# Kirchhoff Migration for Identifying Unknown Targets Surrounded by Random Scatterers
## 4 Simulation Results  
In order to validate the results derived from Theorem 1, we now present a set of simulation results. In this section, we only consider the dielectric permittivity contrast case. For the simulation, \(M=3\) small anomalies, all with the same permittivity \(\varepsilon_{\mathbf{A}}^{(m)}\equiv 5\), permeability \(\mu_{\mathbf{A}}^{(m)}=\mu_{\mathbf{R}}^{(s)}=1\), and radius \(\alpha_{m}\equiv 0.1\) were placed at the following locations: \(\mathbf{x}_{\mathrm{ANO}}^{(1)}=[0.25,0]^{T}\), \(\mathbf{x}_{\mathrm{ANO}}^{(2)}=[-0.4,0.5]^{T}\), and \(\mathbf{x}_{\mathrm{ANO}}^{(3)}=[-0.3,-0.7]^{T}\). In addition, \(S=100\) small scatterers were randomly distributed over \(\Omega=[-1,1]\times[-1,1]\subset\mathbb{R}^{2}\) such that  
\[\mathbf{x}_{\mathrm{RND}}^{(s)}=[\eta_{1}(-1,1),\eta_{2}(-1,1)]^{T},\]  
where the \(\eta_{p}(a,b)\), \(p=1,2\), are arbitrary real values between \(a\) and \(b\). Figure 1 shows the distribution of the three anomalies \(\mathbf{A}_{m}\) and random scatterers \(\mathbf{R}_{s}\). The data \(u_{\infty}(\boldsymbol{\vartheta}_{j},\boldsymbol{\theta}_{l})\) for the MSR matrix \(\mathbb{K}\) was generated by solving the Foldy-Lax formulation to avoid an _inverse crime_ (see [53] for instance). We used a total of \(N=64\) incident and observation directions and wavelengths of \(\lambda=0.7\) and \(\lambda=0.2\) as low and high frequencies, respectively. After obtaining the far-field data, 20 dB Gaussian random noise is added to the unperturbed data through the MATLAB command awgn included in the signal processing package. **Example 1** (Case 1). : _In this Example, we consider the imaging results for examining Case 1. For this, we set the permittivities and radii of the random scatterers as \(\varepsilon_{\mathbf{R}}^{(s)}=\eta_{3}(2,3)\) and \(\sigma_{\mathrm{s}}=\eta_{5}(0.03,0.06)\). Figure 2 shows the distribution of \(\mathbb{K}\)'s normalized singular values and maps of \(\mathfrak{F}_{\mathrm{SUB}}(\mathbf{x})\) and \(\mathfrak{F}_{\mathrm{KIK}}(\mathbf{x})\) for both wavelengths. Note that although the \(3\) singular values were successfully discriminated, it is very hard to identify the anomaly locations from the \(\mathfrak{F}_{\mathrm{SUB}}(\mathbf{x})\) map for the low applied frequency (\(\lambda=0.7\)) due to the presence of huge numbers of artifacts. Fortunately, the three anomalies could be extracted satisfactorily from the \(\mathfrak{F}_{\mathrm{KIK}}(\mathbf{x})\) map at this frequency. At the higher frequency (\(\lambda=0.2\)), selecting the first \(3\) singular values allows the anomaly locations to be identified more clearly from the \(\mathfrak{F}_{\mathrm{SUB}}(\mathbf{x})\) map, supporting the discussion of Case 1. That said, the anomaly locations can also be extracted via Kirchhoff migration at this frequency._  
Figure 1: Distribution of anomalies \(\mathbf{A}_{m}\) (red dots) and random scatterers \(\mathbf{R}_{s}\) (blue ‘\(\times\)’ symbols). **Example 2** (Case 2). : _In this Example, we consider the imaging results for Case 2. For this, we set the permittivities and radii of the random scatterers as \(\varepsilon_{\mathbf{R}}^{(s)}=\eta_{6}(3.5,4.5)\) and \(\sigma_{s}=\eta_{8}(0.05,0.1)\). Based on the Figure 3, it is far from clear that the first \(3\) singular values can be identified as the non-zero values of \(\mathbb{K}\) so the exact threshold of the non-zero singular values cannot be determined. Hence, the \(\mathfrak{F}_{\mathrm{SUB}}(\mathbf{x})\) imaging results are poor at the lower frequency. Although there are some artifacts in the \(\mathfrak{F}_{\mathrm{KIR}}(\mathbf{x})\) map, all the anomaly locations can be accurately identified. In contrast, at the higher frequency, the \(\mathfrak{F}_{\mathrm{SUB}}(\mathbf{x})\) map allows all the anomaly locations to be retrieved very accurately although one random scatterer, located at \([-0.58,-0.42]^{\mathrm{T}}\), is also identified. The permittivity and radius of this random scatterer were \(2.5596\) and \(0.0998\), respectively.