# A Novel Method for Ocean Wind Speed Detection Based on Energy Distribution of Beidou Reflections
## 2 Theoretical Analysis and Methods  
### Theoretical Analysis  
This section presents the analysis of the relationship between GNSS-R waveforms and wind speed to derive the sensitivity of the observable. The shape of GNSS-R waveform can characterize the sea surface roughness: the higher the wind speed is, the rougher the sea surface is and, thus, the extension of the waveform is. This section focuses on the description of the energy distribution of the GNSS-R waveform and defines the bistatic radar model, which describes the total average correlation power of the scattered GNSS signal as a function of the time delay \(\Delta\tau\) and the frequency offset \(\Delta f\) with respect to the time delay and Doppler frequency associate with the nominal specular point on the ocean's surface [30]. \[ \begin{array}{c}\left\langle Y_{r}(\Delta\tau,\Delta f)\right\rangle\propto \underset{D}{\iint}\frac{G_{r}^{2}(\boldsymbol{\rho})}{R_{R_{\boldsymbol{ \rho}}}^{2}(\boldsymbol{\rho})\cdot R_{AC}^{2}(\Delta\tau-\tau(\boldsymbol{ \rho}))}\\ \cdot\left[S(\Delta f-f(\boldsymbol{\rho}))\right]^{2}\cdot o^{0}d\boldsymbol{ \rho}\end{array}  \tag{1}\]where \(\left\langle Y_{\tau}(\cdot)\right\rangle\) is the averaged power waveform; \(D\) is the integration area; \(G_{\tau}\) is the receiver's antenna gain, \(R_{ts}\) is the distance from the transmitter to the reflection point \(\rho\); \(R_{sr}\) is the distance from \(\rho\) to the receiver. \(\Delta\tau\) and \(\Delta f\) are delay offset and Doppler frequency offset, respectively. \(\tau(\rho)\) and \(f(\rho)\) are the delay offset and Doppler frequency shift at the scatter point, respectively. \(R_{AC}\) is the auto-correlation function of the GNSS ranging code defined as \(R_{AC}(\tau)=1-\tau/\tau_{c}\) when \(|\tau|\leq\tau_{c}\), \(R_{AC}(\tau)=0\), elsewhere (\(\tau_{c}\) is the length of one chip of the C/A code). \(|S|\) is the Doppler frequency filter function defined as \(|S|=\left|\sin(\pi f)/(\pi f)\right|\). \(\sigma^{0}\) is the normalized bi-static radar cross section of the sea surface, which is calculated and expressed as a function of the probability density function (PDF) of the slopes based on the geometric optics limit. \[\sigma^{0}=\pi|\Re|^{2}\left(\frac{q}{q_{z}}\right)^{4}p\left(\frac{-q_{\bot} }{q_{z}}\right) \tag{2}\]  
where \(\Re\) is the Fresnel reflection coefficient, \(q\) is the scattering vector, \(P\) is the PDF of the surface slope. In the following simulation, the PDF is assumed to be a 2-D zero-mean Gaussian distribution [31]. The mean square slopes of the up-wind and cross-wind are approximated based on the simplified sea roughness model proposed by [PERSON] [32] as  
\[ \begin{split}\sigma_{u}&=0.45\cdot[0.000+0.00316 \cdot f(u)]\\ \sigma_{c}&=0.45\cdot[0.003+0.00192\cdot f(u)]\end{split}  \tag{3}\]  
where \(\sigma_{u}\) and \(\sigma_{c}\) are the mean square slopes of the up-wind and cross-wind, respectively. \(f(u)\) is the function of the wind speed at the height of 10 m above the sea surface and is calculated as  
\[f(u)=\left\{ \begin{array}{c}u,0<u<3.49\\ 6\ln(u)-4.0,3.49<u<46\\ 0.441u,\ 46<u\end{array} \right.. \tag{4}\]  
To analyze the relationship between the wind speed and energy distribution of the waveform, a simulation is performed with the parameters listed in Table 1.  
Considering the rather small effect of the Doppler frequency spreading in the following experimental scenario, the simulation mainly focuses on the variation of the delay waveform. Figure 1 shows the normalized delay waveforms under different wind conditions. As shown in Figure 1a, we set a constant wind direction of 20 deg and simulate 1-D delay waveforms corresponding to different wind speeds. The extension of the trailing edge shows a dependence on wind speed, while the leading edge of the waveform remains relatively stable with different wind speeds. As shown in Figure 1b, we set a constant wind speed of 6 m/s and simulate the waveforms corresponding to different wind directions. This shows a slight change in the trailing edge of the waveform.