## 3 Problem Statements and _A Priori_ Knowledge Integration  
### Moving Target Steering Vector Mismatch  
As shown in Equation (6), the adaptive clutter cancellation processor is to estimate the moving target steering vector \(\stackrel{{\rightharpoonup}}{{a}}(\psi)\) using the scanning method in the presence of clutter and noise. The goal, of course, is to search the desired steering vector quickly with a high degree of reliability. In reality, the estimation might be obscured by high clutter, noise, and a distorted beampattern, which is always called moving target steering vector mismatch.  
The total available output \(\mathcal{S}\mathcal{N}\)R for the adaptive clutter cancellation can be denoted as [15; 22],  
\[\mathcal{S}\mathcal{N}\mathcal{N}\mathcal{N}\mathcal{N}\mathcal{N}\mathcal{N }\mathcal{N}\mathcal{N}\mathcal{N}\After SAR processing, the definition of \(SCNR\) is somewhat different from the usual definition in STAP (space time adaptive processing) [27]. The \(SCNR\) can be defined as,  
\[SCNR=\frac{(N_{r}N_{a})\frac{P_{T}G_{T}G_{R}\lambda^{2}\sigma_{s}}{(4\pi)^{3}t_{ t}^{4}}}{(N_{r}N_{a})\frac{P_{T}G_{T}G_{R}\lambda^{2}\sigma_{s}^{0}\rho_{a} \rho_{b}\rho_{b}}{(4\pi)^{3}t_{t}^{4}}+K_{B}TB} \tag{9}\]  
where \(\sigma_{s}\) is the radar cross-section (RCS) of the moving targets. \(\sigma_{c}^{0}\) is the mean RCS of the stationary terrain. \(\rho_{a}\) and \(\rho_{r}\) are the SAR azimuth and range resolution. \(r_{t}\) is the target's slant range distance. \(\lambda\) is the radar wavelength. \(P_{T}\) is the radar transmitting power. \(K_{B}\) is the Boltzmann constant. \(G_{T},G_{R}\) are the radar transmitter and receiver antennas gain. \(T\) is the radar receiver noise temperature. \(B\) and \(T\) are the radar receiver bandwidth temperature. \(N_{r}\), \(N_{a}\) are the SAR processing of range samples and azimuth samples.  
In practice, the detectable output \(SCNR\) is sensitive to mismatches between the searched and actual target steering vectors. We define \(\theta\) as the mismatch angle between the searched and actual target steering vector \(\stackrel{{\triangle}}{{a}}\), \(\stackrel{{\rightharpoonup}}{{a}}\)[15; 22; 23; 28; 29; 30].  
\[\cos^{2}(\theta)=\frac{\left|\stackrel{{\triangle}}{{a}}(\psi)^{ H}R^{-1}\stackrel{{\rightharpoonup}}{{a}}(\psi)\right|^{2}}{ \left(\stackrel{{\rightharpoonup}}{{a}}(\psi)^{H}R^{-1} \stackrel{{\rightharpoonup}}{{a}}(\psi)\right)\left(\stackrel{{ \rightharpoonup}}{{a}}(\psi)^{H}R^{-1}\stackrel{{ \rightharpoonup}}{{a}}(\psi)\right)} \tag{10}\]  
The available output \(SCNR\) for target steering vector mismatch can be shown as [15; 22; 28; 29; 30],  
\[SCNR_{out}=\beta^{2}\cdot\stackrel{{\rightharpoonup}}{{a}}(\psi) ^{H}R^{-1}\stackrel{{\rightharpoonup}}{{a}}(\psi)\cdot\cos^{2}( \theta)0\leq\cos^{2}\theta\leq 1 \tag{11}\]  
Note that \(\cos^{2}(\theta)\) partially captures the effect of steering vector mismatch, where \(\stackrel{{\rightharpoonup}}{{a}}\) and \(\stackrel{{\rightharpoonup}}{{a}}\) point in different directions. \(\cos^{2}(\theta)=0\) represents that the estimated steering vectors are orthogonal to the real steering vectors. Similarly, \(\cos^{2}(\theta)=1\) represents that the estimated steering vectors are matched to the real steering vectors. If the filter is steered in the wrong direction, only a fraction of the total output \(SCNR\) is usable. The adaptive search of the target steering vector is a time-consuming process, and the resolution is limited by the sample number, number of satellites, and the baselines' length. Except for some traditional causes, the presence of partially correlated speckle and additive noise has the effect of increasing the variance of the target steering vector.