Search

Full-waveform inversion (FWI) with extended sources first computes wavefields with data-driven source extensions, such that the simulated data in inaccurate velocity models match the observed counterpart well enough to prevent cycle skipping. Then, the source extensions are minimized to update the model parameters. This two-step workflow is iterated until both data and sources are matched. It was recently shown that the source extensions are the least-squares solutions of the recorded scattered data fitting problem. As a result, they are computed by propagating backward in time the deblurred FWI data residuals, where the deblurring operator is the inverse of the damped data-domain Hessian of the scattering-source estimation problem. Estimating the deblurred data residuals is the main computational bottleneck of time-domain extended-source FWI (ES-FWI). To mitigate this issue, we first estimate them when the inverse of the data-domain Hessians is approximated by matching filters in Fourier and short-time Fourier domains. Second, we refine them with conjugate-gradient iterations when necessary. Computing the matching filters and performing one conjugate-gradient iteration each require two simulations per source. Therefore, it is critical to design some workflows that minimize this computational burden. We implement time-domain ES-FWI with the augmented Lagrangian method. Moreover, we further extend its linear regime with a multiscale frequency continuation approach, which is combined with grid coarsening to mitigate the computational burden and regularize the inversion. Finally, we use total-variation regularization to deal with large-contrast reconstruction. We present synthetic cases where different inversion workflows carried out with data-domain Hessians of variable accuracy were assessed with the aim at converging toward accurate solutions while minimizing computational cost.

Full Waveform Inversion can be made immune to cycle skipping by matching the recorded data arbitrarily well from inaccurate subsurface models. To achieve this goal, the simulated wavefields can be computed in an extended search space as the solution of an overdetermined problem aiming at jointly satisfying the wave equation and fitting the data in a least-squares sense. Simply put, the wavefields are computed by solving the wave equation in the inaccurate background model with a feedback term to the data added to the physical source in the right-hand side. Then, the subsurface parameters are updated by canceling out these additional source terms, sometimes called unwisely wave-equation errors, to push the background model toward the true model in the left-hand side wave-equation operator. Although many studies were devoted to these approaches with promising numerical results, their governing physical principles and their relationships with classical FWI don't seem to be understood well yet. The goal of this tutorial is to review these principles in the theoretical framework of inverse scattering theory whose governing forward equation is the Lippmann-Schwinger equation. From this equation, we show how the data-assimilated wavefields embed an approximation of the scattered field generated by the sought model perturbation and how they modify the sensitivity kernel of classical FWI beyond the Born approximation. We also clarify how the approximation with which these wavefields approximate the unknown true wavefields is accounted for in the adjoint source of the parameter estimation problem. The theory is finally illustrated with numerical examples. Understanding the physical principles governing these methods is a necessary prerequisite to assessing their potential and limits and designing relevant heuristics to manage the latter.

Frequency-domain simulation of seismic waves plays an important role in seismic inversion, but it remains challenging in large models. The recently proposed physics-informed neural network (PINN), as an effective deep learning method, has achieved successful applications in solving a wide range of partial differential equations (PDEs), and there is still room for improvement on this front. For example, PINN can lead to inaccurate solutions when PDE coefficients are non-smooth and describe structurally-complex media. In this paper, we solve the acoustic and visco-acoustic scattered-field wave equation in the frequency domain with PINN instead of the wave equation to remove source singularity. We first illustrate that non-smooth velocity models lead to inaccurate wavefields when no boundary conditions are implemented in the loss function. Then, we add the perfectly matched layer (PML) conditions in the loss function of PINN and design a quadratic neural network to overcome the detrimental effects of non-smooth models in PINN. We show that PML and quadratic neurons improve the results as well as attenuation and discuss the reason for this improvement. We also illustrate that a network trained during a wavefield simulation can be used to pre-train the neural network of another wavefield simulation after PDE-coefficient alteration and improve the convergence speed accordingly. This pre-training strategy should find application in iterative full waveform inversion (FWI) and time-lag target-oriented imaging when the model perturbation between two consecutive iterations or two consecutive experiments can be small.

Implementation of the standard full waveform inversion (FWI) poses difficulties as the initial model offsets from the true model. The wavefield reconstruction inversion (WRI) was proposed to mitigate these difficulties by relaxing the wave-equation constraint. In this abstract, working on the nonlinear term in the Hessian matrix of FWI, we develop a new approximate Hessian as an Augmented Gauss-Newton (AGN) Hessian including second-order derivative information. Moreover, we establish an intimate connection between an updating formula which results from approximate solve of the Newton's method with the AGN Hessian on the FWI problem and the WRI method. Our analysis opens new perspectives for developing efficient algorithms for FWI based on the Newton's method and highlights the importance of the nonlinear term in the Hessian matrix, which is ignored in most cases.

Full-waveform inversion (FWI) is a high-resolution and computationally intensive imaging technique to reconstruct unknown parameters in the computational model in which the waves propagate; however, an accurate model of only part of this medium is required for some applications. To decrease the computational burden of such problems, target-oriented FWI was proposed where the redatumed data on the part of the medium or localized solvers for the wave equation are used. On the other hand, the classical formulation of FWI suffers from non-linearity and ill-posedness, which makes FWI sensitive to the initial model, the low-frequency content of the data, and limited illumination. In this study, we propose a localized version of the alternating direction method of multipliers (ADMM)-based FWI method, which was proposed to solve these problems in classical FWI. In our localized FWI or LWI, the medium is decomposed into a few subdomains, where some of them are updated, and the others are kept fixed based on an adaptation of multi-block ADMM, which is a powerful algorithm for solving inverse problems with decomposition and block separability. Numerical tests on the Marmousi model for a time-lapse application confirm the computational efficiency and robustness against background velocity model errors.

Full-waveform inversion (FWI) is a seismic imaging method that provides quantitative inference about subsurface properties with a wavelength-scale resolution. Its frequency-domain formulation is computationally efficient when processing only a few discrete frequencies. However, classical FWI, which is formulated on the reduced-parameter space, requires starting the inversion with a sufficiently-accurate initial model and low frequency to prevent being stuck in local minima due to cycle skipping. FWI with extended search space has been proposed to mitigate this issue. It contains two main steps: first, data-assimilated (DA) wavefields are computed by allowing for wave-equation errors to match the data at receivers closely. Then, subsurface parameters are estimated from these wavefields by minimizing the wave-equation errors. The DA wavefields are the least-squares solution of an overdetermined system gathering the wave and observation equations. The numerical bandwidth of the resulting normal-equation system is two times that of the wave-equation system, which can be a limiting factor for 3D large-scale applications. Therefore, computing highly accurate DA wavefields at a reasonable computational cost is an issue in extended FWI. This issue is addressed here by rewriting the normal system such that its solution can be computed by solving the time-harmonic wave equation several times in sequence. Moreover, the computational burden of multi-right-hand side (RHS) simulations is mitigated with a sketching method. Finally, we solve the time-harmonic wave equation with the convergent Born series method, which conciliates accuracy and computational efficiency. Application of the new extended FWI algorithm on the salt benchmark shows that it reconstructs at a reasonable cost subsurface models that are similar to those obtained with the classical extended FWI.