# Surface Motion for P-Wave Scattering by an Arbitrary-Shaped Canyon in Saturated Half-Space
## 1 Introduction  
The scattering of seismic waves by canyon topography in half-space is a classic model for studying site seismic effects [1, 2]. In the past, over half a century ago, a large number of scholars engaged in this research area and achieved rich research results [3, 4, 5]. These results provide a solid theoretical basis for the study of ground motion characteristics and structural seismic design. The analytical method simplifies the physical model and uses the wave function expansion method to solve the analytical solution of the wave scattering problem. For SH-waves, [PERSON] et al. first used the wave function expansion method [6] to obtain the analytical solution to the SH-wave scattering problem of a semicircular canyon [3] in half-space, and analyzed the influence of wavelength, incident angle, and other factors on canyon response. Since then, different scholars have carried out further research on seismic wave scattering according to its terrain (V-shape canyon [7, 8], semicircular canyon group [9], and a radially multilayered semicylindrical canyon with a rockfill dam [10]). As the research deepens, some scholars have begun to pay attention to the inhomogeneity of the soil. [PERSON] et al. [11] derived an exact solution for a sufficiently inhomogeneous semicylindrical canyonunder oblique incident SH-waves. [PERSON] [12] presented the solution of a linear corrugated orthotropic stratum above a semi-infinite substratum based on a half-space time-domain boundary element method (TD-BEM) subjected to oblique incident plane SH-waves. Similar to [PERSON] [12], based on TD-BEM, [PERSON] et al. [13] successfully obtained the surface motion of an orthotropic Gaussian-shaped sedimentary basin embedded in a linear elastic half-space under transient SH-wave propagation. [PERSON] et al. [14] deals with seismic response evaluation of a finite multilayered geological structure with nonparallel boundaries rested on a semi-infinite elastic half-plane subjected to transient seismic waves. Compared to SH-waves, the analytical solution for P/SV wave scattering by local terrain is complicated due to the coupling of P and SV wave propagation, making it challenging to satisfy stress-free boundary conditions on the half-space surface [15; 16]. For this problem, many scholars have provided relevant methods. [PERSON] and [PERSON] [4; 5] were the first to approximate the horizontal half-space surface by introducing a circular surface with a sufficiently large radius (namely the big-arc method). They obtained an analytical solution for the scattering problem of P and SV waves by circular canyons by combining this method with the wave function expansion method [6]. The essence of this method is to transform the wave problem in the infinite region into a finite region, thus avoiding the problem of satisfying the zero-stress condition on the horizontal ground surface. [PERSON] et al. [16] applied the complex variable function theory [17] and the mirror image method [6] to map the horizontal surface into a semicircle, solve the horizontal surface stress-free boundary conditions on the image plane, and then obtain the analytical solution of the plane wave scattering problem in a semicircular canyon. [PERSON] et al. [18] proposed a wave function combination method based on the Lamb solution [19] and the wave function expansion, and obtained the series solutions of P, SV, and Rayleigh wave propagation and scattering around a semicircular canyon. [PERSON] et al. used the wave potential function [15] that automatically satisfies the surface stress-free boundary condition, and combined it with the method of moments [20] to solve the P-wave scattering problem by an almost circular arbitrary-shaped canyon [21], a shallow arbitrary-shaped canyon [22], and an arbitrary-shaped alluvial valley [23] in elastic half-space. With the development of research, many scholars have extended the scattering problem of P/SV waves from uniform elastic media to saturated porous elastic media.