Learning by example: Fast reliability-aware seismic imaging with normalizing flows

Siahkoohi, Ali; Herrmann, Felix J.

doi:10.1190/segam2021-3581836.1

Cited by 14 publications

(10 citation statements)

References 24 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%

“…While effective in controlled settings, handcrafted priors might introduce unwanted bias to the solution. Recent deep-learning based approaches [20][21][22][23][24][25][26][27][28][29][30][31], on the other hand, learn a prior distribution from available data 1 . While certainly providing a better description of the available prior information when compared to generic handcrafted priors, they may affect the results more seriously when out-of-distribution data is considered, e.g., when the training data is not fully representative of a given scenario.…”

Section: Introductionmentioning

confidence: 99%

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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%

Section: Introductionmentioning

confidence: 99%

Deep Bayesian inference for seismic imaging with tasks

Siahkoohi¹,

Rizzuti²,

Herrmann³

2021

Preprint

Self Cite

View full text Add to dashboard Cite

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“…This amounts to feeding the latent code associated with observed data, i.e., T w1 (y), and Gaussian samples z ∼ p z (z) into the inverse network, T −1 w2 . These samples may be used for Bayesian inference if we have an ideal training dataset [9,[15][16][17]. However, such an assumption is rarely correct in geophysical applications due to Earth's strong heterogeneity [18][19][20], which highlights the importance of devising formulations that are robust to changes in data distribution during inference.…”

Section: Amortized Variational Inferencementioning

confidence: 99%

“…To achieve this, following Orozco et al [5], we train a conditional normalizing flow [NF,6] to capture the conditional distribution of the unknown, given data, i.e., the posterior distribution. The training involves minimizing an amortized variational inference objective [6][7][8][9][10] using existing training pairs in the form of low-fidelity data and model pairs. After training, we are able to capture the low-fidelity posterior distribution for previously unseen seismic data.…”

Section: Introductionmentioning

confidence: 99%

Wave-equation-based inversion with amortized variational Bayesian inference

Siahkoohi¹,

Orozco²,

Rizzuti³

et al. 2022

Preprint

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Solving inverse problems involving measurement noise and modeling errors requires regularization in order to avoid data overfit. Geophysical inverse problems, in which the Earth's highly heterogeneous structure is unknown, present a challenge in encoding prior knowledge through analytical expressions. Our main contribution is a generative-model-based regularization approach, robust to out-ofdistribution data, which exploits the prior knowledge embedded in existing data and model pairs. Utilizing an amortized variational inference objective, a conditional normalizing flow (NF) is pretrained on pairs of low-and high-fidelity migrated images in order to achieve a low-fidelity approximation to the seismic imaging posterior distribution for previously unseen data. The NF is used after pretraining to reparameterize the unknown seismic image in an inversion scheme involving physics-guided data misfit and a Gaussian prior on the NF latent variable. Solving this optimization problem with respect to the latent variable enables us to leverage the benefits of data-driven conditional priors whilst being informed by physics and data. The numerical experiments demonstrate that the proposed inversion scheme produces seismic images with limited artifacts when dealing with noisy and out-of-distribution data.

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“…This linearized imaging problem is challenged by the computationally expensive forward operator as well as presence of measurements noise, linearization errors, modeling errors, and the nontrivial nullspace of the linearized forward Born modeling operator [1][2][3]. These challenges highlight the importance of uncertainty quantification (UQ) in seismic imaging, where instead of finding one seismic image estimate, a distribution of seismic images is obtained that explains the observed data [4], consequently reducing the risk of data overfit and enabling UQ [5][6][7][8][9][10][11][12][13][14][15].…”

Section: Introductionmentioning

confidence: 99%

Velocity continuation with Fourier neural operators for accelerated uncertainty quantification

Siahkoohi¹,

Louboutin²,

Herrmann³

2022

Preprint

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Seismic imaging is an ill-posed inverse problem that is challenged by noisy data and modeling inaccuracies-due to errors in the background squared-slowness model. Uncertainty quantification is essential for determining how variability in the background models affects seismic imaging. Due to the costs associated with the forward Born modeling operator as well as the high dimensionality of seismic images, quantification of uncertainty is computationally expensive. As such, the main contribution of this work is a survey-specific Fourier neural operator surrogate to velocity continuation that maps seismic images associated with one background model to another virtually for free. While being trained with only 200 background and seismic image pairs, this surrogate is able to accurately predict seismic images associated with new background models, thus accelerating seismic imaging uncertainty quantification. We support our method with a realistic data example in which we quantify seismic imaging uncertainties using a Fourier neural operator surrogate, illustrating how variations in background models affect the position of reflectors in a seismic image.

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Learning by example: Fast reliability-aware seismic imaging with normalizing flows

Cited by 14 publications

References 24 publications

Deep Bayesian inference for seismic imaging with tasks

Deep Bayesian inference for seismic imaging with tasks

Wave-equation-based inversion with amortized variational Bayesian inference

Velocity continuation with Fourier neural operators for accelerated uncertainty quantification

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