Training-image based geostatistical inversion using a spatial generative adversarial neural network (1708.04975v2)

Abstract: 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.

• The paper presents a spatial GAN model that integrates multiple-point statistics for effective inversion of complex geologic media.
• The methodology enables rapid generation of 2D and 3D geostatistical realizations for efficient MCMC inversion in high-dimensional parameter spaces.
• The study highlights potential improvements in training stability and direct conditioning data integration, paving the way for advanced subsurface modeling.

An Analytical Overview of "Training-image based geostatistical inversion using a spatial generative adversarial neural network"

The paper "Training-image based geostatistical inversion using a spatial generative adversarial neural network," authored by Eric Laloy et al., introduces a novel approach to performing geostatistical inversion for complex geologic media by leveraging the capabilities of deep neural networks, specifically a type of generative adversarial network (GAN). This paper presents a detailed exploration of the integration of multiple-point statistics (MPS) within a deep learning framework, aiming at efficiently solving high-dimensional inversion problems.

Summary of Methodology

The core advancement described in the paper is the application of a spatial GAN (SGAN) model to generate 2D and 3D geostatistical realizations. The SGAN is trained using a singular training image (TI), which provides a complex representation of the spatial field that might be encountered in geological media. The key innovation lies in the SGAN's ability to define a low-dimensional parameter space that significantly facilitates the inversion process using MCMC methods.

The authors demonstrate that, post-training, the SGAN can produce spatially realistic model realizations rapidly, making it a particularly appealing approach for scenarios where a large number of realizations are necessary, like in MCMC inversion. The approach not only supports the rapid generation of realizations but also allows for incorporation of direct conditioning data within the inversion, which is an enhancement over several traditional methods.

Key Results and Discussion

Several synthetic case studies involving 2D and 3D scenarios were evaluated to demonstrate the SGAN's efficacy. Notably, for unconditional simulation, the SGAN maintained the spatial statistics reflective of the TI, as validated by standard indicators such as the two-point probability and cluster functions.

From an inversion standpoint, the 2D case of steady-state flow and the 3D transient hydraulic tomography elaborated within the paper highlight the SGAN's potency in recovering model realizations that align closely with true geological structures while respecting the observed data. The paper successfully illustrates that the SGAN-based inversion can handle complex constructs of geological prior models while efficiently navigating the probabilistic solution space.

However, the paper notes limitations regarding the direct conditioning to point data during simulation, which remains a challenge for purely geostatistical applications. Training stability and computational cost during the training phase were identified as areas needing further research and optimization. Notwithstanding these limitations, the paper posits that rapid advancements in both deep learning software and hardware could soon mitigate these concerns.

Implications and Future Directions

The practical implications of this research are significant for fields where subsurface characterization is crucial, such as hydrogeology and petroleum engineering. By enabling faster generation of realizations after an initial investment in training, the SGAN-based approach offers a path towards scalable and efficient inversion methodologies. This is particularly relevant when confronted with high-dimensional parameter spaces where traditional MPS-based methods often struggle.

Theoretically, this research provides fertile ground for extending the capabilities of deep generative models in geosciences, especially in the context of transferable methodologies across different types of spatial data (e.g., continuous field data). The proposed methodology also opens up possibilities for hybrid modeling approaches combining traditional statistics-based techniques with modern machine learning frameworks.

Future work could focus on improving training algorithms to ensure stable convergence and robustness against diverse geological scenarios. Exploring integrations with other emerging AI technologies and expanding to continuous TIs could further enhance the model's utility. Additionally, developing strategies for direct conditioning to point data will broaden the applicability of these models in operational settings.

In conclusion, the paper builds a strong case for the application of deep learning models in geostatistical inversion, setting the stage for further innovations in the field of subsurface characterization and modeling.