Application of the Wasserstein metric to seismic signals

Engquist, Björn; Froese, Brittany D.

doi:10.4310/cms.2014.v12.n5.a7

Cited by 191 publications

(218 citation statements)

References 18 publications

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“…Therefore, |x−T k (x)| 2 2 + ψ(x) ∈ C 1,α is one derivative smoother than f and g (and therefore the residual). This is exactly what the preconditioning operator P (with s = −1) did to the residual in the asymptotic regime, for instance, as shown in (13). This shows that W 2 inverse matching has smoothing effect even in the non-asymptotic regime.…”

Section: Wasserstein Iterations In Non-asymptotic Regimesupporting

confidence: 73%

The quadratic Wasserstein metric for inverse data matching

2020

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This work characterizes, analytically and numerically, two major effects of the quadratic Wasserstein (W 2 ) distance as the measure of data discrepancy in computational solutions of inverse problems. First, we show, in the infinite-dimensional setup, that the W 2 distance has a smoothing effect on the inversion process, making it robust against high-frequency noise in the data but leading to a reduced resolution for the reconstructed objects at a given noise level. Second, we demonstrate that, for some finite-dimensional problems, the W 2 distance leads to optimization problems that have better convexity than the classical L 2 andḢ −1 distances, making it a more preferred distance to use when solving such inverse matching problems.

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Section: Wasserstein Iterations In Non-asymptotic Regimesupporting

confidence: 73%

The quadratic Wasserstein metric for inverse data matching

2020

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“…Engquist and Froese () concluded that the Wasserstein distance is insensitive to noise; a numerical example of the original data plus weak Gaussian noise is used to illustrate their conclusion. However, abnormal waveforms may occur between different seismic phases.…”

Section: Quadratic‐wasserstein‐metric‐bsed Seismic Adjoint Tomographymentioning

confidence: 97%

“…The quadratic Wasserstein, which is denoted as W 2 ( f , g ), is given as follows (Engquist & Froese, ; Villani, ):

W^{2} ((), f, g) = \inf_{T_{f, g} \in M} \int_{X} {|x - T_{f, g} (x)|}^{2} f ((), x) normald x,

…”

Section: Quadratic‐wasserstein‐metric‐bsed Seismic Adjoint Tomographymentioning

confidence: 99%

“…If the misfit between the synthetic and observed data is examined from another point of view, a mapping can be considered to exist between them (Ma & Hale, ; Engquist & Froese, ; Métivier et al, ; Yang et al, ). Recently, a novel measure of misfit related to optimal transport was proposed by Engquist and Froese (). According to their analysis, the squared Wasserstein metric is cyclically monotone with respect to shifting, which makes it possible to avoid cycle skipping.…”

Section: Introductionmentioning

confidence: 99%

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Passive Adjoint Tomography of the Crustal and Upper Mantle Beneath Eastern Tibet With a W2‐Norm Misfit Function

Dong

Yang

Niu

2019

Geophysical Research Letters

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Full waveform tomography is an effective method to obtain high‐resolution subsurface velocity structures. It is, however, difficult for the conventional L2‐norm‐based inversion to reach the global minimum because of cycle‐skipping problem. In this study, we investigated the feasibility of using the quadratic Wasserstein metric distance (W2 norm) as the misfit function for passive adjoint tomography. We first derived equations of the Fréchet gradient and adjoint source under W2‐norm misfit function. We then conducted numerical experiments to illustrate the effectiveness of the proposed method in avoiding cycle skipping. We finally applied the adjoint tomography to eastern Tibet and obtained a 3‐D velocity model of the lithosphere. The adjoint tomography revealed two low‐velocity channels beneath the NE and SE margins of the Tibetan plateau, which were also observed by previous studies. Our results are consistent with the lower crust flow model that was proposed to explain the deformation occurring at the two margins.

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“…Our work is motivated in part by [7,8,34], where the authors use the quadratic Wasserstein metric to solve Full-Waveform Inversion (FWI) problems. In particular, it is demonstrated that the quadratic Wasserstein metric, as opposed to the L 2 norm, provides an effective measure of the misfit between given data and computed solution.…”

Section: Prior Workmentioning

confidence: 99%

Diagnosing Forward Operator Error Using Optimal Transport

2019

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We investigate overdetermined linear inverse problems for which the forward operator may not be given accurately. We introduce a new tool called the structure, based on the Wasserstein distance, and propose the use of this to diagnose and remedy forward operator error. Computing the structure turns out to use an easy calculation for a Euclidean homogeneous degree one distance, the Earth Mover's Distance, based on recently developed algorithms. The structure is proven to distinguish between noise and signals in the residual and gives a plan to help recover the true direct operator

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Application of the Wasserstein metric to seismic signals

Cited by 191 publications

References 18 publications

The quadratic Wasserstein metric for inverse data matching

The quadratic Wasserstein metric for inverse data matching

Passive Adjoint Tomography of the Crustal and Upper Mantle Beneath Eastern Tibet With a W2‐Norm Misfit Function

Diagnosing Forward Operator Error Using Optimal Transport

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