# Forward and Inverse Studies on Scattering
## 3 Inverse Problem by Modified BEM
### Formulation of Inverse Reconstruction
 \tag{18}\]  
Figure 8: Reflection amplitudes obtained by BEM and half-space scattering equation respectively. where \(p_{i}(x_{2})\) (\(i=1\), \(2\)) is defined in Equation (13), and \(A^{\rm inc}\) represents the complex amplitude of the incident wave. We use the far-field form of half-space Green's functions (12), and substitute them into Equation (17), and obtain the reflected wave of the same wave number \(\xi_{0}\), as shown by Equation (19):  
\[ \begin{array}{c}u_{n}^{\rm sca}(X,\xi_{0})\approx A^{\rm inc}(\xi_{0})\int_{ \rm V}\biggl{[}\left(-2\xi_{0}^{2}p_{1}^{2}(x_{2})+\frac{d^{2}p_{2}(x_{2})}{dx_ {2}^{2}}p_{2}(x_{2})+\left(\frac{dp_{2}(x_{2})}{dx_{2}}\right)^{2}\right)( \lambda+2\mu)\\ \hskip 142.26378 pt+\left(i\xi_{0}\frac{dp_{1}(x_{2})}{dx_{2}}p_{2}(x_{2}) \right)(\lambda+3\mu)+\left(i\xi_{0}\frac{dp_{2}(x_{2})}{dx_{2}}p_{1}(x_{2}) \right)(3\lambda+\mu)\\ +\left(-2\xi_{0}^{2}p_{2}^{2}(x_{2})+\frac{d^{2}p_{1}(x_{2})}{dx_{2}^{2}}p_{1} (x_{2})+\left(\frac{dp_{1}(x_{2})}{dx_{2}}\right)^{2}\right)\mu\biggr{]} \cdot e^{-i\xi_{0}\cdot 2x_{1}}dV\cdot f(\xi_{0})\cdot p_{n}(X_{2})e^{-i\xi_{0} \cdot X_{1}}\\ (i,j,k,l,n=1,2)\end{array}  \tag{19}\]  
If one takes a thorough observation on Equation (19), he can find that \(p_{n}(X_{2})\exp(-i\xi_{0}\cdot X_{1})\) represents a unitary reflected Rayleigh wave travelling from right to left, and the integral over V is taken with respect to coordinate x of field points, and can be viewed as a complex amplitude of that unitary reflected Rayleigh wave. Also, the integrand is separated into the part in the bracket concerning \(x_{2}\) coordinate, and an exponential item \(\exp(+i\xi_{0}\cdot 2x_{1})\). This inspires us to rewrite the expression of reflected wave into the following form  
\[ \begin{array}{c}u_{n}^{\rm sca}(X,\xi_{0})\approx A^{\rm inc}(\xi_{0})f(\xi _{0})\cdot\int_{\rm V}{\rm Fun}(x_{2},\ \xi_{0})\exp(+i\xi_{0}\cdot 2x_{1})dV\cdot p_{n}(X_{2}) \exp(-i\xi_{0}\cdot X_{1})\\ \hskip 142.26378 pt\equiv A^{\rm ref}(\xi_{0})p_{n}(X_{2})\exp(-i\xi_{0}\cdot X _{1})\end{array}  \tag{20}\]  
where the \({\rm Fun}(x_{2},\ \xi_{0})\) is the expression in the bracket in Equation (19), and the complex amplitude for the reflected wave is termed as \(A^{\rm ref}\). Now we define the reflection coefficient \(C^{\rm ref}(\xi_{0})\) as the ratio between \(A^{\rm ref}(\xi_{0})\) and \(A^{\rm inc}(\xi_{0})\), so that  
\[C^{\rm ref}(\xi_{0})\approx f(\xi_{0})\cdot\int_{\rm V}{\rm Fun}(x_{2},\ \xi_{0})\exp(+i\xi_{0}\cdot 2x_{1})dV \tag{21}\]  
Also, the volume integral is further rewritten as a dual integral:  
\[\int_{\rm V}{\rm Fun}(x_{2},\ \xi_{0})\exp(+i\xi_{0}\cdot 2x_{1})dV=\int_{- \infty}^{\infty}\exp(+i\xi_{0}\cdot 2x_{1})dx_{1}\int_{-d(x_{1})}^{0}{\rm Fun}(x_{2},\ \xi_{0})dx_{2} \tag{22}\]  
where \(d(x_{1})\) is the flaw depth at coordinate \(x_{1}\). The integral range has been extended from minus infinity to infinity, since \(d(x_{1})=0\) in the intact areas. After organization, the reflection coefficient is expressed as  
\[C^{\rm ref}(\xi_{0})\approx\int_{-\infty}^{\infty}e^{+i\xi_{0}\cdot 2x_{1}}dx_{1} \int_{-d(x_{1})}^{0}\left(c_{\rm TT}(\xi_{0})e^{+2R_{\rm IV}x_{2}}+c_{\rm TL}( \xi_{0})e^{+2(R_{\rm IV}+R_{\rm IV})x_{2}}+c_{\rm LL}(\xi_{0})e^{+2R_{\rm IV}x _{2}}\right)dx_{2} \tag{23}\]  
where \(c_{\rm TT}\), \(c_{\rm TL}\), and \(c_{\rm LL}\) are functions of wavenumber \(\xi_{0}\), and are expressed in detail as  
\[ \begin{array}{c}c_{\rm TT}(\xi_{0})=2\mu\frac{\mu_{0}^{4}}{\xi_{0}^{2}}\\ c_{\rm TL}(\xi_{0})=\frac{\xi_{0}^{2}+R_{\rm IV}^{2}}{2\xi_{0}^{2}R_{\rm IV}}( \lambda k_{\rm LO}^{2}(R_{\rm IV}-R_{\rm LO})+\mu(\tau\xi_{0}^{2}R_{\rm IV}-5 \xi_{0}^{2}R_{\rm LO}+4R_{\rm TO}^{3}-R_{\rm TO}^{2}-R_{\rm LO}R_{\rm TO}^{2}+6 R_{\rm LO}^{2}R_{\rm TO}))\\ c_{\rm LL}(\xi_{0})=-(\lambda+2\mu)\frac{\mu_{0}^{4}(\xi_{0}^{2}+R_{\rm TO}^{2}) ^{2}}{2\xi_{0}^{2}R_{\rm LO}}\end{array}