## 2 Direct Scattering Problem and Subspace Migration  
### Two-Dimensional Direct Scattering Problem and Far-Field Pattern  
Here, we briefly introduce a basic concept of the two-dimensional direct scattering problem in the presence of a set of small scatterers and the imaging function of subspace migration. Throughout this paper, we denote \(\Gamma_{m}\), \(m=1,2,\cdots,M\), being a small scatterer embedded in the two-dimensional homogeneous space \(\Omega\subset\mathbb{R}^{2}\) and \(\Gamma\) being the collection of \(\Gamma_{m}\). For this purpose, we assume that every \(\Gamma_{m}\) is a small ball with radius \(\alpha_{m}\) and location \(\mathbf{y}_{m}\).  
Here, we assume that \(\Gamma_{m}\) and \(\Omega\) are completely characterized by their dielectric permittivity at a given angular frequency \(\omega=2\pi f\), where \(f\) denotes the ordinary frequency. Let \(0<\varepsilon_{\mathrm{b}}<\infty\) and \(0<\varepsilon_{m}<\infty\) denote the permittivity of the \(\Omega\) and \(\Gamma_{m}\), respectively. Correspondingly, we can introduce the following piecewise constant of permittivity  
\[\varepsilon(\mathbf{x})=\left\{\begin{array}{ll}\varepsilon_{m}&\text{for} \quad\mathbf{x}\in\Gamma_{m}\\ \varepsilon_{\mathrm{b}}&\text{for}\quad\mathbf{x}\in\Omega\backslash\Gamma \end{array}\right.\]  
With this, we denote \(k=\omega\sqrt{\varepsilon_{\mathrm{b}}l_{\mathrm{b}}}\) as the background wavenumber. Here, \(\mu_{\mathrm{b}}\) is the value of magnetic permeability.  
Here, we consider the plane-wave illumination: \(u_{\mathrm{inc}}(\mathbf{x},\boldsymbol{\theta})=e^{ik\boldsymbol{\theta} \cdot\mathbf{x}}\) as the given incident field with the propagation direction \(\boldsymbol{\theta}\in\mathbb{S}^{1}\) that satisfies the following Helmholtz equation  
\[\
abla\cdot\
abla u_{\mathrm{inc}}(\mathbf{x},\boldsymbol{\theta})+k^{2}u_{ \mathrm{inc}}(\mathbf{x},\boldsymbol{\theta})=0.\]  
Here, \(\mathbb{S}^{d-1}\) denotes the unit circle centered at the origin in \(\mathbb{R}^{d}\). With this, we denote \(u(\mathbf{x},\boldsymbol{\theta})\) as the time-harmonic total field which satisfies the Helmholtz equation  
\[\frac{1}{\mu_{\mathrm{b}}}\
abla\cdot\
abla u(\mathbf{x},\boldsymbol{\theta}) +\omega^{2}\varepsilon(\mathbf{x})u(\mathbf{x},\boldsymbol{\theta})=0\]  
with transmission condition on the boundaries of \(\Gamma_{m}\), \(m=1,2,\cdots,M\). Usually, the \(u(\mathbf{x},\boldsymbol{\theta})\) can be split into the incident field \(u_{\mathrm{inc}}(\mathbf{x},\boldsymbol{\theta})\) and the scattered field \(u_{\mathrm{scat}}(\mathbf{x},\boldsymbol{\theta})\), which satisfies the Sommerfeld radiation condition \[\lim_{|\mathbf{x}|\rightarrow\infty}\sqrt{|\mathbf{x}|}\left(\frac{\partial u_{ \text{scat}}(\mathbf{x},\boldsymbol{\theta})}{\partial|\mathbf{x}|}-iku_{\text{ scat}}(\mathbf{x},\boldsymbol{\theta})\right)=0\quad\text{uniformly in all directions}\quad\boldsymbol{\theta}=\frac{\mathbf{x}}{|\mathbf{x}|}.\]  
Let \(u_{\infty}(\boldsymbol{\theta},\boldsymbol{\theta})\) be the far-field pattern corresponding to the scattered field \(u_{\text{scat}}(\mathbf{x},\boldsymbol{\theta})\) with observation direction \(\boldsymbol{\theta}=\mathbf{x}/|\mathbf{x}|\in\mathbb{S}^{1}\) that satisfies  
\[u_{\text{scat}}(\mathbf{x},\boldsymbol{\theta})=\frac{e^{ik|\mathbf{x}|}}{ \sqrt{|\mathbf{x}|}}u_{\infty}(\boldsymbol{\theta},\boldsymbol{\theta})+O \left(\frac{1}{\sqrt{|\mathbf{x}|}}\right)\quad\text{uniformly in all directions}\quad\boldsymbol{\theta},\quad|\mathbf{x}|\longrightarrow\infty.\]  
Based on [47], the far-field pattern \(u_{\infty}(\boldsymbol{\theta},\boldsymbol{\theta})\) can be represented as an asymptotic expansion formula  
\[u_{\infty}(\boldsymbol{\theta},\boldsymbol{\theta})\approx\frac{k^{2}\pi(1+i) }{4\sqrt{k\pi}}\sum_{m=1}^{M}\alpha_{m}^{2}\bigg{(}\frac{\varepsilon_{m}- \varepsilon_{\text{b}}}{\sqrt{\varepsilon_{\text{b}}\mu_{\text{b}}}}\bigg{)} e^{-ik(\boldsymbol{\theta}-\boldsymbol{\theta})\cdot\mathbf{y}_{m}}. \tag{1}\]