# Kirchhoff Migration for Identifying Unknown Targets Surrounded by Random Scatterers
## Appendix B Derivation of Theorem 1  
In order to explore the mathematical structure of of \(\mathfrak{F}_{\text{KIR}}(\mathbf{x})\), we recall a useful relation derived in [41]. **Lemma A1**. : _For sufficiently large \(N\), \(\mathbf{x}\in\mathbb{R}^{2}\), \(\boldsymbol{\theta}_{n},\boldsymbol{\theta},\boldsymbol{\xi}\in\mathbb{S}^{1}\), the following results hold._  
\[\frac{1}{N}\sum_{n=1}^{N}e^{ik\boldsymbol{\theta}_{n}\cdot \mathbf{x}}\approx\frac{1}{2\pi}\int_{\mathbb{S}^{1}}e^{ik\boldsymbol{\theta} \cdot\mathbf{x}}d\boldsymbol{\theta}=J_{0}(k|\mathbf{x}|)\] \[\frac{1}{N}\sum_{n=1}^{N}(\boldsymbol{\theta}_{n}\cdot \boldsymbol{\xi})e^{ik\boldsymbol{\theta}_{n}\cdot\mathbf{x}}\approx\frac{1} {2\pi}\int_{\mathbb{S}^{1}}(\boldsymbol{\theta}\cdot\boldsymbol{\xi})e^{ik \boldsymbol{\theta}\cdot\mathbf{x}}d\boldsymbol{\theta}=i\left(\frac{\mathbf{ x}}{|\mathbf{x}|}\cdot\boldsymbol{\xi}\right)J_{1}(k|\mathbf{x}|).\]  
Since \(N\) is sufficiently large, applying Lemma A1 yields  
\[\mathbf{W}(\mathbf{x})^{*}\mathbf{E}(\mathbf{x}_{\text{ANO}}^{(m)}) =\frac{1}{N}\left[ \begin{array}{cc}e^{-ik\boldsymbol{\theta}_{1} \cdot\mathbf{x}}\\ e^{-ik\boldsymbol{\theta}_{2}\cdot\mathbf{x}}\\ \vdots\\ e^{-ik\boldsymbol{\theta}_{N}\cdot\mathbf{x}}\end{array} \right]^{T}\left[ \begin{array}{cc}e^{ik\boldsymbol{\theta}_{1}\cdot\mathbf{x}_{\text{ANO}}^{(m )}}&(\boldsymbol{\theta}_{1}\cdot\mathbf{e}_{1})e^{ik\boldsymbol{\theta}_{1} \cdot\mathbf{x}_{\text{ANO}}^{(m)}}&(\boldsymbol{\theta}_{1}\cdot\mathbf{e}_{ 2})e^{ik\boldsymbol{\theta}_{1}\cdot\mathbf{x}_{\text{ANO}}^{(m)}}\\ e^{ik\boldsymbol{\theta}_{2}\cdot\mathbf{x}_{\text{ANO}}^{(m)}}&( \boldsymbol{\theta}_{2}\cdot\mathbf{e}_{1})e^{ik\boldsymbol{\theta}_{2}\cdot \mathbf{x}_{\text{ANO}}^{(m)}}&(\boldsymbol{\theta}_{2}\cdot\mathbf{e}_{2})e^{ ik\boldsymbol{\theta}_{2}\cdot\mathbf{x}_{\text{ANO}}^{(m)}}\\ \vdots&\vdots&\vdots\\ e^{ik\boldsymbol{\theta}_{N}\cdot\mathbf{x}_{\text{ANO}}^{(m)}}&( \boldsymbol{\theta}_{N}\cdot\mathbf{e}_{1})e^{ik\boldsymbol{\theta}_{N}\cdot \mathbf{x}_{\text{ANO}}^{(m)}}&(\boldsymbol{\theta}_{N}\cdot\mathbf{e}_{2})e^{ ik\boldsymbol{\theta}_{N}\cdot\mathbf{x}_{\text{ANO}}^{(m)}}\end{array} \right]\] \[=\left[ \begin{array}{c}\frac{1}{N}\sum_{n=1}^{N}(\boldsymbol{ \theta}_{1}\cdot\mathbf{e}_{1})e^{ik\boldsymbol{\theta}_{1}\cdot(\mathbf{x}_{ \text{ANO}}^{(m)}-\mathbf{x})}\\ \frac{1}{N}\sum_{n=1}^{N}(\boldsymbol{\theta}_{1}\cdot\mathbf{e}_{1})e^{ik \boldsymbol{\theta}_{1}\cdot(\mathbf{x}_{\text{ANO}}^{(m)}-\mathbf{x})}\\ \frac{1}{N}\sum_{n=1}^{N}(\boldsymbol{\theta}_{1}\cdot\mathbf{e}_{2})e^{ik \boldsymbol{\theta}_{1}\cdot(\mathbf{x}_{\text{ANO}}^{(m)}-\mathbf{x})}\end{array}  \right]^{T}\approx\left[ \begin{array}{c}\frac{(k|\mathbf{x}_{\text{ANO}}^{(m )}-\mathbf{x}|)}{|\mathbf{x}_{\text{ANO}}^{(m)}-\mathbf{x}|}\cdot\mathbf{e}_{ 1}\\ i\left(\frac{\mathbf{x}_{\text{ANO}}^{(m)}-\mathbf{x}}{|\mathbf{x}_{\text{ANO}}^ {(m)}-\mathbf{x}|}\cdot\mathbf{e}_{2}\right)J_{1}(k|\mathbf{x}_{\text{ANO}}^ {(m)}-\mathbf{x}|)\end{array} \right]^{T},\]  
we can obtain  
\[\mathbf{W}(\mathbf{x})^{*}\mathbf{E}(\mathbf{x}_{\text{ANO}}^{(m)})\left[ \begin{array}{ccc}\frac{\mathbf{x}_{\text{ANO}}^{(m)}-\mathbf{e}_{0}}{\sqrt {\mathbf{e}_{0}\mu_{0}}}&0&0\\ 0&\frac{\mu_{0}}{\mu_{\text{A}}^{(m)}+\mu_{0}}&0\\ 0&0&\frac{\mu_{0}}{\mu_{\text{A}}^{(m)}+\mu_{0}}\end{array} \right]\approx \left[ \begin{array}{c}\frac{(\mathbf{x}_{\text{A}}^{(m)}-\mathbf{e}_{0}}{ \sqrt{\mathbf{e}_{0}\mu_{0}}})J_{0}(k|\mathbf{x}_{\text{ANO}}^{(m)}-\mathbf{ x}|)\\ i\left(\frac{\mu_{0}}{\mu_{\text{A}}^{(m)}+\mu_{0}}\right)\left(\frac{\mathbf{x}_{ \text{ANO}}^{(m)}-\mathbf{x}}{|\mathbf{x}_{\text{ANO}}^{(m)}-\mathbf{x}|} \cdot\mathbf{e}_{1}\right)J_{1}(k|\mathbf{x}_{\text{ANO}}^{(m)}-\mathbf{x}|)\\ i\left(\frac{\mu_{0}}{\mu_{\text{A}}^{(m)}+\mu_{0}}\right)\left(\frac{\mathbf{x}_{ \text{ANO}}^{(m)}-\mathbf{x}}{|\mathbf{x}_{\text{ANO}}^{(m)}-\mathbf{x}|} \cdot\mathbf{e}_{2}\right)J_{1}(k|\mathbf{x}_{\text{ANO}}^{(m)}-\mathbf{x}|) \end{array}