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%.

中文翻译：

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qP-、qSV-和SH-波下具有3D任意形状冲积谷的分层横向各向同性半空间的表面运动

摘要 提出了一种间接边界元法 (IBEM) 来解决 qP-、qSV- 和 SH-波由嵌入分层横向各向同性 (TI) 半空间中的三维 (3D) 任意形状冲积谷引起的散射问题。 . 首先，通过精确刚度矩阵方法结合傅里叶变换，推导出作用在多层TI半空间内斜面上的均匀分布载荷的3D动态格林函数。接下来，推导出的 3D 格林函数被用作制定特殊 IBEM 的基本解决方案，并且 IBEM 进一步用于研究多层 TI 半空间中受到入射体波影响的 3D 表面不规则的表面运动。与传统的以集中荷载的格林函数为核的边界元法相比，新方法的优点是精度更高，不存在奇异性问题。通过将我们的结果与已发表的结果进行比较，验证了所呈现的 3D Green 函数以及 IBEM 解决方案的准确性。为了进一步证明本方法的普遍适用性和有效性，以单个TI分层半空间中的半椭圆形冲积谷为例，在频域和时域进行数值计算。频域结果表明，材料各向异性对半椭圆形冲积谷地表运动有重要影响。通常，位移幅度随着各向异性程度的增加而减小。各种材料的位移放大光谱有显着差异，这在很大程度上取决于入射角和观察点。时域结果表明，随着各向异性程度的增加，谷内各点的峰值地面加速度（PGA）逐渐减小。与各向同性的情况相比，PGA 的差异可以达到 53.8%。