## 4 Re-Tracking Algorithm Based on Echo Model Fitting  
Re-tracking processing involves parameterizing the echo function curve based on the rough tracking result. It uses a certain algorithm to fit the echo model to the actual waveform to achieve the most consistent state, thereby retrieving the parameters contained in the echo. It can be solved using the hill-climbing algorithm [32; 33] or the least squares (LS) method [34]. However, in certain scenarios the paths defined by the hill-climbing algorithm do not monotonically reduce the distance to the local optimum [35]. Thus, the nonlinear least squares (LS) algorithm is widely used in altimetry parameter estimation. The LS estimator under consideration is defined as  
\[\hat{\theta}_{LS}=\operatorname*{arg\,min}_{\theta}\frac{1}{2}\sum_{k=1}^{K}g _{k}^{2}(\theta), \tag{35}\]  
where \(g_{k}(\theta)=y_{k}-s_{k}(\theta)\) is the residual vector, \(s(\theta)\) represents the multilook echoes, and \(y=(y_{1},\ldots,y_{K})^{T}\) is the echo destroyed by speckle noise. Because \(g_{k}^{2}(\theta)\) is a complicated nonlinear function of SWH and \(\tau\), the optimization problem does not admit a closed-form expression, and the LS estimator (35) is solved using a numerical optimization method. The Levenberg-Marquardt (LM) algorithm is an optimization algorithm used for nonlinear least squares fitting and optimization [12; 22; 36]. It exhibits greater robustness and effectiveness than traditional LS methods. In this paper, a five-parameter iterative LS algorithm based on the LM algorithm is used to estimate the altimetric parameters, as shown in Figure 8.  
The parameter update of the iterative LM algorithm is defined by \(\theta^{(i+1)}=\theta^{(i)}+e^{(i)}\), where \(\theta^{i}\) is the estimate of \(\theta\) at the \(i\)-th iteration. The choice of \(e^{i}\) is based on Taylor expansion (at the first order) of \(g\) in the neighborhood of \(\theta^{i}\)  
\[g\Big{(}\theta^{(i)}+e^{(i)}\Big{)}=g\Big{(}\theta^{(i)}\Big{)}+J\Big{(} \theta^{(i)}\Big{)}e^{(i)} \tag{36}\]  
Figure 7: Effect of \(\psi_{ac}\) on (**a**) the multilook echoes and (**b**) the normalized multilook echoes (Pu = 1, \(\tau\) = 30 gates, SWH = 2 m, and \(\mu=\psi_{ac}=0^{\circ}\)). where \(J=\left[\partial g\Big{(}\theta^{(i)}\Big{)}/\partial\theta_{1},\dots,\partial g \Big{(}\theta^{(i)}\Big{)}/\partial\theta_{I}\right]\) is a partial differential matrix representing the gradient of \(g\). After replacing (36) in (35), the following result is obtained:  
\[G(\theta+e)\simeq L(e)=\frac{1}{2}l(e)^{T}I(e). \tag{37}\]  
The descent direction \(e\) is then obtained by minimizing \(L(e)\). By setting the derivative \(L^{\prime}(e)=J(\theta)^{T}g+J(\theta)^{T}J(\theta)e\) to zero and adding a regularization parameter \(\mu\), leading to  
\[\Big{[}J(\theta)^{T}J(\theta)e+\mu I_{I}\Big{]}e=-J(\theta)^{T}g \tag{38}\]  
where \(I_{I}\) is the identity matrix, this algorithm employs a gradient descent method to update the parameters \(\theta=(\text{SWH},\tau,\mu,\psi_{ac},\psi_{al})\), as follows:  
\[\theta^{(i+1)}=\theta^{(i)}-\Big{[}J^{T}J+\mu_{i}I_{I}\Big{]}^{-1}J^{T}g\Big{(} \theta^{(i)}\Big{)} \tag{39}\]  
where \(\mu_{i}\) controls the convergence speed of the algorithm. Note that the derivatives appearing in \(\frac{\partial g(\theta)}{\partial\theta_{I}}\) can be computed numerically using the finite difference, as follows:  
\[\frac{\partial g(\theta)}{\partial\theta_{j}}\simeq-\frac{s\big{(}\theta_{j}+ \Delta\theta_{j}\big{)}-s\big{(}\theta_{j}\big{)}}{\Delta\theta_{j}}. \tag{40}\]  
It should be noted that the initial value of the iterated parameters in the re-tracking algorithm is \(\theta^{0}=\big{(}\tau^{0},\text{SWH}^{0},\mu^{0},\psi_{ac}^{0},\psi_{al}^{0} \big{)}\), where \(\tau^{0}\) is obtained by rough tracking. Furthermore, the real mis-pointing angles (i.e., the roll and pitch angles) are used as inputs, deviating from their actual values by approximately \(1^{\circ}\).