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Automatic Semivariogram Modeling by Convolutional Neural Network

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Abstract

Modeling the semivariogram to characterize spatial continuity requires expert geostatistical knowledge and domain expertise about the spatial phenomenon of interest. Moreover, although practitioners may have experience in semivariograms, their interpretations may vary due to experimental semivariogram noise and ambiguity. In general, modeling semivariograms remains highly subjective. This paper presents a data-driven, deep learning-based automated semivariogram modeling method known as automatic semivariogram modeling with convolution-based deep learning (ASMC) that improves the utilization of available spatial information to reduce the subjectivity of semivariogram modeling. Training models are generated by sequential Gaussian simulation (SGS) and labeled with their associated semivariogram parameters (i.e., maximum correlation length, aspect ratio of major and minor direction ranges, and azimuth of major direction). ASMC consists of two convolutional neural networks (CNNs). The first CNN model maps the sparse spatial samples to the exhaustive SGS-derived spatial models, and the second CNN maps the SGS spatial model to the semivariogram parameters. Both CNNs are trained with realistic spatial training data, and their validity is also checked with validation data withheld from training. Two-dimensional synthetic, but realistic, case studies demonstrate that the first CNN successfully learns the spatial characteristics among spatial data and generates realistic subsurface model estimates. The second CNN learns the spatial context of the estimated subsurface model and successfully predicts the semivariogram parameters with greater than 96% accuracy. The proposed machine, deep learning-based workflow improves the utilization and objectivity of spatial data in semivariogram-based spatial continuity modeling. With the optimal design of the experiment for training and tuning of model hyperparameters, this method may be generalized for application for a wide range of spatial modeling projects.

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Code and Data availability

We carried out the neural network aspects of our proposed workflow using Tensorflow and performed geostatistical analysis using the GeostatsPy Python package (Pyrcz et al. 2021) reimplementation of GSLIB (Deutsch and Journel 1998). The code and data will be made available via the authors' repositories https://github.com/GeostatsGuy/GeostatsPy and https://github.com/whghdrms.

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Acknowledgements

The authors thank the two reviewers for their critical feedback and their help to improve the quality of the manuscript. The authors also appreciate the support of the DIRECT consortium for subsurface data analytics and machine learning at The University of Texas at Austin. The corresponding author is also thankful to the Hildebrand Department of Petroleum and Geosystems Engineering at The University of Texas at Austin, Texas, USA.

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Authors and Affiliations

  1. Hildebrand Department of Petroleum and Geosystems Engineering, The University of Texas, Austin, TX, USA

    Honggeun Jo & Michael J. Pyrcz

  2. Bureau of Economic Geology, Jackson School of Geosciences, The University of Texas, Austin, TX, USA

    Michael J. Pyrcz

  3. Jackson School of Geosciences, The University of Texas, Austin, TX, USA

    Michael J. Pyrcz

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  1. Honggeun Jo
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  2. Michael J. Pyrcz
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Correspondence to Honggeun Jo.

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Jo, H., Pyrcz, M.J. Automatic Semivariogram Modeling by Convolutional Neural Network. Math Geosci 54, 177–205 (2022). https://doi.org/10.1007/s11004-021-09962-w

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