# Validation of Drifting Buoy Data for Ocean Wave Observation
## Appendix A Relationships of the Wave Parameters  
The JMA drifting buoy-observed wave parameters were estimated through the zero-up crossing method, while the ERA5 wave parameters were estimated from a wave spectrum. The relationship between the wave parameters from the zero-up crossing method and from the wave spectrum was explored.  
The sampling variabilities of the significant wave height and period through the zero-up crossing method were also evaluated. The absolute Fourier amplitudes (square root of spectra) were estimated, where the wave spectrum was the JONSWAP spectrum. The parameter \(\alpha\) in the JONSWAP wave spectrum, which controlled the spectrum amplitude, was \(\alpha=8.1\times 10^{-3}\). The time series of the surface elevations (\(\eta\)) was generated by the inverse Fourier transform (IFFT) with the amplitudes and randomized phases. The time step of the surface elevation generation was 0.125 s. The number of surface elevations generated by the IFFT was \(2^{14}\) (2048 s). The 1024 surface elevations were generated at 0.5 s intervals from the generated surface elevations. The significant wave height (\(H_{1/3}\)) and period (\(T_{1/3}\)) from the 1024 surface elevations were evaluated using the zero-up crossing method. This evaluation of the significant wave heights and periods was conducted \(10^{3}\) times for a given peak wave frequency of the JONSWAP-type wave spectrum. The means and the standard deviations of the significant wave heights and periods were estimated from \(10^{3}\) ensembles of the significant wave heights and periods.  
The sampling variabilities of the GPS wave data were also evaluated. The 1024 surface elevations were generated at 1 s intervals from the generated surface elevations at 0.125 s intervals. The significant wave heights and periods were evaluated from the 1024 surface elevations at 1 s intervals. This evaluation was also conducted \(10^{3}\) times. The mean and the standard deviation of \(H_{1/3}\) and \(T_{1/3}\) were evaluated from the \(10^{3}\) ensembles.  
Figure 14 shows the normalized standard deviations of the significant wave heights and periods by the mean values (\(S_{d}(X)\langle X\rangle^{-1}\), where \(S_{d}(X)=\langle(X-\langle X\rangle)^{2}\rangle^{1/2}\), and \(X\) is \(H_{1/3}\) or \(T_{1/3}\)) as a function of \(H_{0}\) or \(T_{0}\). For example, the normalized standard deviation for the drifting buoy wave height \(S_{d}(H_{1/3})\langle H_{1/3}\rangle^{-1}\) was approximately 0.065 for \(H_{0}\)= 2 m (black solid line in Figure 14). The value of \(S_{d}(H_{1/3})\langle H_{1/3}\rangle^{-1}\) increased with the higher wave heights. The sampling variability of the significant wave heights by the drifting buoys was larger than that by the GPS buoys. The value of \(S_{d}(T_{1/3})\langle T_{1/3}\rangle^{-1}\) was less than 0.03 for \(\langle T_{1/3}\rangle\leq 8\) s (red line, Figure 14). The value of \(S_{d}(T_{1/3})\langle T_{1/3}\rangle^{-1}\) also increased with the longer wave periods.  
Figure 14 shows the ratio of the wave heights as \(\langle H_{1/3}\rangle/H_{0}\)\(-\)1 as a function of \(H_{0}\), where \(H_{0}\)= \(4(M_{0})^{1/2}\). For example, the value of \(\langle H_{1/3}\rangle/H_{0}\)\(-\)1 was approximately \(-\)0.05 for \(H_{0}\) = 2 m for \(H_{0}\) = 2 m (blue solid line, Figure 14), showing \(\langle H_{1/3}\rangle=0.95H_{0}\) or \(H_{0}\)= \((1/0.95)H_{1/3}\). The wave height by the zero-up crossing method (\(H_{1/3}\)) was slightly lower than that of \(H_{0}\). Figure 14 shows the value \(\langle T_{1/3}\rangle/T_{0}\)\(-\)1 as a function of \(T_{0}\), where \(T_{0}\) = \(M_{-1}M_{0}^{-1}\). The value of \(\langle T_{1/3}\rangle/T_{0}\)\(-\)1 was 0.03 for \(T_{0}\)\(\simeq\)7.5 s (green solid line), indicating that \(\langle T_{1/3}\rangle=1.03T_{0}\) or \(T_{0}\) = \((1/1.03)\langle T_{1/3}\rangle\). The value of \(\langle T_{1/3}\rangle\) was slightly longer than \(T_{0}\). The ratio \(\langle T_{1/3}\rangle/T_{0}\) decreased with the longer wave periods.