# Development of a GNSS Buoy for Monitoring Water Surface elevations in Estuaries and Coastal Areas
## Appendix A  
The step-by-step procedures for determining the water surface elevations from the measured accelerations by an ATC sensor areas follows. 1. Determining the Fourier transform of the acceleration The one-sided power spectrum, \(A(f)\), the PSD, \(S_{a}(f)\), and the phase spectrum, \(\varepsilon_{a}(f)\), of acceleration are determined as follows: \[|A(f)|^{2}=G_{n}^{2}+H_{n}^{2},\] (1) \[S_{a}(f)=\frac{|A(f)|^{2}}{\Delta f},\] (2) and: \[\varepsilon_{a}(f)=\tan^{-1}(\frac{H_{n}}{G_{n}}),\] (3) where \(\frac{G_{n}}{2}\) and \(\frac{H_{n}}{2}\) are the real and imaginary parts of the Fourier transform of the acceleration time series data, respectively, and \(\Delta f\) is the frequency resolution. 2. Calculating the phase spectrum of the water surface elevation According to linear wave theory [27], the phase lag between the acceleration of a water particle and the water surface elevation is \(\pi\). The phase spectrum of the water surface elevation, \(\varepsilon_{e}(f)\), is then \[\varepsilon_{e}(f)=\varepsilon_{a}(f)+\pi,\] (4)
3. Filtering the noise in the acceleration signals Because the acceleration signals from the buoys are usually contaminated by noise, [PERSON] [28] proposed an empirical noise correction function for the acceleration obtained by a data buoy. By applying the noise correction function, the noise can be filtered from the PSD of the acceleration. The noise correction function is a linearly decreasing function with respect to frequency and has the following form: \[NC(f)_{LANG}=K\times G[C_{11M}(0.01),C_{11M}(0.02)]\times(F-f)\] (5)where \(K\) is a constant, \(G\) is a function of \(C_{11M}(0.01)\) and \(C_{11M}(0.02)\) and must be determined, \(C_{11M}(0.01)\) and \(C_{11M}(0.02)\) are the PSDs of the acceleration at 0.01 and 0.02 Hz, respectively, \(F\) is a fixed frequency, and \(f\) is the frequency. [PERSON] [28] provided various values of \(K\) and \(F\) for their discus buoys, which had diameters of 3 m, at different locations. \(F\) was set to 0.17 Hz for shallow water and 0.15 Hz for deep water. Because the buoy hull and the mooring line used for the present GNSS buoy are slightly different from those used by [PERSON] [28], [PERSON] et al. [22] modified the noise correction function of [PERSON] [28] and suggested the following form for the buoy used in this work:  
\[NC(f)=\frac{Max[C_{11M}(<0.03)]}{(0.15-f_{MaxC11M})}\times(0.15-f), \tag{10}\]  
where \(Max[C_{11M}(<0.03)]\) is the maximum PSD of the acceleration in the frequency range of 0-0.03 Hz, and \(f_{MaxC11M}\) is the frequency at which \(Max[C_{11M}(<0.03)]\) occurs. Accordingly, the modified PSD of the acceleration can be expressed as follows:  
\[S^{\prime}_{a}(f)=S_{a}(f)-NC(f),\text{ for }f\geq 0.03\text{ Hz},\] (11a) and: \[S^{\prime}_{a}(f)=0,\text{ for }f<0.03\text{ Hz},\] (11b) where \[S_{a}(f)\] is the unmodified PSD of the acceleration. Note from Equation (11b) that signals with frequencies less than 0.03 Hz, which corresponds to a period of 33.3 s, are ignored. Because the periods of typical water waves in the ocean are less than 20 s, ignoring those signals is reasonable. 4. Determining the PSD of the water surface elevations According to [PERSON] and [PERSON] [29], the PSD of the water surface elevations, \(S_{e}(f)\), can be determined from that of the acceleration as follows: \[S_{e}(f)=\frac{1}{\omega^{4}}\times S^{\prime}_{a}(f),\] (12) where \(\omega\) is the angular frequency. 5. Smoothing the PSD of the water surface elevations Data of 512 points with a time interval of 1 s are used in the fast Fourier transform (FFT). The frequency resolution is approximately 0.001953 Hz before smoothing. According to the spectral smoothing technique described in [30], the PSD of the water surface elevation is smoothed using a Bartlett window of 15 points. The degree of freedom of the PSD is 32. After smoothing, the frequency resolution of the PSD becomes 0.03125 Hz. 6.