An efficient method for computing Green's functions for a layered half-space with sources and receivers at close depths

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Abstract
We propose an analytical method to compute efficiently the displacement and stress of static and dynamic Green's functions for viscoelastic layered half-spaces. When source and receiver depths are close, it is difficult to compute Green's functions of the layered half-space, because their integrands, whose variable of integration is the horizontal wavenumber, oscillate with only slowly decreasing amplitude. In particular, when the depths are equal, it is extremely difficult to compute the stress Green's functions, because their integrands oscillate with increasing amplitude. To remedy this problem, we first derive the asymptotic solutions, which converge to the integrands of Green's functions with increasing wavenumber. For this purpose, we modify the generalized R/T (reflection and transmission) coefficient method (Luco and Apsel; 1983) to be completely free from growing exponential terms, which are the obstacles to finding the asymptotic solutions. By subtracting the asymptotic solutions from the integrands of the corresponding Green's functions, we obtain integrands that converge rapidly to zero. We can, therefore, significantly reduce the range of wavenumber integration. Since the asymptotic solutions are expressed by the products of Bessel functions and simple exponential functions, they are analytically integrable. Finally, we obtain accurate Green's functions by adding together numerical and analytical integrations. We first show this asymptotic technique for Green's functions due to point sources, and extend it to Green's functions due to dipole sources. Finally, we demonstrate the validity and efficiency of our method for various cases.

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