Stochastic seismic waveform inversion using generative adversarial networks as a geological prior (1806.03720v1)

Abstract: We present an application of deep generative models in the context of partial-differential equation (PDE) constrained inverse problems. We combine a generative adversarial network (GAN) representing an a priori model that creates subsurface geological structures and their petrophysical properties, with the numerical solution of the PDE governing the propagation of acoustic waves within the earth's interior. We perform Bayesian inversion using an approximate Metropolis-adjusted Langevin algorithm (MALA) to sample from the posterior given seismic observations. Gradients with respect to the model parameters governing the forward problem are obtained by solving the adjoint of the acoustic wave equation. Gradients of the mismatch with respect to the latent variables are obtained by leveraging the differentiable nature of the deep neural network used to represent the generative model. We show that approximate MALA sampling allows efficient Bayesian inversion of model parameters obtained from a prior represented by a deep generative model, obtaining a diverse set of realizations that reflect the observed seismic response.

• The paper integrates GANs as geological priors within a Bayesian inversion framework to address stochastic seismic waveform inversion.
• The methodology employs an approximate MALA with adjoint state techniques for efficient posterior exploration, achieving sub-10% relative error.
• Incorporating well-log data significantly refines the geological models, yielding over 95% per-sample accuracy and enhanced subsurface predictions.

Application of GANs in Stochastic Seismic Waveform Inversion: A Bayesian Approach

The publication under review investigates the integration of Generative Adversarial Networks (GANs) into the framework of stochastic seismic waveform inversion, with a focus on using these networks as geological priors. This methodology is developed within the context of Partial-Differential Equation (PDE) constrained inverse problems, aimed at enhancing the subsurface geological model parameter estimation. Specifically, the framework leverages the representational power of GANs to create a priori models, which predict subsurface geological structures and their petrophysical properties.

Core Methodological Approach

The underlying technique hinges on Bayesian inversion facilitated via an approximate Metropolis-adjusted Langevin Algorithm (MALA). This statistical sampling technique is employed to explore the posterior distributions of model parameters informed by seismic observations. The GAN functions as a stochastic model encapsulating possible geological configurations, realized through spatial distributions of p-wave velocities, amongst other geological facies. Gradients needed for the forward PDE solution are computed through the adjoint state technique, allowing efficient gradient-based optimization in the latent space defined by the GAN.

Through this intricate setup, the authors demonstrate that effective Bayesian inversion is achievable, producing a diverse spectrum of geological realizations aligned with the observed seismic data. This approach not only accommodates seismic data but also offers a mechanism for integrating ancillary information, such as well-log data.

Empirical Findings and Numerical Results

The authors provide a compelling numerical evaluation of their inversion framework within a simplified seismic inversion problem. The resulting ensemble of model parameters evidences consistency with observed seismic data, validated through a match within a relative error margin of less than 10%. Interestingly, incorporating additional information derived from well logs into the GAN's prior significantly tailors the resulting geological models, enhancing both geological plausibility and seismic data fit.

Numerically, the authors underscore the importance of executing a limited number of MALA iterations (typically around 100), given the computational expense associated with solving the forward problem and its corresponding adjoint PDEs. The results highlight the capacity of this approach to decrease seismic data mismatch errors efficiently, demonstrating relative seismic error reductions to levels below 5-7%. Moreover, constraints emanating from bore-hole data were respected with an impressive per-sample accuracy rate exceeding 95%.

Implications and Future Directions

This work implicates several critical advancements in the domain of geophysical inverse problems. The integration of GANs as a stochastic prior within a Bayesian inversion framework represents a meaningful stride toward the computational efficiency required to tackle high-dimensional inverse problems. The authors suggest promising avenues for extending this generative inversion strategy to other domains requiring spatial property model optimization, such as hydrology or materials science.

Furthermore, the differentiable characteristic of the generative model presents opportunities for future research to optimize inversion methods by focusing gradients directly on latent variables rather than the high-dimensional parameter space itself. The application of variational auto-encoders (VAEs) to provide a posterior distribution for these latent variables may also hold potential for further enhancing sampling efficiency and diversity preservation during inversion.

Overall, this paper contributes a novel methodological counterpoint to conventional deterministic and probabilistic seismic inversion approaches, facilitating the generation of plausible geological models that align with multi-source seismic data and integrated bore-hole constraints.