Optimal Transport Based Seismic Inversion:Beyond Cycle Skipping

Engquist, Björn; Yang, Yunan

doi:10.1002/cpa.21990

Cited by 38 publications

(24 citation statements)

References 77 publications

(106 reference statements)

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“…Note that wavefields are not naturally probability distributions. Thus, when we implement the W 2 natural gradient, we normalize the data to be probability densities following [10,11]. As we have discussed in Remark 2.3, the W 2 and Ḣ−1 natural gradients are closely related, which are also reflected in this numerical example as the reconstructions in Figure 5e and Figure 5f are very similar.…”

Section: Full Waveform Inversionsupporting

confidence: 60%

“…We avoid dealing with the nonconvexity by choosing a good initial guess as shown in Figure 5b. One may also choose other objective functions such as the Wasserstein metric to improve the optimization landscape [11]. We follow Subsection 3.3 to carry out the implementation for various natural gradient descent methods since the Jacobian ∂ θ ρ is not explicitly given, and the adjoint-state method has to be applied based on (4.1).…”

Section: Full Waveform Inversionmentioning

confidence: 99%

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Efficient Natural Gradient Descent Methods for Large-Scale Optimization Problems

Nurbekyan¹,

Lei²,

Yang³

2022

Preprint

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We propose an efficient numerical method for computing natural gradient descent directions with respect to a generic metric in the state space. Our technique relies on representing the natural gradient direction as a solution to a standard least-squares problem. Hence, instead of calculating, storing, or inverting the information matrix directly, we apply efficient methods from numerical linear algebra to solve this least-squares problem. We treat both scenarios where the derivative of the state variable with respect to the parameter is either explicitly known or implicitly given through constraints. We apply the QR decomposition to solve the least-squares problem in the former case and utilize the adjoint-state method to compute the natural gradient descent direction in the latter case. As a result, we can reliably compute several natural gradient descents, including the Wasserstein natural gradient, for a large-scale parameter space with thousands of dimensions, which was believed to be out of reach. Finally, our numerical results shed light on the qualitative differences among the standard gradient descent method and various natural gradient descent methods based on different metric spaces in large-scale nonconvex optimization problems.

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Section: Full Waveform Inversionsupporting

confidence: 60%

Section: Full Waveform Inversionmentioning

confidence: 99%

Efficient Natural Gradient Descent Methods for Large-Scale Optimization Problems

Nurbekyan¹,

Lei²,

Yang³

2022

Preprint

Self Cite

View full text Add to dashboard Cite

show abstract

“…Full waveform inversion (FWI) has been receiving wide attention in recent years [9,14,22,32,36,37] due to its high-resolution imaging in geophysical properties. Generally, it can be formulated as a PDE constrained optimization problem in mathematics, which consists of two parts [31]: the forward modeling of seismic wavefield, and the optimization problem searching for suitable model parameters to minimize the mismatch between the predicted and observed seismic signals.…”

Section: Introductionmentioning

confidence: 99%

The Quadratic Wasserstein Metric With Squaring Scaling For Seismic Velocity Inversion

Li¹,

Tang²,

Chen³

et al. 2022

Preprint

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The quadratic Wasserstein metric has shown its power in measuring the difference between probability densities, which benefits optimization objective function with better convexity and is insensitive to data noise. Nevertheless, it is always an important question to make the seismic signals suitable for comparison using the quadratic Wasserstein metric. The squaring scaling is worth exploring since it guarantees the convexity caused by data shift. However, as mentioned in [Commun. Inf. Syst., 2019, 19:95-145], the squaring scaling may lose uniqueness and result in more local minima to the misfit function. In our previous work [J. Comput. Phys., 2018, 373:188-209], the quadratic Wasserstein metric with squaring scaling was successfully applied to the earthquake location problem. But it only discussed the inverse problem with few degrees of freedom. In this work, we will present a more in-depth study on the combination of squaring scaling technique and the quadratic Wasserstein metric. By discarding some inapplicable data, picking seismic phases, and developing a new normalization method, we successfully invert the seismic velocity structure based on the squaring scaling technique and the quadratic Wasserstein metric. The numerical experiments suggest that this newly proposed method is an efficient approach to obtain more accurate inversion results.

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“…A more convex misfit function also implies a more robust solution to the inverse problem when subject to uncertainties in the input parameters. Rigorous mathematical treatment [13] has in fact shown that 1-D quadratic Wasserstein distances (a subset of OT distances) are convex functions with respect to dilation and translation when applied to probability density functions. In order for this to remain valid for generic signals as well, it is however necessary to normalize and positivize them accordingly.…”

Section: Introductionmentioning

confidence: 99%

Bayesian inference under model misspecification using transport-Lagrangian distances: an application to seismic inversion

Scarinci,

Fehler,

Marzouk

2021

Preprint

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Model misspecification constitutes a major obstacle to reliable inference in many inverse problems. Inverse problems in seismology, for example, are particularly affected by misspecification of wave propagation velocities. In this paper, we focus on a specific seismic inverse problem-fullwaveform moment tensor inversion-and develop a Bayesian framework that seeks robustness to velocity misspecification. A novel element of our framework is the use of transport-Lagrangian (TL) distances between observed and model predicted waveforms to specify a loss function, and the use of this loss to define a generalized belief update via a Gibbs posterior. The TL distance naturally disregards certain features of the data that are more sensitive to model misspecification, and therefore produces less biased or dispersed posterior distributions in this setting. To make the latter notion precise, we use several diagnostics to assess the quality of inference and uncertainty quantification, i.e., continuous rank probability scores and rank histograms. We interpret these diagnostics in the Bayesian setting and compare the results to those obtained using more typical Gaussian noise models and squared-error loss, under various scenarios of misspecification. Finally, we discuss potential generalizability of the proposed framework to a broader class of inverse problems affected by model misspecification.

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Optimal Transport Based Seismic Inversion:Beyond Cycle Skipping

Cited by 38 publications

References 77 publications

Efficient Natural Gradient Descent Methods for Large-Scale Optimization Problems

Efficient Natural Gradient Descent Methods for Large-Scale Optimization Problems

The Quadratic Wasserstein Metric With Squaring Scaling For Seismic Velocity Inversion

Bayesian inference under model misspecification using transport-Lagrangian distances: an application to seismic inversion

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