## 3 Semi-Analytical Echo Model of Airborne SAR Altimeters
### Analysis of the Semianalytical Echo Model  
The model aggregates the backscattered energies within the illuminated region to obtain a 2D FISR. The temporal dimension is determined by the propagating circle, while the frequency dimension is defined by the rectangular Doppler beam [22]. Each circle with a radius \(\rho(t)=\sqrt{\hbar ct}\) and each approximate rectangular Doppler beam divides the illuminated area into distinct delay/Doppler bins. Note that the flight path angle \(\mu\) introduces a Doppler frequency shift when the airborne platform flies obliquely (see Figure 3). As \(\mu\) increases, all Doppler beams exhibit a negative Doppler shift \(f_{d,shift}=-2v_{c}h\sin\mu/\lambda r_{xy}\). Additionally, the power within each delay/Doppler bin changes, deviating from a symmetric distribution. Due to disparities in offset range among different delay-Doppler bins after multilooking, the leading edge of the multilook waveforms is correspondingly affected, as shown in Figure 4.  
However, changes in the velocity direction of the airborne platform generally result in antenna mis-pointing angles. Although the mapping relationship is unchanged when considering \(\xi\) and \(\bar{\phi}\), the backscattered energy within the delay-Doppler bins is impacted. In the case where the antenna is vertically fixed to the aircraft, the two mis-pointing angles \(\xi\) and \(\bar{\phi}\) are dependent on the pitch angle \(\psi_{\mathrm{al}}\) and the roll angle \(\psi_{\mathrm{ac}}\) as follows:  
\[\begin{split}\psi_{\mathrm{al}}&=\arctan(\tan(\xi) \sin(\bar{\phi}))\\ \psi_{\mathrm{ac}}&=\arctan(\tan(\xi)\cos(\bar{\phi} )).\end{split} \tag{34}\]  
Figure 5 illustrates the variations in antenna gain across the various mis-pointing angles. It can be observed that the maximum antenna gain is achieved at the nadir point when the antenna mis-pointing angle is zero. As expected, the maximum antenna gain shifts along either the x or y axis as the antenna mis-pointing angle changes (see Figure 5c,d). The corresponding multilook echoes are significantly influenced by the variations in antenna gain. Next, we analyze the impact of antenna gain variation on multilook echoes.  
Figure 4: Effect of flight path angle on (**a**) the multilook echoes and (**b**) the normalized multilook echoes (Pu = 1, \(\tau\) = 30 gates, SWH = 2 m, and \(\psi_{al}=\psi_{ac}=0^{\mathrm{o}}\)).  
We can evaluate the effect of antenna mis-pointing angle on both the multilook and normalized multilook echoes. As shown in Figure 6, increasing the mis-pointing angle along the track leads to a decrease in the amplitude of the multiview echo. However, this has a minimal effect on the normalized multiview echo, indicating that the mis-pointing angle along the track primarily affects the echo's amplitude rather than its shape.  
In addition, Figure 7 demonstrates that the mis-pointing angle across the track has a more pronounced impact on the waveform's amplitude compared to the along-track mis-pointing angle, particularly affecting the trailing edge of the waveform. Note that the mis-pointing echoes obtained by the proposed model are influenced by six parameters \((P_{u},\text{SWH},\tau,\mu,\psi_{ac},\psi_{al})\). The waveform's shape is influenced by both the antenna mis-pointing and flight path angles, which should be considered in parameter estimation.  
Therefore, our focus is on analyzing the effect of the parameter vector \((\text{SWH},\tau,\mu,\psi_{ac},\psi_{al})\) on the normalized echo.