# Forward and Inverse Studies on Scattering
## 1 Introduction
The inverse analysis is an important aspect and fundamental aim for studying the scattering of Rayleigh wave. Many researchers have tried to relate the amplitude and time-of-flight (TOF) of reflected wave with surface-breaking cracks [20; 21; 22; 23] or notches [24; 25], both in experimental and numerical aspects, in order to achieve flaw location and sizing effect. These researchers mainly adopted pitch-catch configurations, and were able to detect the positions and approximate sizes of defects, but lacked further information, such as depth, width, and severity. In order to obtain more information about defects and abnormalities in the near-surface layer, geophysical scientists and engineers have tried tomographic reconstruction approaches using data from scattered Rayleigh wave fields. [PERSON] et al. [26] firstly used Rayleigh wave phase velocities at different frequencies to reconstruct the S-wave velocity profile of multiple elastic layers on top of the half-space. Furthermore, [PERSON] et al. [27] and [PERSON] et al. [28] extended the tomographic approach to the detection of near-surface shallow cavities. [PERSON] et al. [29] used a similar inversion method to reconstruct surface layer stiffness of an ancient sandstone facade. [PERSON] et al. [30] adopted an inversion procedure to detect residual stresses from general Rayleigh wave dispersion of a weakly anisotropic media. These methods stem from wave signal processing theories (in earlier works [27; 28]) or discretized equations of motion (in later works [29; 30]). Through the formulation of an \"operator matrix\" of inverse problem, the researchers try to find an optimum solution of the \"model vector\" that is feasible to the \"data set\". Due to strong nonlinearity of general inverse problems, one needs to perform numerous iterations to achieve an accurate solution. In this paper, we will propose a convenient inverse method to reconstruct the location and geometric properties of surface-opening flaws. The method is subsequent research for inversion method for plate-thinning reconstruction using SH-waves [31; 32] and Lamb waves [33]. We make use of the reflection coefficients of Rayleigh waves obtained from the forward analysis in different frequencies as the input data. These data can also be attained by experiments. By introduction of Born approximation and far-field expressions of half-space Green's functions, and the usage of boundary integral equation over the flaw area, it can be proved that the objective function of flaw depth \(d(x_{1})\) is approximately expressed as an inverse spatial Fourier transform of reflection coefficients in wavenumber domain. This inversion procedure is especially suitable for reconstruction of weak and shallow notches, and can also be used as an initial-step image for deeper and steeper canyons. The paper will be organized as follows: In Section 2, we will discuss about the improved forward problem using modified BEM in Section 2.1, and give some numerical examples and compare with existing results in Section 2.2. In Section 3, we will elaborate on the mathematical formula for inverse reconstruction method for surface-breaking canyon-shaped flaws in Section 3.1, and parametric influences will be discussed in detail in Section 3.2. We will give conclusions in Section 4.