A finite-difference iterative solver of the Helmholtz equation for frequency-domain seismic wave modeling and full waveform inversion

Geophysics ◽  
2020 ◽  
pp. 1-43
Author(s):  
Xingguo Huang ◽  
Stewart Greenhalgh
Keyword(s):  
Finite Difference ◽  
Helmholtz Equation ◽  
Frequency Domain ◽  
Iteration Method ◽  
Krylov Subspace ◽  
Waveform Inversion ◽  
Full Waveform Inversion ◽  
Iterative Solver ◽  
Subspace Iteration ◽  
Full Waveform

We present a finite difference iterative solver of the Helmholtz equation for seismic modeling and inversion in the frequency-domain. The iterative solver involves the shifted Laplacian operator and two-level pre-conditioners. It is based on the application of the pre-conditioners to the Krylov subspace stabilized biconjugate gradient method. A critical factor for the iterative solver is the introduction of a new pre-conditioner into the Krylov subspace iteration method to solve the linear system resulting from the discretization of the Helmholtz equation. This new pre-conditioner is based upon a reformulation of an integral equation-based convergent Born series for the Lippmann-Schwinger equation to an equivalent differential equation. We demonstrate that the proposed iterative solver combined with the novel pre-conditioner when incorporated with the finite difference method accelerates the convergence of the Krylov subspace iteration method for frequency-domain seismic wave modeling. A comparison of a direct solver, a one-level Krylov subspace iterative solver and the proposed two-level iterative solver verified the accuracy and accelerated convergence of the new scheme. Extensive tests in full waveform inversion demonstrate the solver applicability to full waveform inversion applications.

3D elastic frequency-domain iterative solver for full-waveform inversion

2016 ◽  
Author(s):  
Mikhail Belonosov ◽  
Vladimir Tcheverda ◽  
Dmitry Neklyudov ◽  
Victor Kostin ◽  
Maxim Dmitriev
Keyword(s):  
Frequency Domain ◽  
Waveform Inversion ◽  
Full Waveform Inversion ◽  
Iterative Solver ◽  
Full Waveform

Three-dimensional frequency-domain full-waveform inversion with an iterative solver

Geophysics ◽  
2009 ◽  
Vol 74 (6) ◽  
pp. WCC149-WCC157 ◽  
Author(s):  
René-Édouard Plessix
Keyword(s):  
Frequency Domain ◽  
Waveform Inversion ◽  
Three Dimensional ◽  
Iterative Solvers ◽  
Full Waveform Inversion ◽  
Ocean Bottom ◽  
Ocean Bottom Seismometer ◽  
Iterative Solver ◽  
Data Set ◽  
Full Waveform

With the acquisition of wide-aperture seismic data sets, full-waveform inversion is an attractive method for deriving velocity models. Three-dimensional implementations require an efficient solver for the wave equation. Computing 3D time-harmonic responses with a frequency-domain solver is complicated because a large linear system with negative and positive eigenvalues must be solved. Time-domain schemes are an alternative. Nevertheless, existing frequency-domain iterative solvers with an efficient preconditioner are a viable option when full-waveform inversion is formulated in the frequency domain. An iterative solver with a multigrid preconditioner is competitive because of a high-order spatial discretization. Numerical examples illustrated the efficiency of the iterative solvers. Three dimensional full-waveform inversion was then studied in the context of deep-water ocean-bottom seismometer acquisition. Three dimensional synthetic data inversion results showed the behavior of full-waveform inversion with respect to the initial model and the minimum frequency available in the data set. Results on a 3D real ocean-bottom seismometer data set demonstrated the relevance of full-waveform inversion, especially to image the shallow part of the model.

3D finite-difference frequency-domain modeling of visco-acoustic wave propagation using a massively parallel direct solver: A feasibility study

Geophysics ◽  
2007 ◽  
Vol 72 (5) ◽  
pp. SM195-SM211 ◽  
Author(s):  
Stéphane Operto ◽  
Jean Virieux ◽  
Patrick Amestoy ◽  
Jean-Yves L’Excellent ◽  
Luc Giraud ◽  
...  
Keyword(s):  
Wave Propagation ◽  
Finite Difference ◽  
Frequency Domain ◽  
Waveform Inversion ◽  
Staggered Grid ◽  
Full Waveform Inversion ◽  
Domain Modeling ◽  
Acoustic Wave Propagation ◽  
Full Waveform ◽  
Grid Points

We present a finite-difference frequency-domain method for 3D visco-acoustic wave propagation modeling. In the frequency domain, the underlying numerical problem is the resolution of a large sparse system of linear equations whose right-hand side term is the source. This system is solved with a massively parallel direct solver. We first present an optimal 3D finite-difference stencil for frequency-domain modeling. The method is based on a parsimonious staggered-grid method. Differential operators are discretized with second-order accurate staggered-grid stencils on different rotated coordinate systems to mitigate numerical anisotropy. An antilumped mass strategy is implemented to minimize numerical dispersion. The stencil incorporates 27 grid points and spans two grid intervals. Dispersion analysis showsthat four grid points per wavelength provide accurate simulations in the 3D domain. To assess the feasibility of the method for frequency-domain full-waveform inversion, we computed simulations in the 3D SEG/EAGE overthrust model for frequencies 5, 7, and [Formula: see text]. Results confirm the huge memory requirement of the factorization (several hundred Figabytes) but also the CPU efficiency of the resolution phase (few seconds per shot). Heuristic scalability analysis suggests that the memory complexity of the factorization is [Formula: see text] for a [Formula: see text] grid. Our method may provide a suitable tool to perform frequency-domain full-waveform inversion using a large distributed-memory platform. Further investigation is still necessary to assess more quantitatively the respective merits and drawbacks of time- and frequency-domain modeling of wave propagation to perform 3D full-waveform inversion.

Sign in / Sign up

Export Citation Format

Share Document