# Forward and Inverse Studies on Scattering
## 1 Introduction  
The non-destructive testing and structural health monitoring are of great importance in civil, mechanical, and aerospace engineering industries. Corrosions, pittings, and cracks may occur during service years of structures, and cause threats to public life and possession safety. Ultrasonic inspection is one of the fundamental techniques for locating surface and hidden flaws. Compared with traditionally-applied bulk waves, the Rayleigh surface wave has low attenuation and high energy concentration, which makes it suitable for inspection of defects or abnormalities on or close to surface. On the other hand, Rayleigh waves are not dispersive, as compared with other guided waves such as Lamb, SH, and Love waves, thus easier for signal processing [1,2]. Due to the great potential of Rayleigh waves in NDT (Non-Destrctive Testing) applications, it is necessary to examine their scattering and diffraction behaviors, both in the forward and inverse aspects. Researchers have long been interested in forward calculation of scattering phenomena for Rayleigh waves. One of pioneering work can be attributed to [PERSON] and [PERSON]'s paper of calculating the scattered field of Rayleigh waves by surface crack [3], which is followed by [PERSON], who discussed the emission of surface waves from a crack by cyclic loading in detail [4]. Furthermore, [PERSON] and [PERSON] deduced the analytical solution of a scattered Rayleigh wave field by semi-circular canyons [5]. However, theoretical calculations are only applicable for special cases. For more general situations, numerical approaches are required. As a widely applied numerical method, finite elementmethod (FEM) has been applied for Rayleigh wave field simulation as it passes through cracks [6], slots [7], and notches [8] on the flat surface (of a large yet finite solid block). The FEM mesh was performed within a whole elastic solid block [7; 8] or a confined region of an infinite body with absorption boundaries [6]. Although FEM is a sophisticated numerical approach, it is time- and memory-consuming for scattering problems in a half-space, especially when far-field behaviors need to be examined. Thus, some researchers turned to the boundary element method (BEM), in which only the boundaries need to be meshed. Their early works include [PERSON] [9] and [PERSON]'s [10] applications of discrete wavenumber BEM for calculation of scattered Rayleigh wave fields in frequency domains. [PERSON] et al. [11] and [PERSON] et al. [12] used indirect BEM to solve similar problems. For half-space problems, there have been two main modeling patterns: (i) analyzing and meshing the flaw surface only, with usage of half-space Green's functions, in which the interface boundary conditions are satisfied automatically; (ii) analyzing and meshing the flaw surface and the whole interface, with usage of full-space Green's functions. For the former model, the element is fewer, however, the format of Green's functions tends to be complicated, usually not in a closed form, and requires a large amount of resources for contour or numerical integration over complex wavenumber domains [10; 13]. On the other hand, for the latter model, although the whole interface is incorporated, the Green's functions are simpler and easier to apply in the BEM calculation. Thus, many recent researchers used the latter modeling approach to study the scattering of Rayleigh and Lamb waves [14; 15; 16; 17]. But researchers often encounter the problem of \"spurious reflection\" because of the model truncation at far ends [18; 19]. [PERSON] [18] suggested a modified BEM method in which a far-field displacement pattern had been introduced and substituted into the BEM equation set. By enforcing displacement and stress continuous relations, the researchers were able to solve the scattering wave field while suppressing spurious reflections. However, [PERSON] method might not be accurate for calculating far-field reflection and transmission coefficients. In this paper, we will propose an improved modified BEM method.