# Estimation and Analysis of JONSWAP Spectrum Parameter Using Observed Data around Korean Coast

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## Abstract

**:**

## 1. Introduction

## 2. Materials and Methods

#### 2.1. Observation Data

#### 2.2. Method

## 3. Result and Discussion

#### 3.1. Relationship between Significant Wave Height and Wave Period

#### 3.2. Estimation of JONSWAP Spectrum Parameter

- -
- Log-normal distribution$$\mathrm{f}\left({\mathsf{\gamma}}_{c};\mathsf{\mu},\mathsf{\sigma}\right)=\frac{1}{\sqrt{2\pi}\mathsf{\sigma}{\mathsf{\gamma}}_{c}}\mathrm{exp}\left(-\frac{{\left(ln{\mathsf{\gamma}}_{c}-\mathsf{\mu}\right)}^{2}}{2{\mathsf{\sigma}}^{2}}\right)$$

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- Gamma distribution

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- Normal distribution

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- Weibull distribution

## 4. Conclusions

- (1)
- The relationship between the significant wave height and wave period proposed by Goda [27] and SPM [28], which is commonly used, tends to follow the lower limit at all points. In addition, in the calculated relational expression, $\mathsf{\xi}$ and $\tau $ were approximately 2 and 0.5 times different, respectively. When determining the design wave height, it is difficult to use the previously proposed relational formula, and the relationship between the significant wave height and period should be sufficiently understood using the observational data of the target sea area.
- (2)
- PCF estimated using the observation data for all six sites were found to be approximately 40% smaller, with $\overline{{\mathsf{\gamma}}_{c}}=1.13\u20131.42$, than the previously reported PCF, ${\mathsf{\gamma}}_{c}=3.3$. The JONSWAP spectrum can be applied to waves with large height ranges. Therefore, as a result of calculating PCF in the ${H}_{s}\ge 4.0\mathrm{m}$ region, which is considered to be a high-frequency region, $\overline{{\mathsf{\gamma}}_{c}}=1.40\u20132.1$ was obtained, showing a difference of up to 58% from the previously reported value. In addition, as the wave height decreased, the dispersion of ${\mathsf{\gamma}}_{c}$ increased, and ${\mathsf{\gamma}}_{c}$ for each section was distributed between approximately 1.0 and 2.0.
- (3)
- As shown in Figure 5 and Table 3, the probability density distribution of ${\mathsf{\gamma}}_{c}$, calculated by applying the distribution fit test and KL divergence method, showed differences amid the sea areas of the Korean Peninsula. In the case of the East and Yellow coasts of the Korean Peninsula, the gamma and log-normal distributions, respectively, were calculated as the most similar distributions, showing a significant difference from the normal and Weibull distributions.

## Author Contributions

## Funding

## Conflicts of Interest

## Appendix A

## References

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**Figure 2.**Example of the spectrum calculated using FFT transformation with observed WSE data. (

**a**) Example of observed WSE data. (

**b**) Example of wave spectrum.

**Figure 3.**Definition of JONSWAP spectrum parameters and variations in spectrum shape for the PEF and PCF.

**Figure 6.**Comparison of the observed and JONSWAP spectrum with estimated ${\mathsf{\gamma}}_{c}$, ${\mathsf{\gamma}}_{c}=3.3$.

Station Names (Codes) | Longitude | Latitude | Depth (m) | Observation Period | Material |
---|---|---|---|---|---|

Gonghyeonjin (GHJ) | $128\xb0{31}^{\prime}{41.6}^{\u2033}E$ | $38\xb0{21}^{\prime}{41.6}^{\u2033}N$ | 32.0 | 2016.04.29.–2020.11.06. | AWAC (Nortek, Norway) |

Maengbang (MB) | $129\xb0{14}^{\prime}{05.2}^{\u2033}E$ | $37\xb0{24}^{\prime}{00}^{\u2033}N$ | 31.0 | 2013.09.27.–2020.11.05. | |

Hupo (HP) | $129\xb0{29}^{\prime}{03.4}^{\u2033}E$ | $36\xb0{41}^{\prime}{59.1}^{\u2033}N$ | 17.5 | 2015.07.03.–2020.11.06. | |

Ulsan new port (USN) | $129\xb0{22}^{\prime}{52.2}^{\u2033}E$ | $35\xb0{23}^{\prime}{30.44}^{\u2033}N$ | 29.0 | 2017.12.15.–2020.11.10. | Signature ADCP 500 (Nortek, Norway) |

HeMOSU-1 (H1) | $126\xb0{07}^{\prime}{45}^{\u2033}E$ | $35\xb0{27}^{\prime}{55}^{\u2033}N$ | 13.5 | 2013.07.28.–2014.07.06. | Waveguide (Delft, Netherlands) |

HeMOSU-2 (H1) | $126\xb0{12}^{\prime}{45}^{\u2033}E$ | $35\xb0{49}^{\prime}{40}^{\u2033}N$ | 30 | 2013.11.26.–2014.04.23. |

Station | ${\mathit{H}}_{\mathit{s}}\ge 4\mathit{m}$ | $3\mathit{m}\le {\mathit{H}}_{\mathit{s}}<4\mathit{m}$ | $2\mathit{m}\le {\mathit{H}}_{\mathit{s}}<3\mathit{m}$ | $1\mathit{m}\le {\mathit{H}}_{\mathit{s}}<2\mathit{m}$ | ${\mathit{H}}_{\mathit{s}}<1\mathit{m}$ | |
---|---|---|---|---|---|---|

GHJ | $\overline{{\mathsf{\gamma}}_{c}}$ | 1.47 | 1.40 | 1.46 | 1.55 | 1.38 |

$\Delta {\mathsf{\gamma}}_{c}$ | 0.8–2.6 | 0.7–3.0 | 0.45–3.15 | 0.3–5.05 | 0.25–5.85 | |

Ratio | 0.33% | 1.16% | 4.23% | 20.51% | 73.76% | |

MB | $\overline{{\mathsf{\gamma}}_{c}}$ | 1.40 | 1.37 | 1.43 | 1.48 | 1.30 |

$\Delta {\mathsf{\gamma}}_{c}$ | 0.8–2.3 | 0.6–2.65 | 0.45–3.45 | 0.35–4.1 | 0–8.2 | |

Ratio | 0.24% | 0.94% | 5.14% | 23.17% | 70.50% | |

HP | $\overline{{\mathsf{\gamma}}_{c}}$ | 1.45 | 1.40 | 1.42 | 1.45 | 1.20 |

$\Delta {\mathsf{\gamma}}_{c}$ | 0.05–2.25 | 0.05–3.0 | 0.05–3.75 | 0.05–5.15 | 0.05–6.25 | |

Ratio | 0.28% | 1.19% | 6.18% | 27.58% | 64.77% | |

USN | $\overline{{\mathsf{\gamma}}_{c}}$ | 1.46 | 1.55 | 1.38 | 1.36 | 1.11 |

$\Delta {\mathsf{\gamma}}_{c}$ | 0.05–2.4 | 0.5–2.9 | 0.05–3.35 | 0.3–4.7 | 0.05–5.4 | |

Ratio | 0.33% | 0.85% | 2.78% | 21.45% | 74.59% | |

H1 | $\overline{{\mathsf{\gamma}}_{c}}$ | 2.1 | 1.48 | 1.38 | 1.55 | 1.08 |

$\Delta {\mathsf{\gamma}}_{c}$ | 1.9–2.35 | 0.05–4.0 | 0.35–6.05 | 0.35–7.75 | 0.05–10.0 | |

Ratio | 0.01% | 2.12% | 4.06% | 11.29% | 82.52% | |

H2 | $\overline{{\mathsf{\gamma}}_{c}}$ | 1.78 | 1.39 | 1.23 | 1.33 | 1.02 |

$\Delta {\mathsf{\gamma}}_{c}$ | 0.6–3.65 | 0.45–3.90 | 0.35–4.8 | 0.3–6.85 | 0.05–5.9 | |

Ratio | 0.48% | 2.91% | 8.02% | 25.07% | 63.52% |

**Table 3.**Distribution goodness of the fit analysis results according to the probability density for each station.

Distribution Type | Parameter | GHJ | MB | HP | USN | HeMOSU-1 | HeMOSU-2 |
---|---|---|---|---|---|---|---|

Log-Normal Distribution | $\mathsf{\mu}$ | 0.261 | 0.217 | 0.158 | 0.076 | 0.010 | 0.002 |

$\mathsf{\sigma}$ | 0.430 | 0.413 | 0.439 | 0.423 | 0.515 | 0.495 | |

p-value | 0.142 | 0.118 | 0.100 | 0.146 | 0.207 | 0.153 | |

${D}_{KL}$ | 51.725 | 59.639 | 59.000 | 48.254 | 42.393 | 36.541 | |

Gamma Distribution | $k$ | 5.788 | 6.273 | 5.640 | 5.831 | 3.934 | 4.373 |

$\mathsf{\theta}$ | 4.080 | 4.653 | 4.395 | 4.947 | 3.412 | 3.875 | |

p-value | 0.240 | 0.215 | 0.205 | 0.133 | 0.102 | 0.110 | |

${D}_{KL}$ | 33.096 | 35.372 | 48.562 | 44.723 | 63.509 | 68.009 | |

Normal Distribution | $\mathsf{\mu}$ | 1.418 | 1.348 | 1.283 | 1.179 | 1.153 | 1.129 |

$\mathsf{\sigma}$ | 0.599 | 0.543 | 0.543 | 0.506 | 0.636 | 0.571 | |

p-value | 0.013 | 0.019 | 0.019 | 0.002 | 0.0002 | 0.0002 | |

${D}_{KL}$ | 109.593 | 67.135 | 71.013 | 136.757 | 261.708 | 201.619 | |

Weibull Distribution | $k$ | 2.498 | 2.618 | 2.496 | 2.456 | 1.938 | 2.092 |

$\mathsf{\theta}$ | 1.601 | 1.518 | 1.448 | 1.331 | 1.306 | 1.278 | |

p-value | 0.046 | 0.041 | 0.040 | 0.012 | 0.004 | 0.007 | |

${D}_{KL}$ | 119.786 | 76.932 | 77.950 | 146.964 | 194.185 | 212.220 |

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**MDPI and ACS Style**

Lee, U.-J.; Jeong, W.-M.; Cho, H.-Y.
Estimation and Analysis of JONSWAP Spectrum Parameter Using Observed Data around Korean Coast. *J. Mar. Sci. Eng.* **2022**, *10*, 578.
https://doi.org/10.3390/jmse10050578

**AMA Style**

Lee U-J, Jeong W-M, Cho H-Y.
Estimation and Analysis of JONSWAP Spectrum Parameter Using Observed Data around Korean Coast. *Journal of Marine Science and Engineering*. 2022; 10(5):578.
https://doi.org/10.3390/jmse10050578

**Chicago/Turabian Style**

Lee, Uk-Jae, Weon-Mu Jeong, and Hong-Yeon Cho.
2022. "Estimation and Analysis of JONSWAP Spectrum Parameter Using Observed Data around Korean Coast" *Journal of Marine Science and Engineering* 10, no. 5: 578.
https://doi.org/10.3390/jmse10050578