Parametric Spectrum

A study on the characteristics of wave spectra over the seas around Korea by using parametric spectrum method Il-Ju Moon and Im Sang Oh Department of Oceanography, Seoul National University, seoul 151-742, Korea ABSTRACT

The characteristics of wave spectra over the seas around Korea have been studied by using a parametric spectrum method that expresses observed spectra by analytic function. This paper presents a newly developed TMA spectrum called "double-peaked TMA spectrum", which is decomposed into two parts, the low frequency components and the high frequency components of the energy. The proposed spectrum has been applied for the observed 17,750 spectra over the seas around Korea. As a result, this method showed a better fitness of 20% than the previous TMA spectrum (Bouw et al., 1985a) for double-peaked spectra.
A statistical analysis on the parameters of the double-peaked TMA spectrum was carried out. From this results, 25% of the total analyzed spectra are found to be the double-peaked type and the occurrence of this type decreases with increasing significant wave height, and 58% of the double-peaked spectra are found to be the swell-dominated spectra which are dominated by the low frequency peak. This paper also presents the probable spectra which are expected to occur with 95% confidence limits for a given sea severity around Korea.

(Keywords: wave spectrum, parametric spectrum, double-peaked wave spectrum) INTRODUCTION

    The spectral information of power spectrum, which is extracted from wave data, is most commonly used for the studies of ocean wave characteristics and for the design of marine vessels and structures. The individual raw spectrum from wave observations is difficult to apply to the design because it is not a continuous spectrum but a discrete spectrum in frequency domain, and the information collected over a period long enough to ensure statistical representation would yield so many spectra that it would be impossible to use them all. Therefore, they have to be averaged, by season or type. In analyzing spectral information, two methods are widely used. One is the sampling method which is to collect representative spectra during a specified period, the other is statistical method which is to represent the observed spectra by using two parameters, significant wave height and peak frequency. However, sampled spectra would also show a large degree of variability, based in part on natural variations and in part on statistical fluctuations. Furthermore, the shape of wave spectra observed in ocean varies considerably, even though the significant wave heights and peak frequency are the same, depending on the duration and fetch, stage of growth and decay of wind wave, and existence of swell. Therefore, sampled spectra and the representation of wave spectra by two parameters are insufficient to explain all characteristics of wave spectra in a specific area and periods. For example, Fig. 1 shows a variety of shapes of wave spectra, all of which have the same significant wave heights of 1.21 m. As can be seen in the figure, the spectrum 1 has sharp single peak at the lower frequencies, while spectra 2 and 3 have double peaks. Furthermore, two spectra (4 and 5) have the same peak frequency of 0.22. Thus, even though two parameters (significant wave height and peak frequency) are the same, the shape of spectra may be significantly different. Therefore, additional parameters are required for more accurate representation of wave spectra to provide useful spectral information of wave. This makes a need for the parametric spectrum.
    The parametric spectrum is expressed in terms of an analytical curve that preserves the feature of the observed spectra and includes a small number of physically meaningful parameters. Thus, because the parametric spectrum provides a continuous spectral information and more accurate representation of wave spectra, it is widely used to investigate the characteristics of spectra. At the present study, the characteristics of wave spectra over the seas around Korea are investigated by using a newly developed parametric spectrum.
    Many parametric spectra of the form E(f), where E is the energy per unit bandwidth and f the frequency, have been proposed over the years ; among the best known are those of Pierson and Moskowitz (1964), Hasselmann et al. (1973), and Bouws et al. (1985a).
    For fully developed wind waves in the open ocean, Pierson and Moskowitz (1964) proposed a form of the power spectrum (PM spectrum), which shows the fetch-independent form. Hasselmann et al. (1973) proposed the JONSWAP spectrum for the fetched-limited wind waves in the ocean. The effect of the additional factors for the JONSWAP spectrum is allowed for narrower, more peaked spectra which are typical forms of growing wind seas in deep water. Bouws et al. (1985a) suggested a finite water depth spectral shape called TMA spectrum to be applicable to wave conditions in shallow water. TMA spectral form has the additional parameter, water depth, h as well as the four JONSWAP parameters, the Phillips' constant, a, the peak frequency, f_m, the peak enhancement factor, r and the spectral width factor, sigma. The form is expressed as

where

where k is wave number.
    Although all these parametric spectra have some features in common, they differ in their precise shape and in the number and nature of their fitting parameters. It is clearly important to choose a parametric spectrum that combines simplicity with a good fit to observed conditions to provide a spectral climate in an area.
    At the present study, in order to choose a parameter spectrum which has a good fit over the study area, three of the best known form, PM (Pierson and Moskowitz, 1964), JONSWAP (Hasselmann et al., 1973), TMA (Bouw et al., 1985a) spectrum, have been applied for 17,750 spectra observed over the seas around Korea. As a result, the TMA spectrum fitted the observed spectra best, but this spectrum showed a lot of errors in representing measured spectra exhibit two peaks. So this paper presents a newly developed TMA spectrum, which is hereafter called a double-peaked TMA spectrum, to resolve the problem.
    Therefore, the main purposes of this study are to present a newly developed parametric spectrum providing a good fit to measured spectra over the seas around Korea and to investigate the characteristics of wave spectra over the areas through the statistical analysis of the proposed parametric spectrum.
The next section presents wave data and spectra used. The basic concepts of double peaked TMA spectrum adopted here and statistical analysis using the spectral information are given in the following the section. The last section discusses the results of fitting the parametric spectrum to measured ones and investigates the spectral characteristics over the study area.
WAVE DATA AND SPECTRA

The wave data used in the present study were obtained from 10 wave stations over the seas around Korea during a period from 1988 to 1993. Locations of wave observations gauge types, depth of mooring, and observation periods are shown in Fig. 2. Computations of spectra were carried out by FFT method. The number of data points for one subsample is 1024 and Nyquist frequency is 0.5 Hz. Measured wave spectra are corrected by using the band-pass filter to remove the unnecessary frequency band. From this procedure, a total of 17,750 spectra were obtained and used for formulation and statistical analysis of parametric spectrum.

REPRESENTATION OF DOUBLE-PEAKED TMA SPETRA

Spectra with two peaks occur when there is simultaneously swell and wind sea or when a refreshing or a changing direction wind creates a developing wave spectrum. This is a quite common situation over the seas around Korea (Moon, 1994).
Ochi and Hubble (1976) represented spectra with two peaks by a modification of the PM spectrum, and Soares (1984) modeled double-peaked spectra with two JONSWAP types of spectra. However, because the PM and JONSWAP forms have been proposed for deep water, applying them for shallow water can be incorrect. Therefore, in this study we are proposing a newly developed spectral form called double-peaked TMA spectra, which include the depth effect of wave in shallow water as well as spectra with two peaks.
In the development of double-peaked TMA spectrum, the wave spectra are decomposed into two parts, the lower frequency and the higher frequency components (see Fig. 3). Then, each of two components is expressed in a TMA formula (given Equation 1) with five parameters, respectively ; i.e. , the Phillips' constant, the peak frequency, the peak enhancement factor, the spectral width factor, and water depth. Finally, the double-peaked TMA spectra are expressed by a combination of two sets of TMA spectra as given Equation 3.

For each TMA spectra the spectral parameter a, r, fm, and sigma are obtained as the following :

(1) Determination of fm having a maximum energy,

(2) Determination of a by following equation over the range 1.35 fm to 2.0 fm,

(3) Determination of r from the ratio of a peak maximum energy of the observed spectrum to one of the PM spectrum with the same values of fm and a (given in Equation 6)

(4) Determination of sigma by finding the final value from 0.01 to 1 for which the difference between the observed and the theoretical spectra,

is minimized, where

After the first TMA spectrum is calculated, second TMA spectra can be obtained from the difference between first spectra and observed spectra. At this point, in order to classify spectra with two peaks, three criteria are required :

(1) Maximum energy of second TMA spectra should be greater than a third of first TMA spectra.

(2) Distance between frequencies of two spectral peaks should be more than 0.05 Hz

(3) The trough between the two spectral peaks should have an ordinate smaller the lower 90% confidence limit of each peak as suggested by Houmb and Due (1978)

If all above criteria are satisfied, the spectra are double-peaked and then they are expressed by the sum of the first and second TMA spectra. Otherwise, they are single-peaked. Examples of comparisons between observed spectra and parametric spectra are shown in Fig. 4(a) and (b). Fig. 4(a) shows a comparison for the case when swell coexists with wind-generated waves and hence the spectrum has double peaks, while Fig. 4(b) shows a comparison for severe sea in which is partially developed by strong wind, and has a very sharp peak at the lower frequencies in the spectrum. As can be seen in these examples, the double-peaked TMA spectra appear to represent two spectral forms well, while TMA spectra (Bouw et al., 1985a) fit well only for the single peaked spectra.

STATISTICAL ANALYSIS OF PARAMETRERIC SPECTRA

    In the preceding section, the double-peaked TMA spectra that represent well a variety of spectral shapes observed in the ocean were suggested. In this section, a statistical analysis on each parameter of the double-peaked TMA spectrum is carried out so that the characteristics of wave spectra can be investigated and the shapes of spectra for a given sea severity around Korea are established.
    Statistical treatment used in this paper basically follows that of Ochi and Hubble (1976) which provides an adequate representation of the whole data base, and yields an unbiased best estimate with a confidence interval. However, there is a difference between their method and present way in that they use the PM spectrum while the TMA spectrum is used here. In the present way, two sets of TMA spectrum in terms of 5 parameters are defined, respectively, which represent more different spectral shapes observed in shallow sea as well as in deep sea than the ones represented by Ochi and Hubble (1976).
    For this purpose, a total of 17,750 spectra observed over the seas around Korea are classified into ten groups depending on severity as given in Table 1. The histograms of each parameter are expressed in terms of the rate of relative occurrence and then, probability density distributions are estimated from the histograms of each parameter for 10 groups. The normal distribution is assumed for the best fit distribution for these parameters. Comparisons between histogram and probability density function of normal distribution are shown in Fig. 5.
    From this procedure, the mean, upper and lower bound values for a parameter with 95% confidence limits were calculated.

where is the variance and n is the total number. For three values of the parameter, the values of the other parameters are obtained from the averages in the region of +-5% of the selected parameter. For example, the a-value for fm is obtained by taking the average value of a from a sample which belonged to +-5% of fm. Next, five parameters, a, r, fm, sigma_a and sigma_b, are expressed as function of significant wave height by least square curve fitting method (see Fig 6 ). The same procedure in order to derive a set of three spectra associated with fm, is carry out for other four parameters, and thus a total of fifteen spectra can be made for a given sea severity. These spectra are considered as "probable spectra" representing a specified sea to be expected to occur with 95% confidence.
RESULTS AND DISCUSSION

The quality of fitting observed spectra to parametric spectra is characterized by the normalized root-mean-square difference between the parametric and observed spectra, called goodness of fit, which is defined as

where P(fi) is observed spectra and S(fi) is parametric spectra. For 17,750 spectra observed over the seas around Korea, the goodness of fit for three of the best known form, PM (Pierson and Moskowitz, 1964), JONSWAP (Hasselmann et al., 1973), TMA (Bouw et al., 1985a) spectrum, were 0.453, 0.371, and 0356, respectively. The TMA spectrum best fitted the observed spectra, but this spectrum showed a lot of errors in representing measured spectra exhibit two peaks. So the double-peaked TMA spectrum proposed in this study were applied for the same spectral data. The results showed that double-peaked TMA spectrum had a better fitness of 20% than TMA spectrum in representing the observed spectra which exhibit two peaks as can be seen in Table 2, and excellent fitting with increasing significant wave height.
    In the statistical analysis on spectra, the existing spectra have been grouped according to the significant wave height, and the percentages of double-peaked spectra in each wave group have been obtained as shown in Table 3. Double-peaked spectra have been identified according to the criteria mentioned in the previous section. These results show that the percentage of occurrence of the double-peaked spectra decreases with increasing significant wave height. Their occurrences change from about 40% in lower wave groups to 4-7% in higher ones, having an overall average value of 25%.
    In the mean while, if the spectra were inspected more carefully, it was found that there were two types of double-peaked spectra that should be treated separately. The differentiating criterion for the two types was related to which of two peaks dominated (Soares, 1984). One type of spectrum was dominated by the high frequency peak. Such spectrum could have been generated by a low frequency swell system that had traveled a considerable distance loosing much energy before meeting a wind wave system. This type of spectrum is called wind-dominated spectrum. On the other hand, spectra dominated by the low frequency peak could have been generated by a refreshing wind or by a change in wind direction which creates a system of short period waves coexisting with the "old" wave system. When the wind does not continue to drive the old system longer, the wave components become uncoupled and the wave system turns into swell. This type of spectrum is called swell dominated spectrum (Soares, 1984).
    Table 3 presents the relative occurrence of swell dominated spectra among the double-peaked spectra of each wave group. It is probably reasonable to expect swell dominated spectra to occur more often than wind dominated spectra, since wind speed and direction shows a large variability and a new wave system is created whenever the wind changes direction or intensity. However, the present result shows that 58% of the double-peaked spectra are of the swell dominated type over the seas around Korea, which is somewhat higher percentage than expected one. This can be explained that the study areas are easily affected by swell that had traveled from the Pacific Ocean. The results also showed that this percentage of swell-dominated spectra tends to increase with increasing significant wave height.
    From the results of analysis on the parameters of the double-peaked TMA spectrum, probable spectra, which are expected to occur with 95% confidence limits, were established for a given sea severity. The values of parameters for these spectra are expressed in terms of significant wave height, and presented in Table 4. Fig. 7 shows examples of the family of the probable spectra for various significant wave heights obtained from double-peaked spectra of six stations. These spectra can be considered to represent realistic sea spectra for the stations.
CONCLUSIONS

    The characteristics of wave spectra over the seas around Korea have been studied by using a parametric spectrum method that expresses observed spectra by analytic function. In this paper, a newly developed TMA spectrum called a double-peaked TMA spectrum has been applied for the observed 17,750 spectra over the seas around Korea. As a result, the proposed parametric spectra showed a better fitness of 20% than the TMA spectrum (Bouw et al., 1985a) for double-peaked spectra.
    A statistical analysis was carried out on the parameters of the double-peaked TMA spectrum. From the results of analysis, probable spectra, which are expected to occur with 95% confidence limits, were established for a given sea severity around Korea, and the characteristics of waves in analyzed area were studied.
    The characteristics of wave spectra over the study area were shown as following two results. First, 25% of the total analyzed spectra were of the double-peaked type and the occurrence of this type decreased with increasing significant wave height. Second, 58% of the double-peaked spectra were the swell-dominated spectra, which were dominated by the low frequency peak. The occurrence of swell-dominated spectra increased with increasing significant wave height.

ACKNOWLEDGMENTS

This study is supported by the Research Fund of the Meteorological Research Institute, Korea Meteorological Administration (NDE-01-01-04). The authors wish to express their sincere gratitude to Korea Ocean Research & Development Institute (KORDI) for providing the wave data of Korean coasts.

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