# Evaluation of Spaceborne GNSS-R Retrieved Ocean Surface Wind Speed with Multiple Datasets
## Theory and Methods
### Wind Speed Retrieval Algorithm
The number of samples in the different intervals is multiplied by the corresponding scale factor as the numerator of the sample weight coefficients in a certain interval,  
Figure 1: Flow chart of spaceborne GNSS-R wind speed retrieval  
and the sum of the four intervals is used as the denominator of the weight coefficient to compute the weighted value of the discrete points. To eliminate the fluctuation of the empirical model caused by insufficient training samples and system noise and ensure the accuracy of the empirical GMF, first, the maximum probability density bin of the wind speed in the training dataset is calculated at each incident angle because the probability density distribution of the wind speed slightly varies at different incident angle bins, as shown in Figure 3, it presents the PDF of the matched NWP wind for the CYGNSS DDM observables in each incidence angle bin. Then, the corresponding weighted DDM observables were calculated with the wind speed close to its maximum probability density, the observables above or below the wind speed were sequentially computed with the step size of 0.1 m/s, and the discrete GMF was also forced to be a monotone function with wind speed, which means that the GMF values could be same if monotonicity is violated during calculation. Since a single function form was not found to fit the discrete GMF well, we directly chose a piecewise function to obtain the final GMF model as the CYGNSS science operations center. The smoothing function smaller or larger than the piecewise point is shown in Equations (3) and (4), respectively:  
\[\mathrm{obs}=a_{0}+a_{1}u^{-1}+a_{2}u^{-2} \tag{3}\]  
\[\mathrm{obs}=b_{0}+b_{1}u+b_{2}u^{2} \tag{4}\]  
where _obs_ means the DDM observables and \(u\) represents the ground truth wind speed \(U_{10}\). Figure 2: Weighting strategy of observables to form empirical geophysical model function. The nonlinear least-squares fitting was performed for the GMF with wind speeds smaller than the segmented point, a new constraint was added, which requires all coefficients of formula (3) are limited to be non-negative. For discrete points of GMF where the wind speed is greater than the segmented point, the parabolic function fitting means that the values of two functions are equal at the piecewise point, and the first derivatives are equal. Furthermore, new constraints have been added such as limiting the opening of the parabola downward and its axis of symmetry on the left side of the piecewise point to force the established mapping relationship to be smoother and more consistent with the distribution of training data. The smoothing procedure of the discrete GMF is transformed into a nonlinear least square fitting and convex quadratic programming problem. The standard form of the convex quadratic programming is:  
\[ \begin{array}{rl}\min&\frac{1}{2}x^{T}Px+q^{T}\\ s.t.&Gx\leq h\\ Ax=b\end{array}  \tag{5}\]  
One of the important things to apply a piecewise function is to determine the segmentation point. In order to determine the optimal piecewise point, first, the smooth processing is performed at each GMF discrete point to find one with the smallest fitting residual. Then, do the smoothing again in the interval of a step length (0.1 m/s) around the discrete point with a smaller step. Finally, find the best location with the smallest fitting residual as the final transition point to re-smooth the final model. After obtaining the GMF function \(u(\theta,obs_{i})\) (where \(\theta\) denotes the incident angle and \(i\) denotes DDMA or LES) of the spaceborne GNSS-R, more precisely, it is a lookup table. When using the GMF model for the wind speed prediction. The model \(u(\theta,obs_{i})\) is linearly interpolated to the incident angle \(\theta\) corresponding to the observable getting \(u_{\theta}(obs_{i})\) at first, then \(u_{\theta}(obs_{i})\) linearly interpolates again to obtain the inversed wind speed \(u_{\theta,obs_{i}}\) corresponding to the observable. Establishing the separated empirical GMF for both DDMA and LES can benefit quality control.