Surface motion of a layered transversely isotropic half-space with a 3D arbitrary-shaped alluvial valley under qP-, qSV- and SH-waves

Abstract An indirect boundary element method (IBEM) is proposed to solve the scattering of qP-, qSV- and SH-waves by a three-dimensional (3D) arbitrary-shaped alluvial valley embedded in a layered transversely isotropic (TI) half-space. First, the 3D dynamic Green's functions for the uniformly distributed loads acting on an interior inclined plane in a multi-layered TI half-space are derived by means of exact stiffness matrix method combined with the Fourier transform. Next, the deduced 3D Green's functions are used as fundamental solutions to formulate a special IBEM, and the IBEM is further used to investigate the surface motion of a 3D surface irregularities in a multi-layered TI half-space subjected to incident body waves. Compared to the traditional BEM adopting Green's functions for the concentrated loads as the kernel, the new method has the merits of higher precision and none of the problem of singularity. By comparing our results with the published ones, accuracy of the presented 3D Green's functions as well as IBEM solution are verified. To further demonstrate the general applicability and effectiveness of the present method, taking a semi-ellipsoidal alluvial valley in a single TI layered half-space as an example, numerical calculations are performed in both the frequency and time domain. Results in frequency domain show that material anisotropy has an important influence on the surface motion of semi-ellipsoidal alluvial valley. In general, the displacement amplitudes decrease as the degree of anisotropy increases. Displacement amplification spectra are significantly different among the various materials, which strongly depend on the incident angles and the observation points. Results in time domain show that with the increase of the degree of anisotropy, the peak ground acceleration (PGA) of each point inside the valley gradually decreases. Compared with the case of isotropic, the difference of PGA can reach 53.8%.