Anderson accelerated augmented Lagrangian for extended waveform inversion

Geophysics ◽  
2021 ◽  
pp. 1-50
Author(s):  
Kamal Aghazade ◽  
Ali Gholami ◽  
Hossein S. Aghamiry ◽  
Stéphane Operto
Keyword(s):  
Wave Equation ◽  
Large Scale ◽  
Augmented Lagrangian ◽  
Waveform Inversion ◽  
Search Space ◽  
Total Variation Regularization ◽  
Model Parameters ◽  
Extended Space ◽  
Alternating Direction ◽  
Source Dimension

The augmented Lagrangian (AL) method provides a flexible and efficient framework for solving extended-space full-waveform inversion (FWI), a constrained nonlinear optimization problem whereby we seek model parameters and wavefields that minimize the data residuals and satisfy the wave equation constraint. The AL-based wavefield reconstruction inversion, also known as iteratively refined wavefield reconstruction inversion, extends the search space of FWI in the source dimension and decreases sensitivity of the inversion to the initial model accuracy. Furthermore, it benefits from the advantages of the alternating direction method of multipliers (ADMM), such as generality and decomposability for dealing with non-differentiable regularizers, e.g., total variation regularization, and large scale problems, respectively. In practice any extension of the method aiming at improving its convergence and decreasing the number of wave-equation solves would have a great importance. To achieve this goal, we recast the method as a general fixed-point iteration problem, which enables us to apply sophisticated acceleration strategies like Anderson acceleration. The accelerated algorithm stores a predefined number of previous iterates and uses their linear combination together with the current iteration to predict the next iteration. We investigate the performance of the proposed accelerated algorithm on a simple checkerboard model and the benchmark Marmousi II and 2004 BP salt models through numerical examples. These numerical results confirm the effectiveness of the proposed algorithm in terms of convergence rate and the quality of the final estimated model.

scholarly journals Multi-parameter wavefield reconstruction inversion for wavespeed and attenuation with bound constraints and total variation regularization

Geophysics ◽  
2020 ◽  
pp. 1-54 ◽  
Author(s):  
Hossein S. Aghamiry ◽  
Ali Gholami ◽  
Stéphane Operto
Keyword(s):  
Wave Equation ◽  
Attenuation Factor ◽  
Waveform Inversion ◽  
Operator Splitting ◽  
Search Space ◽  
Total Variation Regularization ◽  
Bound Constraints ◽  
The North ◽  
The North Sea ◽  
Attenuation Imaging

Wavefield reconstruction inversion (WRI) extends the search space of Full Waveform Inversion (FWI) by allowing for wave equation errors during wavefield reconstruction to match the data from the first iteration. Then, the wavespeeds are updated from the wavefields by minimizing the source residuals. Performing these two tasks in alternating mode breaks down the nonlinear FWI as a sequence of two linear subproblems, relaying on the bilinearity of the wave equation. We solve this biconvex optimization with the alternating-direction method of multipliers (ADMM) to cancel out efficiently the data and source residuals in iterations and stabilize the parameter estimation with appropriate regularizations.Here, we extend WRI to viscoacoustic media for attenuation imaging. Attenuation reconstruction is challenging because of the small imprint of attenuation in the data and the crosstalks with velocities. To address these issues, we recast the multivariate viscoacoustic WRI as a triconvex optimization and update wavefields, squared slowness, and attenuation factor in alternating mode at each WRI iteration. This requires to linearize the attenuation-estimation subproblem via an approximated trilinear viscoacoustic wave equation. The iterative defect correction embedded in ADMM corrects the errors generated by this linearization, while the operator splitting allows us to tailor ℓ1 regularization to each parameter class. A toy numerical example shows that these strategies mitigate crosstalk artifacts and noise from the attenuation reconstruction. A more realistic example representative of the North sea further shows the ability of the method to jointly reconstruct the wavespeed and the attenuation starting from a very crude attenuation-free initial model, the moderate strength of the crosstalk artifacts and the sensitivity of the multi-parameter reconstruction to noise.

scholarly journals Robust wavefield inversion via phase retrieval

2020 ◽  
Vol 221 (2) ◽  
pp. 1327-1340
Author(s):  
H S Aghamiry ◽  
A Gholami ◽  
S Operto
Keyword(s):  
Wave Equation ◽  
Total Variation ◽  
Phase Retrieval ◽  
Waveform Inversion ◽  
Model Parameters ◽  
Bound Constraints ◽  
Linear Measurements ◽  
Splitting Technique ◽  
Alternating Direction ◽  
Wavefield Inversion

SUMMARY Extended formulation of full waveform inversion (FWI), called wavefield reconstruction inversion (WRI), offers potential benefits of decreasing the non-linearity of the inverse problem by replacing the explicit inverse of the wave-equation operator of classical FWI (the oscillating Green functions) with a suitably defined data-driven regularized inverse. This regularization relaxes the wave-equation constraint to reconstruct wavefields that match the data, hence mitigating the risk of cycle skipping. The subsurface model parameters are then updated in a direction that reduces these constraint violations. However, in the case of a rough initial model, the phase errors in the reconstructed wavefields may trap the waveform inversion in a local minimum leading to inaccurate subsurface models. In this paper, in order to avoid matching such incorrect phase information during the early WRI iterations, we design a new cost function based upon phase retrieval, namely a process which seeks to reconstruct a signal from the amplitude of linear measurements. This new formulation, called wavefield inversion with phase retrieval (WIPR), further improves the robustness of the parameter estimation subproblem by a suitable phase correction. We implement the resulting WIPR problem with an alternating-direction approach, which combines the majorization-minimization (MM) algorithm to linearise the phase-retrieval term and a variable splitting technique based upon the alternating direction method of multipliers (ADMM). This new workflow equipped with Tikhonov-Total variation (TT) regularization, which is the combination of second-order Tikhonov and total variation regularizations and bound constraints, successfully reconstructs the 2004 BP salt model from a sparse fixed-spread acquisition using a 3 Hz starting frequency and a homogeneous initial velocity model.

Optimized Model Parameterization using Compact Full Waveform Inversion

2020 ◽  
Author(s):  
Linan Xu ◽  
Edgar Manukyan ◽  
Hansruedi Maurer
Keyword(s):  
Large Scale ◽  
Waveform Inversion ◽  
Gradient Methods ◽  
Significant Loss ◽  
Inverse Model ◽  
Eigenvalue Decomposition ◽  
Model Parameters ◽  
Full Waveform Inversion ◽  
Full Waveform ◽  
Model Parameterization

Summary Seismic Full Waveform Inversion (FWI) has the potential to produce high-resolution subsurface images, but the computational resources required for realistically sized problems can be prohibitively large. In terms of computational costs, Gauss-Newton algorithms are more attractive than the commonly employed conjugate gradient methods, because the former have favorable convergence properties. However, efficient implementations of Gauss-Newton algorithms require an excessive amount of computer memory for larger problems. To address this issue, we introduce Compact Full Waveform Inversion (CFWI). Here, a suitable inverse model parameterization is sought that allows representing all subsurface features, potentially resolvable by a particular source-receiver deployment, but using only a minimum number of model parameters. In principle, an inverse model parameterization, based on the Eigenvalue decomposition, would be optimal, but this is computationally not feasible for realistic problems. Instead, we present two alternative parameter transformations, namely the Haar and the Hartley transformations, with which similarly good results can be obtained. By means of a suite of numerical experiments, we demonstrate that these transformations allow the number of model parameters to be reduced to only a few percent of the original parameterization without any significant loss of spatial resolution. This facilitates efficient solutions of large-scale FWI problems with explicit Gauss-Newton algorithms.

scholarly journals ADMM-based multiparameter wavefield reconstruction inversion in VTI acoustic media with TV regularization

2019 ◽  
Vol 219 (2) ◽  
pp. 1316-1333 ◽  
Author(s):  
H S Aghamiry ◽  
A Gholami ◽  
S Operto
Keyword(s):  
Wave Equation ◽  
Wave Speed ◽  
Homogeneous Model ◽  
Waveform Inversion ◽  
Search Space ◽  
Bound Constraints ◽  
The North ◽  
Tv Regularization ◽  
Acoustic Media ◽  
Suitable Framework

SUMMARYFull waveform inversion (FWI) is a nonlinear waveform matching procedure, which suffers from cycle skipping when the initial model is not kinematically accurate enough. To mitigate cycle skipping, wavefield reconstruction inversion (WRI) extends the inversion search space by computing wavefields with a relaxation of the wave equation in order to fit the data from the first iteration. Then, the subsurface parameters are updated by minimizing the source residuals the relaxation generated. Capitalizing on the wave-equation bilinearity, performing wavefield reconstruction and parameter estimation in alternating mode decomposes WRI into two linear subproblems, which can be solved efficiently with the alternating-direction method of multiplier (ADMM), leading to the so-called iteratively refined WRI (IR–WRI). Moreover, ADMM provides a suitable framework to implement bound constraints and different types of regularizations and their mixture in IR–WRI. Here, IR–WRI is extended to multiparameter reconstruction for vertical transverse isotropic (VTI) acoustic media. To achieve this goal, we first propose different forms of bilinear VTI acoustic wave equation. We develop more specifically IR–WRI for the one that relies on a parametrization involving vertical wave speed and Thomsen’s parameters δ and ϵ. With a toy numerical example, we first show that the radiation patterns of the virtual sources generate similar wavenumber filtering and parameter cross-talks in classical FWI and IR–WRI. Bound constraints and TV regularization in IR–WRI fully remove these undesired effects for an idealized piecewise constant target. We show with a more realistic long-offset case study representative of the North Sea that anisotropic IR–WRI successfully reconstruct the vertical wave speed starting from a laterally homogeneous model and update the long wavelengths of the starting ϵ model, while a smooth δ model is used as a passive background model. VTI acoustic IR–WRI can be alternatively performed with subsurface parametrizations involving stiffness or compliance coefficients or normal moveout velocities and η parameter (or horizontal velocity).

Full waveform inversion with sparse structure constrained regularization

2018 ◽  
Vol 26 (2) ◽  
pp. 243-257 ◽  
Author(s):  
Zichao Yan ◽  
Yanfei Wang
Keyword(s):  
Large Scale ◽  
Regularization Parameter ◽  
Waveform Inversion ◽  
Difference Operators ◽  
Model Parameters ◽  
Full Waveform Inversion ◽  
Structure Information ◽  
Full Waveform ◽  
Ill Posed ◽  
Scale Optimization

AbstractFull waveform inversion is a large-scale nonlinear and ill-posed problem. We consider applying the regularization technique for full waveform inversion with structure constraints. The structure information was extracted with difference operators with respect to model parameters. And then we establish an {l_{p}}-{l_{q}}-norm constrained minimization model for different choices of parameters p and q. To solve this large-scale optimization problem, a fast gradient method with projection onto convex set and a multiscale inversion strategy are addressed. The regularization parameter is estimated adaptively with respect to the frequency range of the data. Numerical experiments on a layered model and a benchmark SEG/EAGE overthrust model are performed to testify the validity of this proposed regularization scheme.

Elastic reflection traveltime inversion with decoupled wave equation

Geophysics ◽  
2018 ◽  
Vol 83 (5) ◽  
pp. R463-R474 ◽  
Author(s):  
Guanchao Wang ◽  
Shangxu Wang ◽  
Jianyong Song ◽  
Chunhui Dong ◽  
Mingqiang Zhang
Keyword(s):  
Wave Equation ◽  
Wave Velocity ◽  
Waveform Inversion ◽  
P Wave ◽  
Model Parameters ◽  
Local Minima ◽  
S Wave ◽  
Traveltime Inversion ◽  
Velocity Models ◽  
Elastic Reflection

Elastic full-waveform inversion (FWI) updates high-resolution model parameters by minimizing the residuals of multicomponent seismic records between the field and model data. FWI suffers from the potential to converge to local minima and more serious nonlinearity than acoustic FWI mainly due to the absence of low frequencies in seismograms and the extended model domain (P- and S-velocities). Reflection waveform inversion can relax the nonlinearity by relying on the tomographic components, which can be used to update the low-wavenumber components of the model. Hence, we have developed an elastic reflection traveltime inversion (ERTI) approach to update the low-wavenumber component of the velocity models for the P- and S-waves. In our ERTI algorithm, we took the P- and S-wave impedance perturbations as elastic reflectivity to generate reflections and a weighted crosscorrelation as the misfit function. Moreover, considering the higher wavenumbers (lower velocity value) of the S-wave velocity compared with the P-wave case, optimizing the low-wavenumber components for the S-wave velocity is even more crucial in preventing the elastic FWI from converging to local minima. We have evaluated an equivalent decoupled velocity-stress wave equation to ERTI to reduce the coupling effects of different wave modes and to improve the inversion result of ERTI, especially for the S-wave velocity. The subsequent application on the Sigsbee2A model demonstrates that our ERTI method with the decoupled wave equation can efficiently update the low-wavenumber parts of the model and improve the precision of the S-wave velocity.

On efficient frequency-domain full-waveform inversion with extended search space

Geophysics ◽  
2020 ◽  
pp. 1-61
Author(s):  
Hossein S. Aghamiry ◽  
Ali Gholami ◽  
Stéphane Operto
Keyword(s):  
Wave Equation ◽  
Frequency Domain ◽  
Computational Efficiency ◽  
Waveform Inversion ◽  
Search Space ◽  
Full Waveform Inversion ◽  
Lu Decomposition ◽  
Multiple Sources ◽  
Full Waveform ◽  
Convergence Point

Efficient frequency-domain Full-Waveform Inversion (FD-FWI) of wide-aperture data is designed by limiting inversion to few frequencies and by solving the Helmholtz equation with a direct solver to process multiple sources efficiently. Some variants of FD-FWI, which process the wave-equation as a weak constraint, were proposed to increase the computational efficiency or extend the search space. Among them, the contrast-source reconstruction inversion (CSRI) reparametrizes FD-FWI in terms of contrast sources (CS) and contrasts and update them in an alternating mode.This reparametrization allows for one lower-upper (LU) decomposition of the Helmholtz operator to be performed per frequency inversion hence further improving the computational efficiency of FD-FWI.On the other hand, Iteratively-refined Wavefield Reconstruction Inversion (IR-WRI) relies on the alternating-direction method of multipliers (ADMM) to extend the search space by matching the data from the early iterations via an aggressive relaxation of the wave-equation while satisfying it at the convergence point thanks to the defect correction performed by the Lagrange multipliers. In contrast to CSRI, IR-WRI requires to redo one LU decomposition when the subsurface model is updated.In both methods, the CSs or the wavefields are computed by solving in a least-squares sense an overdetermined linear system gathering an observation equation and a wave-equation.A drawback of CSRI is that CSs are estimated approximately with one iteration of a conjugate gradient method, while the wavefields are reconstructed exactly by IR-WRI with a Gauss-Newton method. We combine the benefits of CSRI and IR-WRI to decrease the number of LU decomposition during IR-WRI with a fixed-point (FP) algorithm while preserving its search space extension capability. Application on the 2D complex Marmousi and the BP salt models shows that our FP-based IR-WRI manages to reconstruct these models as accurately as the classical IR-WRI while reducing the number of LU factorizations considerably.

scholarly journals Extended-space full waveform inversion in the time domain with the augmented Lagrangian method

Geophysics ◽  
2021 ◽  
pp. 1-57
Author(s):  
Ali Gholami ◽  
Hossein S. Aghamiry ◽  
Stéphane Operto
Keyword(s):  
Wave Equation ◽  
Augmented Lagrangian ◽  
Augmented Lagrangian Method ◽  
Waveform Inversion ◽  
Normal Equation ◽  
Full Waveform Inversion ◽  
Penalty Parameter ◽  
Time Stepping ◽  
Full Waveform ◽  
The Time Domain

The search space of Full Waveform Inversion (FWI) can be extended via a relaxation of the wave equation to increase the linear regime of the inversion. This wave equation relaxation is implemented by solving jointly (in a least-squares sense) the wave equation weighted by a penalty parameter and the observation equation such that the reconstructed wavefields closely match the data, hence preventing cycle skipping at receivers. Then, the subsurface parameters are updated by minimizing the temporal and spatial source extension generated by the wave-equation relaxation to push back the data-assimilated wavefields toward the physics.This extended formulation of FWI has been efficiently implemented in the frequency domain with the augmented Lagrangian method where the overdetermined systems of the data-assimilated wavefields can be solved separately for each frequency with linear algebra methods and the sensitivity of the optimization to the penalty parameter is mitigated through the action of the Lagrange multipliers.Applying this method in the time domain is however hampered by two main issues: the computation of data-assimilated wavefields with explicit time-stepping schemes and the storage of the Lagrange multipliers capturing the history of the source residuals in the state space.These two issues are solved by recognizing that the source residuals on the right-hand side of the extended wave equation, when formulated in a form suitable for explicit time stepping, are related to the extended data residuals through an adjoint equation.This relationship first allows us to relate the extended data residuals to the reduced data residuals through a normal equation in the data space. Once the extended data residuals have been estimated by solving (exactly or approximately) this normal equation, the data-assimilated wavefields are computed with explicit time stepping schemes by cascading an adjoint and a forward simulation.

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