# Kirchhoff Migration for Identifying Unknown Targets Surrounded by Random Scatterers
## 5 Further Result: Imaging of Extended Dielectric Targets in an Inhomogeneous Medium  
To examine the effectiveness of Kirchhoff migration, additional numerical simulation about the imaging of extended targets completely embedded in an inhomogeneous medium were performed. Following the test configuration in [54], we consider only the permittivity contrast case and set the inhomogeneous domain \(\Omega\) to be a unit circle with a permittivity of \(\varepsilon_{0}=\eta(0.5,1.5)\). For the sake, we set the value of permeability \(\mu(\mathbf{x})=\mu_{0}\equiv 1\) for \(\mathbf{x}\in\Omega\). We assumed the existence of four extended inhomogeneities \(\mathbf{A}_{m}\subset\Omega\) with smooth boundaries \(\partial\mathbf{A}_{m}\) satisfying \(\partial\mathbf{A}_{m}\cap\partial\Omega=\varnothing\) and permittivities \(\varepsilon_{m},m=1,2,3,4\). The shapes of \(\mathbf{A}_{m}\) are shown in the Figure 6, and their boundaries are expressed as  
\[ \begin{array}{lcl}\partial\mathbf{A}_{1}=\left\{[s,t]:11.11(s-0.3)^{2}+25(t- 0.5)^{2}=1\right\},&\varepsilon_{1}=2.5\\ \partial\mathbf{A}_{2}=\left\{[s,t]:53.1250(s-0.5)^{2}-93.75(s-0.5)(t+0.4)+53.1 250(t+0.4)^{2}=1\right\},&\varepsilon_{2}=3\\ \partial\mathbf{A}_{3}=\left\{[s,t]:76.5625(s+0.6)^{2}+81.1898(s+0.6)t+29.6875t ^{2}=1\right\},&\varepsilon_{3}=2\\ \partial\mathbf{A}_{4}=\left\{[s,t]:s^{2}+t^{2}=0.01\right\},&\varepsilon_{4} =0.01.\end{array} \]  
With this configuration, we denote the time-harmonic total field as \(u^{(n)}(\mathbf{x})\), which satisfies the following boundary value problem  
\[\left\{ \begin{array}{cc}\Delta u^{(n)}(\mathbf{x})+\omega^{2}\varepsilon( \mathbf{x})u^{(n)}(\mathbf{x})=0&\text{in}&\Omega\\ \dfrac{\partial u^{(n)}(\mathbf{x})}{\partial\
u(\mathbf{x})}=\dfrac{\partial e ^{\beta\theta_{n}\cdot\mathbf{x}}}{\partial\
u(\mathbf{x})}&\text{on}& \partial\Omega\end{array} \right. \tag{8}\]  
and with transmission condition on the boundary \(\partial\mathbf{A}_{m}\), \(m=1,2,3,4\). Here, \(\
u(\mathbf{x})\) is the unit normal to \(\partial\Omega\) at \(\mathbf{x}\), and \(\theta_{n}\), \(n=1,2,\cdots,N(=128)\) denotes a two-dimensional vector on the \(\partial\Omega\), such that  
\[\theta_{n}=\left[\cos\dfrac{2\pi n}{N},\sin\dfrac{2\pi n}{N}\right]^{T}.\]  
Similarly, \(u_{0}^{(n)}(\mathbf{x})=e^{\beta\theta_{n}\cdot\mathbf{x}}\) denotes the solution of (8) without anomalies and constant \(\varepsilon_{0}\equiv 1\). Figure 5: (Example 4: Case 3) Distribution of normalized singular values, maps of \(\Re_{\text{KIR}}(\mathbf{x})\) and \(\Re_{\text{SUB}}(\mathbf{x})\) for \(\lambda=0.7\) (top line) and \(\lambda=0.2\) (bottom line). Based on [12; 51], \(u^{(n)}(\mathbf{x})-u_{0}^{(n)}(\mathbf{x})\) can be represented as  
\[u^{(n)}(\mathbf{x})-u_{0}^{(n)}(\mathbf{x})=\sum_{m=1}^{4}\int_{ \partial\mathbf{A}_{m}}\mathcal{N}(\mathbf{x},\mathbf{y})\phi^{(n)}(\mathbf{y} )d\mathbf{y}+\mathcal{E}(\varepsilon_{0}), \tag{9}\]  
where \(\phi^{(n)}(\mathbf{y})\) is an unknown density function, \(\mathcal{E}(\varepsilon_{0})\) denotes an error term that is highly depending on the value of \(\varepsilon_{0}\), and \(\mathcal{N}(\mathbf{x},\mathbf{y})\) satisfies  
\[\left\{ \begin{array}{rcl}\Delta\mathcal{N}(\mathbf{x},\mathbf{y})+k^{2} \mathcal{N}(\mathbf{x},\mathbf{y})=-\delta(\mathbf{x},\mathbf{y})&\text{in}& \Omega\\ \frac{\partial\mathcal{N}(\mathbf{x},\mathbf{y})}{\partial\mathbf{\
u}( \mathbf{y})}=0&\text{on}&\partial\Omega.\end{array} \right.\]  
Here, \(\delta\) denotes the dirac delta function. On the basis of (9), we cannot use the boundary measurement data \(u^{(n)}(\mathbf{x})-u_{0}^{(n)}(\mathbf{x})\) directly to design the Kirchhoff and subspace migrations.