## 3 Key Variables in the Analytical Approximation Model
### The Acceleration Vector Term in the Approximate Doppler Rate  
With the expression \(\sigma\left[\Delta\
u\right]\), the standard deviation of the difference between \(\
u\) calculated from state vector measurements and its true value is addressed, and the acceleration vector term can now can be dealt with. As \(\mathbf{R}_{st}\) needs to be analyzed, the traditional yaw steering method [25] is applied here.  
\[\theta_{yaw}=\arctan\left[\frac{\sin i}{N-\cos i}\cos\left(\
u+\omega\right)\right] \tag{30}\]  
where \(N\) is the number of revolutions per day.  
In this approximation model, we replace \(\Delta\mathbf{A}_{st}\) with \(\Delta\mathbf{A}_{s}\) only and ignore the errors in the target acceleration vector while keeping the satellite's part. In the meantime, \(\
u\) is replaced by \(\
u+\Delta\
u\) in the row vector of Equation (14). We can combine Equations (14), (18) and (20) with  
\[\hat{\mathbf{u}}_{a,ECI}=\begin{bmatrix}-\sin\
u&-\cos\
u&0\\ \cos\
u&-\sin\
u&0\\ 0&0&1\end{bmatrix}\begin{bmatrix}\cos\theta_{yaw}&0&-\sin\theta_{yaw}\\ 0&1&0\\ \sin\theta_{yaw}&0&\cos\theta_{yaw}\end{bmatrix}\begin{bmatrix}1&0&0\\ 0&\cos\theta_{L}&\sin\theta_{L}\\ 0&-\sin\theta_{L}&\cos\theta_{L}\end{bmatrix}\begin{bmatrix}0\\ 1\\ 0\end{bmatrix} \tag{31}\]  
where \(\theta_{L}\) is the center off-nadir angle in stripmap SAR. The acceleration vector term results in a Doppler rate difference of \[\Delta f_{r,a} =\frac{2}{\lambda\rho}\Delta\mathbf{A}_{st}\cdot\mathbf{R}_{st}\] \[=k_{a}\left[\cos\theta_{L}+\sqrt{\sin^{2}\theta_{L}\sin^{2}\theta_{ yaw}+\cos^{2}\theta_{L}}\sin\left(\Delta\
u-\arctan\frac{\cos\theta_{L}}{\sin \theta_{L}\sin\theta_{yaw}}\right)\right] \tag{32}\]  
in which  
\[k_{a}=\frac{2\mu\left(1+e\cos\
u\right)^{2}}{\lambda p^{2}} \tag{33}\]  
In common SAR operations, \(\theta_{yaw}\) is usually small. Then, we have the following expansion without cubic and higher-order terms.  
\[\arctan\frac{\cos\theta_{L}}{\sin\theta_{L}\sin\theta_{yaw}} =\frac{\pi}{2}-\arctan\frac{\sin\theta_{L}\sin\theta_{yaw}}{\cos \theta_{L}}\] \[\approx\frac{\pi}{2}-\tan\theta_{L}\sin\theta_{yaw} \tag{34}\]  
With the result from Appendix B and Equations (32)-(34), the expected value and standard deviation of \(\Delta f_{r,a}\) are  
\[\mathbb{E}\left[\Delta f_{r,a}\right]=k_{a}\cos\theta_{L}+k_{a}\sqrt{\sin^{2} \theta_{L}\sin^{2}\theta_{yaw}+\cos^{2}\theta_{L}}\left(1-\frac{\text{Var} \left[\Delta\
u\right]}{2}\right)\left(\frac{1}{2}\tan^{2}\theta_{L}\sin^{2} \theta_{yaw}-1\right) \tag{35}\]  
\[\sigma\left[\Delta f_{r,a}\right]=k_{a}\sqrt{\sin^{2}\theta_{L}\sin^{2}\theta_ {yaw}+\cos^{2}\theta_{L}}\sqrt{\text{Var}\left[\Delta\
u\right]\left(\tan^{2} \theta_{L}\sin^{2}\theta_{yaw}+\frac{\text{Var}\left[\Delta\
u\right]}{2} \right)} \tag{36}\]  
Equation (35) shows an interesting result. Instead of the expected zero value, we have \(\mathbb{E}\left[\Delta f_{r,a}\right]\), which means there will be bias in the expected value of the Doppler rate \(f_{r}\) due to the calculated true anomaly. This bias can be and should be removed when calculating the Doppler parameters for the imaging algorithm.  
However, we need the approximate maximum value of \(\sigma\left[\Delta f_{r,a}\right]\). If we put aside the quadratic term of \(\text{Var}\left[\Delta\
u\right]\) in Equation (36), the leftover part is modulated by \(\text{Var}\left[\Delta\
u\right]\) and \(\sin^{2}\theta_{yaw}\) at each \(\
u\) point. When \(x\) is relatively small, \(\arctan x\propto x\). Combined with Equation (30), we can ascertain that \(\sin^{2}\theta_{yaw}\propto\sin^{2}\left[\cos\left(\
u+\omega\right)\right] \propto\cos^{2}\left(\
u+\omega\right)\).  
and \(\theta_{yaw}=\theta_{yaw,max}\) in Equation (36). Thus, we have  
\[\sigma\left[\Delta f_{r,a}\right]_{max} =k_{a,max}\sqrt{\sin^{2}\theta_{L}\sin^{2}\theta_{yaw,max}+\cos^{2} \theta_{L}}\] \[\cdot\sqrt{\text{Var}\left[\Delta v\right]_{\text{max}}\left(\tan^ {2}\theta_{L}\sin^{2}\theta_{yaw,max}+\frac{\text{Var}\left[\Delta v\right]_{ \text{max}}}{2}\right)} \tag{40}\]