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Stochastic seismic waveform inversion using generative adversarial networks as a geological prior
Lukas Mosser, Olivier Dubrule, Martin J. Blunt
TL;DR
Seismic inverse problems are ill-posed and computationally expensive, making probabilistic ensembles difficult to generate. The paper combines a geological generative model with adjoint-based approximate MALA sampling of latent-variable posteriors, producing diverse inverted realizations that reflect seismic observations.
Problem
Noisy or sparse observations, uncertain forward models, and high computational costs make seismic inversion ill-posed and often limit interpretation to a single model.
Method
The framework combines a lower-dimensional generative model of geological structures with adjoint-derived gradients and approximate MALA sampling of the posterior over latent variables.
Results
The framework evaluates an ensemble of inverted model parameters and finds diverse geological realizations whose variability decreases as the number of seismic sources increases, approaching forward-problem resolution limits.
Takeaways & Limitations
Approximate MALA enables Bayesian inversion with a generative geological prior while retaining an ensemble of realizations consistent with the observed seismic response.
Takeaways & Limitations
Because two- and three-dimensional full-waveform inversion is highly computationally costly, the approximate MALA scheme requires a small number of iterations.
Abstract
from arXiv · showhide
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.
1. Introduction
Seismic inversion is ill-posed and computationally expensive because sparse or noisy observations can match many high-dimensional earth models. The paper proposes a differentiable generative prior with latent-space Bayesian sampling to produce diverse seismic-consistent models while incorporating additional geological information.
- Problem: Sparse or noisy measurements make seismic inversion ill-posed, with numerous model parameters potentially matching the observations.The forward model may also be uncertain.
- Problem: Three-dimensional seismic inversion can involve more than 10^6 parameters and, for large observations, billions of parameters.The adjoint PDE provides gradients of data mismatch for gradient-based parameter modification.
- Problem: High inversion cost means probabilistic ensembles are rarely generated, with interpretation often relying on a single model meeting predefined quality criteria.This limits direct representation of model uncertainty in decision processes.
- Approach: A deep generative model maps lower-dimensional normally distributed latent variables to stochastic geological model-parameter realizations and supplies a differentiable prior.The generated parameters are combined with numerical acoustic-wave modeling to produce synthetic observations.
- Contributions: The framework combines a differentiable latent-variable generative model with a PDE-constrained physical forward problem.This is presented as a central methodological contribution.
- Approach: Adjoint-method gradients in latent space support approximate MALA sampling of the posterior, yielding a diverse ensemble of models matching observed seismic data.The framework also incorporates constraints such as information from an existing bore-hole.
- Contributions: The approach is illustrated on a simple seismic inversion problem and the resulting model-parameter ensemble is evaluated.The paper also identifies integration of geological facies information along one-dimensional bore-holes and extension to other gradient-available inverse problems.
2. Related Work
Prior work established Bayesian and neural-network approaches for seismic inversion, while later studies introduced neural representations of PDE coefficients, surrogate forward models, and generative latent-space inversion. The paper situates its approach within this progression toward learned geological priors and probabilistic inversion.
- Bayesian inversion: Earlier Bayesian seismic-inversion studies used Monte Carlo random walks, Metropolis sampling, or Gibbs sampling to obtain posterior models and assess uncertainty.These works established probabilistic inversion as a general methodology for geophysical problems.
- Neural approaches: Neural networks have been explored as proxy models for expensive geophysical forward and inverse problems because they can approximate complex functions.This motivation follows from the high cost of solving the wave equation.
- Neural approaches: Supervised neural inversion mapped synthetic seismic amplitude responses to depth profiles of acoustic velocity and produced high-resolution approximations.The method was trained on paired synthetic data and velocity models.
- PDE representations: Neural networks have also represented spatially varying PDE coefficients, encoding high-dimensional fields through network weights.This differs from replacing the PDE solution itself with a neural forward model.
- PDE representations: Other studies replaced PDE solutions with neural networks for faster forward computation or mapped seismic features directly to p-wave velocity models.These approaches facilitated inversion or were validated on synthetic examples.
- Generative priors: Cycle-GAN and pretrained-GAN studies addressed seismic inversion through domain transfer or latent-space optimization on synthetic benchmarks.The cycle-GAN mapped between seismic amplitudes and p-wave velocity models, while the pretrained-GAN study optimized latent variables with a quasi-Newton method.
- Graphical formulation: The graphical model represents earth models through m ∼ Gθ(z), generates acoustic observations through a deterministic PDE, and incorporates partial bore-hole information.The latent variables and local geological observations are connected within the Bayesian inversion formulation.
3. Problem Definition
The inversion seeks posterior samples of latent variables that generate geological models consistent with observed seismic data. It combines acoustic-wave forward modeling, adjoint gradients, differentiable generator backpropagation, and approximate MALA sampling, with optional bore-hole facies constraints.
- 3.1. Bayesian Inversion: The Bayesian objective is to sample latent variables z from the posterior conditioned on observed acoustic reflection data dobs.The posterior is defined over latent variables rather than directly over the full spatial model.
- 3.1. Bayesian Inversion: The seismic forward model maps generated geological parameters to predicted observations through numerical solution of the time-dependent acoustic wave equation.The model includes facies, acoustic p-wave velocity, and rock density, while the wave equation uses acoustic p-wave velocity.
- 3.1. Bayesian Inversion: Approximate MALA proposes latent-variable updates using the gradient of the data-mismatch likelihood and adds Gaussian perturbations.The data likelihood is expressed through the L2 mismatch between predicted and observed acoustic data.
- 3.2. Adjoint-State Method: Adjoint-state equations provide gradients of the data mismatch with respect to forward-model parameters, while neural-network backpropagation transfers gradients to latent variables.The adjoint calculation backpropagates mismatch information through the acoustic PDE, and the generator supplies the latent-variable derivatives.
- 3.1. Bayesian Inversion: An optional bore-hole objective incorporates one-dimensional geological facies indicators into posterior sampling alongside the seismic observations.The bore-hole term is weighted separately through ϵ3 when additional geological information is included.
- 3.2. Adjoint-State Method: The numerical setup uses a fourth-order finite-difference acoustic solver, dampening at domain boundaries, surface receivers, and variable source locations.These choices emulate seismic acquisition while preventing lateral boundary reflections.
4. Generative Model
The generative model uses a GAN to represent prior geological structures and spatially varying rock properties through a lower-dimensional latent representation. Its differentiable generator is coupled to the seismic forward operator for inversion.
- Generative prior: A differentiable generator Gθ(z) maps latent vectors to stochastic realizations of spatially varying geological model parameters.Sampling latent vectors produces prior realizations that encode geological knowledge about subsurface structures and properties.
- GAN formulation: The prior is modeled with a GAN comprising a generator that creates samples and a discriminator that distinguishes generated from training examples.The GAN’s probability density is implicitly defined by its training examples.
- GAN formulation: A Wasserstein-GAN with a Lipschitz penalty is used to improve training stability by minimizing Wasserstein distance between generated and real distributions.The penalty coefficient is set to λLP = 200 in the reported training setup.
- Model parameters: In the ground-truth realization, river-channel properties are constant within channels, whereas shale velocity and density vary by layer.The three modeled properties are geological facies, acoustic p-wave velocity Vp, and rock density ρ.
- Model parameters: The generator outputs three channels for facies probability, acoustic p-wave velocity, and rock density.Facies are represented probabilistically, while activation functions constrain the velocity and density output distributions.
- Differentiable coupling: The trained generator and seismic forward operator form a fully differentiable computational graph, with constant-velocity padding accommodating sources and receivers.This graph enables gradients to pass from seismic mismatch through the forward model and generator.
5. Dataset
The dataset uses object-based fluvial geological models to train and test GAN-generated subsurface realizations, with facies, acoustic velocity, and density represented as modeled properties.
- Dataset construction: The training data contain fluvial systems with porous sandstone channels embedded in fine-grained shale, generated using object-based geological models.Channel geometry, locations, widths, and properties are sampled from specified distributions.
- Modeled properties: Each cross-section represents river-channel and shale-matrix facies with acoustic p-wave velocity and density properties.A binary indicator distinguishes the two facies regions.
- Training and testing: 10,000 images were created for GAN training, while 5,000 additional images were reserved to evaluate inversion.Image quality and output distributions were monitored during generative-model training.
- Ground truth: The ground-truth example used to emulate measured seismic data compares geological facies, acoustic p-wave velocity, and rock density distributions.These modeled properties are used to assess how the generated geological models represent the target example.
6. Results
The inversion produces diverse posterior geological ensembles whose uncertainty decreases with seismic coverage and is further localized by bore-hole information, while MALA-approximation sampling reaches low seismic mismatch.
- Prior evaluation: 100 unconditional GAN samples show high variability in geological structures and closely match the training distribution of geophysical properties.The generator is evaluated as a prior before posterior inversion.
- Inversion setup: The inversion accepts models with relative seismic error below 10%, and bore-hole-constrained samples additionally require above 95% facies accuracy.The two inversion cases are acoustic velocity alone, and acoustic velocity combined with geological facies along a bore-hole.
- Seismic-source effects: Increasing acoustic sources from 2 to 27 lowers the standard deviation of the inverted model ensembles overall.Sparse source coverage leaves unsampled regions with greater variability because the prior remains more prevalent there.
- Seismic-source effects: The reduction in variability is only marginal between 9 and 27 sources as the forward problem reaches its resolution limits.This bounds the benefit of adding further sources in the reported experiments.
- Bore-hole constraints: Bore-hole constraints reduce standard deviation around the well and influence lateral channel features, with near-zero variability along the well.The near-zero well variability follows from the per-realization 95% facies-accuracy constraint.
- Sampling behavior: 100 MALA-approximation iterations reduce seismic mismatch to relative errors of 5–7%, while bore-hole constraints require 200 iterations.The step size is reduced linearly, stabilizing the seismic mismatch; the bore-hole case must satisfy both seismic and bore-hole observations.
- Computational limitation: Reducing MALA iterations or using gradient descent can reach small seismic errors but reduces sample diversity.High computational cost in two- and three-dimensional FWI creates pressure to use few iterations.
7. Conclusions
The paper combines a lower-dimensional generative representation of geological structures with adjoint-based acoustic inversion and approximate MALA sampling. The approach is illustrated on a simple geophysical inversion and may extend to other PDE-constrained inverse problems.
- The application addresses computationally intensive inversion performed in the very high-dimensional space of subsurface model properties.
- The method parameterizes geological structures with lower-dimensional latent variables and combines them with numerical acoustic inversion using the adjoint method.
- Approximate MALA uses gradients from the adjoint PDE to sample the posterior over latent variables given seismic-data mismatch.
- The approach may find use in domains where spatial property models control physical systems, including porous-media flow and materials science.
A.1. Generative Model Network Architectures
The appendix describes generator and discriminator architectures for synthetic geological structures and the distributions and activations used to represent their properties.
- The generator and discriminator network architectures are used to create synthetic geological structures.
- Binary geological facies indicators and corresponding p-wave velocities use a bivariate Gaussian representation with hyperbolic tangent activation.
- Rock density is represented with a Gaussian distribution and a soft-plus activation ensures positive numerical values.
- The discriminator is specified as a multi-channel GAN component.
A.2. Samples obtained by optimization in latent space
The appendix presents samples produced by optimizing in latent space under different seismic-source configurations. Cases include increasing source counts and a bore-hole measurement.
- Two acoustic sources are used to obtain samples from latent-space optimization.
- Three acoustic sources are used to obtain samples from latent-space optimization.
- Nine acoustic sources are used to obtain samples from latent-space optimization.
- Twenty-seven acoustic sources are used to obtain samples from latent-space optimization.
- A separate configuration uses two acoustic sources and one bore-hole.