Optimal Transport Based Seismic Inversion: Beyond Cycle Skipping

Engquist, Björn; Yang, Yunan

doi:10.48550/arxiv.2002.00031

Cited by 5 publications

(6 citation statements)

References 71 publications

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“…Typically, the zero-gradient condition is far from enough to guarantee the optimality, especially for FWI. There has been extensive literature on tackling the nonconvexity [9,31]. In this paper, we focus on accelerating the convergence and not addressing the cycle-skipping issues, which is another important research topic by itself.…”

Section: Methodsmentioning

confidence: 99%

Anderson Acceleration for Seismic Inversion

Yang

2020

Preprint

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The state-of-art seismic imaging techniques treat inversion tasks such as FWI and LSRTM as PDE-constrained optimization problems. Due to the large-scale nature, gradient-based optimization algorithms are preferred in practice to update the model iteratively. Higher-order methods converge in fewer iterations but often require higher computational costs, more line search steps, and bigger memory storage. A balance among these aspects has to be considered. We propose using Anderson acceleration (AA), a popular strategy to speed up the convergence of fixed-point iterations, to accelerate the steepest descent algorithm, which we innovatively treat as a fixed-point iteration. Independent of the dimensionality of the unknown parameters, the computational cost of implementing the method can be reduced to an extremely low-dimensional least-squares problem. The cost can be further reduced by a low-rank update. We discuss the theoretical connections and the differences between AA and other well-known optimization methods such as L-BFGS and the restarted GMRES and compare their computational cost and memory demand. Numerical examples of FWI and LSRTM applied to the Marmousi benchmark demonstrate the acceleration effects of AA. Compared with the steepest descent method, AA can achieve fast convergence and provide competitive results with some quasi-Newton methods, making it an attractive optimization strategy for seismic inversion.

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Section: Methodsmentioning

confidence: 99%

Anderson Acceleration for Seismic Inversion

Yang

2020

Preprint

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“…However, the exponential normalization method cannot keep the amplitude ratio with the amplified positive and suppressed negative parts of the seismic data. Especially for data with low SNR in practical applications, the exponential scaling method has a risk of amplifying noise (Engquist & Yang, 2020). Thus, the phase information cannot be extracted exactly.…”

Section: Data Normalization and Gradient Calculationmentioning

confidence: 99%

“…In contrast, the affine scaling strategy is implemented by adding a constant value to the signals. Although it works well for practical applications, this method cannot provide a strict convex misfit function with a relatively wider convergence range compared to the conventional L 2 objec-tive function (Engquist & Yang, 2020). As a common normalization method, an exponentiation strategy can provide good convexity by setting appropriate parameters.…”

Section: Introductionmentioning

confidence: 99%

Full‐waveform inversion for reverse vertical seismic profiling data based on a variant of the optimal transport theory

Liu

2022

Geophysical Prospecting

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Full‐waveform inversion has become a useful tool to construct the details of the geological structure. However, due to the strong nonlinearity of the inverse problem and inaccuracy of the initial model, full‐waveform inversion often faces severe cycle‐skipping problems and thus may exhibit poor inversion results. The optimal transport‐based full‐waveform inversion can mitigate the cycle‐skipping problem. For that, transformations of oscillatory seismic data to distributions need to satisfy positive and mass conservation conditions. In this work, we introduce a new data normalization method based on the Sigmoid function. The convexity of the objective function is improved compared to the traditional L2 norm by setting appropriate parameters. Combined with the adaptive quadratic Wasserstein distance, our approach presents strong adaptability to a poor initial model and can reduce cycle skipping. Numerical examples are provided to validate the method in a reverse vertical seismic profiling geometry, and the inverted models obtained by different normalization methods are also compared.

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“…The Wasserstein metric has been actively studied since the late 20th century [65]. Recently, it is proved to have unique features as a loss function in numerous applied fields, such as image processing [33], machine learning [6], large-scale inverse problems [21] and statistical inference [9], only to mention a few. The research development of optimal transport in applied mathematics is so rapid that we do not attempt to give a comprehensive overview.…”

mentioning

confidence: 99%

“…Second, the basin of attraction for these chaotic dynamics is compactly supported on a low-dimensional subset of R d . The Wasserstein metric is well-defined for densities of different support as it not only considers the local intensity differences but also incorporates the global geometry mismatches [21], which is not the case of Kullback-Leibler divergence. The L p norm also suffers if the supports are disjoint and may produce wrong or random update directions if supports only partially intersect.…”

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confidence: 99%

Optimal Transport for Parameter Identification of Chaotic Dynamics via Invariant Measures

Yang¹,

Nurbekyan²,

Negrini³

et al. 2021

Preprint

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Parameter identification determines the essential system parameters required to build real-world dynamical systems by fusing crucial physical relationships and experimental data. However, the data-driven approach faces main difficulties, such as a lack of observational data, discontinuous or inconsistent time trajectories, and noisy measurements. The ill-posedness of the inverse problem comes from the chaotic divergence of the forward dynamics. Motivated by the challenges, we shift from the Lagrangian particle perspective to the state space flow field's Eulerian description. Instead of using pure time trajectories as the inference data, we treat statistics accumulated from the Direct Numerical Simulation (DNS) as the observable, whose continuous analog is the steady-state probability density function (PDF) of the corresponding Fokker-Planck equation (FPE). We reformulate the original parameter identification problem as a data-fitting, PDE-constrained optimization problem. An upwind scheme based on the finite-volume method that enforces mass conservation and positivity preserving is used to discretize the forward problem. We present theoretical regularity analysis for evaluating gradients of optimal transport costs and introduce three different formulations for efficient gradient calculation. Numerical results using the quadratic Wasserstein metric from optimal transport demonstrate this novel approach's robustness for chaotic dynamical system parameter identification.

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Optimal Transport Based Seismic Inversion: Beyond Cycle Skipping

Cited by 5 publications

References 71 publications

Anderson Acceleration for Seismic Inversion

Anderson Acceleration for Seismic Inversion

Full‐waveform inversion for reverse vertical seismic profiling data based on a variant of the optimal transport theory

Optimal Transport for Parameter Identification of Chaotic Dynamics via Invariant Measures

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