A study of the characteristics of wave spectra over the seas
around Korea by using a parametric spectrum method
 
Il-Ju Moon1 and Im Sang Oh1
 
(received 1998/2/7, revised 1998/4/22, accepted 1998/4/29)
 
ABSTRACT
 

The characteristics of wave spectra over the seas around Korea have been studied by using a parametric spectrum method that expresses observed spectra in an analytic function. This paper presents a newly developed TMA spectrum called the ‘double-peaked TMA spectrum?, which is deconstructed into two parts: the low and high frequency components of the energy. The proposed spectrum was applied for the observed 17,750 spectra over the seas around Korea. As a result, this method showed a 20% better fitness than the previous TMA spectrum (Bouws et al., 1985a) for double-peaked spectra.

A statistical analysis on the parameters of the double-peaked TMA spectrum was carried out. From these results, 25% of the total analyzed spectra were found to be the double-peaked type, and the occurrence of this type decreased with increasing significant wave height; also, 58% of the double-peaked spectra were found to be the swell-dominated spectra that 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)
1 Department of Oceanography, Seoul National University, Seoul 151-742, Korea
 
INTRODUCTION
 
The spectral information of a 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. It is difficult to apply the individual raw spectrum from wave observations to the design because it is not a continuous spectrum. But rather it is a discrete spectrum in the frequency domain; the information collected over a period long enough to ensure statistical representation would yield so many spectra that it would be impossible to use all of them. 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, and the other is the 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 partly on natural variations and partly on statistical fluctuations. Furthermore, the shape of wave spectra observed in the ocean varies considerably despite the significant wave heights and peak frequency being the same depending on the duration and fetch, stage of growth and decay of wind wave, and the existence of swells. Therefore, sampled spectra and the representation of wave spectra by two parameters are insufficient to explain all the characteristics of wave spectra in a specific area and period. 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, spectrum 1 has sharp single peaks at the lower frequencies, while spectra 2 and 3 have double peaks. Furthermore, two spectra (4 and 5) have identical peak frequencies 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, to provide useful spectral information of waves, additional parameters are required for more accurate representations of wave spectra. This creates the need for the parametric spectrum.
  Fig. 1 Variety of wave spectra for a significant wave height of 1.21 m. Spectrum 1, 4 and 5 are the single-peaked spectra, while spectrum 2 and 3 are the double-peaked spectra.
  The parametric spectrum is expressed in terms of an analytical curve that preserves the features of the observed spectra and includes a small number of physically meaningful parameters. Thus, because the parametric spectrum provides continuous spectral information and more accurate representations of wave spectra, it is widely used to investigate the characteristics of spectra. In the present study, the characteristics of wave spectra over the seas around Korea were 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 allows 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 the TMA spectrum, to be applicable to wave conditions in shallow water. The TMA spectral form has the additional parameter h, water depth, as well as the four JONSWAP parameters: the Phillips? constant (a ), the peak frequency (fm), the peak enhancement factor (g ), and the spectral width factor (s ). The form is expressed as:

where

and k is the 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.

In the present study, in order to choose a parameter spectrum having a good fit over the study area, three of the best known forms, PM (Pierson and Moskowitz, 1964), JONSWAP (Hasselmann et al., 1973), and TMA (Bouws et al., 1985a) spectrum, were applied for 17,750 spectra observed over the seas around Korea. As a result, the TMA spectrum best fit the observed spectra, but this spectrum showed several errors in representing measured spectra exhibiting two peaks. This paper thus 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 by using 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 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 the FFT method. The number of data points for one subsample is 1024 and the Nyquist frequency is 0.5 Hz. Measured wave spectra were 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 the parametric spectrum.
 
REPRESENTATION OF DOUBLE-PEAKED TMA SPECTRA
 
Spectra with two peaks occur when there is simultaneous swell and wind sea or when a refreshing or direction-changing 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 on shallow water may be incorrect. Therefore, in this study, we are proposing a newly developed spectral form called the double-peaked TMA spectrum, which includes the depth effect of waves in shallow water as well as spectra with two peaks.
 
 

 

In the development of double-peaked TMA spectra, the wave spectra are deconstructed into two parts: the lower and the higher frequency components (see Fig. 3). Then, each of the two components is expressed in a TMA formula (given Equation 1) with five parameters: 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 in Equation 3:

For each TMA spectra the spectral parameters a , g , fm , and s are obtained as such:

  1. Determination of fm having a maximum energy,
  2. Determination of a by the following equation over the range 1.35 fm to 2.0 fm,
  1. Determination of g from the ratio of the observed spectrum’s peak maximum energy to one of the PM spectra with the same values of fm and a (given in Equation 6).

    2.  
     

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

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

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

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

If all the above criteria are satisfied, the spectra will be double-peaked, and 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 in which a swell coexists with wind-generated waves, and hence the spectrum has double peaks. Fig. 4(b) shows a comparison for severe sea, which is partially developed by strong winds 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 accurately represent two spectral forms, while TMA spectra (Bouws 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 accurately represent 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) in that it 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 the present method in that they use the PM spectrum while the TMA spectrum is used in the present case. In the present method, two sets of TMA spectra are defined in terms of 5 parameters, 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).
 

Fig. 4 Comparisons of observed, TMA, and double peaked TMA spectra (a) for the case having two peaks, (b) for the case having a single peak.   For this purpose, a total of 17,750 spectra observed over the seas around Korea are classed 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 histograms and probability density functions of normal distributions are shown in Fig. 5.

From this procedure, the mean and the 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 , g , fm, s a and s b, are expressed as a function of significant wave height by the least square curve fitting method (see Fig 6). The same procedure is carried out for the other four parameters in order to derive a set of three spectra associated with fm and thus a total of fifteen spectra can be made for a given sea severity. These spectra are considered ‘probable spectra?, representing a specified sea and to be expected to occur with 95% confidence.

 
Fig. 5 Comparison between the histogram of probability density and the normal distribution for five parameters (Group II, in Kangneung).

Fig. 6 Five parameters as a function of significant wave height (by least square curve fitting method).
 

Table 1. Groups of significant wave height used for analysis.

  
Group
Significant wave height (Hs)
Nominal
Range
I
II
III
IV
V
VI
VII
VIII
IX
X
0.2 m
0.6 m
1.0 m
1.4 m
1.8 m
2.2 m
2.6 m
3.0 m
3.4 m
3.8 m
Less than 0.4 m
0.4 m - 0.8 m
0.8 m - 1.2 m
1.2 m - 1.6 m
1.6 m - 2.0 m
2.0 m - 2.4 m
2.4 m - 2.8 m
2.8 m - 3.2 m
3.2 m - 3.6 m
Higher than 3.6 m
 
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 the observed spectra and S(fi) is the parametric spectra. For 17,750 spectra observed over the seas around Korea, the goodness of fit for three of the best known forms, PM (Pierson and Moskowitz, 1964), JONSWAP (Hasselmann et al., 1973), and TMA (Bouws et al., 1985a) spectrum, were 0.453, 0.371, and 0356, respectively. The TMA spectrum best fit the observed spectra, but this spectrum showed several errors in representing measured spectra exhibiting two peaks. So the double-peaked TMA spectra proposed in this study were applied for the same spectral data. The results showed that the double-peaked TMA spectrum had a 20% better fitness than the TMA spectrum in representing the observed spectra which exhibited 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 were grouped according to the significant wave height, and the percentages of double-peaked spectra in each wave group were 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%.
 

Table 2. "goodness of fit" for TMA spectrum (Bouws et al., 1985a) and double-peaked TMA spectrum.  
TMA spectrum (Bouws et al., 1985a)
Double-peaked TMA Spectrum
Single-peaked spectra
0.333
0.333
Double-peaked spectra
0.425
0.345
Average
0.356
0.336
 

In the meanwhile, when 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 whichever of the two peaks was dominant (Soares, 1984). One type of spectrum was dominated by the high frequency peak. Such a spectrum could have been generated by a low frequency swell system that had traveled a considerable distance, losing much energy before meeting a wind wave system. This type of spectrum is called a 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 a 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 show 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 a somewhat higher percentage than the expected one. This can be explained by the fact 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.
 

ST.
KANGNEUNG
DONGHAE
JUJEON
Hs
R1
(number)
R2
R3
R1
(number)
R2
R3
R1
(number)
R2
R3
~ 0.4m 
0.4m ~ 0.8m 
0.8m ~ 1.2m 
1.2m ~ 1.6m 
1.6m ~ 2.0m 
2.0m ~ 2.4m 
2.4m ~ 2.8m 
2.8m ~ 3.2m 
3.2m ~ 3.6m 
3.6m ~ 
 .30(543) 

.27(496) 

.16(266) 

.10(184) 

.08(153) 

.05(82) 

.03(59) 

.012(22) 

.005(9) 

.014(26)
.22
.12
.06
.09
.08
.02
.00
.05
.00
.12
.81
.73
.75
.69
.67
.50
1.0
.00
.67
1.0
 .26(498) 

.30(593) 

.17(329) 

.09(184) 

.08(158) 

.05(88) 

.03(54) 

.01(21) 

.008(15) 
 

 .006(12)
.46
.20
.08
.04
.03
.02
.02
.10
.00
.08
.64
.54
.65
.63
1.0
.50
1.0
.00
1.0
.00
 .17(390) 

.33(756) 

.25(576) 

.11(260) 

.07(171) 

.03(76) 

.016(38) 

.007(17) 

.005(12) 

.007(16)
.49
.29
.19
.18
.16
.09
.08
.06
.25
.19
.67
.57
.61
.40
.52
.86
1.0
.00
.67
.67
(Total)
Mean
(1840)
.13
.77
(1952)
.20
.61
(2312)
.26
.59
 
ST.
YEOSU
HONGDO
EOCHEONGDO
Hs
R1
(number)
R2
R3
R1
(number)
R2
R3
R1
(number)
R2
R3
~ 0.4m 

0.4m ~ 0.8m 

0.8m ~ 1.2m 

1.2m ~ 1.6m 

1.6m ~ 2.0m 

2.0m ~ 2.4m 

2.4m ~ 2.8m 

2.8m ~ 3.2m 

3.2m ~ 3.6m 

3.6m ~ 

 		.09(401) 

.33(1494) 

.29(1286) 

.16(728) 

.07(308) 

.07(118) 

.02(92) 

.009(42) 

.002(9) 

.001(5)

 		.65 

.47 

.33 

.27 

.22 

.19 

.10 

.07 

.11 

.00

 		.51 

.57 

.62 

.54 

.52 

.50 

.56 

1.0 

1.0 

.00

 		.37(727) 

.32(629) 

.14(279) 

.067(133) 

.047(93) 

.024(48) 

.011(22) 

.008(16) 

.009(18) 

.014(27)
.46 
.22 
.09 
.05 
.03 
.02 
.00 
.00 
.00 
.00
 .60 

.56 

.50 

.33 

.33 

.00 

.00 

1.0 

.00 

.00
.35(375) 
.30(323) 
.16(175) 
.10(113) 
.044(48) 
.013(14) 
.015(16) 
.004(4) 
.006(7) 
.006(7)
.50 
.30 
.25 
.10 
.08 
.07 
.00 
.00 
.00 
.00
.57 
.52 
.47 
.55 
.75 
.00 
1.0 
.00 
1.0 
.00
(Total)
Mean
(4483)
.38
.57
(1992)
.25
.58
(1082)
.32
.54
  Table 4. Values of parameters for probable spectra with a single peak in Kangneung, which are expressed in terms of significant wave height (x).  
 
a 
fm
g 
s a
s b
 
A B
A B
A B
A B
A B
Probable 
Spectra
.00125 .79501 
.00135 .83748 
.00082 .98212 
.00110 .95598 
.00077 1.03872 
.00098 .92915 
.00100 .86787 
.00049 1.22836 
.00107 .94860 
.00102 .99688 
.00087 .95084
.18414 -.20648 
.18902 -.18697 
.17180 -.16708 
.18901 -.19939 
.16894 -.14521 
.19199 -.22145 
.19240 -.21139 
.18778 -.20116 
.19586 -.19673 
.20131 -.20263 
.19151 -.21331
2.03174 .06539 
1.85818 .11302 
2.53281 -.01657 
2.46326 -.06156 
2.61692 .01851 
3.43963 -.11424 
3.30958 -.01898 
4.32939 -.26245 
2.77223 -.06146 
2.75875 -.12323 
2.95769 -.05020
.21798 .15853 
.24124 .11307 
.28280 .10652 
.44233 -.41208 
.20073 -.01198 
.22802 .13172 
.22200 -.03596 
.24564 .21855 
.24384 -.01061 
.22610 .26114 
.15100 .30649
.03991 .14957 
.03373 .21235 
.04982 .04847 
.04356 .10453 
.04700 .10596 
.02860 .19144 
.04700 .19497 
.05261 .05200 
.04588 .08487 
.04154 .08092 
.04998 .08973
 
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 an analytic function. In this paper, a newly developed TMA spectrum called a double-peaked TMA spectrum was 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 (Bouws 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 the analyzed area were studied.

The characteristics of wave spectra over the study area were shown as the 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 ? Research on the Forecast System of Tide and Wave over the Seas around Korea Peninsula (1) ?, Meteorological Research Institute, (PD-01-01-04). The authors wish to express their sincere gratitude to the Korea Ocean Research & Development Institute (KORDI) for providing the wave data of Korean coasts.
 
 
 
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