# Directional Extreme Value Models in Wave Energy Applications

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## Abstract

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## 1. Introduction

_{s}tend to occur in groups with a strong correlation of wave conditions in time, hence violating the assumption of independence. In order to ensure that the threshold exceedances are approximately independent so that they can be used to apply the GP distribution model, declustering of exceedances takes place; see, for example, Leadbetter, et al. [7]. This technique stipulates that the exceedances that are separated by fewer than r non-exceeding observations, with r an auxiliary parameter denoting the run length (minimum separation), form a single cluster. Then, the GP fitting is performed only for the largest exceedances from each cluster instead of all the observations exceeding the threshold.

_{s}are usually observed for specific directional sectors depending on the particular characteristics of the location examined (e.g., fetch length, bathymetry). In the context of describing the extremal properties of wave parameters, it is common practice to work with hindcast data sets beside the fact that wave measurements are limited in time (and in space) rendering them inappropriate for extrapolation purposes. Hindcast products have the full information as regards wave conditions, hence mean wave direction can be taken into account and treated as a covariate in order to obtain an integrated and more accurate model for the estimation of the corresponding design values. The significance of directional behavior in works related to offshore/coastal structures for harvesting marine resources has been highlighted by many authors; see, for example, Soukissian [29]; Wei et al. [30]; Soukissian and Karathanasi [31]. Nevertheless, the involvement of directionality, along with other covariates such as space, in extreme value analysis has gained ground mainly the past 15 years, with some sporadic works in the meantime dating back to the early 80′s.

_{s}values above the defined threshold into directional sectors (either fixed or arbitrarily), perform extreme analysis in each sector and then specify design values for a given return period for each sector. However, as stated explicitly by Forristall [42], this sectoring approach leads to inconsistencies with the omni-directional case in terms of design values while it is insufficient at locations where directionality is limited to specific directional sectors. Following the rationale proposed by Robinson and Tawn [37] and Jonathan and Ewans [40], the directional effects vary smoothly so that wave data and their actual behavior in the marine environment are better represented by means of a smooth periodic function.

_{s}taking into account directionality effects. In this context, three methods as regards threshold selection that are widely used in the relevant literature are examined, namely mean excess function, threshold stability and percentiles, along with two common declustering methods, that is, intervals and runs declustering methods. An additional method for declustering extreme data proposed by Soukissian and Kalantzi [43] is also assessed.

_{s}design values is provided. The key features of the most frequently used threshold selection and declustering methods are briefly described along with the DeCA declustering method, which is physically consistent with the wave phenomenon. Section 3 deals with the estimation of the uncertainties of parameters and design values focusing on the bias-corrected and accelerated bootstrap method. In Section 4, the hindcast wave data are presented and statistically analyzed, with respect to H

_{s}and mean sea state direction θ

_{w}, for four locations in the eastern Mediterranean Sea. Section 5 includes some preliminary results regarding the determination of the proposed directional model by considering all the combinations of the methods presented in Section 2, while the final results of parameter estimates and design values along with their uncertainties is presented for a particular combination (based on the maximum number of exceedances) for all locations. The last section includes the concluding remarks of this analysis and suggestions for further research directions.

## 2. Directional Extreme Value Model

#### 2.1. Parameter Estimation

#### 2.2. Design Values for Directional Extreme model

#### 2.3. Methods for Threshold Selection

#### 2.3.1. Mean Excess Plot

#### 2.3.2. Threshold Stability Plot

#### 2.3.3. Percentiles

#### 2.4. Methods for Declustering

- Define clusters of observations in case of consecutive exceedances based on an empirical criterion or parametric models (e.g., Markov chain models, Bartlett-Lewis process).
- Identify the highest value in each cluster, called declustered peaks.
- Assume the declustered peaks are independent and fit GP distribution to these peaks.

#### 2.4.1. Runs Declustering Method

#### 2.4.2. Intervals Declustering Method

#### 2.4.3. Declustering Algorithm (DeCA)

## 3. Uncertainty Quantification

- Step 1: Estimate the unknown parameters $\left({\widehat{\sigma}}_{u},\widehat{\xi}\right)$ of the GP distribution (as functions of $\theta $) from the initial sample using the ML method described above.
- Step 2: Create $r$ (random) samples ${\left\{{s}_{i}^{\left(r\right)}\right\}}_{i=1}^{n}$, $r=1,\dots ,R$, by random resampling with replacement from the initial sample and obtain the estimates $\left({\widehat{\sigma}}_{u}^{*},{\widehat{\xi}}^{*}\right)$.
- Step 3: Repeat step 2 for a large number $R$. The minimum number of bootstrap sample $R$ for the calculation of reliable confidence intervals is 1000, as is suggested by various studies that address modelling of extremes of environmental parameters; see, for example, Kysely [76], Panagoulia, et al. [77] and Soukissian and Tsalis [78].
- Step 4: Estimate the two constants of BCA bootstrap method, ${\widehat{z}}_{0}$ and $\widehat{a}$ for each unknown parameter. Then estimate the lower and upper limits ${\widehat{\sigma}}_{u}^{\left({\alpha}_{1}\right)}$, ${\widehat{\xi}}^{\left({\alpha}_{1}\right)}$ and ${\widehat{\sigma}}_{u}^{\left({\alpha}_{2}\right)}$, ${\widehat{\xi}}^{\left({\alpha}_{2}\right)}$, respectively.

## 4. Description of Wave Data

## 5. Numerical Results

#### 5.1. Preliminary Results

#### 5.2. Final Results

- For Aegean Sea, the dominant sector for extreme wave heights is the northern one, probably attributed to the Etesian winds, which gives extreme values up to 7 m at this sector and lower values characterize the rest directional sectors (e.g., for the sector $\left[50\xb0,310\xb0\right]$ the ${H}_{S}$ value is 4.3 m in the mean) as regards the 50-year return period. Furthermore, the low values of the lower bound of the 97.5% confidence intervals in the north-western sector can be justified by the lack of data obtained from the implementation of the specific combination of methods.
- For Ligurian Sea, the north-eastern sector is characterized by high values of ${H}_{S}$ (5.4 m for the 50-year return period), even though it is the second dominant directional sector for ${H}_{S}$, while the southern sector, with the least amount of extreme data, provides the lowest values (3.6 m).
- For Otranto Strait, the two dominant wave directions (in the south and south-eastern sectors) are translated in two consecutive peaks in the ${H}_{S}$ design value graphs, while the two concave forms (in the north-eastern and western sectors) correspond to the sectors with the minimum amount of extreme data. Let note that the form of the lower bounds differs from the ${H}_{S}$ design value.
- For Sicily Strait, the location with the most intense sea states according to the analyzed hindcast wave data, the second dominant directional sector for ${H}_{S}$ (i.e., the western) is characterized by the highest ${H}_{S}$ design values (8.4 m for the 50-year return period) and the lowest values are observed for the south-eastern sector (5.9 m for the 50-year return period). The largest difference between the lower bounds of the confidence interval and the ${H}_{S}$ design value is close to 6.3 m for the 50-year return period encountered in the south-western sector.

## 6. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

## References

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**Figure 1.**Estimated parameters $\xi $ and ${\sigma}_{u}$ for a 5th order Fourier model with the consideration of the penalty term (dashed line) and without (solid line). Circles represent the estimates from the independent fits of the 45-degree sectors.

**Figure 3.**Bivariate histograms of significant wave height ${H}_{S}$ and mean wave direction ${\theta}_{W}$ for (

**a**) Aegean Sea, (

**b**) Ligurian Sea, (

**c**) Otranto Strait and (

**d**) Sicily Strait. The bin width for ${H}_{S}$ is 0.5m and for ${\theta}_{W}$ is 15°.

**Figure 4.**Plots of mean excess function (left panels) and threshold stability with 95% confidence intervals (right panels) for (

**a**) Aegean Sea, (

**b**) Ligurian Sea, (

**c**) Otranto Strait and (

**d**) Sicily Strait.

**Figure 5.**Estimated parameters $\xi $ and ${\sigma}_{u}$ of the directional extreme value model obtained with the consideration of the penalty term (dashed line) and without (solid line) for (

**a**) Aegean Sea and (

**b**) Otranto Strait. Circles represent the estimates from the independent fits of the 45-degree sectors.

**Figure 6.**Cumulative distribution functions of the directional extreme value model with (blue solid line) and without (green solid line) the penalty term along with the corresponding empirical function (black solid line) for (

**a**) Aegean Sea and (

**b**) Otranto Strait

**Figure 7.**${H}_{S}$ design values for the 50-year return period obtained by the proposed directional model (blue solid line), the Generalized Pareto (GP) distribution without the consideration of directionality (green dashed line) and the independent fits (red circles) for (

**a**) Aegean Sea, (

**b**) Ligurian Sea, (

**c**) Otranto Strait and (

**d**) Sicily Strait.

**Figure 8.**(

**a**) Wave rose of ${H}_{S}$ exceedances and ${H}_{S}$ design values for (

**b**) 50-year and (

**c**) 100-year return period with 97.5% confidence intervals from the bias-corrected and accelerated (BCA) method for Aegean Sea.

**Figure 9.**(

**a**) Wave rose of ${H}_{S}$ exceedances and ${H}_{S}$ design values for (

**b**) 50-year and (

**c**) 100-year return period with 97.5% confidence intervals from BCA method for Ligurian Sea.

**Figure 10.**(

**a**) Wave rose of ${H}_{S}$ exceedances and ${H}_{S}$ design values for (

**b**) 50-year and (

**c**) 100-year return period with 97.5% confidence intervals from BCA method for Otranto Strait.

**Figure 11.**(

**a**) Wave rose of ${H}_{S}$ exceedances and ${H}_{S}$ design values for (

**b**) 50-year and (

**c**) 100-year return period with 97.5% confidence intervals from the BCA method for Sicily Strait.

Area | Latitude, Longitude (°) | Period | Sample Size |
---|---|---|---|

Aegean Sea | 37.75° N, 25.25° E | 1979–2014 | 52,596 |

Ligurian Sea | 43.25° N, 9.75° E | ||

Otranto Strait | 40.25° N, 19.00° E | ||

Sicily Strait | 37.75° N, 12.25° E |

Locations | ${\mathit{m}}_{{\mathit{H}}_{\mathit{S}}}$ (m) | $\mathit{m}\mathit{e}{\mathit{d}}_{{\mathit{H}}_{\mathit{S}}}$ (m) | $\mathbf{m}\mathbf{i}{\mathbf{n}}_{{\mathit{H}}_{\mathit{S}}}$ (m) | $\mathbf{m}\mathbf{a}{\mathbf{x}}_{{\mathit{H}}_{\mathit{S}}}$ (m) | ${\mathit{s}}_{{\mathit{H}}_{\mathit{S}}}$ (m) | $\mathit{C}{\mathit{V}}_{{\mathit{H}}_{\mathit{S}}}$ (%) | $\mathit{S}{\mathit{k}}_{{\mathit{H}}_{\mathit{S}}}$ | $\mathit{K}{\mathit{u}}_{{\mathit{H}}_{\mathit{S}}}$ |
---|---|---|---|---|---|---|---|---|

Aegean Sea | 1.0 | 0.8 | 0.1 | 5.4 | 0.7 | 69.5 | 1.3 | 5.3 |

Ligurian Sea | 0.6 | 0.5 | 0.1 | 5.4 | 0.5 | 80 | 1.8 | 7.6 |

Otranto Strait | 0.5 | 0.3 | 0.0 | 3.8 | 0.4 | 85.5 | 1.9 | 7.7 |

Sicily Strait | 1.0 | 0.8 | 0.1 | 6.4 | 0.7 | 74.4 | 1.7 | 7.1 |

Locations | ${\mathit{m}}_{{\mathit{\theta}}_{\mathit{W}}}$ (rad) | ${\overline{\mathit{R}}}_{{\mathit{\theta}}_{\mathit{W}}}$ | ${\mathit{V}}_{{\mathit{\theta}}_{\mathit{W}}}$ | ${\mathit{s}}_{{\mathit{\theta}}_{\mathit{W}}}$ | $\mathit{S}{\mathit{k}}_{{\mathit{\theta}}_{\mathit{W}}}$ | $\mathit{K}{\mathit{u}}_{{\mathit{\theta}}_{\mathit{W}}}$ |
---|---|---|---|---|---|---|

Aegean Sea | 353.47 | 0.42 | 0.58 | 1.08 | 0.38 | 0.51 |

Ligurian Sea | 272.26 | 0.31 | 0.69 | 1.18 | −0.18 | 0.22 |

Otranto Strait | 240.07 | 0.17 | 0.83 | 1.29 | −0.29 | −0.32 |

Sicily Strait | 287.50 | 0.39 | 0.61 | 1.11 | 0.28 | 0.20 |

**Table 4.**Threshold values of significant wave height (in m) by threshold selection method for the examined locations.

Threshold Selection Method | Aegean Sea | Ligurian Sea | Otranto Strait | Sicily Strait |
---|---|---|---|---|

95th percentile | 2.32 | 1.62 | 1.24 | 2.47 |

Mean excess function | 1.90 | 1.30 | 0.96 | 2.00 |

Threshold stability | 2.10 | 1.50 | 1.00 | 2.10 |

DeCA | 2.61 | 1.89 | 1.25 | 2.66 |

**Table 5.**Number of exceedances of significant wave height for each combination of methods and for all locations.

Threshold Selection Method | Declustering Method | Aegean Sea | Ligurian Sea | Otranto Strait | Sicily Strait |
---|---|---|---|---|---|

95th percentile | Runs | 323 | 340 | 297 | 288 |

Intervals | 671 | 830 | 782 | 669 | |

Mean excess function | Runs | 383 | 374 | 326 | 328 |

Intervals | 1234 | 1303 | 1229 | 1064 | |

Threshold stability | Runs | 365 | 359 | 325 | 322 |

Intervals | 939 | 991 | 1165 | 963 | |

DeCA | DeCA | 197 | 285 | 308 | 233 |

**Table 6.**Order of the Fourier model and value of the weighting constant $w$ (within parenthesis) for each combination of methods and for all locations.

Threshold Selection Method | Declustering Method | Aegean Sea | Ligurian Sea | Otranto Strait | Sicily Strait |
---|---|---|---|---|---|

95th percentile | Runs | 1 (0.20) | 3 (0.24) | 1 (0.13) | 1 (0.06) |

Intervals | 1 (0.03) | 3 (0.18) | 1 (0.12) | 1 (0.01) | |

Mean excess function | Runs | 1 (0.31) | 2 (0.42) | 1 (0.22) | 1 (0.10) |

Intervals | 2 (0.09) | 1 (0.17) | 3 (0.03) | 1 (0.02) | |

Threshold stability | Runs | 1 (0.17) | 3 (0.42) | 1 (0.17) | 1 (0.29) |

Intervals | 1 (0.02) | 1 (0.30) | 1 (0.03) | 3 (0.03) | |

DeCA | DeCA | 1 (0.20) | 3 (0.24) | 1 (0.13) | 1 (0.06) |

Parameter | Estimate | Bootstrap 97.5% CIs |
---|---|---|

${A}_{10}$ | −0.17 | [−0.59, −0.04] |

${A}_{11}$ | 0.10 | [−0.31, 0.20] |

${A}_{21}$ | −0.20 | [−0.54, −0.06] |

${A}_{12}$ | 0.14 | [−0.09, 0.22] |

${A}_{22}$ | 0.06 | [−0.46, 0.32] |

${B}_{10}$ | 0.66 | [0.36, 0.79] |

${B}_{11}$ | −0.02 | [−0.28, 0.10] |

${B}_{21}$ | 0.35 | [−0.16, 0.52] |

${B}_{12}$ | −0.13 | [−0.42, −0.05] |

${B}_{22}$ | 0.00 | [−0.64, 0.18] |

Parameter | Estimate | Bootstrap 97.5% CIs |
---|---|---|

${A}_{10}$ | −0.07 | [−0.29, −0.02] |

${A}_{11}$ | −0.02 | [−0.20, 0.05] |

${A}_{21}$ | 0.07 | [−0.17, 0.16] |

${B}_{10}$ | 0.54 | [0.46, 0.57] |

${B}_{11}$ | 0.16 | [0.01, 0.21] |

${B}_{21}$ | −0.08 | [−0.20, −0.01] |

Parameter | Estimate | Bootstrap 97.5% CIs |
---|---|---|

${A}_{10}$ | −0.24 | [−0.87, −0.11] |

${A}_{11}$ | 0.00 | [−0.31, 0.09] |

${A}_{21}$ | 0.15 | [−0.44, 0.31] |

${A}_{12}$ | 0.16 | [−0.64, 0.28] |

${A}_{22}$ | 0.16 | [−0.24, 0.30] |

${A}_{13}$ | 0.15 | [−0.43, 0.32] |

${A}_{23}$ | −0.07 | [−0.26, 0.00] |

${B}_{10}$ | 0.51 | [0.23, 0.54] |

${B}_{11}$ | −0.14 | [−0.35, −0.09] |

${B}_{21}$ | 0.00 | [−0.20, 0.10] |

${B}_{12}$ | −0.01 | [−0.23, 0.07] |

${B}_{22}$ | −0.15 | [−0.33, −0.06] |

${B}_{13}$ | −0.03 | [−0.27, 0.05] |

${B}_{23}$ | 0.09 | [−0.08, 0.15] |

Parameter | Estimate | Bootstrap 97.5% CIs |
---|---|---|

${A}_{10}$ | 0.00 | [−0.52, 0.08] |

${A}_{11}$ | −0.16 | [−1.05, −0.04] |

${A}_{21}$ | −0.02 | [−0.71, 0.13] |

${B}_{10}$ | 0.71 | [0.27, 0.76] |

${B}_{11}$ | 0.37 | [−0.45, 0.47] |

${B}_{21}$ | −0.09 | [−0.96, 0.01] |

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## Share and Cite

**MDPI and ACS Style**

Karathanasi, F.; Soukissian, T.; Belibassakis, K. Directional Extreme Value Models in Wave Energy Applications. *Atmosphere* **2020**, *11*, 274.
https://doi.org/10.3390/atmos11030274

**AMA Style**

Karathanasi F, Soukissian T, Belibassakis K. Directional Extreme Value Models in Wave Energy Applications. *Atmosphere*. 2020; 11(3):274.
https://doi.org/10.3390/atmos11030274

**Chicago/Turabian Style**

Karathanasi, Flora, Takvor Soukissian, and Kostas Belibassakis. 2020. "Directional Extreme Value Models in Wave Energy Applications" *Atmosphere* 11, no. 3: 274.
https://doi.org/10.3390/atmos11030274