# Forward and Inverse Studies on Scattering
## 3 Inverse Problem by Modified BEM  
### Formulation of Inverse Reconstruction  
We now try to reconstruct the position and geometric properties of the canyon-shaped flaw. In order to quantitatively depict the flaw's geometric characteristics, an objective function \(d(x_{1})\) is defined in \(x_{1}\) domain, which means the flaw depth at coordinate \(x_{1}\), whereas in the intact area, its value is zero. The scattering Equation (11) in half-space is adopted for inversion. In this paper, a shallow flaw is considered, which can be thought of as a weak scatterer. In this case, the Born approximation [36] can be used for linearization, which replaces the total wave field by the incident wave field on flaw boundaries. e.g., \(u_{1}^{\rm tot}\approx u_{1}^{\rm inc}\). Then Equation (11) becomes  
\[u_{n}^{\rm sca}(X)\approx\int_{\Gamma_{1}^{-}\cup\Gamma_{1}^{+}}C_{ijkl}n_{j}( x)\frac{\partial C_{ln}^{\rm H}(x,X)}{\partial x_{k}}u_{n}^{\rm inc}(x)d\Gamma(x) \quad(i,j,k,l,n=1,\,2) \tag{15}\]  
It is noticed that, on the original half-space surface \(\Gamma_{1}^{\prime}\) (see the dotted line in Figure 1), the traction induced by half-space Green functions is zero. Thus, the surface for integration can be extended to \(\Gamma_{1}^{-}\cup\Gamma_{1}^{+}\cup\Gamma_{1}^{\prime}\). Furthermore, after substitution of Lame constant \(\lambda\) and shear modulus \(\mu\), the scattering equation is expanded as  
\[ \begin{array}{l}u_{n}^{\rm sca}(X)\approx\int_{\Gamma_{1}^{-}\cup\Gamma_{1}^ {+}\cup\Gamma_{1}^{\prime}}\biggl{[}\biggl{(}n_{2}\frac{\partial C_{ln}^{\rm H }}{\partial x_{1}}u_{1}^{\rm inc}+n_{2}\frac{\partial C_{ln}^{\rm H}}{ \partial x_{2}}u_{2}^{\rm inc}\biggr{)}(\lambda+2\mu)+\biggl{(}n_{2}\frac{ \partial C_{ln}^{\rm H}}{\partial x_{1}}u_{2}^{\rm inc}+n_{1}\frac{\partial C_ {ln}^{\rm H}}{\partial x_{1}}u_{1}^{\rm inc}\biggr{)}\lambda\\ +\biggl{(}n_{2}\frac{\partial C_{ln}^{\rm H}}{\partial x_{1}}u_{1}^{\rm inc}+n _{1}\frac{\partial C_{ln}^{\rm H}}{\partial x_{2}}u_{2}^{\rm inc}+n_{2}\frac{ \partial C_{ln}^{\rm H}}{\partial x_{2}}u_{1}^{\rm inc}+n_{1}\frac{\partial C _{ln}^{\rm H}}{\partial x_{1}}u_{2}^{\rm inc}\biggr{)}\mu\biggr{]}d\Gamma(x) \quad(i,j,k,l,n=1,2)\end{array}  \tag{16}\]  
Since surface \(\Gamma_{1}^{-}\cup\Gamma_{1}^{+}\cup\Gamma_{1}^{\prime}\) is closed, we can perform [PERSON]'s theorem to the flawed region and obtain  
\[ \begin{array}{l}u_{n}^{\rm sca}(X)\approx\int_{V}\biggl{[}\biggl{(}\frac{ \partial^{2}C_{ln}^{\rm H}}{\partial x_{1}^{2}}u_{1}^{\rm inc}+\frac{\partial C _{ln}^{\rm H}}{\partial x_{1}}\frac{\partial u_{1}^{\rm inc}}{\partial x_{2}} +\frac{\partial^{2}C_{ln}^{\rm H}}{\partial x_{2}}u_{2}^{\rm inc}+\frac{ \partial C_{ln}^{\rm H}}{\partial x_{2}}\frac{\partial u_{2}^{\rm inc}}{ \partial x_{2}}\biggr{)}(\lambda+2\mu)\\ +\biggl{(}\frac{\partial^{2}C_{ln}^{\rm H}}{\partial x_{1}\partial x_{2}}u_{2} ^{\rm inc}+\frac{\partial C_{ln}^{\rm H}}{\partial x_{1}}\frac{\partial u_{1} ^{\rm inc}}{\partial x_{2}}-\frac{\partial^{2}C_{ln}^{\rm H}}{\partial x_{1} \partial x_{2}}u_{1}^{\rm inc}+\frac{\partial C_{ln}^{\rm H}}{\partial x_{2} }\frac{\partial u_{1}^{\rm inc}}{\partial x_{1}}\biggr{)}\lambda\\ +\biggl{(}\frac{\partial^{2}C_{ln}^{\rm H}}{\partial x_{1}\partial x_{2}}u_{1} ^{\rm inc}+\frac{\partial C_{ln}^{\rm H}}{\partial x_{1}}\frac{\partial u_{1} ^{\rm inc}}{\partial x_{2}}+\frac{\partial^{2}C_{ln}^{\rm H}}{\partial x_{1} }u_{2}^{\rm inc}+\frac{\partial C_{ln}^{\rm H}}{\partial x_{2}}\frac{\partial u _{1}^{\rm inc}}{\partial x_{1}}\biggr{)}\mu\biggr{]}d\Gamma(x)\quad(i,j,k,l,n= 1,2)\end{array}  \tag{17}\]  
It is assumed that a pure incident Rayleigh wave mode is sent from left to right, and the reflected wave is observed at the far field on the left side. Thus, the incident wave with wavenumber \(\xi_{0}\) can be expressed as  
\[ \begin{array}{l}u_{1}^{\rm inc}(x,\xi_{0})=A^{\rm inc}(\xi_{0})p_{1}(x_{2})e ^{+i\xi_{0}x_{1}}\\ u_{2}^{\rm inc}(x,\xi_{0})=A^{\rm inc}(\xi_{0})p_{2}(x_{2})e^{+i\xi_{0}x_{1}} \end{array}