Comparison of the theory with field data

It is very difficult to check the theory developed in this paper. The main difficulty is that the probability of maximum wave height depends on both $mu_4$ and $N$. It means that a huge amount of data including spectral information such as the BF Index, and statistical parameters such as $mu_4$, etc are required to verify the theory. Unfortunately, from the present operational observations not all these parameters have been obtained and archived. Therefore, we will only try to check the dependence of the maximum wave on $mu_4$ qualitatively using the available field data set.

The observed data was originally collected by the Tokyo Electric Power Company using an ultra sonic wave gauge at a depth of 30m, off the coast of the Pacific Ocean. The length of each record was 20min and the data were collected every hour from March 1 to the end of June in 2001. The wave statistics such as $H_{max}$, $H_{1/3}$, $T_{1/3}$, $N$, $mu_3$, and $mu_4$ were operationally calculated and archived. Note that the water depth of 30m is relatively shallow water. Therefore, to eliminate shallow water effects³, the data are excluded if the dimensionless water depth $k_ph$ is less than 2.0 (it corresponds to $T_{1/3}ge 8$s). The total number of valid data was about 2546.

Figure shows the direct comparison between $H_{max}/H_{1/3}$ and $mu_4$. The linear correlation between $H_{max}/H_{1/3}$ and $mu_4$ is only 0.73. However, it is well known that $H_{max}/H_{1/3}$ not only depends on $mu_4$ but also on $N$. Thus, the data are stratified according to $mu_4$ and $N$ and are compared with the theory. Figure shows the direct comparison of the maximum wave height distribution between observed data and theory for the $N=150-200$ bin. The histogram shows the observed PDF of the maximum wave height, while the solid line and the dashed line indicate Eq.() and Rayleigh theory, respectively. The number of wave records in each category is indicated by the `sample' number. For fixed number of waves, the maximum wave height distribution according to Rayleigh theory is constant, although the observed data shows a clear dependence of the PDF on $mu_4$. The peak of observed PDF is lower than Rayleigh theory for $mu_4<3$ but becomes higher than Rayleigh theory for $mu_4>3$. The maximum wave height distribution predicted by Eq.() qualitatively agrees with the observed data, although it slightly underestimates.

Next, we discuss the general behavior of the PDF of maximum wave height in the nonlinear wave field, by showing the ensemble averaged $H_{max}/H_{1/3}$ of each bin as a function of $mu_4$ and $N$ in Figure . The brackets $<>$ indicate the ensemble averaged value. Figure (a) is observed data and (b) is the expected value of Eq.() through numerical integration. The dependence of $H_{max}/H_{1/3}$ on $N$ is weaker than expected from Eq.(). This is because the length of observed time series was fixed to 20min, so we cannot discuss the dependence of $H_{max}/H_{1/3}$ on number of waves in detail. On the other hand, the dependence of $langle H_{max}/H_{1/3} rangle$ on $mu_4$ is clear. The theoretically predicted $langle H_{max}/H_{1/3} rangle$ is underestimated compared to the observed data but it agrees with the observed data in a qualitative sense. The observed $langle H_{max}/H_{1/3} rangle$ monotonically increases for increasing $mu_4$, but for high values of kurtosis the theoretically estimated value of $langle H_{max}/H_{1/3} rangle$ is lower.

Figure shows the comparison between observed data and theory of freak wave occurrence frequency, $P_{freak}$. To eliminate statistical fluctuations, the observed data is excluded if the number of samples is less than 20. The observed $P_{freak}$ clearly increases as $mu_4$ is increased. However, there is no clear dependence of $P_{freak}$ on $N$ while according to theory there should be. The total number of wave trains is 2546 but this is still not sufficient to examine the validity of theory completely. Hence, more data will be required to verify the theory quantitatively.