## 6 Conclusions  
In order to perform complex navigation tasks, high-precision altimetry on highly mobility platforms is attracting growing attention. However, the waveform re-tracking methods proposed in previous works cannot be applied to highly mobility platforms, which may have a large antenna mis-pointing angle and significant vertical movement. To improve the estimation accuracy of waveform re-tracking, in this paper we propose a novel semi-analytical waveform model and signal processing method for SAR altimeters with vertical movement and large antenna mis-pointing angles. Our main conclusions are as follows:  
1. An analytical expression that considers mis-pointing angles, circular antenna patterns, and vertical velocity is introduced for the echo model. The proposed model shows that the across-track mis-pointing angle affects the shape and the amplitude of the altimetric echo, whereas the along-track mis-pointing angle mainly affects the amplitude of the echo. When we consider \(\xi\leq 20^{\circ}\) (\(\xi<\theta_{\rm 3 dB}/2\)), the expansion order \(m=4\) is adequate to achieve the required level of error, which means that the global NQE is less than \(10^{-10}\).
2. A novel delay compensation method based on sinc interpolation is proposed to obtain the multilook echo in order to optimally handle non-integer delays and maintain the signal frequency characteristics. Compared with the generalized model, the proposed semi-analytic model is closer to the fully numerical waveform at different values of mis-pointing angle \(\psi_{ac}\) and \(\psi_{al}\), especially when the roll angle \(\psi_{ac}\) is large.  
Figure 14: RMSE of (**a**) \(\tau\) and (**b**) SWH versus \(\psi_{al}\) for the G-PRA and PRA algorithms (Pu = 1, \(\tau\) = 30 gates, SWH = 2 m, \(\psi_{ac}=0^{0}\)).  
Figure 15: RMSE of (**a**) \(\tau\) and (**b**) SWH versus \(\psi_{ac}\) for G-PRA and PRA algorithms (Pu = 1, \(\tau\) = 30 gates, SWH = 2 m, \(\psi_{al}=0^{0}\)).  
3. A five-parameter estimation strategy using the least squares procedure is proposed for SAR altimeters with vertical movement. The performance of the proposed model is evaluated using simulated data for typical airborne scenarios, analyzing the influence of different mis-pointing angles and flight path angle on parameter estimation. When the mis-pointing angles are within 10 degrees, the RMSE of \(\tau\) obtained by the re-tracking method fitted by the proposed model is less than 0.2 m, and increases more slowly compared to the one fitted by the generalized model.  
In February 2024, a more in-depth airborne radar altimeter experiment will be conducted in southern China. The model and re-tracking methods proposed in this paper will be validated against real data collected during this experiment. In addition, future research will focus on exploring the optimal performance of the estimation strategies under various antenna mis-pointing angles; it is worth considering a further modification to the strategy to involve fewer parameters in iterative updates.  
Conceptualization, Q.W. and W.J.; methodology, Q.W. and X.L.; software, Q.W. and X.L.; validation, Q.W., W.J. and X.L.; formal analysis, Q.W.; investigation, Q.W. and X.L.; resources, G.J.; data curation, X.L.; writing--original draft preparation, Q.W.; writing--review and editing, X.L., B. H. and W.J.; visualization, Q.W. and B.H.; supervision, G.J.; project administration, W.J.; funding acquisition, W.J. and G.J. All authors have read and agreed to the published version of the manuscript.  
This research was funded by the Fund of National Key Lab of Microwave Imaging Technology, Beijing 100190, China.  
Not applicable.  
Data Availability Statement: Data are contained within the article.  
The authors declare no conflict of interest.