# Kirchhoff Migration for Identifying Unknown Targets Surrounded by Random Scatterers
## 2 Two-Dimensional Direct Scattering Problems  
In this section, we briefly survey two-dimensional direct scattering problems and introduce an asymptotic expansion formula that holds true in the presence of small anomalies. For a more detailed discussion of this topic, we recommend [51]. Let \(\mathbf{A}_{m}\) be anomalies with small diameters \(\alpha_{m}\), \(m=1,2,\cdots,M\), expressed as  
\[\mathbf{A}_{m}=\mathbf{x}_{\mathrm{ANO}}^{(m)}+\alpha_{m}\mathbf{B}_{m},\]  
where \(\mathbf{x}_{\mathrm{ANO}}^{(m)}\) denotes the location of \(\mathbf{A}_{m}\) and the \(\mathbf{B}_{m}\) are simply connected smooth domains containing the origin. For simplicity, we let \(\Lambda\) be the set of all \(\mathbf{A}_{m}\). Analogously, let \(\mathbf{R}_{s}\) be random scatterers with small radii \(\sigma_{s},s=1,2,\cdots,S\), which is given by  
\[\mathbf{R}_{s}=\mathbf{x}_{\mathrm{RND}}^{(s)}+\sigma_{s}\mathbf{B}_{s},\]  
where \(\mathbf{x}_{\mathrm{RND}}^{(s)}\) denotes the location of \(\mathbf{R}_{s}\) and \(\mathrm{Y}\) is the set of all \(\mathbf{R}_{s}\). In this paper, we assume that the \(\mathbf{A}_{m}\) and \(\mathbf{R}_{s}\) are characterized by their dielectric permittivities and magnetic permeabilities at a given positive angular frequency \(\omega=2\pi f\). For simplicity, we define the piecewise-constant electric permittivity \(\varepsilon(\mathbf{x})\) and magnetic permeability \(\mu(\mathbf{x})\) such that  
\[\varepsilon(\mathbf{x})=\left\{ \begin{array}{ll}\varepsilon_{\mathbf{A}}^{ (m)}&\text{for}\quad\mathbf{x}\in\mathbf{A}_{m}\\ \varepsilon_{\mathbf{R}}^{(s)}&\text{for}\quad\mathbf{x}\in\mathbf{R}_{s}\\ \varepsilon_{0}&\text{for}\quad\mathbf{x}\in\mathbb{R}^{2}\backslash(\overline {\Lambda}\cup\overline{\mathrm{Y}})\end{array} \right.\quad\text{and}\quad \mu(\mathbf{x})=\left\{ \begin{array}{ll}\mu_{\mathbf{A}}^{(m)}&\text{for} \quad\mathbf{x}\in\mathbf{A}_{m}\\ \mu_{\mathbf{R}}^{(s)}&\text{for}\quad\mathbf{x}\in\mathbf{R}_{s}\\ \mu_{0}&\text{for}\quad\mathbf{x}\in\mathbb{R}^{2}\backslash(\overline{ \Lambda}\cup\overline{\mathrm{Y}}),\end{array} \right.\]  
respectively. With this, we set the wavenumber \(k=\omega\sqrt{\varepsilon_{0}\mu_{0}}=2\pi/\lambda\), where \(\lambda\) denotes the wavelength that satisfies \(\lambda\gg\alpha_{m}\) and \(\lambda\gg\sigma_{s}\) for all \(m\) and \(s\). Throughout this paper, we consider plane-wave illumination. For a given fixed wavenumber \(k\), \(u_{\mathrm{inc}}(\mathbf{x},\boldsymbol{\theta})=e^{ik\boldsymbol{\theta} \cdot\mathbf{x}}\) denotes a plane-wave incident field with propagation direction \(\boldsymbol{\theta}\in\mathbb{S}^{1}\), where \(\mathbb{S}^{1}\) denotes the two-dimensional unit circle. Let \(u(\mathbf{x},\boldsymbol{\theta})\) be the time-harmonic total field that satisfies the Helmholtz equation  
\[\
abla\cdot\left(\frac{1}{\mu(\mathbf{x})}\
abla u(\mathbf{x},\boldsymbol{ \theta})\right)+\omega^{2}\varepsilon(\mathbf{x})u(\mathbf{x},\boldsymbol{ \theta})=0\]with transmission conditions on the boundaries of \(\mathbf{A}_{m}\) and \(\mathbf{R}_{s}\). With this, we let \(u_{\text{scat}}(\mathbf{x},\boldsymbol{\theta})=u(\mathbf{x},\boldsymbol{\theta} )-u_{\text{inc}}(\mathbf{x},\boldsymbol{\theta})\) be the scattered field that satisfies the Sommerfeld radiation condition  
\[\lim_{|\mathbf{x}|\to\infty}\sqrt{|\mathbf{x}|}\left(\frac{\partial u_{\text{ scat}}(\mathbf{x},\boldsymbol{\theta})}{\partial|\mathbf{x}|}-iku_{\text{scat}}( \mathbf{x},\boldsymbol{\theta})\right)=0\]  
uniformly in all directions \(\boldsymbol{\vartheta}=\mathbf{x}/|\mathbf{x}|\in\mathbb{S}^{1}\).