## 1 Introduction
If SAR altimetry adheres to the same five assumptions as [PERSON]'s model, the convolution model of traditional radar altimeters can be directly extended to it [21]. In analytical modeling of SAR altimeters, the sinc function obtained by coherent along-track accumulation is generally regarded as the sub-beam pattern after beam sharpening. Then, the sub-beam pattern is multiplied by the antenna pattern to establish the echo model according to the CA [16]. This extends the average flat pulse response in the average echo model to a two-dimensional description of the illumination response of different Doppler frequencies of sub-beams [21; 22; 23]. A rectangular function is used to approximate the sub-beam pattern; a semi-analytical model of the SAR altimeter echo was proposed based on the geometric approximation of the delay-Doppler bin in [22]. Their model was later enhanced by incorporating the effect of the antenna mis-pointing angle [12]. Furthermore, a Gaussian function was employed to approximate the sub-beam pattern, and the echo model included small antenna mis-pointing angles [23]. A closed-form expression for the SAR altimeter waveform was formulated in [24] by expressing the waveform in terms of a set of basis functions. An innovative processing scheme known as fully focused SAR (FF-SAR) was proposed in [25] to maximize the altimeter's along-track resolution to the theoretical limit of half the antenna length, proving valuable in SAR imaging and radar altimetry. A new analytical derivation of a delay-Doppler map model for FF-SAR has been proposed, including vertical wave particle motion and a more general surface representation [26]. Unfocused SAR (UF-SAR) retracking methods(e.g., the physical SAMOSA model) have been utilized to retrack fully-focused SAR (FF-SAR) waveforms in [27], ultimately improving data quality for SAR altimetry in coastal regions. A novel coastal retracking algorithm for SAR altimetry was presented in [28] that employed an adaptive interference masking scheme to mitigate interfering signals from reflective coastal targets, significantly increasing the number of valid records in the coastal zone without compromising the quality of the estimated significant wave height. Airborne altimeters frequently encounter skewed flight conditions and larger antenna mis-pointing angles, which is due to several reasons. Airborne platforms such as aircraft and drones are subject to constant changes in flight dynamics and conditions. Unlike satellites, which have a relatively stable trajectory in space, airborne vehicles often have to adjust their course and altitude to accommodate weather conditions, air traffic, and mission objectives. Such adjustments can lead to vertical movement. In addition, the maneuvers required for aircraft to follow specific survey paths or avoid obstacles can result in off-nadir pointing, leading to significant antenna mis-pointing angles. It must be emphasized that existing research has predominantly focused on altimeters in horizontal flight scenarios, and as such is not suitable for airborne scenarios. For instance, satellite altimeters such as CryoSat-2 generally consider mis-pointing angles of less than 1 degree barring specific maneuvering instances [29]. However, these angles can greatly exceed 1 degree during oblique flights due to substantial pitch and roll angles. Furthermore, mechanical vibrations and the inherent instability of aircraft flight can contribute to greater mis-pointing angles. Despite advancements in flight control systems, it remains challenging to maintain perfect alignment of the radar beam with the vertical axis [30]. All these factors are inherent to airborne operations, and present substantial challenges for high-accuracy altimetry based on echo data from airborne altimeters. In this paper, we present a strategy for high-accuracy altimetry using airborne altimeters. We propose a generalized semi-analytical echo model for the re-tracking process of SAR altimetry with consideration of vertical movement and large antenna mis-pointing angles.