# Multivariate Hybrid Modelling of Future Wave-Storms at the Northwestern Black Sea

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## Abstract

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## 1. Introduction

## 2. Study Area

## 3. Methods

#### 3.1. First Step: Process-Based Dynamical Modelling

#### 3.2. Second Step: The Statistical Model

#### 3.2.1. Definition of Wave-Storms and Their Components

#### 3.2.2. Generalized Pareto Distribution: Univariate Distribution-Function

#### 3.2.3. Copulas: The Joint-Dependence Structure

#### 3.3. Third Step: Validation of the Non-Stationary Statistical Model

#### 3.4. Fourth Step: Comparison of the Different GCMs

## 4. Results

#### 4.1. RCP4.5

#### 4.2. RCP8.5

## 5. Discussion

#### 5.1. RCP4.5

#### 5.2. RCP8.5

#### 5.3. Applicability of the Results

## 6. Conclusions

## Acknowledgments

## Author Contributions

## Conflicts of Interest

## Abbreviations

D | total wave-storm duration |

E | total wave-storm energy |

EA | East Atlantic Pattern |

GCM | general (atmospheric) circulation model |

GPD | generalized Pareto distribution |

HAC | hierarchical Archimedean copula |

${H}_{p}$ | significant wave-height at the peak of the wave-storm |

NAO | North Atlantic Oscillation |

PACF | partial autocorrelation function |

RCM | regional (atmospheric) circulation model |

SC | Scandinavian Pattern |

SWAN | Simulating WAves Nearshore (spectral wave-model) |

${T}_{p}$ | peak wave-period at the peak of the wave-storm |

VGAM | vectorial generalized additive model |

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**Figure 1.**(

**i**) Map of the Black Sea. The computational domain for the Simulating WAves Nearshore (SWAN) model is enclosed by a green rectangle (see first step of the proposed methodology), whereas the northwestern Black Sea is enclosed by a red rectangle (see the second step of the same methodology). (

**ii**) Map of the northwestern Black Sea. Nodes from the statistical model are in red and are numbered from north to south. The right figure shows wave-roses at points

**A**and

**B**(orange dots) from the map of the northwestern Black Sea [29]. The bar on the right-bottom shows the wave height-ranges at the wave-rose.

**Figure 2.**Flow-chart of the different steps in the proposed hybrid methodology. This methodology is analogous for each climate change scenario (Representative Concentration Pathways, RCP, 4.5 and 8.5). CNRM-CM5 is the General (Atmospheric) Circulation Model used to build the statistical model. ERA-interim is the reanalysis used to validate the statistical model. Each variable is fit by a generalized Pareto distribution (GPD) function and their joint probability structure is characterized by a series of hierarchical Archimedean copulas (HAC). Rectangles are the results, rhombuses are the methods. Different colors are only intended to separate different stages of the process. Elements inside the “input” box have been obtained from external sources. The side analyses are not included in the graph, for the sake of clarity.

**Figure 3.**(

**a**) Schematic view of the definition of a wave-storm. The blue thick line represents a wave, the red thin line represents a wave-storm threshold of value ${h}_{0}$. (

**b**) Hierarchical Archimedean copula (HAC)-structure-tree at node 13, in the period 2001–2050, under the RCP4.5 scenario. The circles contain the names of the storm components, which are bound together by a dependence-parameter, written in the rectangles ($\theta \in [0,\infty )$). This HAC-structure-tree comes from the stationary statistical model. It is representative of all nodes, at each time-window of 50 years and in both emission scenarios. Then, it is selected to represent the dependence-structure of the whole non-stationary statistical model.

**Figure 4.**Location parameter ${x}_{0}$ of the generalized Pareto distribution (GPD) that models (

**a**) E, (

**b**) ${H}_{p}$, (

**c**) ${T}_{p}$, and (

**d**) D at selected nodes, under the RCP4.5 scenario. The Eastern Atlantic Pattern (EA) was used in the vectorial generalized additive model (VGAM) to predict the location parameter ${x}_{0}$ of E and D. The first time-derivative of the North Atlantic Oscillation (NAO) was used in the VGAM to predict the location parameter ${x}_{0}$ of ${H}_{p}$. The Scandinavian Pattern (SC) was used in the VGAM to predict the location parameter ${x}_{0}$ of ${T}_{p}$. The time series are approximated by a straight line, which helps to see the trend of the location parameters ${x}_{0}$ in each case.

**Figure 5.**Scale-parameter $\beta $ of the GPD that models (

**a**) E, (

**b**) ${H}_{p}$, (

**c**) ${T}_{p}$, and (

**d**) D at selected nodes, under the RCP4.5 scenario. The EA was used in the VGAM to predict the scale parameter $\beta $ of ${H}_{p}$ and D. The first time-derivative of EA was used in the VGAM to predict the scale parameter $\beta $ of ${T}_{p}$. The scale parameter $\beta $ of E is not affected by the selected climate indices. The time series are approximated by a straight line, which helps to see the trend of the scale parameter $\beta $ in each case.

**Figure 6.**The dependence-parameter $\tau $ of (

**a**) all the storm components and (

**b**) the subset E-D, under the RCP4.5 scenario. Selected nodes serve to represent the results. Note that the scales for $\tau $ are different for (a) and (b).

**Figure 7.**Location parameter ${x}_{0}$ of the GPD that models (

**a**) E, (

**b**) ${H}_{p}$, (

**c**) ${T}_{p}$, and (

**d**) D at selected nodes, under the RCP8.5 scenario. The second time-derivative of EA was used in the VGAM to predict the location parameter ${x}_{0}$ of ${H}_{p}$. The NAO was used in the VGAM to predict the location parameter ${x}_{0}$ of ${T}_{p}$. The location parameter ${x}_{0}$ of E and D are not affected by the selected climate indices. The time series are approximated by a straight line, which helps to see the trend of the location parameters ${x}_{0}$ at each case.

**Figure 8.**Scale parameter $\beta $ of the GPD that models (

**a**) E, (

**b**) ${H}_{p}$, (

**c**) ${T}_{p}$, and (

**d**) D at selected nodes, under the RCP8.5 scenario. The SC was used in the VGAM to predict the scale parameter $\beta $ of ${H}_{p}$. The EA was used in the VGAM to predict the scale parameter $\beta $${T}_{p}$. The first time-derivative of SC was used in the VGAM to predict the scale parameter $\beta $ of D. The scale parameter $\beta $ of E is not affected by the selected climate indices. The time series are approximated by a straight line, which helps to see the trend of the scale parameter $\beta $ at each case.

**Figure 9.**The dependence-parameter $\tau $ of (

**a**) all the storm components and (

**b**) the subset E-D, under the RCP8.5 scenario. Selected nodes serve to represent the results. Note that the scales for $\tau $ are different for (a,b).

**Table 1.**List of general circulation models (GCM) employed. The GCM CNRM-CM5 is used to build the non-stationary statistical model, whereas the other GCMs are compared to CNRM-CM5. CNRM-CM5 is shown in boldface to ease its search in the list.

GCM | Latitude | Longitude |
---|---|---|

Grid Size (${}^{\circ}$) | Grid Size (${}^{\circ}$) | |

CMCC-CM | 0.7484 | 0.7500 |

CMCC-CMS | 3.7111 | 3.7500 |

CNRM-CM5 | 1.4008 | 1.4063 |

FGOALS-G2 | 2.7906 | 2.8125 |

GFDL-CM3 | 2.0000 | 2.5000 |

GFDL-ESM2G | 2.0225 | 2.0000 |

GFDL-ESM2M | 2.0225 | 2.5000 |

HadGEM2-AO | 1.2500 | 1.8750 |

HadGEM2-CC | 1.2500 | 1.8750 |

HadGEM2-ES | 1.2500 | 1.8750 |

INM-CM4 | 1.5000 | 2.0000 |

IPSL-CM5A-LR | 1.8947 | 3.7500 |

IPSL-CM5B-LR | 1.8947 | 3.7500 |

IPSL-CM5A-MR | 1.2676 | 2.5000 |

MIROC-ESM | 2.7906 | 2.8125 |

MIROC-ESM-CHEM | 2.7906 | 2.8125 |

MIROC5 | 1.4008 | 1.4063 |

MPI-ESM-LR | 1.8653 | 1.8750 |

MPI-ESM-MR | 1.8653 | 1.8750 |

**Table 2.**Summary table of the Discussion section. $d\left(\xb7\right)$ and ${d}^{2}\left(\xb7\right)$ are the first and second time-derivatives, respectively.

Variable or Test | Parameter | RCP4.5 | RCP8.5 |
---|---|---|---|

Main Covariate | |||

Estimated storminess | 27–35 storms/year | 23–32 storms/year | |

Storminess | ${x}_{0}$ | None | None |

Wave-storm threshold | ${x}_{0}$ | $d\mathrm{SC}+{d}^{2}\mathrm{SC}$ | ${d}^{2}\mathrm{EA}$ |

E | ${x}_{0}$ | EA | None |

$\beta $ | None | None | |

${H}_{p}$ | ${x}_{0}$ | $d\mathrm{NAO}$ | ${d}^{2}\mathrm{EA}$ |

$\beta $ | EA | SC | |

${T}_{p}$ | ${x}_{0}$ | SC | NAO |

$\beta $ | $d\mathrm{EA}$ | EA | |

D | ${x}_{0}$ | EA | None |

$\beta $ | EA | $d\mathrm{SC}$ | |

${\tau}_{root}$ | $0.55$–$0.65$ | $0.45$–$0.65$ | |

${\tau}_{(E,D)}$ | $0.82$–$0.87$ | $0.82$–$0.86$ | |

HAC is non-stationary? | Yes | Yes | |

Validated for 1979–2016? | Yes | Yes |

© 2018 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 (http://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Lin-Ye, J.; García-León, M.; Gràcia, V.; Ortego, M.I.; Stanica, A.; Sánchez-Arcilla, A.
Multivariate Hybrid Modelling of Future Wave-Storms at the Northwestern Black Sea. *Water* **2018**, *10*, 221.
https://doi.org/10.3390/w10020221

**AMA Style**

Lin-Ye J, García-León M, Gràcia V, Ortego MI, Stanica A, Sánchez-Arcilla A.
Multivariate Hybrid Modelling of Future Wave-Storms at the Northwestern Black Sea. *Water*. 2018; 10(2):221.
https://doi.org/10.3390/w10020221

**Chicago/Turabian Style**

Lin-Ye, Jue, Manuel García-León, Vicente Gràcia, M. Isabel Ortego, Adrian Stanica, and Agustín Sánchez-Arcilla.
2018. "Multivariate Hybrid Modelling of Future Wave-Storms at the Northwestern Black Sea" *Water* 10, no. 2: 221.
https://doi.org/10.3390/w10020221