Articles | Volume 16, issue 23
https://doi.org/10.5194/gmd-16-6987-2023
© Author(s) 2023. This work is distributed under
the Creative Commons Attribution 4.0 License.
the Creative Commons Attribution 4.0 License.
https://doi.org/10.5194/gmd-16-6987-2023
© Author(s) 2023. This work is distributed under
the Creative Commons Attribution 4.0 License.
the Creative Commons Attribution 4.0 License.
GeoINR 1.0: an implicit neural network approach to three-dimensional geological modelling
Michael Hillier
CORRESPONDING AUTHOR
Geological Survey of Canada, Natural Resources Canada, 601 Booth Street, Ottawa, ON K1A 0E8, Canada
Florian Wellmann
Computational Geoscience and Reservoir Engineering (CGRE), RWTH Aachen University, Mathieustr. 30, 52074 Aachen, Germany
Eric A. de Kemp
Geological Survey of Canada, Natural Resources Canada, 601 Booth Street, Ottawa, ON K1A 0E8, Canada
Boyan Brodaric
Geological Survey of Canada, Natural Resources Canada, 601 Booth Street, Ottawa, ON K1A 0E8, Canada
Ernst Schetselaar
Geological Survey of Canada, Natural Resources Canada, 601 Booth Street, Ottawa, ON K1A 0E8, Canada
Karine Bédard
Geological Survey of Canada, Natural Resources Canada, 490 rue de la Couronne, Quebec City, QC G1K 9A9, Canada
Related authors
Marion N. Parquer, Eric A. de Kemp, Boyan Brodaric, and Michael J. Hillier
Geosci. Model Dev., 18, 71–100, https://doi.org/10.5194/gmd-18-71-2025, https://doi.org/10.5194/gmd-18-71-2025, 2025
Short summary
Short summary
This is a proof-of-concept paper outlining a general approach to how 3D geological models would be checked to be geologically
reasonable. We do this with a consistency-checking tool that looks at geological feature pairs and their spatial, temporal, and internal polarity characteristics. The idea is to assess if geological relationships from a specific 3D geological model match what is allowed in the real world from the perspective of geological principles.
Friedrich Carl, Peter Achtziger-Zupančič, Jian Yang, Marlise Colling Cassel, and Florian Wellmann
EGUsphere, https://doi.org/10.5194/egusphere-2025-3203, https://doi.org/10.5194/egusphere-2025-3203, 2025
This preprint is open for discussion and under review for Solid Earth (SE).
Short summary
Short summary
A method for shape quantification based on geometrical parameters is proposed alongside a set of regular geometries established as geomodeling benchmarks. Dimensions, gradient and curvature data is obtained on cross-sections. Data analyses provide insight into the main geometrical characteristics of the benchmark models and visualizes geometrical dis-/similarities between bodies. The method and benchmarks are usable in geomodeling workflows and structural comparisons based on sparse data.
Denise Degen, Moritz Ziegler, Oliver Heidbach, Andreas Henk, Karsten Reiter, and Florian Wellmann
Solid Earth, 16, 477–502, https://doi.org/10.5194/se-16-477-2025, https://doi.org/10.5194/se-16-477-2025, 2025
Short summary
Short summary
Obtaining reliable estimates of the subsurface state distributions is essential to determine the location of, e.g., potential nuclear waste disposal sites. However, providing these is challenging since it requires solving the problem numerous times, yielding high computational cost. To overcome this, we use a physics-based machine learning method to construct surrogate models. We demonstrate how it produces physics-preserving predictions, which differentiates it from purely data-driven approaches.
Marion N. Parquer, Eric A. de Kemp, Boyan Brodaric, and Michael J. Hillier
Geosci. Model Dev., 18, 71–100, https://doi.org/10.5194/gmd-18-71-2025, https://doi.org/10.5194/gmd-18-71-2025, 2025
Short summary
Short summary
This is a proof-of-concept paper outlining a general approach to how 3D geological models would be checked to be geologically
reasonable. We do this with a consistency-checking tool that looks at geological feature pairs and their spatial, temporal, and internal polarity characteristics. The idea is to assess if geological relationships from a specific 3D geological model match what is allowed in the real world from the perspective of geological principles.
Denise Degen, Daniel Caviedes Voullième, Susanne Buiter, Harrie-Jan Hendricks Franssen, Harry Vereecken, Ana González-Nicolás, and Florian Wellmann
Geosci. Model Dev., 16, 7375–7409, https://doi.org/10.5194/gmd-16-7375-2023, https://doi.org/10.5194/gmd-16-7375-2023, 2023
Short summary
Short summary
In geosciences, we often use simulations based on physical laws. These simulations can be computationally expensive, which is a problem if simulations must be performed many times (e.g., to add error bounds). We show how a novel machine learning method helps to reduce simulation time. In comparison to other approaches, which typically only look at the output of a simulation, the method considers physical laws in the simulation itself. The method provides reliable results faster than standard.
Mohammad Moulaeifard, Simon Bernard, and Florian Wellmann
Geosci. Model Dev., 16, 3565–3579, https://doi.org/10.5194/gmd-16-3565-2023, https://doi.org/10.5194/gmd-16-3565-2023, 2023
Short summary
Short summary
In this work, we propose a flexible framework to generate and interact with geological models using explicit surface representations. The essence of the work lies in the determination of the flexible control mesh, topologically similar to the main geological structure, watertight and controllable with few control points, to manage the geological structures. We exploited the subdivision surface method in our work, which is commonly used in the animation and gaming industry.
Michał P. Michalak, Lesław Teper, Florian Wellmann, Jerzy Żaba, Krzysztof Gaidzik, Marcin Kostur, Yuriy P. Maystrenko, and Paulina Leonowicz
Solid Earth, 13, 1697–1720, https://doi.org/10.5194/se-13-1697-2022, https://doi.org/10.5194/se-13-1697-2022, 2022
Short summary
Short summary
When characterizing geological/geophysical surfaces, various geometric attributes are calculated, such as dip angle (1D) or dip direction (2D). However, the boundaries between specific values may be subjective and without optimization significance, resulting from using default color palletes. This study proposes minimizing cosine distance among within-cluster observations to detect 3D anomalies. Our results suggest that the method holds promise for identification of megacylinders or megacones.
Eric A. de Kemp
Geosci. Model Dev., 14, 6661–6680, https://doi.org/10.5194/gmd-14-6661-2021, https://doi.org/10.5194/gmd-14-6661-2021, 2021
Short summary
Short summary
This is a proof of concept and review paper of spatial agents, with initial research focusing on geomodelling. The results may be of interest to others working on complex regional geological modelling with sparse data. Structural agent-based swarming behaviour is key to advancing this field. The study provides groundwork for research in structural geology 3D modelling with spatial agents. This work was done with NetLogo, a free agent modelling platform used mostly for teaching complex systems.
Alexander Schaaf, Miguel de la Varga, Florian Wellmann, and Clare E. Bond
Geosci. Model Dev., 14, 3899–3913, https://doi.org/10.5194/gmd-14-3899-2021, https://doi.org/10.5194/gmd-14-3899-2021, 2021
Short summary
Short summary
Uncertainty is an inherent property of any model of the subsurface. We show how geological topology information – how different regions of rocks in the subsurface are connected – can be used to train uncertain geological models to reduce uncertainty. More widely, the method demonstrates the use of probabilistic machine learning (Bayesian inference) to train structural geological models on auxiliary geological knowledge that can be encoded in graph structures.
Stephanie Thiesen, Diego M. Vieira, Mirko Mälicke, Ralf Loritz, J. Florian Wellmann, and Uwe Ehret
Hydrol. Earth Syst. Sci., 24, 4523–4540, https://doi.org/10.5194/hess-24-4523-2020, https://doi.org/10.5194/hess-24-4523-2020, 2020
Short summary
Short summary
A spatial interpolator has been proposed for exploring the information content of the data in the light of geostatistics and information theory. It showed comparable results to traditional interpolators, with the advantage of presenting generalization properties. We discussed three different ways of combining distributions and their implications for the probabilistic results. By its construction, the method provides a suitable and flexible framework for uncertainty analysis and decision-making.
Cited articles
Atzmon, M. and Lipman, Y.: Sal: Sign agnostic learning of shapes from raw data, in: Proceedings of the IEEE/CVF Conference on Computer Vision and Pattern Recognition, Seattle, WA, USA, 2565–2574, https://doi.org/10.1109/CVPR42600.2020.00264, 2020.
Bédard, K., Marsh, A., Hillier, M., Music, T.: 3D geological model of the Western Canadian Sedimentary Basin in Saskatchewan, Canada, Geological Survey of Canada, Open File 8969, https://doi.org/10.4095/331747, 2023.
Bi, Z., Wu, X., Geng, Z., and Li, H.: Deep relative geologic time: a deep learning method for simultaneously interpreting 3- D seismic horizons and faults, J. Geophys. Res.-Sol. Ea., 126, e2021JB021882, https://doi.org/10.1029/2021JB021882, 2021.
Bi, Z., Wu, X., Li, Z., Chang, D., and Yong, X.: DeepISMNet: three-dimensional implicit structural modeling with convolutional neural network, Geosci. Model Dev., 15, 6841–6861, https://doi.org/10.5194/gmd-15-6841-2022, 2022.
Boisvert, J. B., Manchuk, J. G., and Deutsch, C. V.: Kriging in the Presence of Locally Varying Anisotropy Using Non-Euclidean Distances, Math. Geosci., 41, 585–601, https://doi.org/10.1007/s11004-009-9229-1, 2009.
Calcagno, P., Chilès, J. P., Courrioux, G., and Guillen, A.: Geological modelling from field data and geological knowledge: part I. Modelling method coupling 3D potential-field interpolation and geological rules, Phys. Earth Planet. Int., 171, 147–157, https://doi.org/10.1016/j.pepi.2008.06.013, 2008.
Carr, J. C., Beatson, R. K., Cherrie, J. B., Mitchell, T. J., Fright, W. R., McCallum, B. C., and Evans, T. R.: Reconstruction and representation of 3D objects with radial basis functions, in: ACM SIGGRAPH 2001, Computer graphics proceedings. ACM Press, New York, 67–76, https://doi.org/10.1145/383259.383266, 2001.
Caumon, G.: Towards stochastic time-varying geological modeling, Math. Geosci., 42, 555–569, https://doi.org/10.1007/s11004-010-9280-y, 2010.
Caumon, G., Collon-Drouaillet P., Carlier Le de Veslud, C., Viseur, S., and Sausse, J.: Surface-based 3D modelling of geological structures, Math. Geosci., 41, 927–945, https://doi.org/10.1007/s11004-009-9244-2, 2009.
Caumon, G., Gray, G., Antoine, C., and Titeux, M.-O.: Three-Dimensional Implicit Stratigraphic Model Building From Remote Sensing Data on Tetrahedral Meshes: Theory and Application to a Regional Model of La Popa Basin, NE Mexico, IEEE T. Geosci. Remote, 51, 1613–1621, https://doi.org/10.1109/TGRS.2012.2207727, 2012.
Cowan, E., Beatson, R., Ross, H., Fright, W., McLennan, T., Evans, T., Carr, J., Lane, R., Bright, D., Gillman, A., Oshust, P., and Titley, M.: Practical implicit geological modelling, 5th Int. Min. Geol. Conf., 8, 89–99, 2003.
Davies, T., Nowrouzezahrai, D., and Jacobson, A.: On the Effectiveness of Weight-Encoded Neural Implicit 3D Shapes, arXiv [preprint], https://doi.org/10.48550/arXiv.2009.09808,17 January 2021.
de Kemp, E. A., Corrigan, D., St-Onge, M. R.: Evaluating the potential for three-dimensional structural modelling of the Archean and Paleoproterozoic rocks of central Baffin Island, Nunavut, Geological Survey of Canada, Current Research, 2001-C24, 22, https://doi.org/10.4095/212255, 2001.
de Kemp, E. A. and Sprague, K. B.: Interpretive Tools for 3-D Structural Geological Modeling Part I: Bézier-Based Curves, Ribbons and Grip Frames, Geoinformatica 7, 55–71, https://doi.org/10.1023/A:1022822227691, 2003.
de Kemp, E. A., Jessell, M. W., Aillères, L., Schetselaar, E. M., Hillier, M., Lindsay, M. D., and Brodaric, B.: Earth model construction in challenging geologic terrain: Designing workflows and algorithms that makes sense, in: Proceedings of Exploration'17: Sixth DMEC – Decennial International Conference on Mineral Exploration, edited by: Tschirhart, V. and Thomas, M. D., Integrating the Geosciences: The Challenge of Discovery, Toronto, Canada, 21–25 October 2017, 419–439, 2017.
de la Varga, M. and Wellmann, F.: Structural geologic modeling as an inference problem: a Bayesian perspective, Interpretation, 4, 1–16, https://doi.org/10.1190/INT-2015-0188.1, 2016.
de la Varga, M., Schaaf, A., and Wellmann, F.: GemPy 1.0: open-source stochastic geological modeling and inversion, Geosci. Model Dev., 12, 1–32, https://doi.org/10.5194/gmd-12-1-2019, 2019.
Dubrule, O. and Kostov, C.: An interpolation method taking into account inequality constraints: I. Methodology, Math. Geosci., 18, 33–51, https://doi.org/10.1007/BF00897654, 1986.
Emmert-Streib, F., Yang, Z., Feng, H., Tripathi, S., and Dehmer, M.: An introductory review of deep learning for prediction models with big data, Fr. Art. Int., 3, 4, https://doi.org/10.3389/frai.2020.00004, 2020.
Frank, T., Tertois, A.-L. L., and Mallet, J.-L. L.: 3D-reconstruction of complex geological interfaces from irregularly distributed and noisy point data, Comput. Geosci., 33, 932–943, https://doi.org/10.1016/j.cageo.2006.11.014, 2007.
Glorot, X. and Bengio, Y.: Understanding the difficulty of training deep feedforward neural networks, in: Proceedings of the thirteenth international conference on artificial intelligence and statistics, 249–256, http://proceedings.mlr.press/v9/glorot10a.html (last access: 17 November 2023), 2010.
Gonçalves, Í. G., Kumaira, S., and Guadagnin, F.: A machine learning approach to the potential-field method for implicit modeling of geological structures, Comput. Geosci., 103, 173–182, https://doi.org/10.1016/j.cageo.2017.03.015, 2017.
Gropp, A., Yariv, L., Haim, N., Atzmon, M., and Lipman, Y.: Implicit geometric regularization for learning shapes, arXiv [preprint], https://doi.org/10.48550/arXiv.2002.10099, 9 July 2020.
Grose, L., Ailleres, L., Laurent, G., Armit, R., and Jessell, M.: Inversion of geological knowledge for fold geometry, J. Struct. Geol., 119, 1–14, https://doi.org/10.1016/j.jsg.2018.11.010, 2019.
Grose, L., Ailleres, L., Laurent, G., Caumon, G., Jessell, M., and Armit, R.: Modelling of faults in LoopStructural 1.0, Geosci. Model Dev., 14, 6197–6213, https://doi.org/10.5194/gmd-14-6197-2021, 2021a.
Grose, L., Ailleres, L., Laurent, G., and Jessell, M.: LoopStructural 1.0: time-aware geological modelling, Geosci. Model Dev., 14, 3915–3937, https://doi.org/10.5194/gmd-14-3915-2021, 2021b.
He, K., Zhang, X., Ren, S., and Sun, J.: Delving deep into rectifiers: Surpassing human-level performance on imagenet classification, in: Proceedings of the IEEE International Conference on Computer Vision, Santiago, Chile, 1026–1034, https://doi.org/10.1109/ICCV.2015.123, 2015.
Hillier, M. J., Schetselaar, E. M., de Kemp, E. A., and Perron, G.: Three-dimensional modelling of geological surfaces using generalized interpolation with radial basis functions, Math. Geosci., 46, 931–951, https://doi.org/10.1007/s11004-014-9540-3, 2014.
Hillier, M., de Kemp, E. A., and Schetselaar, E. M.: Implicitly modelled stratigraphic surfaces using generalized interpolation, in: AIP conference proceedings, 1738, 050004, International Conference of Numerical Analysis and Applied Mathematics, 22–28 September 2015, Rhodes, Greece, https://doi.org/10.1063/1.4951819, 2016.
Hillier, M., Wellmann, F., Brodaric, B., de Kemp, E. A., and Schetselaar, E.: Three-Dimensional Structural Geological Modeling Using Graph Neural Networks, Math. Geosci., 53, 1725–1749, https://doi.org/10.1007/s11004-021-09945-x, 2021.
Hillier, M., Wellmann, F., de Kemp, E. A., Brodaric, B., Schetselaar, E., and Bédard, K.: MichaelHillier/GeoINR: GeoINR 1.0: an implicit neural network approach to three-dimensional geological modelling, Zenodo [code and data set], https://doi.org/10.5281/zenodo.8352541, 2023.
Hornik, K., Stinchcombe, M., and White, H.: Multilayer feedforward networks are universal approximators, Neural Networks, 2, 359–366, https://doi.org/10.1016/0893-6080(89)90020-8, 1989.
Irakarama, M., Laurent, G., Renaudeau, J., and Caumon G.: Finite Difference Implicit Structural Modeling of Geological Structures, Math. Geosci. 53, 785–808, https://doi.org/10.1007/s11004-020-09887-w, 2021.
Jacot, A., Gabriel, F., and Hongler, C.: Neural tangent kernel: Convergence and generalization in neural networks, arXiv [preprint], https://doi.org/10.48550/arXiv.1806.07572, 10 February 2020.
Jessell, M. W., Ailleres, L., and de Kemp, E. A.: Towards an integrated inversion of geoscientific data: What price of geology?, Tectonophysics, 490, 294–306, https://doi.org/10.1016/j.tecto.2010.05.020, 2010.
Kirkwood, C., Economou, T., Pugeault, N., and Odbert, H.: Bayesian Deep Learning for Spatial Interpolation in the Presence of Auxiliary Information, Math. Geosci., 54, 507–531, https://doi.org/10.1007/s11004-021-09988-0, 2022
Lajaunie, C., Courrioux, G., and Manuel, L.: Foliation fields and 3D cartography in geology; principles of a method based on potential interpolation, Math. Geol., 29, 571–584, https://doi.org/10.1007/BF02775087, 1997.
Laurent, G., Ailleres, L., Grose, L., Caumon, G., Jessell, M., and Armit, R.: Implicit modeling of folds and overprinting deformation, Earth Planet. Sc. Lett., 456, 26–38, https://doi.org/10.1016/j.epsl.2016.09.040, 2016.
Li, H., Xu, Z., Taylor, G., Studer, C., and Goldstein, T.: Visualizing the loss landscape of neural nets, arXiv [preprint], https://doi.org/10.48550/arXiv.1712.09913, 7 November 2018.
Liaw, R., Liang, E., Nishihara, R., Moritz, P., Gonzalez, J. E., and Stoica, I.: Tune: A research platform for distributed model selection and training, arXix [preprint], https://doi.org/10.48550/arXiv.1807.05118, 13 July 2018.
Lindsay, M. D., Aillères, L., Jessel, M. W., de Kemp, E. A., and Betts, P. G.: Locating and quantifying geological uncertainty in three-dimensional models: Analysis of the Gippsland Basin, southeastern Australia, Technophysics, 546, 10–27, https://doi.org/10.1016/j.tecto.2012.04.007, 2012.
Llanas, B., Lantarón, S., and Sáinz, F. J.: Constructive Approximation of Discontinuous Functions by Neural Networks, Neural Process. Lett., 27, 209–226, https://doi.org/10.1007/s11063-007-9070-9, 2008.
Loshchilov, I. and Hutter, F.: Decoupled weight decay regularization, arXix [preprint], https://doi.org/10.48550/arXiv.1711.05101, 4 January 2017.
Ma, L., Liu, Y., Zhang, X., Ye, Y., Yin, G., and Johnson, B. A.: Deep learning in remote sensing applications: A meta-analysis and review, ISPRS J. Photogramm., 152, 166–177, https://doi.org/10.1016/j.isprsjprs.2019.04.015, 2019.
Mallet, J.-L.: Discrete smooth interpolation in geometric modelling, Computer-Aided Design, 24, 178–191, https://doi.org/10.1016/0010-4485(92)90054-E, 1992.
Mallet, J.-L.: Discrete modeling for natural objects, Math. Geol., 29, 199–219, https://doi.org/10.1007/BF02769628, 1997.
March, A. and Love, M.: Regional Stratigraphic Framework of the Phanerozoic in Saskatchewan; Saskatchewan Phanerozoic Fluids and Petroleum Systems Project; Sask. Ministry of the Economy, Saskatchewan Geological Survey, Open File 2014-1, https://publications.saskatchewan.ca/#/products/79907 (last access: 17 November 2023), 2014.
Mildenhall, B., Srinivasan, P. P., Tancik, M., Barron, J. T., Ramamoorthi, R., and Ng, R.: Nerf: Representing scenes as neural radiance fields for view synthesis, Commun. ACM, 65, 99–106, https://doi.org/10.1145/3503250, 2021.
Park, J. J., Florence, P., Straub, J., Newcombe, R., and Lovegrove, S.: DeepSDF: Learning Continuous Signed Distance Functions for Shape Representation, in: Proceedings of the IEEE/CVF Conference on Computer Vision and Pattern Recognition, 165–174, https://openaccess.thecvf.com/content_CVPR_2019/html/Park_DeepSDF_Learning_Continuous_Signed_Distance_Functions_for_Shape_Representation_CVPR_2019_paper.html (last access: 17 November 2023), 2019.
Perol, T., Gharbi, M., and Denolle, M.: Convolutional neural network for earthquake detection and location, Sci. Adv., 4, e1700578, https://doi.org/10.1126/sciadv.1700578, 2018.
Pizzella, L., Alais, R., Lopez, S., Freulon X., and Rivoirard, J.: Taking Better Advantage of Fold Axis Data to Characterize Anisotropy of Complex Folded Structures in the Implicit Modeling Framework, Math. Geosci., 54, 95–130, https://doi.org/10.1007/s11004-021-09950-0, 2022.
Renaudeau, J., Malvesin, E., Maerten, F., and Caumon, G.: Implicit Structural Modeling by Minimization of the Bending Energy with Moving Least Squares Functions, Math. Geosci., 51, 693–724, https://doi.org/10.1007/s11004-019-09789-6, 2019.
Rodríguez, J. D., Pérez, A., and Lozano, J. A.: Sensitivity Analysis of k-Fold Cross Validation in Prediction Error Estimation, IEEE T. Pattern Anal., 32, 569–575, https://doi.org/10.1109/TPAMI.2009.187, 2009.
Ross, Z. E., Meier, M. A., and Hauksson E.: P Wave Arrival Picking and First-Motion Polarity Determination With Deep Learning, J. Geophys. Res.-Sol., 123, 5120–2129, https://doi.org/10.1029/2017JB015251, 2018.
Santa, F. D. and Pieraccini, S.: Discontinuous neural networks and discontinuity learning, J. Comput. Appl. Math., 419, 114678, https://doi.org/10.1016/j.cam.2022.114678, 2023.
Scott, D. J., St-Onge, M. R., and Corrigan, D.: Geology, Dewar Lakes, Nunavut, Geological Survey of Canada, Open File 4201, 2 sheets; 1 CD-ROM, map: scale 1:100000, https://doi.org/10.4095/213226, 2002.
Shi, Y., Wu, X., and Fomel, S.: SaltSeg: Automatic 3D salt segmentation using a deep convolutional neural network, Interpretation, 7, SE113–SE122, https://doi.org/10.1190/INT-2018-0235.1, 2019.
Shi, C. and Wang, Y.: Non-parametric machine learning methods for interpolation of spatially varying non-stationary and non-Gaussian geotechnical properties, Geosci. Front., 12, 339–350, https://doi.org/10.1016/j.gsf.2020.01.011, 2021
Sides, E. J.: Geological modelling of mineral deposits for prediction in mining, Geol. Rundsch., 86, 342–353, https://doi.org/10.1007/s005310050145, 1997.
Sitzmann, V., Martel, J., Bergman, A., Lindell, D., and Wetzstein, G.: Implicit Neural Representations with Periodic Activation Functions, Adv. Neur. In., 33, 7462–7473, 2020.
Sprague, K. B. and de Kemp, E. A.: Interpretive Tools for 3-D Structural Geological Modelling part II: Surface Design from Sparse Spatial Data, GeoInformatica 9, 5–32, https://doi.org/10.1007/s10707-004-5620-8, 2005.
St-Charles, P. -L., Rousseau, B., Ghosn, J., Nantel J.-P, Bellefleur, G., and Schetselaar, E.: A Multi-Survey Dataset and Benchmark for First Break Picking in Hard Rock Seismic Exploration, in: Proc. Neurips 2021 Workshop on Machine Learning for the Physical Sciences (ML4PS), https://ml4physicalsciences.github.io/2021/files/NeurIPS_ML4PS_2021_3.pdf (last access: 17 November 2023), 2021.
St-Onge, M. R., Scott, D. J., and Corrigan, D.: Geology, Straits Bay, Nunavut, Geological Survey of Canada, Open File 4200, map: scale 1:100000, https://doi.org/10.4095/213225, 2002.
Sullivan, C. B. and Kaszynski, A. A.: PyVista: 3D plotting and mesh analysis through a streamlined interface for the Visualization Toolkit (VTK), J. Open Source Softwa., 4, 1450, https://doi.org/10.21105/joss.01450, 2019.
Tancik, M., Srinivasan, P. P., Mildenhall, B., Fridovich-Keil, S., Raghavan, N., Singhal, U., Ramamoorthi, R., Barron, J. T., and Ng, R.: Fourier Features Let Networks Learn High Frequency Functions in Low Dimensional Domains, arXiv [preprint], https://doi.org/10.48550/arXiv.2006.10739, 18 June 2020.
Taubin, G.: Distance approximations for rasterizing implicit curves, ACM T. Graphic., 13, 3-42, https://doi.org/10.1145/174462.174531, 1994.
von Harten, J., de la Varga, M., Hillier, M., and Wellmann, F.: Informed Local Smoothing in 3D Implicit Geological Modeling, Minerals, 11, 1281, https://doi.org/10.3390/min11111281, 2021.
Wang, D. and Chen, G.: Seismic Stratum Segmentation Using an Encoder-Decoder Convolutional Neural Network, Math. Geo., 53, 1355–1374, https://doi.org/10.1007/s11004-020-09916-8, 2021.
Wang, P. S., Liu, Y., Yang, Y. Q., and Tong, X.: Spline positional encoding for learning 3d implicit signed distance fields, arXiv [preprint], https://doi.org/10.48550/arXiv.2106.01553, 28 October 2021.
Wang, S., Cai, Z., Si, X., and Cui, Y.: A Three-Dimensional Geological Structure Modeling Framework and Its Application in Machine Learning, Math. Geosci., 55, 163–200, https://doi.org/10.1007/s11004-022-10027-9, 2022.
Wellmann, F. and Caumon, G.: 3-D Structural geological models: Concepts, methods, and uncertainties, Adv. Geophys., 59, 1–121, https://doi.org/10.1016/bs.agph.2018.09.001, 2018.
Wellmann, J. F. and Regenauer-Lieb, K.: Uncertainties have a meaning: Information entropy as a quality measure for 3-D geological models, Technophysics, 526, 207–216, https://doi.org/10.1016/j.tecto.2011.05.001 , 2012.
Wu, X., Liang, L., Shi, Y., and Fomel, S.: FaultSeg3D: using synthetic datasets to train an end-to-end convolutional neural network for 3D seismic fault segmentation, Geophysics, 84, IM35– IM45, https://doi.org/10.1190/geo2018-0646.1, 2019.
Wu, Y., Lin, Y., Zhou, Z., Bolton, D. C., Liu, J., and Johnson, P.: DeepDetect: A cascaded region-based densely connected network for seismic event detection, IEEE T. Geosci. Remote, 57, 62–75, https://doi.org/10.1109/TGRS.2018.2852302, 2018.
Yang, L., Hyde, D., Grujic, O., Scheidt, C., and Caers, J.: Assessing and visualizing uncertainty of 3D geological surfaces using level sets with stochastic motion, Comput. Geosci., 122, 54–67, https://doi.org/10.1016/j.cageo.2018.10.006, 2019.
Zhuang, F., Qi, Z., Duan, K., Xi, D., Zhu, Y., Zhu, H., Xiong, H., and He, Q.: A Comprehensive Survey on Transfer Learning, Proc. IEEE, 109, 43–76, https://doi.org/10.1109/JPROC.2020.3004555, 2021.
Short summary
Neural networks can be used effectively to model three-dimensional geological structures from point data, sampling geological interfaces, units, and structural orientations. Existing neural network approaches for this type of modelling are advanced by the efficient incorporation of unconformities, new knowledge inputs, and improved data fitting techniques. These advances permit the modelling of more complex geology in diverse geological settings, different-sized areas, and various data regimes.
Neural networks can be used effectively to model three-dimensional geological structures from...