## 3 Method  
### The Retracking Functional Form  
To accurately fit the waveform, two variants of the Brown model are selected as the basis for waveform retracking: the Brown model with the second-order Bessel function [28] and the Brown model based on the roughness of the sea surface [29].  
The Brown model with the second-order Bessel function fits the ocean-like waveforms in this study. Compared with the first-order Bessel function Brown model, this model can more accurately fit the waveforms with a larger mispointing angle. The specific description of this model is as follows:  
\[W(t)=a_{g}P_{u}[1+erf(u_{1})]\cdot\exp(-v_{1})-\frac{a_{g}-P_{u}}{2}[1+erf(u_ {2})]\cdot\exp(-v_{2})+T_{n} \tag{1}\]  
Figure 2: The study areas and the ground track of HY-2B altimeter (red lines respectively indicate the trajectory of pass 323 (**A**) and the trajectory of pass 196 (**B**), the yellow blocks are the tide gauge stations and the shadow represents coastal land).  
where  
\[\begin{array}{ll}a_{\xi}=\exp(-\frac{4\sin^{2}\xi}{\gamma})&\gamma=\sin^{2}( \theta_{0})\frac{1}{2\ln(2)}\\ erf(x)=\frac{2}{\sqrt{\chi}}\int_{0}^{x}e^{-t^{2}}dt\\ u_{1}=\frac{t-\tau-a_{1}\sigma_{\xi}^{2}}{\sqrt{2}\sigma_{c}}&v_{1}=\alpha_{1}(t -\tau-\frac{a_{1}\sigma_{\xi}^{2}}{2})&\alpha_{1}=\delta-\frac{\beta^{2}}{8} \\ u_{2}=\frac{t-\tau-a_{2}\sigma_{c}^{2}}{\sqrt{2}\sigma_{c}}&v_{1}=\alpha_{2}(t -\tau-\frac{a_{2}\sigma_{c}^{2}}{2})&\alpha_{2}=\delta\\ \delta=\frac{4\epsilon}{\gamma h}\cos(2\xi)&\beta=\frac{4}{\gamma}(\frac{ \epsilon}{h})^{1/2}\sin(2\xi)&h=H(1+H/\text{R}_{e})\\ \sigma_{c}^{2}=\sigma_{s}^{2}+\sigma_{p}^{2}&\sigma_{s}=\frac{SWH}{2c}\\ \end{array}\]  
Considering the existence of specular waveforms and waveforms with a small trailing edge slope (the sharpening waveform shown in Figure 3), which are caused by a calm sea surface, the Brown model based on sea surface roughness is applied. The specific description of this model is as follows:  
\[W_{\text{s}}(t)=\frac{a_{\xi}P_{\text{u}}}{2}[1+erf(u)]\cdot\exp(-v) \tag{2}\]  
where  
\[\begin{array}{ll}u=\frac{t-\tau-a\sigma_{c}^{2}}{\sqrt{2}\sigma_{c}}&v=a \Big{(}t-\tau-\frac{a\sigma_{c}^{2}}{2}\Big{)}\\ \alpha=\frac{4\epsilon}{Th}&\Gamma=\frac{4\gamma\text{-}mss}{4\cdot miss+\gamma }\\ \end{array}\]  
the parameters of the two models are same, where \(\text{a}_{\xi}\) is calculated using the mispointing angle \(\xi\). The value of \(\gamma\) depends on the antenna beam width \(\theta_{0}\). \(\text{P}_{\text{u}}\) is the amplitude of the waveform. t is the echo traveling time. \(\tau\) is the midpoint of the leading edge. c is the speed of light. h is the corrected height calculated from the satellite orbit height H and the Earth curvature \(\text{R}_{e}\). \(\sigma_{\text{c}}\) is the composite backscattering coefficient. \(\sigma_{\text{p}}\) is the corresponding width of the radar point target, and the HY-2B altimeter uses 0.513\(tr\), where \(tr\) is the continuous power gate interval time equal to 3.125 ns. \(\Gamma\) is a parameter defined by \(\gamma\) and the mean square slope (mss) of the reflected surface and is used to determine the slope of the waveform trailing edge. When the mss value is close to infinity, the Brown model based on sea surface roughness is transformed into the first-order Bessel function Brown model. For the Brown model with a second-order Bessel function, the main fitting parameters are \(\tau\), SWH, \(\text{P}_{\text{u}}\), and \(\xi\), (referred to as the WLS4), while the fitted parameters of the Brown model based on sea surface roughness are \(\tau\), SWH, \(\text{P}_{\text{u}}\), and \(\Gamma\)(referred to as the MB4). The iterative method of the model adopts weight least squares fitting based on the Nelder-Mead algorithm [30].  
Figure 3: The examples of real waveform with small mispointing angle (The **left** is the sharpening waveform and the **right** is the specular waveform).