We have developed 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 preconditioners. It is based on the application of the preconditioners to the Krylov subspace stabilized biconjugate gradient method. A critical factor for the iterative solver is the introduction of a new preconditioner into the Krylov subspace iteration method to solve the linear equation system resulting from the discretization of the Helmholtz equation. This new preconditioner is based on a reformulation of an integral equation-based convergent Born series for the Lippmann-Schwinger equation to an equivalent differential equation. We have determined that our iterative solver combined with the novel preconditioner 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 our two-level iterative solver verified the accuracy and accelerated convergence of the new scheme. Extensive tests in full-waveform inversion demonstrate the solver’s applicability to such problems.

中文翻译：

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亥姆霍兹方程的有限差分迭代求解器，用于频域地震波建模和全波形反演

我们已经开发了Helmholtz方程的有限差分迭代求解器，用于频域中的地震建模和反演。迭代求解器包含移位的拉普拉斯算子和两级前置条件。它基于预处理器在Krylov子空间稳定双共轭梯度法中的应用。迭代求解器的关键因素是在Krylov子空间迭代方法中引入了一种新的预处理器，以求解由亥姆霍兹方程离散化而产生的线性方程组。这种新的预处理器基于将Lippmann-Schwinger方程的基于积分方程的收敛Born级数重新公式化为等效的微分方程。我们已经确定，当与有限差分方法结合使用与新型预处理器结合的迭代求解器时，可以加快Krylov子空间迭代方法在频域地震波建模中的收敛速度。通过比较直接求解器，一级Krylov子空间迭代求解器和我们的二级迭代求解器，验证了新方案的准确性和加速的收敛性。全波形反演中的大量测试证明了求解器对此类问题的适用性。