# Forward and Inverse Studies on Scattering
## 2 Forward Problem by Modified BEM  
### Formulation of Modified BEM  
The problem to be solved is shown in Figure 1. An artificial canyon-shaped flaw has been cut on the surface of an isotropic elastic half-space. The incident Rayleigh wave propagates from left to the right side, and the reflected wave is observed at the far end of the left side. For accurately computing the scattering wave field, we divide the forward analysis into two major steps, as shown in Figure 2.  
(i) A time-harmonic incident Rayleigh wave field in the intact half-space is considered, and the correspondent stress \(\sigma_{ij}^{\mathrm{inc}}\) is calculated along the curve \(\Gamma_{1}^{-}\), which is actually the flaw surface. (ii) Since the actual traction on \(\Gamma_{1}^{-}\) is zero, the stress \(\sigma_{ij}^{\mathrm{inc}}\) is released by exerting a compensation traction \(f_{i}^{\mathrm{sca}}=-\sigma_{ij}^{\mathrm{inc}}n_{j}\) along \(\Gamma_{1}^{-}\). The scattering problem is transformed into the question of solving the wave field in the flawed half-space subjected to load \(f_{i}^{\mathrm{sca}}\). Figure 1: Problem configuration. Figure 2: Two steps of calculating scattering wave field, (**a**) Step 1; (**b**) Step 2. Then, the displacement in frequency domain of scattering wave field at an arbitrary point can be expressed in the form of boundary integral equation (BIE), as shown by Equation (1) [34]. \[c_{pi}(X)u_{i}^{\text{sca}}(X,\Omega)=\int_{\Gamma}\big{[}G_{ip}(x,X,\Omega)t_{i} ^{\text{sca}}(x,\Omega)-T_{ip}(x,X,\Omega)u_{i}^{\text{sca}}(x,\Omega)\big{]}d \Gamma(x)(i,p=1,2) \tag{1}\]  
where \(u_{i}^{\text{sca}}\) and \(t_{i}^{\text{sca}}\) are displacement and boundary traction, respectively. \(G_{ip}(x,X,\Omega)\) and \(T_{ip}(x,X,\Omega)\) are fundamental solutions representing the \(i\)-th component of the displacement and stress at point \(x\) due to a unit time-harmonic concentrated load of angular frequency \(\Omega\) exerted in the p-th direction at point \(X\) in the elastic full-space. The coefficient \(c_{pi}(X)=\delta_{pi}\) for \(X\in V\), \(c_{pi}(X)=\delta_{pi}/2\) for point \(X\in\Gamma\) which is smooth locally, \(c_{pi}(X)=0\) for \(X\
otin V\cup\Gamma\). It should be noted that here the boundary \(\Gamma\) includes both the flaw area and the intact infinite surface, as the full-space fundamental solutions have been chosen. Now the whole surface is divided into six parts, \(\Gamma_{\infty}^{-}\) and \(\Gamma_{\infty}^{+}\) tending to infinity on the left and right side, respectively; \(\Gamma_{0}^{-}\) and \(\Gamma_{0}^{+}\) representing the near-field boundaries; \(\Gamma_{1}^{-}\) and \(\Gamma_{1}^{+}\) the left and right half of the flaw surface, respectively. Consequently, the scattered wave field is rewritten from Equation (1):  
\[ \begin{split} c_{pi}(X)u_{i}^{\text{sca}}(X,\Omega)=-\int_{\Gamma_ {\infty}^{+}\cup\Gamma_{\infty}^{+}}T_{ip}(x,X,\Omega)u_{i}^{\text{sca}}(x, \Omega)d\Gamma(x)-\int_{\Gamma_{0}\cup\Gamma_{0}^{+}}T_{ip}(x,X,\Omega)u_{i}^{ \text{sca}}(x,\Omega)d\Gamma(x)\\ +\int_{\Gamma_{1}\cup\Gamma_{1}^{+}}[G_{ip}(x,X,\Omega)t_{i}^{ \text{sca}}(x,\Omega)-T_{ip}(x,X,\Omega)u_{i}^{\text{sca}}(x,\Omega)]d\Gamma( x),\ \ \ (i,p=1,2)\end{split}  \tag{2}\]  
where the traction-free boundary condition on \(\Gamma_{0}^{-}\cup\Gamma_{0}^{+}\cup\Gamma_{\infty}^{-}\cup\Gamma_{\infty}^{+}\) has been taken into account. For traditional BEM, only the near-field boundaries \(\Gamma_{0}^{-}\cup\Gamma_{0}^{+}\cup\Gamma_{1}^{-}\cup\Gamma_{1}^{+}\) are meshed, thus, the first item on the right hand side of Equation (2) is omitted. This treatment can cause spurious reflected waves at the truncation points which do not attenuate in the 2D half-space, which are unwanted. It is noticed that, if the truncation point is far enough from the flaw area, the displacement fields on the infinite boundaries will follow a Rayleigh wave pattern travelling out of the flaw region [18], as expressed by  
\[