Mathematical Geosciences最新文献

{"title":"Causal Discovery and Deep Learning Algorithms for Detecting Geochemical Patterns Associated with Gold-Polymetallic Mineralization: A Case Study of the Edongnan Region","authors":"Zijing Luo, Renguang Zuo","doi":"10.1007/s11004-024-10153-6","DOIUrl":"https://doi.org/10.1007/s11004-024-10153-6","url":null,"abstract":"<p>The identification of mineral deposit footprints by processing geochemical survey data constitutes a crucial stage in mineral exploration because it provides valuable and substantial information for future prospecting endeavors. However, the selection of appropriate pathfinder elements and the recognition of their anomalous patterns for determining metallogenic favorability based on geochemical survey data remain challenging tasks because of the complex interactions among different geochemical elements and the highly nonlinear and heterogeneous characteristics of their spatial distribution patterns. This study investigated the application of a causal discovery algorithm and deep learning models to identify geochemical anomaly patterns associated with mineralization. Using gold-polymetallic deposits in the Edongnan region of China as a case study, stream sediment samples containing concentrations of 39 elements were collected and preprocessed using a centered log-ratio transformation, addressing the closure effect of compositional data. The combination of the synthetic minority oversampling technique, Tomek link algorithm, and causal discovery algorithm to explore the potential associations and influences among geochemical elements provides new insights into the selection of pathfinder elements. Regarding the problem of identifying anomalous spatial distribution patterns in pathfinder elements and considering that the formation of mineral deposits is the result of various geological processes interacting under specific spatiotemporal conditions, we proposed a hybrid deep learning model called VAE-CAPSNET-GAN, which combines a variational autoencoder (VAE), capsule network (CAPSNET), and generative adversarial network (GAN). The model was designed to capture the spatial distribution characteristics of pathfinder elements and the spatial coupling relationships between mineral deposits and geochemical anomalies, enabling the recognition of geochemical anomaly patterns related to mineralization. The results showed that, compared to the VAE model, which also uses reconstruction error as the anomaly detection principle, VAE-CAPSNET-GAN exhibited superior performance in identifying known mineral deposits and delineating anomalous areas aligned more closely with the established metallogenic model. Furthermore, this weakens the impact of overlapping information. Multiple outcomes indicated that an integrated analytical framework combining a causal discovery algorithm with deep learning models can provide valuable clues for further delineating prospects.</p>","PeriodicalId":51117,"journal":{"name":"Mathematical Geosciences","volume":"43 1","pages":""},"PeriodicalIF":2.6,"publicationDate":"2024-07-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141740500","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"地球科学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Non-Gaussian Ensemble Optimization","authors":"Mathias M. Nilsen, Andreas S. Stordal, Patrick N. Raanes, Rolf J. Lorentzen, Kjersti S. Eikrem","doi":"10.1007/s11004-024-10148-3","DOIUrl":"https://doi.org/10.1007/s11004-024-10148-3","url":null,"abstract":"<p>Ensemble-based optimization (EnOpt), commonly used in reservoir management, can be seen as a special case of a natural evolution algorithm. Stein’s lemma gives a new interpretation of EnOpt. This interpretation enables us to study EnOpt in the context of general mutation distributions. In this paper, a non-Gaussian generalization of EnOpt (GenOpt) is proposed, where the control gradient is estimated using Stein’s lemma, and the mutation distribution is updated separately via natural evolution. For the multivariate case, a Gaussian copula is used to represent dependencies between the marginals. The correlation matrix is also iteratively optimized. It is shown that using beta distributions as marginals in the GenOpt algorithm addresses the truncation problem that sometimes arises when applying EnOpt on bounded optimization problems. The performance of the proposed optimization algorithm is evaluated on several test cases. The experiments indicate that GenOpt is less dependent on the chosen hyperparameters, and it is able to converge more quickly than EnOpt on a reservoir management test case.</p>","PeriodicalId":51117,"journal":{"name":"Mathematical Geosciences","volume":"65 1","pages":""},"PeriodicalIF":2.6,"publicationDate":"2024-06-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141550394","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"地球科学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"A Method to Extract Image Features and Lineaments Based on a Multi-hillshade Continuous Wavelet Transform","authors":"Man Hyok Song, Jin Gyong Ho, Chol Kim, Yong O. Chol, Song Lyu","doi":"10.1007/s11004-024-10146-5","DOIUrl":"https://doi.org/10.1007/s11004-024-10146-5","url":null,"abstract":"<p>This paper presents a new method for extracting the image features and lineaments related to local extrema of an image or a digital elevation model (DEM) such as ridges and valleys based on the continuous wavelet transform (CWT) of a set of variously illuminated hillshades. The method originates from the principle that a hillshade can exactly reflect the lineaments nearly perpendicular to the illumination direction of the hillshade, but not other ones. The method consists of four steps: (1) preparation of a set of differently illuminated hillshades of the input data, (2) detection of directional edges nearly perpendicular to the illumination direction from each hillshade based on the CWT, (3) a combination of multidirectional edges into an omnidirectional feature image, and (4) identification of lineaments through linkage and linearization of image feature lines. CWT coefficients of each hillshade are used to calculate the gradient and its direction of the hillshade. For each hillshade, directional edge pixels where the gradient direction is parallel to the illumination direction are selectively detected to form accurate and solitary image feature lines related to local extrema of the input data. Directional edges of each hillshade are easily classified into positive and negative edges using the hillshade gradient. As they have similar directions, they are easily linked to form longer line raster objects, which are converted into vector objects, that is, directional lineaments. The multidirectional edges and lineaments given from all the hillshades are combined to form an omnidirectional feature image and a group of omnidirectional lineaments. Its application to real DEMs shows the demonstrated advantages of the proposed method over other methods and the similarity between detected lineaments and fault lines in the study area.</p>","PeriodicalId":51117,"journal":{"name":"Mathematical Geosciences","volume":"43 1","pages":""},"PeriodicalIF":2.6,"publicationDate":"2024-06-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141256075","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"地球科学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Addressing Configuration Uncertainty in Well Conditioning for a Rule-Based Model","authors":"Oscar Ovanger, Jo Eidsvik, Jacob Skauvold, Ragnar Hauge, Ingrid Aarnes","doi":"10.1007/s11004-024-10144-7","DOIUrl":"https://doi.org/10.1007/s11004-024-10144-7","url":null,"abstract":"<p>Rule-based reservoir models incorporate rules that mimic actual sediment deposition processes for accurate representation of geological patterns of sediment accumulation. Bayesian methods combine rule-based reservoir modelling and well data, with geometry and placement rules as part of the prior and well data accounted for by the likelihood. The focus here is on a shallow marine shoreface geometry of ordered sedimentary packages called bedsets. Shoreline advance and sediment build-up are described through progradation and aggradation parameters linked to individual bedset objects. Conditioning on data from non-vertical wells is studied. The emphasis is on the role of ‘configurations’—the order and arrangement of bedsets as observed within well intersections in establishing the coupling between well observations and modelled objects. A conditioning algorithm is presented that explicitly integrates uncertainty about configurations for observed intersections between the well and the bedset surfaces. As data volumes increase and model complexity grows, the proposed conditioning method eventually becomes computationally infeasible. It has significant potential, however, to support the development of more complex models and conditioning methods by serving as a reference for consistency in conditioning.</p>","PeriodicalId":51117,"journal":{"name":"Mathematical Geosciences","volume":"51 1","pages":""},"PeriodicalIF":2.6,"publicationDate":"2024-06-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141195979","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"地球科学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Generalized Solution for Double-Porosity Flow Through a Graded Excavation Damaged Zone","authors":"Kristopher L. Kuhlman","doi":"10.1007/s11004-024-10143-8","DOIUrl":"https://doi.org/10.1007/s11004-024-10143-8","url":null,"abstract":"<p>Prediction of flow to boreholes or excavations in fractured low-permeability rocks is important for resource extraction and disposal or sequestration activities. Analytical solutions for fluid pressure and flowrate, when available, are powerful, insightful, and efficient tools enabling parameter estimation and uncertainty quantification. A flexible porous media flow solution for arbitrary physical dimensions is derived and extended to double porosity for converging radial flow when permeability and porosity decrease radially as a power law away from a borehole or opening. This distribution can arise from damage accumulation due to stress relief associated with drilling or mining. The single-porosity graded conductivity solution was initially found for heat conduction, the arbitrary dimension flow solution comes from hydrology, and the solution with both arbitrary dimension and graded permeability distribution appeared in reservoir engineering. These existing solutions are combined and extended here to two implementations of the double-porosity conceptual model, for both a simpler thin-film mass transfer and more physically realistic diffusion between fracture and matrix. This work presents a new specified-flowrate solution with wellbore storage for the simpler double-porosity model, and a new, more physically realistic solution for any wellbore boundary condition. A new closed-form expression is derived for the matrix diffusion solution (applicable to both homogeneous and graded problems), improving on previous infinite series expressions.</p>","PeriodicalId":51117,"journal":{"name":"Mathematical Geosciences","volume":"12 1","pages":""},"PeriodicalIF":2.6,"publicationDate":"2024-05-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140927233","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"地球科学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Principal Component Analysis for Distributions Observed by Samples in Bayes Spaces","authors":"Ivana Pavlů, Jitka Machalová, Raimon Tolosana-Delgado, Karel Hron, Kai Bachmann, Karl Gerald van den Boogaart","doi":"10.1007/s11004-024-10142-9","DOIUrl":"https://doi.org/10.1007/s11004-024-10142-9","url":null,"abstract":"<p>Distributional data have recently become increasingly important for understanding processes in the geosciences, thanks to the establishment of cost-efficient analytical instruments capable of measuring properties over large numbers of particles, grains or crystals in a sample. Functional data analysis allows the direct application of multivariate methods, such as principal component analysis, to such distributions. However, these are often observed in the form of samples, and thus incur a sampling error. This additional sampling error changes the properties of the multivariate variance and thus the number of relevant principal components and their direction. The result of the principal component analysis becomes an artifact of the sampling error and can negatively affect the subsequent data analysis. This work presents a way of estimating this sampling error and how to confront it in the context of principal component analysis, where the principal components are obtained as a linear combination of elements of a newly constructed orthogonal spline basis. The effect of the sampling error and the effectiveness of the correction is demonstrated with a series of simulations. It is shown how the interpretability and reproducibility of the principal components improve and become independent of the selection of the basis. The proposed method is then applied on a dataset of grain size distributions in a geometallurgical dataset from Thaba mine in the Bushveld complex.</p>","PeriodicalId":51117,"journal":{"name":"Mathematical Geosciences","volume":"42 1","pages":""},"PeriodicalIF":2.6,"publicationDate":"2024-05-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140883589","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"地球科学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Geologically Constrained Convolutional Neural Network for Mineral Prospectivity Mapping","authors":"Fanfan Yang, Renguang Zuo","doi":"10.1007/s11004-024-10141-w","DOIUrl":"https://doi.org/10.1007/s11004-024-10141-w","url":null,"abstract":"<p>Various deep learning algorithms (DLAs) have been successfully employed for mineral prospectivity mapping (MPM) to support mineral exploration, due to their superior nonlinear extraction capabilities. DLAs algorithms are typically purely data-driven approaches that may ignore the geological domain knowledge. This renders the predictive results inconsistent with the mineralization mechanism and results in poor interpretation. In this study, a geologically constrained convolutional neural network (CNN) that involves soft and hard geological constraints was proposed for mapping gold polymetallic mineralization potential in western Henan Province of China. A penalty term based on the controlling equation of the spatial coupling relationship between the ore-controlling strata and gold deposits was constructed as a soft constraint to guide the CNN model training according to additional prior geological knowledge. In addition, domain knowledge related to mineralization processes and a geochemical indicator were simultaneously embedded as hard constraints in the feature extractor and classifier of the CNN, respectively, to control the model training based on the mineralization mechanism. The comparative experiments demonstrated that the geologically constrained CNN was superior to other models, thus indicating that the coupling of data and domain knowledge is effective for MPM and further improves the rationality and interpretability of the obtained results.</p>","PeriodicalId":51117,"journal":{"name":"Mathematical Geosciences","volume":"22 1","pages":""},"PeriodicalIF":2.6,"publicationDate":"2024-04-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140812500","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"地球科学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"And the 2024 Krumbein Medalist of the IAMG is…","authors":"Eric Grunsky","doi":"10.1007/s11004-024-10140-x","DOIUrl":"https://doi.org/10.1007/s11004-024-10140-x","url":null,"abstract":"","PeriodicalId":51117,"journal":{"name":"Mathematical Geosciences","volume":"6 1","pages":""},"PeriodicalIF":2.6,"publicationDate":"2024-04-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140588575","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"地球科学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Estimating Rock Composition from Replicate Geochemical Analyses: Theory and Application to Magmatic Rocks of the GeoPT Database","authors":"Maxime Keutgen De Greef, Gert Jan Weltje, Irène Gijbels","doi":"10.1007/s11004-024-10138-5","DOIUrl":"https://doi.org/10.1007/s11004-024-10138-5","url":null,"abstract":"<p>Chemical analyses of powdered rocks by different laboratories often yield varying results, requiring estimation of the rock’s true composition and associated uncertainty. Challenges arise from the peculiar nature of geochemical data. Traditionally, major and trace elements have been measured using different methods, resulting in chemical analyses where the sum of the parts fluctuates around 1 rather than precisely totaling 1. Additionally, all chemical analyses contain an undisclosed mass fraction representing undetected chemical elements. Because of this undisclosed and unknown mass fraction, geochemical data represent a particular kind of compositional data in which closure to unity is not guaranteed. We argue that chemical analyses exist in the hypercube while being sampled from a true composition residing in the simplex. Therefore, we propose an algorithm that generates random chemical analyses by simulating the data acquisition protocol in geochemistry. Using the algorithm’s output, we measure the bias and mean squared error (MSE) of various estimators of the true mean composition. Additionally, we explore the impact of missing values on estimator performance. Our findings reveal that the optimized binary log-ratio mean, a new estimator, exhibits the lowest MSE and bias. It performs well even with up to 70% missing values, in contrast to other classical estimators such as the arithmetic mean or the geometric mean. Applying our approach to the GeoPT database, which contains replicate analyses of igneous rocks from numerous geochemical laboratories, we introduce an outlier detection technique based on the Mahalanobis distance between a laboratory’s logit coordinates and the optimized mean estimate. This enables a probabilistic ranking of laboratories based on the atypicality of their performance. Finally, we offer an accessible R implementation of our findings through the GitHub repository linked to this paper [subject classification numbers: 10 (compositions) 85 (statistics)].</p>","PeriodicalId":51117,"journal":{"name":"Mathematical Geosciences","volume":"56 1","pages":""},"PeriodicalIF":2.6,"publicationDate":"2024-04-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140588583","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"地球科学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"2.5D Hexahedral Meshing for Reservoir Simulations","authors":"","doi":"10.1007/s11004-023-10106-5","DOIUrl":"https://doi.org/10.1007/s11004-023-10106-5","url":null,"abstract":"<h3>Abstract</h3> <p>We present a new method for generating pure hexahedral meshes for reservoir simulations. The grid is obtained by extruding a quadrangular mesh, using ideas from the latest advances in computational geometry, specifically the generation of semi-structured quadrangular meshes based on global parameterization. Hexahedral elements are automatically constructed to smoothly honor the geometry of input features (domain boundaries, faults, and horizons), thus making it possible to be used for multiple types of physical simulations on the same mesh. The main contributions are as follows: the introduction of a new semi-structured hexahedral meshing workflow producing high-quality meshes for a wide range of fault systems, and the study and definition of weak verticality on triangulated surface meshes. This allows us to design better and more robust algorithms during the extrusion phase along non-vertical faults. We demonstrate (i) the simplicity of using such hexahedral meshes generated using the proposed method for coupled flow-geomechanics simulations with state-of-the-art simulators for reservoir studies, and (ii) the possibility of using such semi-structured hexahedral meshes in commercial structured flow simulators, offering an alternative gridding approach to handle a wider family of fault networks without recourse to the stair-step fault approximation. </p>","PeriodicalId":51117,"journal":{"name":"Mathematical Geosciences","volume":"56 1","pages":""},"PeriodicalIF":2.6,"publicationDate":"2024-04-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140588388","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"地球科学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}