Abstract:

Born scattering series is often limited by weak scattering assumptions when solving strongly seismic scattering problems, resulting in slow convergence or divergence. A simple and effective way is to improve the iterative algorithm using numerical analysis. One such method is the generalized successive over-relaxation (GSOR) iterative method, which can be applied to solve the Lippmann-Schwinger (L-S) equation and obtain the desired convergent Born scattering series. However, in strongly heterogeneous media, the GSOR iterative method may also face the challenge of slow convergence speed while calculating the high-frequency Green’s function. In this paper, the complex wavenumber Green’s function is utilized with the GSOR iterative method to numerically solve the L-S equation of the Green’s function. The complex wavenumber has imaginary components that enable localizing the energy of the background Green’s function and exponential decay, reducing the singularity of the background Green’s function. To reduce the condition number of the coefficient matrix, we further introduce the preconditioning operator and provide a preconditioned generalized successive over-relaxation (Pre-GSOR) iteration format. The convergent iteration series is obtained by selecting an appropriate damping factor and preconditioning operator. Then it is used to calculate the numerical Green’s function in the seismic strongly scattering media. Numerical results indicate that the Pre-GSOR iteration method for the complex wavenumber L-S equations can produce simulation results consistent with those obtained by direct methods for real wavenumber L-S equations. The condition number of the coefficient matrix in the Pre-GSOR iterative method for the complex wavenumber L-S equation is only 10% of the original condition number at high frequencies. Under the same number of iterations, the normalized convergence residual obtained by this method can be reduced by more than three orders of magnitude. The new method exhibits lower convergence error, better convergence, and strong adaptability to high frequencies, effectively mitigating the convergence stagnation problem encountered by the generalized over-relaxation iterative method for the real wavenumber L-S equation in strongly scattering media.

Key words: seismic scattering wave field, Green’s function, generalized over-relaxation iteration, preconditioning operator