# Kirchhoff Migration for Identifying Unknown Targets Surrounded by Random Scatterers
## 3 Structure and Properties of Kirchhoff Migration
### Various Properties of Kirchhoff Migration  
Identified structure (6) allows us to examine how the locations of the anomalies \(\mathbf{A}_{m}\) are identified using Kirchhoff migration. It is worth mentioning that unlike to the several researches, the applied wave number \(k\), i.e., angular frequency \(\omega\) must be sufficiently large: for small values of \(k\), the results in Theorem 1 no longer hold and cannot be used to identify \(\mathbf{A}_{m}\), we refer to (Figure 7, [49]) for an example of low-frequency imaging. In the following discussion, we consider following three possible cases. **Case 1**. : _Assume that \(\alpha_{m}\gg\sigma_{s}\,\varepsilon_{\mathbf{A}}^{(m)}\gg\varepsilon_{\mathbf{R}}^ {(s)}\), and \(\mu_{\mathbf{A}}^{(m)}\ll\mu_{\mathbf{R}}^{(s)}\) for all m and s. Then, the terms associated with \(\mathbf{x}_{\mathrm{RND}}^{(s)}\) are dominated by the ones associated with \(\mathbf{x}_{\mathrm{ANO}}^{(m)}\). Thus, \(\mathfrak{F}_{\mathrm{KIR}}(\mathbf{x})\) will become_  
\[\mathfrak{F}_{\mathrm{KIR}}(\mathbf{x})\approx N\left|\sum_{m=1}^ {M}\alpha_{m}^{2}|\mathbf{B}_{m}|\left(\frac{\varepsilon_{\mathbf{A}}^{(m)}- \varepsilon_{0}}{\sqrt{\varepsilon_{0}\mu_{0}}}\right)J_{0}(k|\mathbf{x}_{ \mathrm{ANO}}^{(m)}-\mathbf{x}|)^{2}\right.\] \[-\left.\sum_{m=1}^{M}\alpha_{m}^{2}|\mathbf{B}_{m}|\left(\frac{ \mu_{0}}{\mu_{\mathbf{A}}^{(m)}+\mu_{0}}\right)\left\{\left(\frac{\mathbf{x} _{\mathrm{ANO}}^{(m)}-\mathbf{x}}{|\mathbf{x}_{\mathrm{ANO}}^{(m)}-\mathbf{x }|}\cdot\mathbf{e}_{1}\right)^{2}+\left(\frac{\mathbf{x}_{\mathrm{ANO}}^{(m)} -\mathbf{x}}{|\mathbf{x}_{\mathrm{ANO}}^{(m)}-\mathbf{x}|}\cdot\mathbf{e}_{2} \right)^{2}\right\}J_{1}(k|\mathbf{x}_{\mathrm{ANO}}^{(m)}-\mathbf{x}|)^{2} \right|.\]  
_On the basis of [49], the first \(M\) singular vectors, \(\mathbf{U}_{1},\mathbf{U}_{2},\cdots,\mathbf{U}_{M}\), will be associated with the vectors \(\mathbf{W}(\mathbf{x}_{\mathrm{ANO}}^{(m)})\), \(m=1,2,\cdots,M\), and remaining singular vectors will be associated with the vectors \(\mathbf{W}(\mathbf{x}_{\mathrm{RND}}^{(s)})\), \(s=1,2,\cdots,S\). Thus, the singular vectors will satisfy_  
\[\tau_{1}\geq\tau_{2}\geq\cdots\geq\tau_{M}\gg\tau_{M+1}\geq\tau_{M+2}\geq \cdots\geq\tau_{M+S}\geq\cdots\geq\tau_{N}\]  
_so that it will be possible to discriminate the first \(M\) singular values. In this case, the imaging function of subspace migration \(\mathfrak{F}_{\mathrm{SUB}}(\mathbf{x})\) becomes (see [52] for instance)_  
\[\mathfrak{F}_{\mathrm{SUB}}(\mathbf{x}) =\left|\sum_{m=1}^{3M}\langle\mathbf{W}(\mathbf{x}),\mathbf{U}_{ n}\rangle\langle\mathbf{W}(\mathbf{x}),\overline{\mathbf{\
abla}}_{n}\rangle\right| \tag{7}\] \[\approx\left|\sum_{m=1}^{M}\left\{I_{0}(k|\mathbf{x}_{\mathrm{ANO }}^{(m)}-\mathbf{x}|)^{2}-\left[\left(\frac{\mathbf{x}_{\mathrm{ANO}}^{(m)}- \mathbf{x}}{|\mathbf{x}_{\mathrm{ANO}}^{(m)}-\mathbf{x}|}\cdot\mathbf{e}_{1} \right)^{2}+\left(\frac{\mathbf{x}_{\mathrm{ANO}}^{(m)}-\mathbf{x}}{|\mathbf{ x}_{\mathrm{ANO}}^{(m)}-\mathbf{x}|}\cdot\mathbf{e}_{2}\right)^{2}\right]J_{1}(k| \mathbf{x}_{\mathrm{ANO}}^{(m)}-\mathbf{x}|)^{2}\right\}\right|.\]  
_Hence, subspace migration will provide better results to the Kirchhoff migration. However, if the wrong singular values are selected, the subspace migration results will actually be worse than those of the Kirchhoff migration._  
**Case 2**. : _Assume that \(\alpha_{m}\geq\sigma_{s}\,\varepsilon_{\mathbf{A}}^{(m)}\geq\varepsilon_{ \mathbf{R}}^{(s)}\), and \(\mu_{\mathbf{A}}^{(m)}\leq\mu_{\mathbf{R}}^{(s)}\) for all m and s. Then, even though the first \(M\) singular vectors, \(\mathbf{U}_{1},\mathbf{U}_{2},\cdots,\mathbf{U}_{M}\) will be associated with the vectors \(\mathbf{W}(\mathbf{x}_{\mathrm{ANO}}^{(m)})\), \(m=1,2,\cdots,M\), it will still be very hard to discriminate the first \(M\) singular values.