## 5 Results
### Analysis of Re-Tracking Precision  
This section evaluates the performance of the proposed re-tracking algorithms for the numerical waveforms affected by speckle noise. To simulate the real echo, speckle noise is added to the multilook waveform (33) following the approach described in [12; 14; 22]. Consequently, the noisy echo \(y(t)\) can be obtained by  
\[y(t)=\sum_{k=1}^{N}P(t_{r}-2\delta_{r}(f_{d})/c,k)q(t_{r}-2\delta_{r}(f_{d})/c,k), \tag{42}\]  
where \(q(t_{r}-2\delta_{r}(f_{d})/c,k)\) is gamma-distributed, multiplicatively independent, and identically distributed speckle noise expressed as \(\Gamma(L,1/L)\). The normalized echo used for parameter estimation depends on the unknown parameter vector \(\theta_{5}=(\text{SWH},\,\tau,\mu,\psi_{ac},\psi_{al})^{T}\). The effectiveness of the strategy can be evaluated through the simulation of multilook echoes. The root mean square error (RMSE) of the altimetric parameters is adopted to evaluate the algorithm's performance. All results presented in this section have been averaged using NMC = 500 Monte Carlo runs. The RMSE of the \(i\)-th altimetric parameter \(\theta_{i}\) is defined as  
\[\text{RMSE}(\theta_{i})=\sqrt{\frac{1}{N}\sum_{j=1}^{N}\left[\theta_{i}-\hat{ \theta}_{i}(j)\right]^{2}},i=1,\ldots,J, \tag{43}\]  
where \(\theta_{i}\) is the true parameter, \(\hat{\theta}_{i}(j)\) is the estimated parameter for the \(j\)-th waveform, and \(N=500\) is the Number of Monte Carlo simulations. To verify the effectiveness of the proposed model, a generalized delay-Doppler altimetric model is used for comparison. We replace \(\psi_{ac}\) and \(\psi_{al}\) with zeros and estimate the two remaining parameters \(\left(\text{SWH},\tau\right)^{T}\) using the Levenberg-Marquardt algorithm. The resulting estimation strategy is denoted as G-PRA (indicating the generalized altimetric model with two unknown parameters used in the parameterized re-tracking algorithm). The estimations provided by G-PRA need to be compared with those obtained with the proposed algorithm, which is denoted as PRA. The first group of experiments corresponds to level flight in the absence of mis-pointing while considering different SWH; the other altimetric parameters are set as \((P_{u},\tau,\mu,\psi_{ac},\psi_{al})^{T}=(1,30\text{ gates},0^{0},0^{0},0^{0})^{T}\). The parameter RMSEs obtained by the algorithm are shown in Figure 13, and the figure shows similar performance for the G-PRA and PRA algorithms. Thus, the results demonstrate that there is not much performance reduction in estimating \(\tau\) and SWH in the absence of mis-pointing. Additionally, the RMSE(SWH) is a decreasing function of SWH. The second set of experiments aimed to investigate the performance of parameter estimation for oblique flight with varying mis-pointing angles \(\psi_{al}\) and \(\psi_{ac}\). The remaining altimetric parameters are set as \(\left(\text{SWH},P_{u},\tau,\mu\right)^{T}=\left(2\text{ m},1,30\text{ gates},6^{0}\right)^{T}\). Figure 14 demonstrates that the proposed PRA algorithm leads to a smaller \(\text{RMSE}(\tau)\) compared to G-PRA in the presence of \(\psi_{ac}\). As \(\psi_{ac}\) increases, the difference in \(\text{RMSE}(\tau)\) and \(\text{RMSE}(\text{SWH})\) obtained by the two algorithms becomes larger. However, the improvement is less in the presence of \(\psi_{al}\), as shown in Figure 15. This property can be explained by the fact that the cross-track mis-pointing angle has a greater influence on the waveform shape compared to \(\psi_{al}\). In addition, It can be observed that as the mis-pointing angle increases, the RMSE of \(\tau\) obtained by the PRA algorithm initially remains at a low level and then gradually increases. This may be attributed to the non-negligible error in model approximation. The mis-pointing angle in airborne scenarios is typically not greater than 10 degrees, indicating that the estimation accuracy of this method falls within an acceptable range.