Source-linked AI summary
Training-image based geostatistical inversion using a spatial generative adversarial neural network
Eric Laloy, Romain Hérault, Diederik Jacques, Niklas Linde
TL;DR
MPS-based probabilistic inversion is computationally prohibitive for high-dimensional complex geological models. The paper introduces a training-image-based spatial GAN whose low-dimensional latent representation enables MCMC inversion, and its case studies show rapid posterior exploration in 2D and realistic data-fitting realizations in 3D.
Problem
MPS-based probabilistic inversion is slow, often incompletely explores the posterior, and model reduction can lose the geological realism of complex training-image priors.
Method
The spatial GAN learns 2D and 3D geostatistical realizations from training images and performs MCMC inversion in its low-dimensional latent space.
Results
The 2D inversion rapidly explores the posterior, while the 3D case recovers realizations that fit the data near the target level and visually resemble the true model.
Takeaways & Limitations
After training, SGAN can generate large realizations quickly, supporting repeated simulation and probabilistic inversion for complex geological media.
Takeaways & Limitations
The 3D transient hydraulic-tomography posterior was not appropriately sampled, and direct conditioning to point data is unavailable for pure geostatistical simulation.
Abstract
from arXiv · showhide
Probabilistic inversion within a multiple-point statistics framework is often computationally prohibitive for high-dimensional problems. To partly address this, we introduce and evaluate a new training-image based inversion approach for complex geologic media. Our approach relies on a deep neural network of the generative adversarial network (GAN) type. After training using a training image (TI), our proposed spatial GAN (SGAN) can quickly generate 2D and 3D unconditional realizations. A key characteristic of our SGAN is that it defines a (very) low-dimensional parameterization, thereby allowing for efficient probabilistic inversion using state-of-the-art Markov chain Monte Carlo (MCMC) methods. In addition, available direct conditioning data can be incorporated within the inversion. Several 2D and 3D categorical TIs are first used to analyze the performance of our SGAN for unconditional geostatistical simulation. Training our deep network can take several hours. After training, realizations containing a few millions of pixels/voxels can be produced in a matter of seconds. This makes it especially useful for simulating many thousands of realizations (e.g., for MCMC inversion) as the relative cost of the training per realization diminishes with the considered number of realizations. Synthetic inversion case studies involving 2D steady-state flow and 3D transient hydraulic tomography with and without direct conditioning data are used to illustrate the effectiveness of our proposed SGAN-based inversion. For the 2D case, the inversion rapidly explores the posterior model distribution. For the 3D case, the inversion recovers model realizations that fit the data close to the target level and visually resemble the true model well.
1 Introduction
Complex geological priors make MPS-based probabilistic inversion slow and prone to incomplete posterior exploration. The paper introduces a spatial GAN that uses a low-dimensional latent representation to support faster 2D and 3D inversion while retaining training-image information.
- Research gap: MPS simulations honor complex geological priors from large gridded 2D or 3D training images, but probabilistic inversion remains slow and posterior exploration is often incomplete.Existing model-reduction methods can also lose the geological realism encoded by the training image.
- Related approach: Deep learning can learn parametric representations of data and, for some models, map learned representations back to the original data space.The paper treats training images as data and motivates low-dimensional representations for subsurface models.
- Proposed approach: The proposed spatial GAN supports MPS-based geostatistical inversion in both 2D and 3D spatial domains.It extends an earlier 2D spatial GAN to three dimensions.
- Proposed approach: SGAN inversion uses a low-dimensional latent space, where high-dimensional subsurface realizations are generated from a random noise vector and sampled with MCMC.This parameterization makes the inversion parameters equal to the latent variables.
- Proposed approach: Compared with the earlier VAE-based approach, SGAN training uses a single training image, supports binary and multi-categorical data, and is fully 3D.The paper also emphasizes a spatially structured latent representation for the generated model domain.
- Evaluation: The paper evaluates SGAN simulation quality before testing synthetic 2D steady-state and 3D transient groundwater-flow inversion.These evaluations are presented as necessary because realization quality conditions the quality of the subsequent inversion.
2 Methods
The SGAN uses convolutional generator and discriminator networks to learn spatial realizations from training images while providing a low-dimensional latent parameterization for inversion. Training alternates discriminator and generator optimization, and the approach supports spatial-quality assessment and MCMC-based posterior exploration.
- Spatial GAN architecture: The proposed SGAN extends a 2D architecture to 3D using direct and transposed convolutional layers.The network is a deep neural network designed for spatial data.
- Convolutional architecture: Convolutional layers encode spatially local information through local filters and stacked feature maps.Increasing the number of feature maps enriches the representation of the input data.
- Spatial GAN architecture: A spatially structured latent array is transformed by the generator into a 2D or 3D grayscale realization.Each latent element controls a partially overlapping region of the generated model domain.
- Dimensionality reduction: 125 latent variables can generate a 129 × 129 × 129 realization, yielding more than 4 orders of magnitude dimensionality reduction.This example uses dp = 5 and q = 1; q = 3 still gives more than 3 orders of magnitude reduction.
- GAN training: During GAN training, the discriminator distinguishes TI patches from generated realizations while the generator learns to make generated samples appear real.The discriminator and generator are optimized in alternating steps with opposing objectives.
- Generation quality assessment: The method evaluates generated categorical realizations using facies proportions, pairwise probabilities, and connectivity probabilities across spatial directions.For 3D cases, the assessment includes six metric combinations for each indicator and facies.
3 Geostatistical Simulation Results
The SGAN generated 2D and 3D categorical realizations that generally matched the training-image patterns and spatial statistics, while enabling much faster realization generation than DS after training. Some discrepancies arose from dependent TI patches, convolutional settings, and filtering or thresholding choices.
- 2D binary channelized TI: SGAN realizations for the binary channelized TI visually matched the TI and generally reproduced its probability and connectivity statistics.The TI matrix proportion was 0.74, compared with 0.73 averaged across 100 realizations.
- 2D braided-river TI: The braided-river SGAN realizations fairly resembled the TI and showed relatively good correspondence in probability and connectivity statistics.Average facies fractions were 0.546, 0.147, and 0.307 for facies 0, 1, and 2, compared with TI fractions of 0.596, 0.119, and 0.285.
- 2D braided-river TI: The braided-river realizations had a larger statistical spread and recurring small probability-function peaks, likely related to dependent patches and convolutional or filtering settings.The TI-patch-to-TI size ratio was approximately 1/2, making the TI patches strongly dependent.
- SGAN versus DS: After training, SGAN generated one braided-river realization in 0.1 s, more than 10,000 times faster than DS and with more consistent probability and connectivity statistics.The comparison used DS realizations with maximum neighboring-node settings of 30 and 75.
- Computational cost: Training required substantial time, but the trained SGAN generated a 129 × 129 × 129 realization in 12 s including post-processing, with generation itself requiring about 5 s.Training for 50 epochs took about 12 hours on the reported hardware.
- 3D fold-aquifer TI: The 3D fold-aquifer realizations showed patterns similar to the TI and relatively well matched its probability and connectivity statistics.The realizations slightly over-represented broken channels and rarely contained small isolated patches; the TI patches were highly dependent, with a size ratio of about 1/2.
4 Inverse Problems
The inversion case studies apply SGAN-generated low-dimensional models to 2D steady-state flow and 3D transient hydraulic tomography, with and without direct conditioning data. The 2D chains rapidly explore the posterior, while the challenging 3D case reaches data misfits near the target level within the stated computational budget.
- 2D steady-state flow: The 2D steady-state flow case models a 125 × 125 channelized aquifer with hydraulic conductivities of 1 × 10−2 m/s for channels and 1 × 10−4 m/s for matrix.
- 2D steady-state flow: After approximately 96,000 total iterations, the 2D chains jointly sample the posterior, and sampled realizations closely resemble the true model with limited posterior variability.
- 2D steady-state flow: The unconditional 2D inversion satisfies the Gelman–Rubin criterion, R̂ ≤1.2, for every sampled parameter after 387,200 total iterations.
- Direct conditioning: Direct facies data from 49 well-location cells are incorporated by weighting direct and indirect likelihood terms through σx, with σx = 0.5 selected as a tradeoff.
- Direct conditioning: With direct conditioning, convergence occurs after 35,300 iterations per chain, while 88% of the last 160 realizations honor all 49 conditioning data.
- 3D transient hydraulic tomography: The 3D hydraulic tomography case uses a 30 × 61 × 61 aquifer, 1,568 measurements, and a relatively large signal-to-noise ratio of 22, making it challenging.
- 3D transient hydraulic tomography: After 50,000 iterations per chain, 16 DREAM(ZS) chains converge toward data misfits of 0.0101 m–0.0103 m, close to the target level.
5 Discussion
The discussion highlights SGAN’s computational efficiency, dimensional compression, and broader simulation potential, while identifying conditioning, training time, sampling, and simulation-quality limitations. Continuous-TI experiments are described as promising but not fully demonstrated.
- Architecture and compression: SGAN’s low-dimensional representation can exceed a 10,000-fold compression ratio for a 3D binary channelized aquifer.
- Limitations: The 3D transient hydraulic tomography posterior was not appropriately sampled within 50,000 iterations per chain, and longer runs or more chains would likely help.
- Efficiency and uncertainty quantification: SGAN generates 1–2 million-cell 3D realizations within seconds after training, although training itself takes several hours.For a braided-river TI, training took 6 hours and generation required 0.1 s per realization on the used CPU.
- Efficiency and uncertainty quantification: For the braided-river comparison, SGAN produced more consistent realizations and required more than 10,000 times less computation per realization than DeeSse.
- Limitations: Pure geostatistical simulation cannot directly condition SGAN realizations on point data, while the proposed latent-space optimization workaround adds computational overhead.
- Continuous training images: Continuous-TI training is presented as possible but not demonstrated in the main studies; a 375-dimensional latent vector required about 12 hours of GPU training.
6 Conclusion
The paper presents SGAN as a training-image-based approach for generating 2D and 3D geologic realizations and enabling probabilistic inversion through a very low-dimensional representation. Its case studies combine fast post-training generation with inversion applications in steady-state flow and transient hydraulic tomography.
- SGAN generates 2D and 3D unconditional realizations from categorical training images and, in principle, continuous training images.
- The method captures training-image multiple-point statistics using a very low-dimensional representation, supporting efficient probabilistic inversion.
- Training takes 3–12 hours, whereas a 3D binary channelized model with more than 2 million voxels can be generated in about 12 seconds.
- Synthetic 2D steady-state flow and 3D transient hydraulic tomography studies illustrate the SGAN-based inversion approach.