# Kirchhoff Migration for Identifying Unknown Targets Surrounded by Random Scatterers
## Appendix A Basic Idea of Kirchhoff Migration  
Here, a detailed description of Kirchhoff migration is discussed. For this, let us recall the MSR matrix \(\mathbb{K}\):  
\[\mathbb{K}=\left[ \begin{array}{cccc}u_{\infty}(\mathbf{\vartheta}_{1},\mathbf{ \theta}_{1})&u_{\infty}(\mathbf{\vartheta}_{1},\mathbf{\theta}_{2})&\cdots&u_{\infty} (\mathbf{\vartheta}_{1},\mathbf{\theta}_{N})\\ u_{\infty}(\mathbf{\vartheta}_{2},\mathbf{\theta}_{1})&u_{\infty}(\mathbf{\vartheta}_{2}, \mathbf{\theta}_{2})&\cdots&u_{\infty}(\mathbf{\vartheta}_{2},\mathbf{\theta}_{N})\\ \vdots&\vdots&\ddots&\vdots\\ u_{\infty}(\mathbf{\vartheta}_{N},\mathbf{\theta}_{1})&u_{\infty}(\mathbf{\vartheta}_{N}, \mathbf{\theta}_{2})&\cdots&u_{\infty}(\mathbf{\vartheta}_{N},\mathbf{\theta}_{N})\\ \end{array} \right].\]  
Then, since each element \(u_{\infty}(\mathbf{\vartheta}_{j},\mathbf{\theta}_{l})\) of \(\mathbb{K}\) is  
\[u_{\infty}(\mathbf{\vartheta}_{j},\mathbf{\theta}_{l})=\sum_{m=1}^{M} \alpha_{m}^{2}|\mathbf{B}_{m}|\left(\frac{\varepsilon_{\mathbf{A}}^{(m)}- \varepsilon_{0}}{\sqrt{\varepsilon_{0}\mu_{0}}}+\frac{2\mu_{0}}{\mu_{ \mathbf{A}}^{(m)}+\mu_{0}}\mathbf{\theta}_{j}\cdot\mathbf{\theta}_{l}\right)e^{ik(\bm {\theta}_{j}+\mathbf{\theta}_{l})\cdot\mathbf{x}_{\mathrm{ANO}}^{(m)}}\\ +\sum_{s=1}^{S}\sigma_{s}^{2}|\mathbf{B}_{s}|\left(\frac{ \varepsilon_{\mathbf{R}}^{(s)}-\varepsilon_{0}}{\sqrt{\varepsilon_{0}\mu_{0}} }+\frac{2\mu_{0}}{\mu_{\mathbf{R}}^{(s)}+\mu_{0}}\mathbf{\theta}_{j}\cdot\mathbf{ \theta}_{l}\right)e^{ik(\mathbf{\theta}_{j}+\mathbf{\theta}_{l})\cdot\mathbf{x}_{ \mathrm{RND}}^{(s)}},\]\(\mathbb{K}\) can be represented as  
\[\mathbb{K}=N\sum_{m=1}^{M}\alpha_{m}^{2}|\mathbf{B}_{m}|\mathbf{E}( \mathbf{x}_{\text{ANO}}^{(m)})\left[ \begin{array}{ccc}\dfrac{\epsilon_{ \mathbf{A}}^{(m)}-\epsilon_{0}}{\sqrt{\epsilon_{0}\mu_{0}}}&0&0\\ 0&\dfrac{2\mu_{0}}{\mu_{\mathbf{A}}^{(m)}+\mu_{0}}&0\\ 0&0&\dfrac{2\mu_{0}}{\mu_{\mathbf{A}}^{(m)}+\mu_{0}}\end{array} \right]\mathbf{E }(\mathbf{x}_{\text{ANO}}^{(m)})^{T}\\ +N\sum_{s=1}^{S}\sigma_{s}^{2}|\mathbf{B}_{s}|\mathbf{E}(\mathbf{x}_{ \text{RND}}^{(s)})\left[ \begin{array}{ccc}\dfrac{\epsilon_{\mathbf{R}}^{(s)}- \epsilon_{0}}{\sqrt{\epsilon_{0}\mu_{0}}}&0&0\\ 0&\dfrac{2\mu_{0}}{\mu_{\mathbf{R}}^{(s)}+\mu_{0}}&0\\ 0&0&\dfrac{2\mu_{0}}{\mu_{\mathbf{R}}^{(s)}+\mu_{0}}\end{array} \right] \mathbf{E}(\mathbf{x}_{\text{RND}}^{(s)})^{T}, \tag{10}\]  
where  
\[\mathbf{E}(\mathbf{x}):=\dfrac{1}{\sqrt{N}}\left[ \begin{array}{ccc}e^{ik \boldsymbol{\theta}_{1}\cdot\mathbf{x}}&(\boldsymbol{\theta}_{1}\cdot\mathbf{e }_{1})e^{ik\boldsymbol{\theta}_{1}\cdot\mathbf{x}}&(\boldsymbol{\theta}_{1} \cdot\mathbf{e}_{2})e^{ik\boldsymbol{\theta}_{1}\cdot\mathbf{x}}\\ e^{ik\boldsymbol{\theta}_{2}\cdot\mathbf{x}}&(\boldsymbol{\theta}_{2}\cdot \mathbf{e}_{1})e^{ik\boldsymbol{\theta}_{2}\cdot\mathbf{x}}&(\boldsymbol{ \theta}_{2}\cdot\mathbf{e}_{2})e^{ik\boldsymbol{\theta}_{2}\cdot\mathbf{x}}\\ \vdots&\vdots&\vdots\\ e^{ik\boldsymbol{\theta}_{N}\cdot\mathbf{x}}&(\boldsymbol{\theta}_{N}\cdot \mathbf{e}_{1})e^{ik\boldsymbol{\theta}_{N}\cdot\mathbf{x}}&(\boldsymbol{ \theta}_{N}\cdot\mathbf{e}_{2})e^{ik\boldsymbol{\theta}_{N}\cdot\mathbf{x}} \end{array} \right]\in\mathbb{C}^{N\times 3} \tag{11}\]  
with \(\mathbf{e}_{1}=[1,0]^{T}\) and \(\mathbf{e}_{2}=[0,1]^{T}\). Based on this representation, we can introduce the basic concept of Kirchhoff migration as follows.