# Research on Wind Speed Inversion Method for X-Band Networked SAR Satellite
## 3 Methods  
A cost function was established using the XMOD2. When establishing the cost function, multiple sets of SAR data with different incident angles were used. The more data there are, the more accurate the inversion results. In this study, SAR data from three satellites were simulated, and the cost function established is the same as that in (5).  
\[\begin{array}{c}J_{\text{cost}}(\theta,\Phi,u)=(\sigma_{1}^{\text{m}}(\theta_ {1},\Phi_{1},u)-\sigma_{1}^{0}(\theta_{1},\Phi_{1},u))^{2}\\ \qquad\qquad+(\sigma_{2}^{\text{m}}(\theta_{2},\Phi_{2},u)-\sigma_{2}^{0}( \theta_{2},\Phi_{2},u))^{2}\\ \qquad\qquad+(\sigma_{3}^{\text{m}}(\theta_{3},\Phi_{3},u))-\sigma_{3}^{0}( \theta_{3},\Phi_{3},u))^{2}\,,\end{array} \tag{5}\]  
In the above equations, \(\sigma_{1}^{\text{m}}\), \(\sigma_{2}^{\text{m}}\), and \(\sigma_{3}^{\text{m}}\) are the backscatter coefficients calculated by the wind speed inversion model, while \(\sigma_{1}^{0}\), \(\sigma_{2}^{\text{b}}\), and \(\sigma_{3}^{0}\) are the backscatter coefficients obtained by the radiation calibration of SAR images.  
After obtaining the cost function, there are two ways to obtain a unique result. The first way is to find the minimum value of the cost function, and the corresponding wind speed is the inverted wind speed. In order to minimize the cost function, the partial derivatives of the cost function with respect to wind speed and relative wind direction are respectively obtained, as shown in (6). Since the wind speed inversion model is a relatively complex function, (5) and (6) are also complex functions. To reduce the computational burden, this paper chose a double-precision search method to obtain the minimum value of the target cost function to obtain the fuzzy solutions of the sea surface wind speed and wind direction. The search process was divided into two main steps: a rough search and a fine search. In the process of rough search, the wind speed of 0-30 m/s was substituted into (6) with the interval of 1 m/s and the wind direction of 0-360\({}^{\circ}\) with the interval of 10\({}^{\circ}\) to obtain the value of partial derivative. In the calculation results, some adjacent results were of different signs, indicating that there was a point with 0 partial derivative between adjacent two groups of data, namely, the extreme value point. Then, a fine search was conducted within the range where the extreme point existed, that is, the wind speed was set at an interval of 0.1 m/s, and the wind direction was set at an interval of 1\({}^{\circ}\) and substituted into (6). The value closest to 0 in the results of each group was saved, and the corresponding wind speed and wind direction were substituted into (5) to calculate the cost function. The wind speed corresponding to the minimum value of the cost function is the wind speed inversion result. This method does not rely on wind direction data, and can obtain wind speed.  
\[\begin{array}{c}\frac{\partial\int_{\text{cost}}(\theta,\Phi,u)}{\partial u }=\ \sum\limits_{i=1}^{3}2\left(\sigma_{i}^{\text{m}}-\sigma_{i}^{0}\right) \frac{\sigma_{i}^{\text{m}}}{\partial u},\\ \frac{\partial\int_{\text{cost}}(\theta,\Phi,u)}{\partial\cos(\Phi)}=\ \sum\limits_{i=1}^{3}2\left( \sigma_{i}^{\text{m}}-\sigma_{i}^{0}\right)\frac{\sigma_{i}^{0}}{\alpha\cos( \Phi)}\end{array} \tag{6}\]  
In the second way, accurate wind direction data was used as a reference to screen the final results. According to the extreme value of the cost function, an appropriate screening threshold was set, and the wind vector corresponding to the screening result was the preliminary result. Finally, the wind direction data were used to eliminate the fuzzy solution to obtain a unique solution. The advantage of this method is that when the difference between the actual SAR backscatter coefficient and the backscatter coefficient calculated by XMOD2 is small, the extracted wind speed is more accurate.