# Uncertainties of Estimating Extreme Significant Wave Height for Engineering Applications Depending on the Approach and Fitting Technique—Adriatic Sea Case Study

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Methodology

_{s}) for 50-year and 100-year return periods.

_{s}data were extracted from the underlying database. The ID approach used all H

_{s}data points, although a lower cut-off threshold can sometimes be introduced [5]. The AM approach implied that only the yearly H

_{s}maximums were to be used for the analysis. Finally, the POT method meant that the analysis was based on the maximum values (peaks) of storm events that are mathematically defined as an uninterrupted number of H

_{s}values that exceeded a certain threshold.

_{s}datasets for each approach from the underlying database, the probability of H

_{S}

_{, i}not exceeding the value H

_{S}

_{, i}was assigned to each data point.

_{S}

_{, i}< H

_{S}

_{, i}}, with H

_{S}

_{, i}being the lower limit of the bin), was evaluated by:

_{i}is the cumulative number of observation lower than H

_{s, i}and N is the total number of observations.

_{S, i}< H

_{S, i}} was evaluated according to Gringorten [15] and Goda [16] as:

_{S}

_{, i}< H

_{S}

_{, i}} for small datasets. However, when they were tested and compared to the AM-dataset of 23 data points, they proved to produce very small differences that were not considered significant.

#### 2.1. The Initial Distribution Approach—Three Parameter Weibull Distribution

#### 2.2. The Annual Maximum Approach—Gumbel Distribution

_{s}

_{,AM}, were extracted. Thus, 23 maximums were extracted from the database. Such a dataset, according to the extreme value theory, corresponds well with the generalized extreme value (GEV) distribution. Moreover, if the parent distribution is the Weibull distribution, GEV can be reduced to the Gumbel (Fisher–Tippett Type I, FT-I) distribution given by its CDF in Equation (4):

#### 2.3. The Peak-Over-Threshold Approach—Exponential Distribution

_{0}is the chosen threshold. Aside from the exponential distribution, the Pareto theoretical distribution can also be applied—but with caution [7].

#### 2.4. Fitting Techniques

#### 2.5. Estimation of Significant Wave Height at Specified Return Periods

_{s}

^{RP}) is the probability of exceeding a certain H

_{s}in a specified return period:

_{R}is the return period in years. Attention should be paid when selecting Δt, as different intervals are encountered in different databases. For example, within the WorldWave atlas (WWA), the sea state duration reporting interval is assumed to be 6 h. DNV–GL [7] suggests using 3-h intervals. When using data from the Global Wave Statistics (GWS), users are advised to select sea state duration of 3 h. However, this is theoretically inconsistent, as the recording interval of each sea state in GWS is basically unknown.

_{h0}is the yearly cluster rate estimated by the average number of storms/peak excesses per year. If the underlying database has no recording gaps, as was the case in this study, the yearly cluster rate is calculated in a straightforward way by of dividing the total number of identified peaks with the number of years in the database.

## 3. Results

#### 3.1. The Initial Distribution Approach

_{s}values, a deviation was evident between the empirical and the theoretical results. Consistency at higher values was especially important, since this part of the curve was extrapolated for evaluating H

_{s}at return periods longer than the database duration. The calculated H

_{s}values for the 23-year return periods are compared later with the maximum empirical recorded significant wave height from the database that reads H

_{s}

_{,max}= 5.26 m.

_{s}went to the value of the location parameter. This can be seen from the log likelihood function.

_{min}, ln(x

_{min}− θ) goes to −∞; and if β is less than 1, then (β − 1)ln(x

_{min}− α) goes to +∞. Thus, it is always possible “to find” a larger likelihood value, and convergence cannot be achieved. To avoid this issue, the lower bound of the shape parameter was set to 1. The method reached a solution, though it did so with a poorer fit compared to the LSM method. To verify this issue, ocean wave scatter diagrams available in DNV–GL (2017) were checked. It was found that the shape parameter of the three-parameter-Weibull distribution never showed values lower than 1 for the defined global nautical zones. The shape parameter was also calculated for several other locations in the Adriatic, converging to values around or lower than 1. This was, therefore, a specific problem of the local sea basin discarding the MLE fitting technique within the ID approach for the Adriatic Sea.

#### 3.2. Annual (Monthly) Extreme Approach

#### 3.3. The Peak-Over-Threshold Approach

_{s}of each storm. The identified storms, and their respective H

_{s}peaks, should be independent variables so that the number of events follows a Poisson distribution, and, therefore, the interarrival time follows an exponential distribution. There are various strategies found in literature to ensure that the storms are mutually independent, e.g., [23]. A simple strategy is to define the minimum interarrival period, which is the minimum time separation between selected events, to calculate peak-to-peak (instead of threshold down-crossing to the next up-crossing). Another matter is the selection of the threshold, which is a trade-off between bias and variance: Too low a threshold is likely to violate the asymptotic basis of the model, leading to bias; too high a threshold will generate fewer excesses with which to estimate the model, leading to high variance [22]. Thus, the minimum interarrival time and the threshold were varied in order to find the optimal values. The exponential distribution shape parameter α and the respective 50-year significant wave height estimate Hs

^{RP}are presented in Figure 5.

_{0}= 2.5 m, the minimum interarrival time was set to 48 h (peak-to-peak), and the minimum duration time was set to 6 h. Similar values have been used in other extreme significant wave height studies, especially for comparable regions in the Mediterranean Sea [24]. The chosen values also corresponded to our understanding of the specific storm behavior in the Adriatic Sea, which is heavily influenced by the surrounding mountain topography. The significant wave height threshold of 2.5 m corresponded to sea state 5 (“rough”) and above according to the Douglas sea state scale dominantly used by maritime professionals. Figure 5 also clearly indicates how small differences in the selected extreme event/storm definition parameters could significantly vary the extreme estimates.

_{s}, identified excesses of the threshold, and their respective peaks. It can be noticed (e.g., the beginning of February 2014) how only the larger of the two successive peaks attributed to the same storm event was extracted in order to satisfy the variable independence requirement.

_{s}> H

_{s}) was 323.2 records, i.e., 80.8 days. The exponential distribution parameter α, calculated by the fitting techniques methods, is presented in Table 6.

_{s,max}= 5.26 m, was not captured well by the fit line, but it laid bellow the theoretical line. As with previous approaches, GOF values and residual plots did not indicate notable differences to suggest that one technique yielded better results than the other.

#### 3.4. Extreme Significant Wave Height Estimation

_{s}. Results are presented in Table 7, Table 8 and Table 9.

## 4. Discussion on Accuracy Evaluation Including Spatial Considerations

_{0}= 3.2 m, while the minimum interarrival time (defined simplified as peak-to-peak) remained 48 h. Minimum storm duration was defined as 6 h, corresponding to one record in the underlaying database. Considering the chosen high threshold (H

_{s}> 2.5 (3.2) m), six hours was considered to be a representative storm duration in the Adriatic Sea. The same minimum storm duration has been used in other studies for the Mediterranean Sea [24].

_{s,ref}. Thus, the mutual trend of other approaches and techniques, H

_{s,i}

^{RP}, was calculated according to:

_{s}

^{RP=23y}= 5.28 m (stand. dev. = 0.25); H

_{s}

^{RP=50y}= 5.75 m (stand. dev. = 0.24); and H

_{s}

^{RP=100y}= 6.17 m (stand. dev. = 0.22).

_{s}), which is especially important for extrapolation to long return periods. This could be countered by employing other goodness-of-fit tests, such as the Anderson–Darling test, which is devised to give heavier weight to the tails.

## 5. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

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**Figure 1.**(

**a**) Wave database locations in the Adriatic Sea; (

**b**) available wave and wind physical parameters at each location.

**Figure 2.**Long-term Weibull distribution of significant wave height for years 1993–2015 at 13.5° E–45.00° N. Straight line represents best-fit candidate distribution (LSM method). Parameter “loc” on the vertical axis stands for the location parameter Θ. Complete range (

**a**). Higher values (

**b**).

**Figure 4.**Gumbel distribution fit of yearly maximums for period 1993–2015 at 13.5° E–45.0° N. Straight line represents best-fit with the LSM method.

**Figure 5.**Scale parameter α of the exponential distribution (

**a**) and the Hs estimate for the 50-year return period (

**b**) depending on threshold and the minimum interarrival time (defined as peak-to-peak) for the MLE fitting technique).

**Figure 8.**Dissipation of significant wave estimates depending on the method and fitting technique; absolute values for the north (

**a**), central (

**b**) and south (

**c**) location; relative difference between obtained results at the north (

**d**), central (

**e**) and south (

**f**) location in the Adriatic Sea.

**Table 1.**Sea state table: peak period T

_{p}and significant wave height H

_{s}, 1993–2016, at 13.5° E–45.0° N. The values are color shaded white to red, with red intensity denoting higher number of occurrence.

Tp/Hs | 0.0–0.5 | 0.5–1.0 | 1.0–1.5 | 1.5–2.0 | 2.0–2.5 | 2.5–3.0 | 3.0–3.5 | 3.5–4.0 | 4.0–4.5 | 4.5–5.0 | 5.0–5.5 | 5.5–6.0 | 6.0–6.5 | Sum |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|

0–1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |

1–2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |

2–3 | 10,422 | 2508 | 59 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 12,989 |

3–4 | 3584 | 4500 | 982 | 91 | 3 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 9160 |

4–5 | 1302 | 1628 | 1027 | 480 | 119 | 6 | 1 | 0 | 0 | 0 | 0 | 0 | 0 | 4563 |

5–6 | 1213 | 905 | 491 | 313 | 202 | 79 | 20 | 2 | 0 | 0 | 0 | 0 | 0 | 3225 |

6–7 | 565 | 602 | 257 | 122 | 55 | 47 | 15 | 12 | 5 | 0 | 0 | 0 | 0 | 1680 |

7–8 | 164 | 164 | 117 | 81 | 39 | 17 | 10 | 2 | 5 | 5 | 3 | 0 | 0 | 607 |

8–9 | 78 | 52 | 14 | 12 | 12 | 9 | 4 | 3 | 0 | 1 | 0 | 0 | 0 | 185 |

9–10 | 19 | 14 | 5 | 2 | 3 | 1 | 0 | 2 | 0 | 0 | 0 | 0 | 0 | 46 |

10–11 | 12 | 5 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 17 |

11–12 | 11 | 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 12 |

12–13 | 4 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 4 |

13–14 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |

Sum | 17,374 | 10,379 | 2952 | 1101 | 433 | 159 | 50 | 21 | 10 | 6 | 3 | 0 | 0 | 32,488 |

**Table 2.**Initial distribution (ID) approach estimated parameters for three-parameter-Weibull distribution. LSM: least squares method; MoM; method of moments; and MLE: maximum likelihood estimator.

Parameters | LSM | MoM | MLE |
---|---|---|---|

Θ, location | 0.260 | 0.250 * | 0.250 |

α, scaling | 0.412 | 0.433 | 0.444 |

β, shape | 0.903 | 0.943 | 1.000 ** |

GOF | LSM | MoM | MLE |
---|---|---|---|

SSE | 0.0024 | 0.0025 | 0.0045 |

R-square | 0.9992 | 0.9992 | 0.9985 |

Adjusted R-square | 0.9992 | 0.9992 | 0.9985 |

RMSE | 0.0050 | 0.0050 | 0.0068 |

Month | Year | ||||||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|

1993 | 1994 | 1995 | 1996 | 1997 | 1998 | 1999 | 2000 | 2001 | 2002 | 2003 | 2004 | 2005 | 2006 | 2007 | 2008 | 2009 | 2010 | 2011 | 2012 | 2013 | 2014 | 2015 | |

1 | 5.26 | 3.07 | 4.11 | 1.94 | 1.54 | 2.21 | 3.08 | 1.75 | 2.94 | 1.52 | 2.99 | 2.44 | 2.21 | 2.21 | 1.62 | 1.67 | 1.92 | 1.33 | 1.58 | 1.22 | 1.72 | 3.06 | 2.53 |

2 | 2.74 | 3.1 | 1.52 | 2.16 | 1.54 | 1.67 | 2.99 | 1.07 | 3.97 | 2.1 | 2.41 | 2.84 | 1.72 | 1.73 | 1.52 | 1.35 | 2.31 | 3.09 | 2.17 | 2.34 | 3.07 | 2.85 | 2.51 |

3 | 3.87 | 1.7 | 2.82 | 1.28 | 1.12 | 2.08 | 2.73 | 2.62 | 2.73 | 2.12 | 1.71 | 3.03 | 1.32 | 1.95 | 2.7 | 2.29 | 3.38 | 2.42 | 2.42 | 1.42 | 2.74 | 2.29 | 2.29 |

4 | 1.52 | 2.47 | 1.49 | 1.84 | 2.47 | 1.87 | 3.08 | 2.33 | 2.79 | 3.53 | 2.02 | 1.38 | 2.75 | 1.46 | 1.27 | 1.81 | 3.81 | 1.81 | 1.05 | 1.47 | 1.52 | 1.54 | 1.57 |

5 | 1.57 | 3.57 | 1.47 | 1.15 | 1.47 | 1.62 | 1.36 | 1.63 | 1.42 | 1.87 | 1.75 | 1.81 | 1.1 | 2.07 | 1.28 | 1.49 | 1.1 | 1.95 | 1.62 | 1.89 | 2.74 | 1.4 | 1.49 |

6 | 1.67 | 2.26 | 2.07 | 1.71 | 1.58 | 1.28 | 1.29 | 1.41 | 2.05 | 2.53 | 0.96 | 1.5 | 1.51 | 1.23 | 1.49 | 0.96 | 1.71 | 1.09 | 1.28 | 1.36 | 0.91 | 1.16 | 1.55 |

7 | 2.68 | 0.92 | 0.83 | 1.18 | 1.29 | 1.29 | 1.65 | 2.74 | 2.37 | 1.95 | 1.46 | 1.27 | 1.15 | 0.8 | 1.9 | 1.2 | 1.63 | 1.41 | 1.23 | 1.52 | 0.81 | 1.36 | 1.74 |

8 | 2.36 | 1.29 | 1.17 | 1.02 | 1.33 | 2.1 | 1.47 | 0.94 | 2.41 | 1.63 | 2.15 | 1.13 | 0.85 | 2.01 | 0.88 | 0.96 | 1.2 | 1.55 | 1.25 | 1.04 | 1.73 | 1.24 | 1.44 |

9 | 2.6 | 2.34 | 1.73 | 1.87 | 1.99 | 2.18 | 2.84 | 2.29 | 2.58 | 1.6 | 1.97 | 1.81 | 1.97 | 2.36 | 1.6 | 1.44 | 1.68 | 1.3 | 1.17 | 1.49 | 1.35 | 1.64 | 1.69 |

10 | 3.98 | 2.82 | 1.12 | 3.19 | 2.05 | 1.87 | 3.29 | 2.71 | 1.42 | 2.52 | 3.02 | 1.78 | 1.5 | 1.57 | 1.92 | 2.29 | 2.11 | 2.33 | 1.88 | 2.57 | 1.9 | 1.48 | 2.11 |

11 | 2.71 | 4.26 | 1.86 | 3.89 | 2.66 | 2.73 | 3.16 | 4.05 | 2.62 | 4.43 | 3.11 | 3.02 | 2.63 | 1.77 | 2.09 | 3.57 | 2.63 | 2.83 | 2.31 | 4.56 | 2.34 | 2.82 | 1.74 |

12 | 3.03 | 4.68 | 2.47 | 2.57 | 2.15 | 2.86 | 3.73 | 1.97 | 3.29 | 2.71 | 2.94 | 1.96 | 2.45 | 1.77 | 1.91 | 3.34 | 2.18 | 2.85 | 2.07 | 1.79 | 3.21 | 2.46 | 1.02 |

Year_Max | 5.26 | 4.68 | 4.11 | 3.89 | 2.66 | 2.86 | 3.73 | 4.05 | 3.97 | 4.43 | 3.11 | 3.03 | 2.75 | 2.36 | 2.7 | 3.57 | 3.81 | 3.09 | 2.42 | 4.56 | 3.21 | 3.06 | 2.53 |

Parameters | LSM | MoM | MLE |
---|---|---|---|

α | 3.1098 | 3.1097 | 3.0979 |

β | 0.6463 | 0.6265 | 0.6404 |

Parameters | LSM | MoM/MLE |
---|---|---|

α | 0.5931 | 0.5952 |

**Table 7.**Initial distribution approach—significant wave height estimates for 23-, 50-, and 100-year return periods.

TR [years] | Hs^{RP}–LSM [m] | Hs^{RP}–MoM [m] | Hs^{RP}–MLE [m] |
---|---|---|---|

23 | 5.77 | 5.44 | 4.87 |

50 | 6.24 | 5.86 | 5.22 |

100 | 6.65 | 6.23 | 5.53 |

**Table 8.**Annual maximum (extreme value) approach—significant wave height estimates for 23-, 50-, and 100-year return periods.

TR [years] | Hs^{RP}–LSM [m] | Hs^{RP}–MoM [m] | Hs^{RP}–MLE [m] |
---|---|---|---|

23 | 5.11 | 5.05 | 5.08 |

50 | 5.63 | 5.55 | 5.60 |

100 | 6.08 | 5.99 | 6.04 |

**Table 9.**Peak-over-threshold approach—significant wave height estimates for 23-, 50-, and 100-year return periods.

TR [years] | Hs^{RP}–LSM [m] | Hs^{RP}–MoM/MLE [m] |
---|---|---|

23 *** | 5.23 | 5.24 |

50 | 5.69 | 5.70 |

100 | 6.10 | 6.11 |

_{s}recorded in the underlying database was 5.26 m—this value “should” correspond to the estimated values for the 23-year return period. It is possible to notice that the variations, i.e., uncertainties were significant.

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**MDPI and ACS Style**

Katalinić, M.; Parunov, J.
Uncertainties of Estimating Extreme Significant Wave Height for Engineering Applications Depending on the Approach and Fitting Technique—Adriatic Sea Case Study. *J. Mar. Sci. Eng.* **2020**, *8*, 259.
https://doi.org/10.3390/jmse8040259

**AMA Style**

Katalinić M, Parunov J.
Uncertainties of Estimating Extreme Significant Wave Height for Engineering Applications Depending on the Approach and Fitting Technique—Adriatic Sea Case Study. *Journal of Marine Science and Engineering*. 2020; 8(4):259.
https://doi.org/10.3390/jmse8040259

**Chicago/Turabian Style**

Katalinić, Marko, and Joško Parunov.
2020. "Uncertainties of Estimating Extreme Significant Wave Height for Engineering Applications Depending on the Approach and Fitting Technique—Adriatic Sea Case Study" *Journal of Marine Science and Engineering* 8, no. 4: 259.
https://doi.org/10.3390/jmse8040259