# Microwave Staring Correlated Imaging Based on Unsteady Aerostat Platform
## 2 MSCI Method Based on Unsteady Aerostat Platform
### Imaging Equation  
Since the whole imaging region \(S\) has been divided into \(M=P\times Q\) discrete grids. The scattering coefficient of the \(m\)-th grid is \(\sigma(\mathbf{r}_{m})\). At the beginning of the \(l\)-th pulse, each transmitter simultaneously transmits independent and stochastic signal. All signals are superimposed in \(S\) to generate TSSRF. The radiation field at \(\mathbf{r}_{m}\) can be expressed as \[E^{inc}(t_{l,}\mathbf{r}_{m})=\sum_{n=1}^{N}D_{l}\left(m\right)\frac{F_{n}\left( \mathbf{\hat{R}}_{n}\right)s_{n}(t_{l}-\left|\mathbf{r}_{m}-\mathbf{r}^{n}\left( t_{l,0}\right)\right|/c\left.\right)}{4\pi\left|\mathbf{r}_{m}-\mathbf{r}^{n} \left(t_{l,0}\right)\right|}, \tag{14}\]  
where \(\mathbf{\hat{R}}_{n}=[\mathbf{r}_{m}-\mathbf{r}^{n}\left(t_{l,0}\right)]/ \left|\mathbf{r}_{m}-\mathbf{r}^{n}\left(t_{l,0}\right)\right|\). \(F_{n}\left(\mathbf{\hat{R}}_{n}\right)\) denotes the radiation pattern of the \(n\)-th transmitter antenna. \(t_{l,0}=(l-1)T\) denotes the initial time of the \(l\)-th pulse. The radiation field interacts with the targets and the received echo can be expressed as  
\[E^{sca}(t_{l})=\sum_{m=1}^{M}\sigma(\mathbf{r}_{m})\frac{E^{inc}(t_{l}-\frac{ \left|\mathbf{r}^{\prime}(t_{l})\right|-\mathbf{r}_{m}}{c},\mathbf{r}_{m})}{4 \pi\left|\mathbf{r}^{s}\left(t_{l}\right)-\mathbf{r}_{m}\right|}F_{s}\left( \mathbf{\hat{R}}_{s}\right)+n\left(t_{l}\right), \tag{15}\]  
where \(\mathbf{\hat{R}}_{s}=[\mathbf{r}_{m}-\mathbf{r}^{s}\left(t_{l}\right)]/| \mathbf{r}_{m}-\mathbf{r}^{s}\left(t_{l}\right)|\). \(F_{s}\left(\mathbf{\hat{R}}_{s}\right)\) denotes the radiation pattern of the receiver antenna. \(n\left(t_{l}\right)\) denotes the additive noise. Considering the round-trip propagation of the electromagnetic field in the free space, the modified radiation field is defined as  
\[E^{rad}(t_{l},\mathbf{r}_{m})=\sum_{n=1}^{N}\left\{\frac{F_{s}\left(\mathbf{ \hat{R}}_{s}\right)F_{n}\left(\mathbf{\hat{R}}_{n}\right)s_{n}\left[t_{l}- \left(\left|\mathbf{r}_{m}-\mathbf{r}^{n}\left(t_{l,0}\right)\right|+\left| \mathbf{r}^{s}\left(t_{l}\right)-\mathbf{r}_{m}\right|\right)/c\left.\right] }{\left(4\pi\right)^{2}\left|\mathbf{r}_{m}-\mathbf{r}^{n}\left(t_{l,0}\right) \right|\left|\mathbf{r}^{s}\left(t_{l}\right)-\mathbf{r}_{m}\right|}D_{l} \left(m\right)\right\}. \tag{16}\]  
Let \(t_{l,k}\), \(l=1,2,\cdots,L\) be the sampling time in the \(l\)-th pulse, thus the imaging equation in the matrix vector form can be written as  
\[\mathbf{E}^{sca}=\mathbf{E}^{rad}\cdot\boldsymbol{\sigma}+\mathbf{n}, \tag{17}\]  
where \(\mathbf{E}^{sca}=[E^{sca}(t_{1,k}),E^{sca}(t_{2,k}),\cdots,E^{sca}(t_{L,k})]^ {T}\) is the echo vector, \(\boldsymbol{\sigma}=[\sigma\left(\mathbf{r}_{1}\right),\sigma\left(\mathbf{r}_ {2}\right),\cdots,\sigma\left(\mathbf{r}_{M}\right)]^{T}\) is the scattering coefficient vector, \(\mathbf{n}=[n\left(t_{1,k}\right),n\left(t_{2,k}\right),\cdots n\left(t_{L,k} \right)]^{T}\) is the noise vector, \(\mathbf{E}^{rad}\) is the modified radiation field matrix with \(\left[\mathbf{E}^{rad}\right]_{lm}=E^{rad}\left(t_{l,k},\mathbf{r}_{m}\right)\). The scattering coefficient vector \(\boldsymbol{\sigma}\) can be reconstructed by the correlated processing between \(\mathbf{E}^{sca}\) and \(\mathbf{E}^{rad}\), which can be described as  
\[\boldsymbol{\hat{\sigma}}=\zeta\left[\mathbf{E}^{rad},\mathbf{E}^{sca}\right], \tag{18}\]  
where \(\zeta\) denote the correlated operator. Common correlated imaging algorithms include Pseudo-Inverse algorithm, Tikhonov regularization, TV regularization, and sparse reconstruction algorithms, such as Orthogonal Matching Pursuit, sparse Bayesian learning, etc. This paper adopts Tikhonov regularization algorithm because it is robust to noise and does not require a priori of the target.