# Comparative Analysis of Environmental Contour Approaches to Estimating Extreme Waves for Offshore Installations for the Baltic Sea and the North Sea

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## Abstract

**:**

## 1. Introduction

## 2. Data

#### 2.1. Observations

#### 2.2. Hindcast Data

## 3. Methods

#### 3.1. I-FORM

#### 3.2. I-FORM with PCA

#### 3.3. 2D POT

## 4. Results

#### 4.1. Implementing Each Method

#### 4.1.1. I-FORM

#### 4.1.2. I-FORM with PCA

#### 4.1.3. 2D POT

#### 4.2. Comparing the Different Data Sets and Different Methods

## 5. Discussion

## 6. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 1.**Workflow for determining the characteristic load, with the environmental data as the starting point. The aspects of this workflow that are treated in this paper—observational and hindcast data, and the contour approach—are highlighted.

**Figure 5.**Goodness-of-fit plots for ${H}_{s}$ in the I-FORM method, showing both the 3-parameter Weibull fit (gray dotted line) and the hybrid fit (black dashed line). The hindcast data and observations at Tyne Tees are shown in red in their respective plots. In order to facilitate a comparison for the higher waves, a plot showing only the highest 5% of ${H}_{s}$ is given for both data sets.

**Figure 6.**Sensitivity analysis for the hybrid fit (Equation (2)) for ${H}_{s}$ in the I-FORM method for 50-year contours. As explained in the text, a percentile of ${H}_{s}$ is chosen which affects the choice of the threshold $\eta $. The region with the largest contour variability is enlarged to more clearly show the colors. Only data from Ölands Södra Grund is shown.

**Figure 7.**In subfigure (

**a**) data from NDBC 46022 [61] are used to obtain contours to be compared with Figure 11b in Eckert-Gallup et al. [27], here reprinted in subfigure (

**b**) with permission from Elsevier, Copyright 2016. In the two plots, the black solid lines show the I-FORM with PCA, and the black dashed lines show the I-FORM with 3-parameter Weibull fit for ${H}_{s}$. Note that the data density coloring scheme differ slightly for the two plots and thus no comparison regarding data density can be made.

**Figure 8.**The 2D Peaks-Over-Threshold method. In subfigure (

**a**), the data are shown in the $(s,{H}_{s})$-plane and the 95-percentiles are drawn. The 50-year return level curve is shown in blue. In subfigure (

**b**), the data has been transformed back to the $({T}_{z},{H}_{s})$-plane.

**Figure 9.**A threshold plot utilized in choosing the thresholds for the 2D POT method. The upper two subplots show how the Generalized Pareto-parameter $\xi $ varies with the choice of threshold for the hindcast data at Dowsing, both for the steepness (${\xi}_{x}$) and for the wave height (${\xi}_{y}$). Correspondingly the lower two subplots show how the parameter $\sigma $ varies with threshold.

**Figure 10.**Comparing 50-year environmental contours for three different methods: (1) I-FORM with 3-parameter Weibull fit for ${H}_{s}$, with $\beta $ inflated using ${\alpha}_{0}^{2}=0.2$ (gray solid line); (2) I-FORM with a hybrid fit for ${H}_{s}$, without inflation (black solid line); and (3) 2D Peaks-Over-Threshold (blue solid line). The threshold percentiles for both ${H}_{s}$ and the steepness are also shown (black dotted lines).

**Figure 11.**Comparing 50-year environmental contours for two methods: (1) I-FORM with a hybrid fit for ${H}_{s}$, without inflation (black solid line) and (2) I-FORM with principal component analysis (PCA) (red dashed line).

**Figure 12.**Comparing 50-year environmental contours for the observations (solid lines) and the model data (dashed lines). Two methods are shown: (1) I-FORM with 3-parameter Weibull fit for ${H}_{s}$, with $\beta $ inflated using ${\alpha}_{0}^{2}=0.2$ (blue lines) and (2) I-FORM with a hybrid fit for ${H}_{s}$, without inflation (black lines).

**Figure 13.**Number of independent (decorrelated) sea states which exceed the contour lines in the region above the 95-percentiles of ${H}_{s}$ and steepness s. Also shown is the expected—in a statistical sense—number of events for the 2D POT method. The return period which is used in this study, 50 years, is highlighted with a vertical gray line.

**Table 1.**Properties of the four sites considered in this study. The water depth and the distance to the closest coast are shown. The mean significant wave height is denoted $\overline{{H}_{s}}$ and is given for the observational and hindcast data sets. Also shown is the data availability, in years, for the observations.

Site | Lat | Lon | Depth [m] | Distance [km] | $\overline{{H}_{s}}$ (obs.) [m] | $\overline{{H}_{s}}$ (hind.) [m] | Availability [years] |
---|---|---|---|---|---|---|---|

Ölands Södra Grund | 56.0667° N | 16.6833° E | 38 | 22 | 1.03 | 1.09 | 16.7 |

Väderöarna | 58.4833° N | 10.9333° E | 72 | 18 | 1.12 | 1.05 | 12.6 |

Tyne Tees | 54.9190° N | 0.7487° W | 66 | 37 | 1.34 | 1.51 | 10.8 |

Dowsing | 53.5318° N | 1.0538° E | 22 | 56 | 1.23 | 1.36 | 15.4 |

**Table 2.**Comparison between the 3-parameter Weibull fit and the hybrid fit (Equation (2)) for the I-FORM method. The maximum value of ${H}_{s}$ along the 50-year contour is shown. Also shown is the maximum steepness s, conditioned on ${H}_{s}>{H}_{s,\mathrm{max}}/2$ where ${H}_{s,\mathrm{max}}$ is the maximum value of ${H}_{s}$ for the observations at each site.

Max H_{s} (m) | Cond. max s | ||||
---|---|---|---|---|---|

Site | Data | 3-Wbl | Hybrid | 3-Wbl | Hybrid |

Ölands Södra Grund | obs. hind. | 7.61 7.80 | 7.88 9.12 | 0.099 0.109 | 0.093 0.103 |

Väderöarna | obs. hind. | 9.97 8.25 | 10.32 9.38 | 0.105 0.104 | 0.098 0.099 |

Tyne Tees | obs. hind. | 9.07 9.10 | 10.58 13.94 | 0.121 0.147 | 0.112 0.133 |

Dowsing | obs. hind. | 7.51 7.87 | 7.13 11.71 | 0.138 0.164 | 0.125 0.145 |

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## Share and Cite

**MDPI and ACS Style**

Wrang, L.; Katsidoniotaki, E.; Nilsson, E.; Rutgersson, A.; Rydén, J.; Göteman, M. Comparative Analysis of Environmental Contour Approaches to Estimating Extreme Waves for Offshore Installations for the Baltic Sea and the North Sea. *J. Mar. Sci. Eng.* **2021**, *9*, 96.
https://doi.org/10.3390/jmse9010096

**AMA Style**

Wrang L, Katsidoniotaki E, Nilsson E, Rutgersson A, Rydén J, Göteman M. Comparative Analysis of Environmental Contour Approaches to Estimating Extreme Waves for Offshore Installations for the Baltic Sea and the North Sea. *Journal of Marine Science and Engineering*. 2021; 9(1):96.
https://doi.org/10.3390/jmse9010096

**Chicago/Turabian Style**

Wrang, Linus, Eirini Katsidoniotaki, Erik Nilsson, Anna Rutgersson, Jesper Rydén, and Malin Göteman. 2021. "Comparative Analysis of Environmental Contour Approaches to Estimating Extreme Waves for Offshore Installations for the Baltic Sea and the North Sea" *Journal of Marine Science and Engineering* 9, no. 1: 96.
https://doi.org/10.3390/jmse9010096