# Statistical Modeling and Applications of Joint Distributions for Significant Wave Height, Spectral Peak Period, and Peak Direction of Propagation: A Case Study in the Norwegian Sea

^{1}

^{2}

^{*}

## Abstract

**:**

^{−2}and 10

^{−4}. Various approaches can be employed for this purpose, with preference given to statistical long-term analysis, which involves aggregating the exceedance probabilities of all potential sea states contributing to the exceedance of the target extremes. A joint model encompassing important metocean parameters such as wind, waves, and current is often necessary. This study specifically focuses on waves and wave-induced responses. In characterizing short-term sea state conditions, significant wave height (${H}_{s}$), spectral peak period (${T}_{p}$) and peak direction of propagation (${\mathsf{\Phi}}_{p}$) are identified as the most important sea state characteristics. The objective of this work is to present the results of the joint model for the three sea state parameters, i.e., ${H}_{s}$, ${T}_{p}$ and ${\mathsf{\Phi}}_{p}$, at an offshore site in the Norwegian Sea. The conditional modeling approach is applied using long-term hindcast data, and different statistical models are discussed for fitting the marginal and conditional distributions. The fitted parameters for all directional sectors are provided, offering a comprehensive representation of the joint model for direct use in long-term response analysis. Two case studies are included to illustrate the application of the fitted joint model in long-term response analyses. The case studies identify the governing wave directions and the most important combinations of short-term sea state characteristics regarding the estimation of long-term extreme responses.

## 1. Introduction

^{−2}. This value is multiplied by a safety factor, ${\gamma}_{e}$, typically 1.3–1.4. For ALS control, ${x}_{e}$ is often defined as the value exceeded by a probability of 10

^{−4}per year. In most cases, the safety factors are set to 1 in connection with the ALS control.

## 2. Statistical Assessment of Action and Action Effects

#### 2.1. Description of Short-Term Sea States for Response Estimation

#### 2.2. Long-Term Response Analysis

## 3. Joint Probabilistic Modeling of Sea State Characteristics

#### 3.1. Database and Reference Site

#### 3.2. Conditional Distribution of ${H}_{s}$ Given ${\Phi}_{p}$

^{−2}annual probability value. Regarding the estimation of the 10

^{−4}annual probability value, it will be more uncertain. It is, however, expected that the values estimated from the fitted 3-parameter Weibull will be on the safe side. This is good for the ALS design since the partial safety factor is usually 1 for the ALS limit state. What is presented here is representative for most sectors, but there are exceptions for the coastal sectors (Sectors 2–4). This will be further discussed in Section 3.4. The fitted parameters for the 3-parameter Weibull distribution for all sectors are given in Table 2.

#### 3.3. Conditional Distribution of ${T}_{p}$ Given ${H}_{s}$ and ${\Phi}_{p}$

#### 3.4. Discussion of Joint Model for Response Analyses

## 4. Case Studies

#### 4.1. Case Study on the Wave Crest Height

#### 4.2. Case Study on the Airgap of a Semi-Submersible

^{−2}accounting for all directions reads 17.22 m, while the value corresponding to annual probability of 10

^{−4}reads 22.65 m. The still water airgap for the reference platform is given as 20 m. This means there will be no wave-deck impacts at the ULS level, while a 2–3 m submergence of cellar deck bottom must be expected for the western corner at ALS level.

## 5. Conclusions

- The studies reveal that the characteristic response for design is governed by the three to four critical sectors with a direction width of 30°. Furthermore, the results indicate that sea states with a significant wave height lower than 8 m do not significantly impact the estimated ULS and ALS extreme responses. This suggests that it may not be necessary to develop a distribution model for all sectors. Instead, it is recommended to focus on a joint probabilistic model covering the worst sectors, which can be represented by a width of 90°–120°, and includes all sea states with a significant wave height above 8 m.
- In the second case study, when the structure is oriented with the most unfavorable heading against the worst incoming weather, the directional sensitivity of the extreme responses narrows. However, it is still advisable to focus on a wider design sector spanning 90°–120° around the worst direction. This ensures that the structural design remains appropriate for all directions, accounting for potential variations in weather conditions.

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## Appendix A

**Figure A2.**Fitted 3-parameter Weibull distribution for ${H}_{s}$ for sectors 1, 3, 6, 8, 10 and 12: [345° 15°], [45° 75°], [135° 165°] [195° 225°], [255° 285°] and [315° 345°] (the results are plotted in the Gumbel probability paper).

**Figure A3.**Fitting of the Lognormal parameters for sectors 1, 3, 6 and 10: [345° 15°], [45° 75°], [135° 165°] and [255° 285°].

**Figure A4.**Fitted Lognormal and 3-parameter Weibull distributions for ${T}_{p}$ of various ${H}_{s}$ classes for sector 9 [225° 255°] (the results are plotted in the Gumbel probability paper).

## References

- NORSOK. NORSOK Standard—Action and Action Effects (N-003); Standards Norway: Oslo, Norway, 2017. [Google Scholar]
- Naess, A.; Moan, T. Probabilistic design of offshore structures. In Handbook of Offshore Engineering; Elsevir: Amsterdam, The Netherlands, 2005; pp. 197–277. [Google Scholar]
- Sagrilo, L.V.S.; Naess, A.; Doria, A.S. On the long-term response of marine structures. Appl. Ocean Res.
**2011**, 33, 208–214. [Google Scholar] [CrossRef] - Mei, C.C.; Stiassnie, M.A.; Yue, D.K.P. Theory and Applications of Ocean Surface Waves: Part 1: Linear Aspects; World Scientific Publishing Co Pte Ltd.: Singapore, 2005. [Google Scholar]
- Haver, S.; Patiño, J. Airgap assessment of semi-submersible accounting for simultaneous occurrence of wind sea and swell. In Proceedings of the International Conference on Offshore Mechanics and Arctic Engineering, Glasgow, UK, 9–14 June 2019; American Society of Mechanical Engineers: Glasgow, Scotland, 2019. [Google Scholar]
- Vikenes, O.K. Assessment of Necessary Air Gap of Semi-Submersible Accounting for Simultaneous Occurrence of Wind, Wind Sea and Swell Sea. Master’s Thesis, Norwegian University of Science and Technology, Trondheim, Norway, 2018. [Google Scholar]
- Prince-Wright, R. Maximum Likelihood Models of Joint Environmental Data for TLP Design (No. CONF-950695-); American Society of Mechanical Engineers: New York, NY, USA, 1995. [Google Scholar]
- Bitner-Gregersen, E.M.; Haver, S. Joint environmental model for reliability calculations. In Proceedings of the First International Offshore and Polar Engineering Conference, Edinburgh, UK, 11–16 August 1991; OnePetro: Edinburgh, UK, 1991. [Google Scholar]
- Der Kiureghian, A.; Liu, P.L. Structural reliability under incomplete probability information. J. Eng. Mech.
**1986**, 112, 85–104. [Google Scholar] [CrossRef] - Bitner-Gregersen, E.M.; Guedes Soares, C.; Machado, U.; Cavaco, P. Comparison of different approaches to joint environmental modelling. In Proceedings of the 17th International Conference on Offshore Mechanics and Arctic Engineering, Lisbon, Portugal, 5–9 July 1998; ASME: New York, NY, USA, 1998. [Google Scholar]
- Bitner-Gregersen, E.M.; Waseda, T.; Parunov, J.; Yim, S.; Hirdaris, S.; Ma, N.; Soares, C.G. Uncertainties in Long-Term Wave Modelling. Mar. Struct.
**2022**, 84, 103217. [Google Scholar] [CrossRef] - Li, L.; Gao, Z.; Moan, T. Joint distribution of environmental condition at five European offshore sites for design of combined wind and wave energy devices. J. Offshore Mech. Arct. Eng.
**2015**, 137, 31901. [Google Scholar] [CrossRef] - Johannessen, K.; Meling, T.S.; Hayer, S. Joint distribution for wind and waves in the northern north sea. In Proceedings of the eleventh international offshore and polar engineering conference, Stavanger, Norway, 17–22 June 2001. [Google Scholar]
- ISSC. Report of ISSC Committee I.1 (Environmental Conditions). In Proceedings of the 6th International Ship Structures Congress, Boston, MA, USA, 3–5 April 1976. [Google Scholar]
- Hasselmann, K.; Barnett, T.P.; Bouws, E.; Carlson, H.; Cartwright, D.E.; Enke, K.; Ewing, J.A.; Gienapp, A.; Hasselmann, D.E.; Kruseman, P.; et al. Measurements of Wind-Wave Growth and Swell Decay during the Joint North Sea Wave Project (JONSWAP); Ergaenzungsheft zur Deutschen Hydrographischen Zeitschrift, Reihe A: Hamburg, Germany, 1973. [Google Scholar]
- DNVGL. Recommended Practice—Environmental Conditions and Environmental Loads (DNV-RP-C205); DNVGL: Bærum, Norway, 2014. [Google Scholar]
- Li, L.; Haver, S.; Eltervaag, A. Predicting long-term extreme responses using two approaches with a case study of a jacket structure. Adv. Anal. Des. Mar. Struct.
**2023**, 1, 539–549. [Google Scholar] - Reistad, M.; Breivik, Ø.; Haakenstad, H.; Aarnes, O.J.; Furevik, B.R.; Bidlot, J.R. A high-resolution hindcast of wind and waves for the North Sea, the Norwegian Sea, and the Barents Sea. J. Geophys. Res.
**2011**, 116, C05019. [Google Scholar] [CrossRef] - Haakenstad, H.; Breivik, Ø.; Reistad, M.; Aarnes, O.J. NORA10EI: A revised regional atmosphere-wave hindcast for the North Sea, the Norwegian Sea and the Barents Sea. Int. J. Climatol.
**2020**, 40, 4347–4373. [Google Scholar] [CrossRef] - Bruserud, K.; Haver, S. Comparison of wave and current measurements to NORA10 and NoNoCur hindcast data in the northern North Sea. Ocean Dyn.
**2016**, 66, 823–838. [Google Scholar] [CrossRef] - Bitner-Gregersen, E.M. Joint probabilistic description for combined seas. In Proceedings of the International Conference on Offshore Mechanics and Arctic Engineering, Halkidiki, Greece, 12–17 June 2005; ASME: New York, NY, USA, 2005; pp. 169–180. [Google Scholar]
- Forristall, G. Wave Crest Distribution: Observations and Second Order Theory. J. Phys. Oceanogr.
**2000**, 30, 1931–1943. [Google Scholar] [CrossRef] - DNVGL. Offshore Technical Guidance—Prediction of Air Gap for Column Stabilized Units (DNVGL-OTG-13); DNVGL: Bærum, Norway, 2019. [Google Scholar]
- Tromans, P.S.; Vandersohuren, L. Response based design conditions in the North Sea: Application of a new method. In Proceedings of the Offshore Technology Conference, OTC 7683, Houston, TX, USA, 1–4 May 1995. [Google Scholar]
- Stanisic, D.; Efthymiou, M.; Kimiaei, M.; Zhao, W. Design loads and long term distribution of mooring line response of a large weathervaning vessel in a tropical cyclone environment. Mar. Struct.
**2018**, 61, 361–380. [Google Scholar] [CrossRef]

**Figure 2.**Fitted 3-parameter Weibull distribution in Gumbel probability paper for sector 9 [225° 255°]. (The fitted parameters are given in Table 2).

**Figure 3.**Fitting of the Lognormal parameters for Sector [225° 255°] (the fitted parameters refer to Table 4).

**Figure 4.**Fitted Lognormal distribution for conditional distribution of ${T}_{p}$ for two high ${H}_{s}$ for Sector [225° 255°] plotted in the Gumbel probability paper.

**Figure 5.**Comparison of fitted 3-parameter Weibull distribution for omni-directional Hs in the Gumbel probability paper using two methods.

**Figure 6.**Histogram of the relative contributions to exceedance probability of ${c}_{ULS}$ and ${c}_{ALS}$ for the various sectors.

**Figure 7.**Relative contributions from different sea states to exceed extreme crest heights corresponding to ${q}_{0.01}$ (ULS) and ${q}_{0.0001}$ (ALS) for Sector 10, respectively. The red and blue lines are the 100-year and 10,000-year contour lines of ${H}_{s}$ and ${T}_{p}$ obtained from the joint distributions.

**Figure 10.**Histogram for the relative contributions to exceedance probability of ${R}_{P,ULS}$ and ${R}_{P,ALS}$ for the 12 sectors.

**Figure 11.**Relative contributions from different sea states in Sector 10 to exceed the extreme relative surface elevations corresponding to ${q}_{0.01}$ (ULS) and ${q}_{0.0001}$ (ALS), respectively. The red and blue lines are the 100-year and 10,000-year contour lines of ${H}_{s}$ and ${T}_{p}$ obtained from the joint distributions.

**Table 1.**Main characteristics of the 12 directional sectors for the reference site and the probability mass function for the sectors, ${p}_{{\phi}_{i}}$.

$\mathbf{Sector}{\mathit{\phi}}_{\mathit{i}}$ | Sector’s Range | Sector’s Midpoint | ${\mathit{p}}_{{\mathit{\phi}}_{\mathit{i}}}$ |
---|---|---|---|

1 | [345° 15°] | 0° | 0.1094 |

2 | [15° 45°] | 30° | 0.1790 |

3 | [45° 75°] | 60° | 0.0263 |

4 | [75° 105°] | 90° | 0.0046 |

5 | [105° 135°] | 120° | 0.0069 |

6 | [135° 165°] | 150° | 0.0267 |

7 | [165° 195°] | 180° | 0.0162 |

8 | [195° 225°] | 210° | 0.1094 |

9 | [225° 255°] | 240° | 0.2904 |

10 | [255° 285°] | 270° | 0.1369 |

11 | [285° 315°] | 300° | 0.0453 |

12 | [315° 345°] | 330° | 0.0490 |

**Table 2.**Estimated parameters of 3-parameter Weibull distribution for marginal distribution of ${H}_{s}$ for each sector (see Equation (12)), and corresponding estimated ${H}_{s}$ extremes. The last row provides the extremes by combining all sectors (omni-directional results).

$\mathbf{Sector}{\mathit{\phi}}_{\mathit{i}}$ | ${\mathit{a}}_{3\mathit{w}}$ | ${\mathit{b}}_{3\mathit{w}}$ | ${\mathit{c}}_{3\mathit{w}}$ | ${\mathit{H}}_{\mathit{s}}$_{,1-year} (m) | ${\mathit{H}}_{\mathit{s}}$_{,100-year} (m) | ${\mathit{H}}_{\mathit{s}}$_{,10,000-year} (m) |
---|---|---|---|---|---|---|

1 | 1.99 | 1.38 | 0.78 | 8.29 | 11.66 | 14.98 |

2 | 1.39 | 1.19 | 0.85 | 7.80 | 11.26 | 14.88 |

3 | 1.26 | 1.22 | 0.77 | 5.32 | 8.33 | 11.38 |

4 | 1.44 | 1.56 | 0.63 | 3.57 | 5.72 | 7.62 |

5 | 1.98 | 1.56 | 0.84 | 5.23 | 8.11 | 10.69 |

6 | 2.31 | 1.65 | 0.68 | 6.68 | 9.43 | 11.94 |

7 | 2.47 | 1.47 | 0.91 | 7.56 | 11.40 | 14.99 |

8 | 2.75 | 1.46 | 0.68 | 10.27 | 14.27 | 18.14 |

9 | 2.36 | 1.31 | 0.73 | 11.37 | 15.78 | 20.25 |

10 | 2.32 | 1.25 | 0.71 | 11.04 | 16.07 | 21.21 |

11 | 2.35 | 1.25 | 0.61 | 9.58 | 14.80 | 20.08 |

12 | 2.37 | 1.37 | 0.75 | 8.91 | 13.11 | 17.22 |

All sectors | - | - | - | 12.36 | 16.75 | 21.29 |

**Table 3.**The mean, ${m}_{ln{T}_{p}}$, and variance, ${\sigma}_{ln{T}_{p}}^{2}$, of $ln\left({T}_{p}\right)$ for Sector 9 [225° 255°] (see Equation (13)).

${\mathit{H}}_{\mathit{s}}$ (m) | <1 | 1–2 | 2–3 | 3–4 | 4–5 | 5–6 | 6–7 | 7–8 | 8–9 | 9–10 | 10–11 | >11.0 |
---|---|---|---|---|---|---|---|---|---|---|---|---|

${m}_{ln{T}_{p}}$ | 2.12 | 2.25 | 2.37 | 2.47 | 2.53 | 2.57 | 2.61 | 2.64 | 2.68 | 2.71 | 2.73 | 2.79 |

${\sigma}_{ln{T}_{p}}^{2}$ | 0.06 | 0.06 | 0.06 | 0.04 | 0.04 | 0.03 | 0.02 | 0.02 | 0.01 | 0.01 | 0.01 | 0.01 |

**Table 4.**Estimated parameters of Lognormal distribution for conditional distribution of ${T}_{p}$ given ${H}_{s}$ and ${\mathsf{\Phi}}_{p}$ (see Equations (14) and (15)).

$\mathbf{Sector}{\mathit{\phi}}_{\mathit{i}}$ | ${\mathit{a}}_{1,\mathit{i}}$ | ${\mathit{a}}_{2,\mathit{i}}$ | ${\mathit{a}}_{3,\mathit{i}}$ | ${\mathit{b}}_{1,\mathit{i}}$ | ${\mathit{b}}_{2,\mathit{i}}$ | ${\mathit{b}}_{3,\mathit{i}}$ |
---|---|---|---|---|---|---|

1 | 0.95 | 1.05 | 0.21 | 0.001 | 0.05 | −0.25 |

2 | 0.57 | 1.44 | 0.15 | 0.001 | 0.05 | −0.22 |

3 | 1.41 | 0.42 | 0.42 | 0.001 | 0.05 | −0.33 |

4 | 1.52 | 0.15 | 0.94 | 0.001 | 0.18 | −1.43 |

5 | −0.02 | 1.75 | 0.11 | 0.001 | 0.03 | −0.11 |

6 | 1.34 | 0.48 | 0.39 | 0.001 | 0.06 | −0.50 |

7 | 1.29 | 0.57 | 0.37 | 0.001 | 0.04 | −0.16 |

8 | 1.32 | 0.77 | 0.24 | 0.001 | 0.06 | −0.23 |

9 | 1.51 | 0.70 | 0.24 | 0.001 | 0.07 | −0.18 |

10 | 1.49 | 0.65 | 0.26 | 0.001 | 0.08 | −0.23 |

11 | 1.53 | 0.46 | 0.40 | 0.001 | 0.04 | −0.21 |

12 | 0.07 | 1.90 | 0.14 | 0.001 | 0.04 | −0.21 |

**Table 5.**The 100-year (ULS) and 10,000-year (ALS) extreme crest heights for various sectors (see Equation (9)). The last row provides the extremes by combining all sectors (omni-directional results) (see Equation (10)).

$\mathbf{Sector}{\mathit{\phi}}_{\mathit{i}}$ | ${\mathit{K}}_{1\mathit{Y},}\left(\mathit{i}\right)$ | 100-Year (m) | 10,000-Year (m) |
---|---|---|---|

1 | 319.4 | 13.1 | 17.4 |

2 | 522.7 | 12.6 | 17.2 |

3 | 76.8 | 9.57 | 13.5 |

4 | 13.4 | 6.61 | 8.89 |

5 | 20.1 | 8.32 | 11.2 |

6 | 78.0 | 11.0 | 14.4 |

7 | 47.3 | 12.8 | 17.4 |

8 | 319.4 | 16.2 | 21.4 |

9 | 848.0 | 17.6 | 23.5 |

10 | 399.7 | 17.9 | 24.5 |

11 | 132.3 | 16.1 | 22.3 |

12 | 143.1 | 14.4 | 19.6 |

All sectors | 2920.25 | 19 | 25.5 |

Disclaimer/Publisher’s Note: The statements, opinions and data contained in all publications are solely those of the individual author(s) and contributor(s) and not of MDPI and/or the editor(s). MDPI and/or the editor(s) disclaim responsibility for any injury to people or property resulting from any ideas, methods, instructions or products referred to in the content. |

© 2023 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Pires Vieira Serta, C.; Haver, S.; Li, L.
Statistical Modeling and Applications of Joint Distributions for Significant Wave Height, Spectral Peak Period, and Peak Direction of Propagation: A Case Study in the Norwegian Sea. *J. Mar. Sci. Eng.* **2023**, *11*, 2372.
https://doi.org/10.3390/jmse11122372

**AMA Style**

Pires Vieira Serta C, Haver S, Li L.
Statistical Modeling and Applications of Joint Distributions for Significant Wave Height, Spectral Peak Period, and Peak Direction of Propagation: A Case Study in the Norwegian Sea. *Journal of Marine Science and Engineering*. 2023; 11(12):2372.
https://doi.org/10.3390/jmse11122372

**Chicago/Turabian Style**

Pires Vieira Serta, Clarissa, Sverre Haver, and Lin Li.
2023. "Statistical Modeling and Applications of Joint Distributions for Significant Wave Height, Spectral Peak Period, and Peak Direction of Propagation: A Case Study in the Norwegian Sea" *Journal of Marine Science and Engineering* 11, no. 12: 2372.
https://doi.org/10.3390/jmse11122372