# S-Band Doppler Wave Radar System
## 2 Coherent Wave Measuring Method  
According to the Bragg scattering theory and the composite surface scattering theory, backscattering occurs when electromagnetic wave grazes onto the sea surface [16; 21]. The middle- and large-scale waves on the sea surface cause multiple modulations of capillary waves. Those modulations make the central frequency of the echoes shift and the bandwidth extend [22; 23; 24; 25], and the wave parameters can be deduced from the reflection of these influences in the echoes. The radial velocity measured by Doppler radar is the sum of the scattered intrinsic velocity and any surface current component [5]. The expression for the radial velocity \(V_{r}\) at a point, under the action of a single long ocean wave with amplitude, angular frequency, wavenumber A(f), \(\omega\), and K (radial and transverse components \(k_{r}\) and \(k_{t}\)) respectively, is:  
\[V_{r}(R,t)=u\cos\theta+w\sin\theta \tag{1}\]  
\[u=U_{0}\cos(k_{r}R+k_{t}y^{\prime}-\omega t)\cos(\alpha_{r}-\alpha)+V_{D}+V_{S} \tag{2}\]  
\[w=U_{0}\sin(k_{r}R+k_{t}y^{\prime}-\omega t)\tanh(Kh)(1-\frac{V_{D}+V_{S}}{V}) \tag{3}\]  
\[U_{0}=A(f)\omega\coth(Kh) \tag{4}\]  
where \(u\) and \(w\) are the horizontal and vertical components of the orbital velocity, respectively; \(U_{0}\) is the orbital velocity; \(\theta\) is the radar grazing angle; \(\alpha\) is the wave azimuth(tan\({}^{-1}\)\(k_{t}/k_{r}\)); \(\alpha_{r}\) is the radar azimuth; \(V_{D}\) is the surface current velocity; \(V_{S}\) is the effective scatterer velocity; \(R\) is the distance between radar and the observed ocean area; \(y^{\prime}\) represents the axis that is vertical to \(R\); \(V\) is the long-wave phase velocity; and h is the water depth. The illustration of those symbols is shown in Figure 1.  
The velocity spectrum G(\(\omega\),\(\theta\),\(a_{r}\)) is obtained by applying the Fourier transformation to \(V_{r}\). Then, the relation between the velocity spectrum and wave height spectrum is used to deduce the wave height spectrum \(E(\omega)\) as follows:  
\[G(\omega,\theta,a_{r})=FT(V_{r}(R,t)) \tag{5}\]  
\[E(\omega)=G(\omega,\theta,a_{r})/[\omega^{2}\Gamma(\omega,\theta,a_{r})] \tag{6}\]  
where \(\Gamma\) is the relation between the velocity spectrum and wave height spectrum [5; 26]:  
\[\Gamma(\omega,\theta,a_{r})=\int_{-\pi}^{\pi}S(\omega,\alpha)[\cos^{2}\theta \coth^{2}(Kh)\cos^{2}(\alpha-a_{r})+\sin^{2}\theta]d\alpha \tag{7}\]  
where \(S(\omega,\alpha)\) is the normalized spreading function. The phases from two adjacent range cells acquired simultaneously are compared to solve the 180-degree ambiguity [27]. Assuming that the power density spectra of the velocity sequence of two adjacent range cells acquired with the same antenna are \(V_{1}(\omega)\) and \(V_{2}(\omega)\) and \(V_{1}(\omega)\) is farther from the radar, then the cross spectrum of \(V_{1}(\omega)\) and \(V_{2}(\omega)\) is as follows:  
\[\phi_{12}(\omega)=V_{1}(\omega)\times V_{2}^{*}(\omega) \tag{8}\]  
where \(V_{2}^{*}(\omega)\) is the complex conjugate of \(V_{2}(\omega)\). A relation is observed between \(V_{1}(\omega)\) and \(V_{2}(\omega)\) because of the adjacency of the two range cells:  
\[V_{2}(\omega)=V_{1}(\omega)\times e^{\hat{y}k\Delta r} \tag{9}\]  
where \(k\) is the wavenumber and \(\Delta r\) the range between two range cells. Either the approaching waves or the receding waves can contribute to the velocity spectrum. Hence, we assume that the expression of the velocity spectrum of the first range cell is as follows:  
\[u=U_{0}\cos(k_{r}R+k_{t}y^{\prime}-\omega t)\cos(\alpha_{r}-\alpha)+V_{D}+V_{S} \tag{2}\]0  
where \(B^{+}\) and \(B^{-}\) correspond to the approaching waves or the receding waves, respectively.