applications, Monte Carlo Calculations in Nuclear Medicine, Second Edition: Applications in Diagnostic Imaging covers the applications of Monte Carlo calculations in nuclear medicine and critically reviews them from a diagnostic perspective. Like the first edition, this book explains the Monte Carlo method and the principles behind SPECT and PET imaging, introduces the reader to some Monte Carlo software currently in use, and gives the reader a detailed idea of some possible applications of Monte Carlo in current research in SPECT and PET.
New chapters in this edition cover codes and applications in pre-clinical PET and SPECT. The book explains how Monte Carlo methods and software packages can be applied to evaluate scatter in SPECT and PET imaging, collimation, and image deterioration. A guide for researchers and students developing methods to improve image resolution, it also demonstrates how Monte Carlo techniques can be used to simulate complex imaging systems.
Many traditional multivariate techniques such as ordination, clustering, classification and discriminant analysis are now routinely used in most fields of application. However, the past decade has seen considerable new developments, particularly in computational ...
The book provides an up-to-date description of the methods used for fitting experimental data, or to estimate model parameters, and to unify these methods into the Inverse Problem Theory. The first part of the book deals with problems and describes Maximum likelihood, Monte Carlo, Least squares, and Least absolute values methods. The second part deals with inverse problems involving functions. Theoretical concepts are emphasized, and the author has all the useful formulas listed, with many special cases included. The book serves as a reference manual.
In such fields as geology and biology, a common problem is that of modelling complex surfaces that are defined by data of various types. Classical modelling techniques based on Bézier and spline interpolations can account for only some of these types of data. The paper proposes a different approach that is based on the discrete smooth interpolation method. In this approach, surfaces are modelled as 2D graphs whose node locations are determined for a wide variety of heterogeneous data.
Bell System Technical Journal, also pp. 623-656 (October)
Markov chain Monte Carlo (MCMC) is a technique for estimating by simulation the expectation of a statistic in a complex model. Successive random selections form a Markov chain, the stationary distribution of which is the target distribution. It is particularly useful for the evaluation of posterior distributions in complex Bayesian models. In the Metropolis–Hastings algorithm, items are selected from an arbitrary “proposal” distribution and are retained or not according to an acceptance rule. The Gibbs sampler is a special case in which the proposal distributions are conditional distributions of single components of a vector parameter. Various special cases and applications are considered.
Keywords:
Metropolis–Hastings;
Gibbs sampler;
stationary distribution;
proposal distribution;
hybrid chain;
reversible jump;
hierarchical;
missing data
We examine the general non-linear inverse problem with a finite number of parameters. In order to permit the incorporation of any a priori information about parameters and any distribution of data (not only of gaussian type) we propose to formulate the problem not using single quantities (such as bounds, means, etc) but using probability density functions for data and parameters. We also want our formulation to allow for the incorporation of theoretical errors, i.e. non-exact theoretical relationships between data and parameters (due to discretization, or incomplete theoretical knowledge); to do that in a natural way we propose to define general theoretical relationships also as probability density functions. We show then that the inverse problem may be formulated as a problem of combination of information: the experimental information about data, the a priori information about parameters, and the theoretical information. With this approach, the general solution of the non-linear inverse problem is unique and consistent (solving the same problem, with the same data, but with a different system of parameters does not change the solution).-Authors
In this paper we present a new and efficient Bayesian method for non-linear three-dimensional large-scale structure inference. We employ a Hamiltonian Monte Carlo (HMC) sampler to obtain samples from a multivariate highly non-Gaussian lognormal Poissonian density posterior given a set of observations. The HMC allows us to take into account the non-linear relations between the observations and the underlying density field which we seek to recover. As the HMC provides a sampled representation of the density posterior any desired statistical summary, such as the mean, mode or variance, can be calculated from the set of samples. Further, it permits us to seamlessly propagate non-Gaussian uncertainty information to any final quantity inferred from the set of samples. The developed method is extensively tested in a variety of test scenarios, taking into account a highly structured survey geometry and selection effects. Tests with a mock galaxy catalogue based on the Millennium Run show that the method is able to recover the filamentary structure of the non-linear density field. The results further demonstrate the feasibility of non-Gaussian sampling in high-dimensional spaces, as required for precision non-linear large-scale structure inference. The HMC is a flexible and efficient method, which permits for simple extension and incorporation of additional observational constraints. Thus, the method presented here provides an efficient and flexible basis for future high-precision large-scale structure inference.
Interpolating physical properties in the subsurface is a recurrent problem in geology. In sedimentary geology, the geometry of the layers is generally known with a precision much superior to that which one can reasonably expect for the properties. The geometry of the layers is affected by folding and faulting since the time of deposition, whereas the distribution of properties is, to a certain extent, determined at the time of deposition. As a consequence, it may be wise to model first the geometry of the layers and then, simplify the geologic equation by removing the influence of that geometry. Inspired from the work of H. E. Wheeler on Time-Stratigraphy, we define, mathematically, a new space where all horizons are horizontal planes and where faults, if any, have disappeared. We surmise that this new space, however approximative, is better to model physical properties of the subsurface whatever the subsequent interpolation method used. The proposed mathematical framework also provides solutions to complex problems such as determination of strains resulting from tectonic events and up-scaling of permeabilities on structured and unstructured 3D grids.
A modeling method that takes into account known points on a geological interface and plane orientation data such as stratification
or foliation planes is described and tested. The orientations data do not necessarily belong to one of the interfaces but
are assumed to sample the main anisotropy of a geological formation as in current geological situations. The problem is to
determine the surfaces which pass through the known points on interfaces and which are compatible with the orientation data.
The method is based on the interpolation of a scalar field defined in the space the gradient in which is orthogonal to the
orientations, given that some points have the same but unknown scalar value (points of the same interface), and that scalar
gradient is known on the other points (foliations). The modeled interfaces are represented as isovalues of the interpolated
field. Preliminary two-dimensional tests carried-out with different covariance models demonstrate the validity of the method,
which is easily transposable in three dimensions.
The inversion of multiple geoscientific datasets to define structures in hard rock terrains has for the most part been limited to geophysical methods supported by rock property constraints. In this paper we expand the range of constraints available to geological modellers by defining a series of misfit functions that are used to define the quality of a given geological model with respect to the available geological data. These misfit functions encompass both spatial and temporal observations, and include primary data such as the location of an outcrop or bore hole location of a given rock type, the orientation of its different generations of foliations, as well as secondary geologic observations such as age relationships. When we combine geologic conditions with geophysical misfit functions, which already exist for many geophysical measures, we have the potential to better constrain two- and 3D models of the Earth. As a first test, a subset of these misfit functions are applied to a set of synthetic three-dimensional volumetric models built with an implicit surface modelling scheme. By perturbing the input structural data within a narrow range, we can simulate a range of significantly different three-dimensional models, for which we can then apply both geological and geophysical misfit functions. Several joint geological/geophysical inversion schemes may be developed based on this methodology, with the ultimate goal being an inversion scheme that simultaneously minimizes both geological and geophysical misfit.
This paper develops a Monte Carlo simulation method for solving option valuation problems. The method simulates the process generating the returns on the underlying asset and invokes the risk neutrality assumption to derive the value of the option. Techniques for improving the efficiency of the method are introduced. Some numerical examples are given to illustrate the procedure and additional applications are suggested.
Best known in our circles for his key role in the renaissance of low- density parity-check (LDPC) codes, David MacKay has written an am- bitious and original textbook. Almost every area within the purview of these TRANSACTIONS can be found in this book: data compression al- gorithms, error-correcting codes, Shannon theory, statistical inference, constrained codes, classification, and neural networks. The required mathematical level is rather minimal beyond a modicum of familiarity with probability. The author favors exposition by example, there are few formal proofs, and chapters come in mostly self-contained morsels richly illustrated with all sorts of carefully executed graphics. With its breadth, accessibility, and handsome design, this book should prove to be quite popular. Highly recommended as a primer for students with no background in coding theory, the set of chapters on error-correcting codes are an excellent brief introduction to the elements of modern sparse-graph codes: LDPC, turbo, repeat-accumulate, and fountain codes are de- scribed clearly and succinctly. As a result of the author's research on the field, the nine chapters on neural networks receive the deepest and most cohesive treatment in the book. Under the umbrella title of Probability and Inference we find a medley of chapters encompassing topics as varied as the Viterbi algorithm and the forward-backward algorithm, Monte Carlo simu- lation, independent component analysis, clustering, Ising models, the saddle-point approximation, and a sampling of decision theory topics. The chapters on data compression offer a good coverage of Huffman and arithmetic codes, and we are rewarded with material not usually encountered in information theory textbooks such as hash codes and efficient representation of integers. The expositions of the memoryless source coding theorem and of the achievability part of the memoryless channel coding theorem stick closely to the standard treatment in (1), with a certain tendency to over- simplify. For example, the source coding theorem is verbalized as: " i.i.d. random variables each with entropy can be compressed into more than bits with negligible risk of information loss, as ; conversely if they are compressed into fewer than bits it is virtually certain that informa- tion will be lost." Although no treatment of rate-distortion theory is offered, the author gives a brief sketch of the achievability of rate with bit- error rate , and the details of the converse proof of that limit are left as an exercise. Neither Fano's inequality nor an operational definition of capacity put in an appearance. Perhaps his quest for originality is what accounts for MacKay's pro- clivity to fail to call a spade a spade. Almost-lossless data compres- sion is called "lossy compression;" a vanilla-flavored binary hypoth-
While the prediction of observations is a forward problem, the use of actual observations to infer the properties of a model is an inverse problem. Inverse problems are difficult because they may not have a unique solution. The description of uncertainties plays a central role in the theory, which is based on probability theory. This book proposes a general approach that is valid for linear as well as for nonlinear problems. The philosophy is essentially probabilistic and allows the reader to understand the basic difficulties appearing in the resolution of inverse problems. The book attempts to explain how a method of acquisition of information can be applied to actual real-world problems, and many of the arguments are heuristic.
Prompted by recent developments in inverse theory, Inverse Problem Theory and Methods for Model Parameter Estimation is a completely rewritten version of a 1987 book by the same author. In this version there are lots of algorithmic details for Monte Carlo methods, least-squares discrete problems, and least-squares problems involving functions. In addition, some notions are clarified, the role of optimization techniques is underplayed, and Monte Carlo methods are taken much more seriously. The first part of the book deals exclusively with discrete inverse problems with a finite number of parameters while the second part of the book deals with general inverse problems.
While the forward problem has (in deterministic physics) a unique solution, the inverse problem does not. As an example, consider measurements of the gravity field around a planet: given the distribution of mass inside the planet, we can uniquely predict the values of the gravity field around the planet (forward problem), but there are different distributions of mass that give exactly the same gravity field in the space outside the planet. Therefore, the inverse problem — of inferring the mass distribution from observations of the gravity field — has multiple solutions (in fact, an infinite number).
We make an analogy between images and statistical mechanics systems. Pixel gray levels and the presence and orientation of edges are viewed as states of atoms or molecules in a lattice-like physical system. The assignment of an energy function in the physical system determines its Gibbs distribution. Because of the Gibbs distribution, Markov random field (MRF) equivalence, this assignment also determines an MRF image model. The energy function is a more convenient and natural mechanism for embodying picture attributes than are the local characteristics of the MRF. For a range of degradation mechanisms, including blurring, nonlinear deformations, and multiplicative or additive noise, the posterior distribution is an MRF with a structure akin to the image model. By the analogy, the posterior distribution defines another (imaginary) physical system. Gradual temperature reduction in the physical system isolates low energy states (``annealing''), or what is the same thing, the most probable states under the Gibbs distribution. The analogous operation under the posterior distribution yields the maximum a posteriori (MAP) estimate of the image given the degraded observations. The result is a highly parallel ``relaxation'' algorithm for MAP estimation. We establish convergence properties of the algorithm and we experiment with some simple pictures, for which good restorations are obtained at low signal-to-noise ratios.
Automated reliability assessment is essential for systems that entail dynamic adaptation based on runtime mission-specific requirements. One approach along this direction is to monitor and assess the system using machine learning-based software defect prediction techniques. Due to the dynamic nature of software data collected, Instance-based learning algorithms are proposed for the above purposes. To evaluate the accuracy of these methods, the paper presents an empirical analysis of four different real-time software defect data sets using different predictor models.
The results show that a combination of 1R and Instance-based learning along with Consistency-based subset evaluation technique provides a relatively better consistency in achieving accurate predictions as compared with other models. No direct relationship is observed between the skewness present in the data sets and the prediction accuracy of these models. Principal Component Analysis (PCA) does not show a consistent advantage in improving the accuracy of the predictions. While random reduction of attributes gave poor accuracy results, simple Feature Subset Selection methods performed better than PCA for most prediction models. Based on these results, the paper presents a high-level design of an Intelligent Software Defect Analysis tool (ISDAT) for dynamic monitoring and defect assessment of software modules.
All geological models are subject to several kinds of uncertainty. These can be classified into three different types: data imprecision and quality, inherent randomness and incomplete knowledge. With our approach, we address uncertainties introduced by input data quality in complex three-dimensional (3D) models of subsurface structures (geological models). As input data, we consider parameters of geological structures, i.e. formation and fault boundary points and orientation measurements. Our method consists of five steps: construction of an initial geological model with an implicit potential-field method, assignment of probability distributions to data positions and orientation measurements, simulation of several input data sets, construction of several model realisations based on these simulated data sets and finally the visualisation and analysis of the uncertainties. We test our approach in two generic models, a simple graben setting and a complex dome structure. The first model shows that our approach can evaluate uncertainties from different structures and their interaction. Furthermore, it indicates that the final uncertainty of the model is not simply the sum of all input data uncertainties but complex interactions exist. The second example demonstrates that our approach can handle full three-dimensional settings with overturned surfaces. Results of the uncertainty simulation can be visualised and analysed in several ways, ranging from borehole histograms to uncertainty maps. For complex 3D visualisation and analysis, we use indicator functions. When we apply these, we can treat the visualisation of uncertainties in complex settings in a self-consistent manner. With our simulation method, it is possible to analyse and visualise the uncertainties directly introduced by imprecision in the input data. Our approach is intuitive and straight-forward and suitable in both simple and complex geological settings. It enables detailed insights into the model quality, even for the non-expert. In cases where a geological model is the basis for geophysical simulation, it opens-up the way to geological data-driven ensemble modelling and inversion.
A general method, suitable for fast computing machines, for investigating such properties as equations of state for substances consisting of interacting individual molecules is described. The method consists of a modified Monte Carlo integration over configuration space. Results for the two-dimensional rigid-sphere system have been obtained on the Los Alamos MANIAC and are presented here. These results are compared to the free volume equation of state and to a four-term virial coefficient expansion.
A general method, suitable for fast computing machines, for investigating
such properties as equations of state for substances consisting of
interacting individual molecules is described. The method consists
of a modified Monte Carlo integration over configuration space. Results
for the two-dimensional rigid-sphere system have been obtained on
the Los Alamos MANIAC and are presented here. These results are compared
to the free volume equation of state and to a four-term virial coefficient
expansion. The Journal of Chemical Physics is copyrighted by The
American Institute of Physics.