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: (i) the solution implicitly for the volume-scattering wave with high accuracy to model subtle effects of volume heterogeneities; (ii) 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; (iii)the solution explicitly for the volume-scattering wave using the first-order Born approximation for smooth volume heterogeneities; and (iv)the solution explicitly for the volume-scattering wave using the second-order/high-order Born approximation for practical volume heterogeneities. These solutions are tested for dimensionless frequency responses to a heterogeneous alluvial valley where the velocity is perturbed randomly in the range of ca 5-20 per cent, which is not rare in most complex near-surface areas. Numerical experiments indicate that several times of site amplification can be expected as a result of heterogeneities introduced in a homogeneous valley. The test also confirms that the first-order Born approximation to the volume-scattering wave is strictly valid for velocity perturbation less than 10 per cent and approximately used for up to 15 per cent for general applications. The second-order Born approximation to the volume-scattering wave is strictly valid for velocity perturbation less than 15 per cent and approximately used for up to 20 per cent for general applications. © 2004 RAS.