A Bayesian-based approach to improving acoustic Born waveform inversion of seismic data for viscoelastic media

Muhumuza, Kenneth; Roininen, Lassi; Huttunen, Janne M. J.; Lähivaara, Timo

doi:10.1088/1361-6420/ab8f81

Cited by 3 publications

(4 citation statements)

References 42 publications

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“…The BAE approach was used to compensate for the linearisation of the forward problem in [22,23], both of which employed the Born approximation…”

Section: Example 1: Inverse Medium Scatteringmentioning

confidence: 99%

“…Of specific relevance to the current study are the works [20][21][22][23][24], in which the accurate nonlinear parameter-to-observable map is replaced by an approximate linear counterpart while accounting for the model error using the BAE approach. Specifically, in [24] the inverse problem governed by the acoustic wave equation (modelling photoacoustic tomography) is formulated and solved for the initial pressure.…”

Section: Introductionmentioning

confidence: 99%

“…The induced additional uncertainties are accounted for by the BAE approach. On the other hand, in [22,23], the linear Born approximation is used to solve inverse scattering problems with high contrast materials, while in [20,21] nonlinear parameter-to-observable maps related to diffuse optical tomography and electrical impedance tomography (EIT), respectively, are replaced with their respective Fréchet derivatives.…”

Section: Introductionmentioning

confidence: 99%

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On global normal linear approximations for nonlinear Bayesian inverse problems

Nicholson¹,

Petra²,

Villa³

et al. 2023

Inverse Problems

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The replacement of a nonlinear parameter-to-observable mapping with a linear (aﬃne) approximation is often carried out to reduce the computational costs associated with solving large-scale inverse problems governed by partial diﬀerential equations (PDEs). In the case of a linear parameter-to-observable mapping with normally distributed additive noise and a Gaussian prior measure on the parameters, the posterior is Gaussian. However, substituting an accurate model for a (possibly well justiﬁed) linear surrogate model can give misleading results if the induced model approximation error is not accounted for. To account for the errors, the Bayesian approximation error approach can be utilised, in which the ﬁrst and second order statistics of the errors are computed via sampling. The most common linear approximation is carried out via linear Taylor expansion, which requires the computation of (Fréchet) derivatives of the parameter-to-observable mapping with respect to the parameters of interest. In this paper, we prove that the (approximate) posterior measure obtained by replacing the nonlinear parameter-to-observable mapping with a linear approximation is in fact independent of the choice of the linear approximation when the Bayesian approximation error approach is employed. Thus, somewhat non-intuitively, employing the zero-model as the linear approximation gives the same approximate posterior as any other choice of linear approximations of the parameter-to-observable model. The independence of the linear approximation is demonstrated mathematically and illustrated with two numerical PDE-based problems: an inverse scattering type problem and an inverse conductivity type problem.

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“…The BAE approach was used to compensate for the linearisation of the forward problem in [22,23], both of which employed the Born approximation…”

Section: Example 1: Inverse Medium Scatteringmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

On global normal linear approximations for nonlinear Bayesian inverse problems

Nicholson¹,

Petra²,

Villa³

et al. 2023

Inverse Problems

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“…where h is discretization step in both coordinate directions, and λ is the regularization parameter. These priors have been used for X-ray tomography in oil and gas industry (Mendoza et al, 2019) and subsurface imaging (Muhumuza et al, 2020).…”

Section: E10-6mentioning

confidence: 99%

Enhancing industrial X-ray tomography by data-centric statistical methods

Suuronen

Emzir

Lasanen

et al. 2020

DCE

Self Cite

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X-ray tomography has applications in various industrial fields such as sawmill industry, oil and gas industry, as well as chemical, biomedical, and geotechnical engineering. In this article, we study Bayesian methods for the X-ray tomography reconstruction. In Bayesian methods, the inverse problem of tomographic reconstruction is solved with the help of a statistical prior distribution which encodes the possible internal structures by assigning probabilities for smoothness and edge distribution of the object. We compare Gaussian random field priors, that favor smoothness, to non-Gaussian total variation (TV), Besov, and Cauchy priors which promote sharp edges and high- and low-contrast areas in the object. We also present computational schemes for solving the resulting high-dimensional Bayesian inverse problem with 100,000–1,000,000 unknowns. We study the applicability of a no-U-turn variant of Hamiltonian Monte Carlo (HMC) methods and of a more classical adaptive Metropolis-within-Gibbs (MwG) algorithm to enable full uncertainty quantification of the reconstructions. We use maximum a posteriori (MAP) estimates with limited-memory BFGS (Broyden–Fletcher–Goldfarb–Shanno) optimization algorithm. As the first industrial application, we consider sawmill industry X-ray log tomography. The logs have knots, rotten parts, and even possibly metallic pieces, making them good examples for non-Gaussian priors. Secondly, we study drill-core rock sample tomography, an example from oil and gas industry. In that case, we compare the priors without uncertainty quantification. We show that Cauchy priors produce smaller number of artefacts than other choices, especially with sparse high-noise measurements, and choosing HMC enables systematic uncertainty quantification, provided that the posterior is not pathologically multimodal or heavy-tailed.

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