Parameterizing uncertainty by deep invertible networks: An application to reservoir characterization

Rizzuti, Gabrio; Siahkoohi, Ali; Witte, Patrick; Herrmann, Felix J.

doi:10.1190/segam2020-3428150.1

Cited by 17 publications

(10 citation statements)

References 20 publications

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“…Typically these samples are obtained by applying a series of learned nonlinear functions to random realizations from a canonical distribution. Early work on variational inference [26,30,31,44,[113][114][115][116][117][118] shows encouraging results, which opens enticing new perspectives on uncertainty quantification in the field of wave-equation based inversion.…”

Section: Discussionmentioning

confidence: 99%

Deep Bayesian inference for seismic imaging with tasks

Siahkoohi¹,

Rizzuti²,

Herrmann³

2021

Preprint

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We propose to use techniques from Bayesian inference and deep neural networks to translate uncertainty in seismic imaging to uncertainty in tasks performed on the image, such as horizon tracking. Seismic imaging is an ill-posed inverse problem because of unavoidable bandwidth and aperture limitations, which that is hampered by the presence of noise and linearization errors. Many regularization methods, such as transform-domain sparsity promotion, have been designed to deal with the adverse effects of these errors, however, these methods run the risk of biasing the solution and do not provide information on uncertainty in the image space and how this uncertainty impacts certain tasks on the image. A systematic approach is proposed to translate uncertainty due to noise in the data to confidence intervals of automatically tracked horizons in the image. The uncertainty is characterized by a convolutional neural network (CNN) and to assess these uncertainties, samples are drawn from the posterior distribution of the CNN weights, used to parameterize the image. Compared to traditional priors, in the literature it is argued that these CNNs introduce a flexible inductive bias that is a surprisingly good fit for many diverse domains in imaging. The method of stochastic gradient Langevin dynamics is employed to sample from the posterior distribution. This method is designed to handle large scale Bayesian inference problems with computationally expensive forward operators as in seismic imaging. Aside from offering a robust alternative to maximum a posteriori estimate that is prone to overfitting, access to these samples allow us to translate uncertainty in the image, due to noise in the data, to uncertainty on the tracked horizons. For instance, it admits estimates for the pointwise standard deviation on the image and for confidence intervals on its automatically tracked horizons.

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Section: Discussionmentioning

confidence: 99%

Deep Bayesian inference for seismic imaging with tasks

Siahkoohi¹,

Rizzuti²,

Herrmann³

2021

Preprint

Self Cite

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“…Because of their sequential nature, MCMC sampling methods require a large number of sampling steps to perform accurate Bayesian inference [7], which reduces their applicability to large-scale problems due to the costs associated with the forward operator [4,5,[8][9][10][11][12][13][14]. As an alternative, variational inference methods [15][16][17][18][19][20][21][22][23] approximate the posterior distribution with a surrogate and easy-to-sample distribution. By means of this approximation, sampling is turned into an optimization problem, in which the parameters of the surrogate distribution are tuned in order to minimize the divergence between the surrogate and posterior distributions.…”

Section: Introductionmentioning

confidence: 99%

“…Thus, the variational inference optimization problem must be solved again for every new set of observations. Solving this optimization problem may require numerous iterations [19,20], which may not be feasible in inverse problems with computationally costly forward operators, such as seismic imaging.…”

Section: Introductionmentioning

confidence: 99%

Reliable amortized variational inference with physics-based latent distribution correction

Siahkoohi¹,

Rizzuti²,

Orozco³

et al. 2022

Preprint

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Bayesian inference for high-dimensional inverse problems is challenged by the computational costs of the forward operator and the selection of an appropriate prior distribution. Amortized variational inference addresses these challenges where a neural network is trained to approximate the posterior distribution over existing pairs of model and data. When fed previously unseen data and normally distributed latent samples as input, the pretrained deep neural network-in our case a conditional normalizing flow-provides posterior samples with virtually no cost. However, the accuracy of this approach relies on the availability of high-fidelity training data, which seldom exists in geophysical inverse problems because of the highly heterogeneous structure of the Earth. In addition, accurate amortized variational inference requires the observed data to be drawn from the training data distribution. As such, we offer a solution that increases the resilience of amortized variational inference when faced with data distribution shifts, e.g., changes in the forward model or prior distribution. Our method involves a physics-based correction to the conditional normalizing flow latent distribution to provide a more accurate approximation to the posterior distribution for the observed data at hand. To accomplish this, instead of a standard Gaussian latent distribution, we parameterize the latent distribution by a Gaussian distribution with an unknown mean and diagonal covariance. These unknown quantities are then estimated by minimizing the Kullback-Leibler divergence between the corrected and true posterior distributions. While generic and applicable to other inverse problems, by means of a seismic imaging example, we show that our correction step improves the robustness of amortized variational inference with respect to changes in number of source experiments, noise variance, and shifts in the prior distribution. This approach provides a seismic image with limited artifacts and an assessment of its uncertainty with approximately the same cost as five reverse-time migrations.

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“…They have been used to solve inverse problems in medicine (Ardizzone et al., 2018), astrophysics (Osborne et al., 2019), optical imaging (Adler et al., 2019; Moran et al., 2018) and morphology (Sahin & Gurevych, 2020). INNs have also been used to solve a variational problem to parameterize uncertainty for reservoir characterization (Rizzuti et al., 2020), and the idea of auxiliary variables has also been used in seismic full waveform inversion (Huang et al., 2018). In this study, we use INNs to solve seismic tomographic inverse problems.…”

Section: Introductionmentioning

confidence: 99%

Bayesian Geophysical Inversion Using Invertible Neural Networks

Zhang

Curtis

2021

JGR Solid Earth

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Geoscientists build models of the subsurface to understand properties and processes in the Earth's interior. The models are usually parameterized in some way, so to constrain the models, we must solve a parameter estimation problem. Data are recorded which provide constraints, together with prior knowledge. However, since the physical relationships between parameters and data usually predict data given the parameters (known as the forward calculation) but not the reverse, the solution must be found using inverse theory.Geophysical inverse problems usually have nonunique solutions due to noise in the data, to nonlinearity of the physical relationships between model parameters and data, and to fundamentally unconstrained combinations of parameters. Uncertainties in parameter estimates must therefore be quantified to interpret inversion results correctly. Unfortunately, estimating uncertainty in nonlinear inverse problems can be computationally expensive, and the cost increases both with the number of parameters, and with the computational cost of the forward calculation. In this study, we therefore solve two different types of seismic tomography problems which each have fewer than 100 parameters, and have relatively rapid forward functions (each evaluation takes on the order of seconds). This allows us to evaluate solutions sufficiently accurately to thoroughly test a new method of Geophysical inversion.Geophysical inverse problems are traditionally solved by linearizing (approximating) the nonlinear physics, and using optimization methods which seek a model that minimizes the misfit between observed and predicted data (Aki & Lee, 1976;Dziewonski & Woodhouse, 1987;Iyer & Hirahara, 1993). However, despite their wide applications, linearized procedures cannot produce accurate estimates of uncertainty because data uncertainty distributions can deviate substantially from analytic forms of probability such as Gaussians or Uniform distributions that can be propagated through linearized methods efficiently, and since the distribution of possible models post inversion can be strongly affected by nonlinearity in the physics (Bo-

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Parameterizing uncertainty by deep invertible networks: An application to reservoir characterization

Cited by 17 publications

References 20 publications

Deep Bayesian inference for seismic imaging with tasks

Deep Bayesian inference for seismic imaging with tasks

Reliable amortized variational inference with physics-based latent distribution correction

Bayesian Geophysical Inversion Using Invertible Neural Networks

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