Uncertainty quantification in imaging and automatic horizon tracking – A Bayesian deep-prior based approach

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

doi:10.1190/segam2020-3417560.1

Cited by 20 publications

(10 citation statements)

References 27 publications

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“…Compared to the MAP estimate, the conditional mean, which corresponds to the minimum-variance estimate [85], is less prone to overfitting [86]. This was confirmed empirically for seismic imaging [54,55]. In the experimental sections below, we will provide further evidence of advantages the conditional mean offers compared to MAP estimation.…”

Section: Conditional Mean Estimationmentioning

confidence: 85%

“…The need for repeated evaluations of the forward operator, the correlation between consecutive samples [88], and the high dimensionality of the problem are the chief computational challenges for these methods. Despite these difficulties, MCMC methods have been applied successfully in imaging problems including [6,8,9,54,55,89,90].…”

Section: Sampling Via Stochastic Gradient Langevin Dynamicsmentioning

confidence: 99%

“…In this work, we take advantage of a novel prior recently deployed in computer vision and geophysics [48][49][50][51][52][53][54][55][56][57][58], known as the deep prior, which utilizes the inductive bias [59] of untrained convolutional neural networks (CNNs) as a prior. This approach is tantamount to restricting feasible models to the range of an untrained CNN with a fixed input and randomly initialized weights.…”

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: Conditional Mean Estimationmentioning

confidence: 85%

Section: Sampling Via Stochastic Gradient Langevin Dynamicsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Deep Bayesian inference for seismic imaging with tasks

Siahkoohi¹,

Rizzuti²,

Herrmann³

2021

Preprint

Self Cite

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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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“…MCMC methods sample the posterior distribution via a series of random walks in the probability space where the posterior probability density function (PDF) needs to be evaluated or approximated at each step-e.g., via stochastic gradient Langevin dynamics [3]. These sampling methods typically require many steps to traverse the probability space [4][5][6][7][8][9][10][11], which fundamentally limits their applicability to large-scale problems due to costs associated with the forward operator [12][13][14][15]. Alternatively, variational inference methods [16] approximate the posterior distribution with a parametric and easy-to-sample distribution.…”

Section: Introductionmentioning

confidence: 99%

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

Siahkoohi¹,

Herrmann²

2021

Preprint

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Uncertainty quantification provides quantitative measures on the reliability of candidate solutions of ill-posed inverse problems. Due to their sequential nature, Monte Carlo sampling methods require large numbers of sampling steps for accurate Bayesian inference and are often computationally infeasible for large-scale inverse problems, such as seismic imaging. Our main contribution is a data-driven variational inference approach where we train a normalizing flow (NF), a type of invertible neural net, capable of cheaply sampling the posterior distribution given previously unseen seismic data from neighboring surveys. To arrive at this result, we train the NF on pairs of low-and high-fidelity migrated images. In our numerical example, we obtain high-fidelity images from the Parihaka dataset and low-fidelity images are derived from these images through the process of demigration, followed by adding noise and migration. During inference, given shot records from a new neighboring seismic survey, we first compute the reverse-time migration image. Next, by feeding this low-fidelity migrated image to the NF we gain access to samples from the posterior distribution virtually for free. We use these samples to compute a high-fidelity image including a first assessment of the image's reliability. To our knowledge, this is the first attempt to train a conditional network on what we know from neighboring images to improve the current image and assess its reliability.

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Uncertainty quantification in imaging and automatic horizon tracking – A Bayesian deep-prior based approach

Cited by 20 publications

References 27 publications

Deep Bayesian inference for seismic imaging with tasks

Deep Bayesian inference for seismic imaging with tasks

Velocity continuation with Fourier neural operators for accelerated uncertainty quantification

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

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