DISCRETE WAVENUMBER SOLUTIONS TO NUMERICAL WAVE PROPAGATION IN PIECEWISE HETEROGENEOUS MEDIA - I. THEORY OF TWO-DIMENSIONAL SH CASE

Abstract

A semi-analytical, semi-numerical method of seismogram synthesis is presented for piecewise heterogeneous media resulting from an arbitrary source. The method incorporates the discrete wavenumber Green's function representation into the boundary-volume integral equation numerical techniques. The presentation is restricted to 2-D antiplane motion (SH waves). To model different parts of the media to a necessary accuracy, the incident, boundary-scattering and volume-scattering waves are separately formulated in the discrete wavenumber domain and handled flexibly at various accuracies using approximation methods. These waves are accurately superposed through the generalized Lippmann-Schwinger integral (GLSI) equation. The full-waveform boundary method is used for the boundary-scattering wave to accurately simulate the reflection/transmission across strong-contrast boundaries. Meanwhile for volume heterogeneities, the following four flexible approaches have been developed in the numerical modelling scheme present here, with a great saving of computing time and memory: <list> <li> <p align=justify>the solution implicitly for the volume-scattering wave with high accuracy to model subtle effects of volume heterogeneities; </li> <li> <p align=justify>the solution semi-explicitly for the volume-scattering wave using the average Fresnel-radius approximation to volume integrations to reduce numerical burden by making the coefficient matrix sparser; </li> <li> <p align=justify>the solution explicitly for the volume-scattering wave using the first-order Born approximation for smooth volume heterogeneities; and </li> <li> <p align=justify>the solution explicitly for the volume-scattering wave using the second-order/high-order Born approximation for practical volume heterogeneities.