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.

scholarly journals Improving full-waveform inversion by wavefield reconstruction with the alternating direction method of multipliers

Geophysics ◽  
2019 ◽  
Vol 84 (1) ◽  
pp. R125-R148 ◽  
Author(s):  
Hossein S. Aghamiry ◽  
Ali Gholami ◽  
Stéphane Operto
Keyword(s):  
Wave Equation ◽  
Penalty Method ◽  
Augmented Lagrangian Method ◽  
Waveform Inversion ◽  
Full Waveform Inversion ◽  
Penalty Parameter ◽  
Full Waveform ◽  
Alternating Direction ◽  
Highly Nonlinear ◽  
Equation Constraint

Full-waveform inversion (FWI) is an iterative nonlinear waveform matching procedure subject to wave-equation constraint. FWI is highly nonlinear when the wave-equation constraint is enforced at each iteration. To mitigate nonlinearity, wavefield-reconstruction inversion (WRI) expands the search space by relaxing the wave-equation constraint with a penalty method. The pitfall of this approach resides in the tuning of the penalty parameter because increasing values should be used to foster data fitting during early iterations while progressively enforcing the wave-equation constraint during late iterations. However, large values of the penalty parameter lead to ill-conditioned problems. Here, this tuning issue is solved by replacing the penalty method by an augmented Lagrangian method equipped with operator splitting (iteratively refined WRI [IR-WRI]). It is shown that IR-WRI is similar to a penalty method in which data and sources are updated at each iteration by the running sum of the data and source residuals of previous iterations. Moreover, the alternating direction strategy exploits the bilinearity of the wave-equation constraint to linearize the subsurface model estimation around the reconstructed wavefield. Accordingly, the original nonlinear FWI is decomposed into a sequence of two linear subproblems, the optimization variable of one subproblem being passed as a passive variable for the next subproblem. The convergence of WRI and IR-WRI is first compared with a simple transmission experiment, which lies in the linear regime of FWI. Under the same conditions, IR-WRI converges to a more accurate minimizer with a smaller number of iterations than WRI. More realistic case studies performed with the Marmousi II and the BP salt models indicate the resilience of IR-WRI to cycle skipping and noise, as well as its ability to reconstruct with high-fidelity, large-contrast salt bodies and subsalt structures starting the inversion from crude initial models and a 3 Hz starting frequency.

Decoupled Fréchet kernels based on a fractional viscoacoustic wave equation

Geophysics ◽  
2021 ◽  
pp. 1-42
Author(s):  
Guangchi Xing ◽  
Tieyuan Zhu
Keyword(s):  
Wave Equation ◽  
Fractional Laplacian ◽  
Waveform Inversion ◽  
Full Waveform Inversion ◽  
Physical Processes ◽  
Full Waveform ◽  
Adjoint State ◽  
The Finite Difference Method ◽  
Adjoint State Method ◽  
Laplacian Operators

We formulate the Fréchet kernel computation using the adjoint-state method based on a fractional viscoacoustic wave equation. We first numerically prove that both the 1/2- and the 3/2-order fractional Laplacian operators are self-adjoint. Using this property, we show that the adjoint wave propagator preserves the dispersion and compensates the amplitude, while the time-reversed adjoint wave propagator behaves identically as the forward propagator with the same dispersion and dissipation characters. Without introducing rheological mechanisms, this formulation adopts an explicit Q parameterization, which avoids the implicit Q in the conventional viscoacoustic/viscoelastic full waveform inversion ( Q-FWI). In addition, because of the decoupling of operators in the wave equation, the viscoacoustic Fréchet kernel is separated into three distinct contributions with clear physical meanings: lossless propagation, dispersion, and dissipation. We find that the lossless propagation kernel dominates the velocity kernel, while the dissipation kernel dominates the attenuation kernel over the dispersion kernel. After validating the Fréchet kernels using the finite-difference method, we conduct a numerical example to demonstrate the capability of the kernels to characterize both velocity and attenuation anomalies. The kernels of different misfit measurements are presented to investigate their different sensitivities. Our results suggest that rather than the traveltime, the amplitude and the waveform kernels are more suitable to capture attenuation anomalies. These kernels lay the foundation for the multiparameter inversion with the fractional formulation, and the decoupled nature of them promotes our understanding of the significance of different physical processes in the Q-FWI.

Laplace-domain wave-equation modeling and full waveform inversion in 3D isotropic elastic media

2014 ◽  
Vol 105 ◽  
pp. 120-132 ◽  
Author(s):  
Woohyun Son ◽  
Sukjoon Pyun ◽  
Changsoo Shin ◽  
Han-Joon Kim
Keyword(s):  
Wave Equation ◽  
Waveform Inversion ◽  
Equation Modeling ◽  
Full Waveform Inversion ◽  
Elastic Media ◽  
Laplace Domain ◽  
Full Waveform
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