# Development of a GNSS Buoy for Monitoring Water Surface elevations in Estuaries and Coastal Areas
## Appendix B  
This appendix presents the methods of determining the directional wave spectrum and the peak wave direction from the \(\eta uv\) data provided by the GNSS buoy, where \(\eta\) denotes the altitude, \(u\) and \(v\) are the velocities in the east and north directions of the GNSS antenna, respectively. The velocities in the east and north directions are also referred to as the \(x\)- and \(y\)-velocity components, respectively. [PERSON] and [PERSON] [29] described the relationship between the directional spectrum and the cross-power spectrum for any pair of wave properties-such as \(\eta\), \(u\), and \(v\). [PERSON] et al. [31] expressed this relationship as follows:  
\[\Phi_{mn}(f)=\int_{-\pi}^{\pi}I_{m}(f,\theta)I_{n}^{s}(f,\theta)\{\cos[k(x_{mn} \cos\theta+y_{mn}\sin\theta)]-i\sin[k(x_{mn}\cos\theta+y_{mn}\sin\theta)]\}S(f, \theta)d\theta \tag{11}\]  
where \(\Phi_{mn}(f)\) is the cross-power spectrum between the \(m\)-th and \(n\)-th wave properties, \(I_{m}(f,\theta)\) is the transfer function from the water surface elevation to the \(m\)-th wave property, \(\theta\) is the wave propagation angle measured from the \(x\)-axis and increases in the counterclockwise direction, the asterisk (*) denotes the conjugate complex, \(k\) is the wavenumber, \(x_{mn}=x_{n}-x_{m}\), \(y_{mn}=y_{n}-y_{m}\), \((x_{m},y_{m})\) is the location of the wave probe for the \(m\)-th wave property, \(i\) is the imaginary unit, and \(S(f,\theta)\) is the directional wave spectrum. For the \(\eta\mu\
u\) method, the vertical elevation (\(\eta\)) and the velocities in the east and north directions (\(u\) and \(v\), respectively) are measured by the same GNSS antenna. Accordingly, \(x_{mn}=0\) and \(y_{mn}=0\). Thus, Equation (10) can be simplified as follows:  
\[\Phi_{mn}(f)=\int_{-\pi}^{\pi}I_{m}(f,\theta)I_{n}^{s}(f,\theta)S(f,\theta)d\theta, \tag{12}\]  
with:  
\[I_{m}(f,\theta)=h_{m}(f)\cos^{\alpha}\theta\sin^{\beta}\theta, \tag{13}\]  
where the function \(h_{m}\) and parameters \(\alpha\) and \(\beta\) vary with the wave properties and are provided in Hashimoto and Konbune [29]. When Equation (13) is substituted into Equation (12), the cross-spectra between the various wave properties can be obtained as follows:  
\[\Phi_{\eta u}(f)=\int_{-\pi}^{\pi}S(f,\theta)h_{u}(f)\cos\theta\,d\theta=C_{ \eta u}(f), \tag{14}\]  
\[\Phi_{\eta v}(f)=\int_{-\pi}^{\pi}S(f,\theta)h_{v}(f)\sin\theta\,d\theta=C_{ \eta v}(f), \tag{15}\]  
\[\Phi_{uu}(f)=\int_{-\pi}^{\pi}S(f,\theta)h_{u}^{2}(f)\cos^{2}\theta\,d\theta= C_{uu}(f), \tag{16}\]  
\[\Phi_{uv}(f)=\int_{-\pi}^{\pi}S(f,\theta)h_{u}(f)h_{v}(f)\cos\theta\sin\theta \,d\theta=C_{uv}(f), \tag{17}\]  
and:  
\[\Phi_{vv}(f)=\int_{-\pi}^{\pi}S(f,\theta)h_{v}^{2}(f)\sin^{2}\theta\,d\theta= C_{vv}(f), \tag{18}\]  
where \(C_{mn}(f)\) is the co-spectrum between the \(m\)-th and \(n\)-th wave properties. The directional wave spectrum, \(S(f,\theta)\), can be expressed as follows:  
\[S(f,\theta)=D(f,\theta)\Phi_{\eta\eta}(f)=D(f,\theta)C_{\eta\eta}(f), \tag{19}\]  
where \(D(f,\theta)\) is the directional spreading function. Note that according to [PERSON] and [PERSON] [29], both \(h_{u}(f)\) and \(h_{v}(f)\) are real and the directional wave spectrum is also real.