Convergence analyses of different modeling schemes for generalized Lippmann–Schwinger integral equation in piecewise heterogeneous media

             

Convergence analyses of different modeling schemes for generalized Lippmann–Schwinger integral equation in piecewise heterogeneous media

Abstract

For wave propagation simulation in piecewise heterogeneous media, Gaussian-elimination-based full-waveform solutions to the generalized Lippmann–Schwinger integral equation (GLSIE) are highly accurate, but involved with extremely time-consuming computations because of the very large size of the resulting boundary–volume integral equation matrix to be inverted. Several flexible approximations to the GLSIE are scaled in an iterative way to adapt numerical solutions to the smoothness of heterogeneous media in terms of incident wavelengths, with a great saving of computing time and memory. Among various typical iterative schemes to the GLSIE matrix, the generalized minimal residual method (GMRES) is an efficient approach to reduce the computational intensity to some degree. The most efficient approximation can be obtained using a Born series, as an alternative iterative solution, to both the boundary-scattering and volume-scattering waves, leading to the Born-series approximation (BSA) scheme and the improved Born-series approximation (IBSA) scheme. These iteration schemes are validated by dimensionless frequency responses to a heterogeneous semicircular alluvial valley, and then applied to a heterogeneous multilayered model by calculating synthetic seismograms to evaluate approximation accuracies. Numerical experiments, compared with the full-waveform numerical solution, indicate that the convergence rates of these methods decrease gradually with increasing velocity perturbations. The comparison also shows that the BSA scheme has a faster convergence than the GMRES method for velocity perturbations less than 10 percent, but converges slowly and even hardly achieves convergence for velocity perturbations greater than 15 percent. The IBSA scheme gives a superior performance over the other methods, with the least iterations to achieve the necessary convergence.

Keywords

  • Generalized Lippmann–Schwinger integral equation;
  • Piecewise heterogeneous media;
  • Born series approximation;
  • The GMRES method;
  • Convergence analyses

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