Scattering of elastic waves by a 3D anisotropic basin

Abstract

SUMMARYScattering of elastic waves by a three-dimensional transversely isotropic basin of arbitrary shape embedded in a half-space is considered using an indirect boundary integral equation approach. The unknown scattered waves are expressed in terms of point sources distributed on the so-called auxiliary surfaces. The sources are expressed in terms of the full-space Green's functions with their intensities determined from the requirement that the boundary and the continuity conditions are to be satis"ed in the least-squares sense. Steady-state results were obtained for incident plane pseudo-P-, SH-, SV-, and Rayleigh waves.Using the Radon transform the Green's functions are obtained in the form of "nite integrals over a unit sphere or a unit circle which can be numerically evaluated very e$ciently.Detailed analysis of the method includes the discussion on the shape of the auxiliary surfaces and the distribution of the collocation points and sources. The convergence criteria is de"ned in terms of transparency tests, isotropic limit test, and minimization of a certain norm. The isotropic limit tests show excellent agreement with the isotropic results available in literature.For anisotropic materials the numerical results are given for a semispherical basin. The results show that presence of an anisotropic basin may result in signi"cant ampli"cation of surface motion atop the basin. While the amplitude of peak surface motion may be similar to the corresponding isotropic results, the di!erence in the displacement patterns may be quite di!erent between the two. Therefore, this study clearly demonstrates that material anisotropy may be very important for accurate assessment of surface ground motion ampli"cation atop basins.

Scattering of elastic waves by a general anisotropic basin. Part 1: a 2D model

“…The free-ÿeld already satisÿes the free-surface condition and therefore does not appear in (18). However, since the scattered wave ÿelds (15) and (16) are linear combinations of the full-space Green functions, g (J ) ik , they do not automatically satisfy the stress-free boundary conditions.…”

“…However, the choice of auxiliary curves, source location, and collocation points ( Figure 3) necessary for evaluation of the scattered wave ÿeld has to be determined through careful parametric study of the problem. In this work the procedure suggested by Zheng and Dravinski [18] is adopted to determine these key parameters which are: • 1 ; 2 ; 1 ; 2 ; -size and position of the auxiliary curves, where 1 ¡1 and 2 ¿1 are scaling parameters used to generate the auxiliary surfaces (1) and (2) from the interface [18] while 1 ; 2 ; and are vertical o sets deÿned by Figure 3; • M , L 1 and L 2 -number of source points along the auxiliary curves (1) and (2) ; • N -number of collocation points along the interface between the half-space and the basin where the continuity of displacement and traction ÿelds is imposed; • P, Q -the number of collocation points along the half-space surface, outside and inside the basin, respectively, where the stress-free boundary conditions are imposed; • size of the half-space surface where discretization by P occurs.…”