## 6 Methodology of Retracking  
Equation (9) could be written as  
\[\langle|Y(\tau)|^{2}\rangle=|Y(\tau)|^{2}\otimes\frac{1}{N_{\text{incoh}}}\sum_ {i=1}^{N_{\text{incoh}}}\delta(\tau-i\Delta\tau) \tag{18}\]  
where \(|Y(\tau)|^{2}\) is the pure waveform; \(\Delta\tau\) is the change of delay difference between adjacent waveform samples; \(\otimes\) is the convolution operator; \(\delta(\tau)\) is the Dirac function; and \(\frac{1}{N_{\text{incoh}}}\sum_{i=1}^{N_{\text{incoh}}}\delta(\tau-i\Delta\tau)\) is called as the point spread function (PSF). By measured waveform and PSF, the pure waveform could be estimated using the deconvolution, such as in [17; 18] where the constrained least squares (CLS) and the truncated singular value decomposition (TSVD) were used to reconstruct scattering coefficient (\(\sigma^{0}\)) from Delay-Doppler Map (DDM), respectively. In fact, many reconstruction methods have been proposed to solve deconvolution in image processing, however, unfortunately, either those methods are unsuitable to solve the problem with the existing of the large noise level, or their performance have dependence on the choice of the method parameter. The parameter \(p\) of (TSVD) in [17] and \(\gamma\) of CLS in [18] are key point to control the reconstruction results. In this paper, Least Square Matching (LSM) is used to estimated the pure waveform as  
\[\{\hat{W}S,\hat{A},\hat{\tau_{0}}\}=\arg\min\left\{\langle|Y_{\text{M}}(\tau)| ^{2}\rangle-\hat{A}\langle|Y(\tau)|^{2}\rangle\right\} \tag{19}\]  
where \(\langle|Y_{\text{M}}(\tau)|^{2}\rangle\) represents the measured waveform. This method aims to find proper wind speed \(\hat{W}S\) and delay \(\hat{\tau_{0}}\) to minimize the root mean square error between averaged waveform and modeled one by Equation (9) or (18) minimum. It is noted that this method has the same procedure with the processing of retrieving wind speed using Least Square Matching (LSM), the only difference is that this method considered the impact of the time-varying geometry configuration of GNSS-R. The estimated \(\tau_{0}\) is relative to the delay used in altimeter. As known, the Z-V model is time- and resource-consuming,  
Figure 7: Sensitivity of (**a**) peak waveform; (**b**) leading edge slope; and (**c**) trailing edge slope on wind speed before and after retracking under the condition of the moving direction of LEO and GNSS satellite are both 45\({}^{\circ}\); and the incidence angle is 30\({}^{\circ}\). hence, it is difficult to be used in fast processing. The alternative approach is to find an analytic function to replace Model (2) as  
\[f(\tau)=(\beta_{1}e^{\beta_{4}Q}+\beta_{5}e^{\beta_{6}Q})P\left(\frac{\tau-\beta_ {2}}{\beta_{3}}\right) \tag{20}\]  
where \(\beta_{1}\sim\beta_{6}\) are fitted parameters; and  
\[Q=\left\{ \begin{array}{ll}0,&\tau<\beta_{2}-0.5\beta_{3}\\ \tau-(\beta_{2}-0.5\beta_{3}),&\tau\geq\beta_{2}-0.5\beta_{3}\end{array} \right.\]  
\[P(\tau)=\frac{1}{\sqrt{2\pi}}\int_{-\infty}^{\tau}e^{-\frac{\tau^{2}}{\tau}}\,dq\]  
Figure 8 presents simulated delay waveform and the fitted result using analytic Function (20), from which it is seen that by finding proper \(\beta_{1}\sim\beta_{6}\), the delay waveform could be replaced well by (20). Therefore, Equation (19) is changed to  
\[\{\hat{\beta}_{1},\cdots,\hat{\beta}_{6}\}=\arg\,\min\left\{\langle|Y(\tau)|^ {2}\rangle-f(\tau)\right\} \tag{21}\]  
The retracking processing of distorted waveform on-ground is outlined in Figure 9, in which the detailed steps are implemented as follow:  
1. using estimated Doppler difference between direct and reflected signals to produce the DDCR and PSF;
2. developing model of the distorted waveform using convolution Equation (18) and the initial coefficients of the Model (20);
3. fitting the distorted model above with the measured waveform using nonlinear least square to obtain the optimal coefficients of (20);
4. reconstructing the pure waveform using the Model (20) and the estimated coefficients above.