# Forward and Inverse Studies on Scattering
## 2 Forward Problem by Modified BEM
### Formulation of Modified BEM
\begin{split} u_{i}^{\text{sca}}(x,\Omega)\approx& R^{-}( \Omega)\widetilde{u}_{i}^{-}(x,\Omega),& x\in\Gamma_{\infty}^{-} \\ u_{i}^{\text{sca}}(x,\Omega)\approx& R^{+}( \Omega)\widetilde{u}_{i}^{+}(x,\Omega),& x\in\Gamma_{\infty}^{+} \end{split}  \tag{3}\]  
where \(\widetilde{u}_{i}^{\pm}\). is the displacement field of the unitary Rayleigh wave travelling in positive (\(+\)) or negative (\(-\)) directions, and \(R^{\pm}\) the coefficient to be solved. Equation (3) makes it possible to take the far-field integrals in Equation (2) into consideration. Then, for any point on the boundary which is locally smooth, we have  
\[ \begin{split}\tfrac{1}{2}u_{p}^{\text{sca}}(X,\Omega)+R^{-}( \Omega)A_{p}^{-}(X,\Omega)+R^{+}(\Omega)A_{p}^{+}(X,\Omega)+\int_{\Gamma_{0} \cup\Gamma_{0}^{+}\cup\Gamma_{1}^{-}\cup\Gamma_{1}^{+}}T_{ip}(x,X,\Omega)u_{i} ^{\text{sca}}(x,\Omega)d\Gamma(x)\\ =&\int_{\Gamma_{0}\cup\Gamma_{0}^{+}}G_{ip}(x,X,\Omega )t_{i}^{\text{sca}}(x,\Omega)d\Gamma(x),&(i,p=1,2).\end{split}  \tag{4}\]  
where the correction terms \(A_{p}^{\pm}\) having the definition \(A_{p}^{-}(X,\Omega)=\int_{\Gamma_{\infty}^{-}}T_{ip}(x,X,\Omega)\widetilde{u }_{i}^{-}(x,\Omega)d\Gamma(x)\) and \(A_{p}^{+}(X,\Omega)=\int_{\Gamma_{\infty}^{+}}T_{ip}(x,X,\Omega)\widetilde{u }_{i}^{+}(x,\Omega)d\Gamma(x)\). They are constants. In paper [18], the coefficients \(R^{\pm}\) are not explicit in the modified BEM formulation, consequently, the reflection coefficients obtained from \(x_{1}\) and \(x_{2}\) components of displacement might not coincide with each other. As an improvement, we will try to solve \(R^{\pm}\) together with displacement fields. The correction terms \(A_{p}^{\pm}\) involve the integral over an infinite boundary, however, by applying the reciprocity theorem, the integral can be transferred to a bounded region [18; 35]. \[ \begin{split} A_{p}^{+}(X,\Omega)=& c_{pi}(X) \widetilde{u}_{i}^{+}(X,\Omega)-\int\limits_{\Gamma_{0}^{+}}T_{ip}(x,X,\Omega) \widetilde{u}_{i}^{+}(x,\Omega)d\Gamma(x)\\ &+\int_{\Gamma_{1}^{+}\cup\Gamma_{2}}[G_{ip}(x,X,\Omega) \widetilde{t}_{i}^{+}(x,\Omega)-T_{ip}(x,X,\Omega)\widetilde{u}_{i}^{+}(x, \Omega)]d\Gamma(x)(i,p=1,2)\end{split}  \tag{5}\]\[A_{p}^{-}(X,\Omega) =c_{pi}(X)\widetilde{u_{i}^{-}}(X,\Omega)-\int_{\Gamma_{1}^{+}}T_{ip} (x,X,\Omega)\widetilde{u_{i}^{-}}(x,\Omega)d\Gamma(x) \tag{6}\] \[\quad+\int_{\Gamma_{1}^{+}\cup\Gamma_{2}}\bigl{[}G_{ip}(x,X,\Omega) \widetilde{t}_{i^{-}}^{-}(x,\Omega)-T_{ip}(x,X,\Omega)\widetilde{u}_{i}^{-}(x,\Omega)\bigr{]}d\Gamma(x)(i,p=1,2)\]  
where \(\Gamma_{2}\) represents an auxiliary vertical boundary of 2-3 Rayleigh wavelengths, see Figure 2. \(\widetilde{t}_{i}^{+}=-\widetilde{\sigma}_{i1}^{+}n_{1}\), and \(\widetilde{t}_{i}^{-}=\widetilde{\sigma}_{i1}^{-}n_{1}\) are defined on \(\Gamma_{2}\) with \(n_{1}\) the unit normal vector pointing in positive x1-direction. \(\widetilde{\sigma}_{i1}^{\pm}\) is the stress field of the unitary Rayleigh wave propagating in positive or negative x1-directions, respectively. By substitution of Equations (5) and (6) into Equation (4), we can obtain an equation set with the displacement and coefficients \(R^{\pm}\) as the undetermined variables. Here, we adopt the constant elements (see Figure 3) in the BEM discretization, in which one element is represented by the only node in the middle, and obtain the matrix representation in the form of  
\[[\mathbf{T}]\{\mathrm{u}^{\mathrm{sca}}\}=[\mathbf{G}]\{\mathrm{t}^{\mathrm{ sca}}\} \tag{7}\]  
Here, we assume that for the 1 st or the \(N\)-th node, the displacements are not independent, and can be expressed as \(R^{-}(\Omega)\widetilde{u_{i}^{-}}(x_{1},\Omega)\) and \(R^{+}(\Omega)\widetilde{u}_{i}^{+}(x_{N},\Omega)\), respectively, where \(x_{1}\) and \(x_{N}\) are coordinates of the 1 st and the \(N\)-th node, respectively. Thus, the submatrices concerning the 1 st and the \(N\)-th elements in \([\mathbf{T}]\) are combined with the correction terms \(A_{p}^{\pm}\), and placed as the first and last columns.