A reduced-order variational Bayesian approach for efficient subsurface imaging

Urozayev, Dias; Ait‐El‐Fquih, Boujemaa; Hoteit, Ibrahim; Peter, Daniel

doi:10.1093/gji/ggab507

Cited by 11 publications

(3 citation statements)

References 60 publications

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“…Instead of increasing the number of particles which may be computationally intractable, one may try to reduce the dimensionality of the problem. For example, other parameterizations that require fewer parameters to represent the model may be used, such as Voronoi cells (Bodin & Sambridge 2009;Zhang et al 2018b), wavelet parameterization (Hawkins & Sambridge 2015), Johnson-Mehl tessellation (Belhadj et al 2018), Delaunay and Clough-Tocher parameterizations (Curtis & Snieder 1997) or discrete cosine transforms (Urozayev et al 2022). In addition, other SVGD variants which project the high dimensional parameter space into a lower dimensional space may be used to improve the results, for example, projected SVGD (Chen & Ghattas 2020) or sliced SVGD (Gong et al 2020).…”

Section: Discussionmentioning

confidence: 99%

3D Variational Full-Waveform Inversion

Zhang

Zhou

Lomas

et al. 2022

Preprint

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Seismic full-waveform inversion (FWI) produces high resolution images of the subsurface by exploiting information in full seismic waveforms, and has been applied at global, regional and industrial spatial scales. FWI is traditionally solved by using optimization, in which one seeks a best model by minimizing the misfit between observed waveforms and model predicted waveforms. Due to the nonlinearity of the physical relationship between model parameters and waveforms, a good starting model is often required to produce a reasonable model. In addition, the optimization methods cannot produce accurate uncertainty estimates, which are required to better interpret the results.To estimate uncertainties more accurately, nonlinear Bayesian methods have been deployed to solve the FWI problem. Monte Carlo sampling is one such algorithm but it is computationally expensive, and all Markov chain Monte Carlo-based methods are difficult to parallelise fully. Variational inference provides an efficient, fully parallelisable alternative methodology. This is a class of methods that optimize an approximation to a probability distribution describing post-inversion parameter uncertainties. Both Monte Carlo and variational full waveform inversion (VFWI) have been applied previously to solve 2D FWI problems, but neither of them have been applied to 3D FWI. In this study we apply the VFWI method to a 3D FWI problem. Specifically we use Stein variational gradient descent (SVGD) method to solve the 3D Bayesian FWI problem and to obtain an optimised set of samples of the full posterior probability distribution. The aim of this study is to explore performance of the method in 3D, to assess the computational requirements and to provide useful information for practitioners. Our results demonstrate that the 3D VFWI is practical, at least for small problems, and can be applied to image the subsurface in reality.

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Section: Discussionmentioning

confidence: 99%

3D Variational Full-Waveform Inversion

Zhang

Zhou

Lomas

et al. 2022

Preprint

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“…(2020). Since then variational methods have been applied to a variety of problems including travel time tomography (Levy, Laloy, & Linde, 2022; X. Zhang & Curtis, 2020a; X. Zhao et al., 2021), seismic denoising (Siahkoohi et al., 2021, 2023), seismic amplitude inversion (Zidan et al., 2022), earthquake hypocenter inversion (Smith et al., 2022), slip distribution inversion (Sun et al., 2023), full waveform inversion in 2D (Urozayev et al., 2022; W. Wang et al., 2023; X. Zhang & Curtis, 2020b) and in 3D (Lomas et al., 2023; X. Zhang et al., 2023), and survey or experimental design (Strutz & Curtis, 2024). In addition, various types of neural networks produce probabilistic outputs and can be considered variational methods (Bishop, 1994), and these have been applied to subsurface imaging problems for more than two decades (Bloem et al., 2023; Cao et al., 2020; Devilee et al., 1999; de Wit et al., 2013; Earp & Curtis, 2020; Earp et al., 2020; Hansen & Finlay, 2022; Käufl et al., 2014, 2016; Lubo‐Robles et al., 2021; Meier et al., 2007a, 2007b; A. K. Ray & Biswal, 2010; Shahraeeni & Curtis, 2011; Shahraeeni et al., 2012; Siahkoohi et al., 2022; X. Zhang & Curtis, 2021b; X. Zhao et al., 2021).…”

Section: Introductionmentioning

confidence: 99%

Bayesian Inversion, Uncertainty Analysis and Interrogation Using Boosting Variational Inference

Zhao,

Curtis

2024

JGR Solid Earth

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Geoscientists use observed data to estimate properties of the Earth's interior. This often requires non‐linear inverse problems to be solved and uncertainties to be estimated. Bayesian inference solves inverse problems under a probabilistic framework, in which uncertainty is represented by a so‐called posterior probability distribution. Recently, variational inference has emerged as an efficient method to estimate Bayesian solutions. By seeking the closest approximation to the posterior distribution within any chosen family of distributions, variational inference yields a fully probabilistic solution. It is important to define expressive variational families so that the posterior distribution can be represented accurately. We introduce boosting variational inference (BVI) as a computationally efficient means to construct a flexible approximating family comprising all possible finite mixtures of simpler component distributions. We use Gaussian mixture components due to their fully parametric nature and the ease with which they can be optimized. We apply BVI to seismic travel time tomography and full waveform inversion, comparing its performance with other methods of solution. The results demonstrate that BVI achieves reasonable efficiency and accuracy while enabling the construction of a fully analytic expression for the posterior distribution. Samples that represent major components of uncertainty in the solution can be obtained analytically from each mixture component. We demonstrate that these samples can be used to solve an interrogation problem: to assess the size of a subsurface target structure. To the best of our knowledge, this is the first method in geophysics that provides both analytic and reasonably accurate probabilistic solutions to fully non‐linear, high‐dimensional Bayesian full waveform inversion problems.

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“…More specifically, we introduce a new iterative filtering scheme to sample ensemble members for the state and the deterministic parameters using an EnKF‐OSAS and derive closed forms for the posterior pdf of the stochastic parameter and associated PM estimate. The derivation of such a scheme is founded on the introduction of posterior independence between the state and the deterministic parameters on the one hand and the stochastic parameters on the other hand, using the (functional) variational Bayesian (VB) optimization, that is, according to the Kullback–Leibler divergence (KLD) minimization criterion (Smidl and Quinn, 2006; Blei et al ., 2017; Urozayev et al ., 2022). On top of its online nature, the proposed filtering algorithm, called VB‐EnKF‐OSAS, exploits the advantages of the OSAS‐based ensemble filtering to mitigate inconsistency issues of the joint EnKF, and also provides a full posterior distribution for the stochastic parameters, in contrast with the ML approach, which only provides point estimates.…”

Section: Introductionmentioning

confidence: 99%

A variational Bayesian approach for ensemble filtering of stochastically parametrized systems

Ait‐El‐Fquih

Subramanian

Hoteit

2023

Quart J Royal Meteoro Soc

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Modern climate models use both deterministic and stochastic parametrization schemes to represent uncertainties in their physics and inputs. This work considers the problem of estimating the involved parameters of such systems simultaneously with their state through data assimilation. Standard state‐parameter filtering schemes cannot be applied to such systems, owing to the posterior dependence between the stochastic parameters and the “dynamical” augmented state, defined as the state augmented by the deterministic parameters. We resort to the variational Bayesian (VB) approach to break this dependence, by approximating the joint posterior probability density function (pdf) of the augmented state and the stochastic parameters with a separable product of two marginal pdfs that minimizes the Kullback–Leibler divergence. The resulting marginal pdf of the augmented state is then sampled using a one‐step‐ahead smoothing‐based ensemble Kalman filter (EnKF‐OSAS), whereas a closed form is derived for the marginal pdf of the stochastic parameters. The proposed approach combines the effectiveness of the OSAS filtering approach to mitigate inconsistency issues that often arise with the joint ensemble Kalman filter (EnKF), with the advantage of obtaining a full posterior pdf for the stochastic parameters, which is not possible with the traditional maximum‐likelihood method. We demonstrate the relevance of the proposed approach through extensive numerical experiments with a one‐scale Lorenz‐96 model, which includes a stochastic parametrization representing subgrid‐scale effects.

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A reduced-order variational Bayesian approach for efficient subsurface imaging

Cited by 11 publications

References 60 publications

3D Variational Full-Waveform Inversion

3D Variational Full-Waveform Inversion

Bayesian Inversion, Uncertainty Analysis and Interrogation Using Boosting Variational Inference

A variational Bayesian approach for ensemble filtering of stochastically parametrized systems

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