# Modeling of Future Extreme Storm Surges at the NW Mediterranean Coast (Spain)

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## Abstract

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## 1. Introduction

## 2. Study Area

## 3. Theoretical Background and Methods

#### 3.1. The Process-Based Model

#### 3.2. Statistical Model

#### 3.2.1. Commonalities of the Two Strategies

#### 3.2.2. Differences of the Two Strategies for Modeling Dependence

#### 3.3. Validation of the Non-Stationary Statistical Model

^{th}, 20

^{th}quantiles). There were elements from both the observed ${S}_{p}$ and the model ${S}_{p}^{\ast}$ in each category of quantiles. Two vectors can be defined as:

## 4. Results

#### 4.1. Process-Based Model

#### 4.2. Non-Stationary Statistical Models

## 5. Discussion

## 6. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

## Abbreviations

AIC | Akaike Information Criterion |

CMIP5 | Coupled Model Intercomparison Project 5 |

$EAWR$ | East Atlantic Pattern |

GCM | General Circulation Model |

GPD | Generalized Pareto Distribution |

HYPSE | Hydrostatic Padua Surface Elevation Model |

$NAO$ | North Atlantic Oscillation |

Probability Distribution Function | |

RCM | Regional Circulation Model |

REDMAR | RED de MAReográfos (Spanish Acronym for Puertos del Estado Tidal Monitoring Network) |

$SC$ | Scandinavian Pattern |

SWH | Significant Wave Height |

VGAM | Vectorial Generalized Additive Model |

VGLM | Vectorial Generalized Linear Model |

## References

- Bolaños, R.; Jordà, G.; Cateura, J.; López, J.; Puigdefàbregas, J.; Gómez, J.; Espino, M. The XIOM: 20 years of a regional coastal observatory in the Spanish Catalan coast. J. Mar. Syst.
**2009**, 77, 237–260. [Google Scholar] [CrossRef] [Green Version] - Soomere, T.; Pindsoo, K. Spatial variability in the trends in extreme storm surges and weekly-scale high water levels in the eastern Baltic Sea. Cont. Shelf Res.
**2016**, 115, 53–64. [Google Scholar] [CrossRef] - Brown, J.D.; Spencer, T.; Moeller, I. Modeling storm surge flooding of an urban area with particular reference to modeling uncertainties: A case study of Canvey Island, United Kingdom. Water Resour. Res.
**2007**, 43. [Google Scholar] [CrossRef] [Green Version] - Hinkel, J.; Lincke, D.; Vafeidis, A.T.; Perrette, M.; Nicholls, R.J.; Tol, R.S.J.; Marzeion, B.; Fettweis, X.; Ionescu, C.; Levermann, A. Coastal flood damage and adaptation costs under 21st century sea-level rise. Proc. Natl. Acad. Sci. USA
**2014**, 111, 3292–3297. [Google Scholar] [CrossRef] [PubMed] [Green Version] - Zhang, H.; Cheng, W.C.; Qiu, X.X.; Feng, X.B.; Gong, W.P. Tide-surge interaction along the east coast of the Leizhou Peninsula, South China Sea. Cont. Shelf Res.
**2017**, 142, 32–49. [Google Scholar] [CrossRef] - Ullmann, A.; Pirazzoli, P.A.; Tomasin, A. Sea surges in Camargue: Trends over the 20th century. Cont. Shelf Res.
**2007**, 27, 922–934. [Google Scholar] [CrossRef] - Staneva, J.; Wahle, K.; Koch, W.; Behrens, A.; Fenoglio-Marc, L.; Stanev, E.V. Coastal flooding: Impact of waves on storm surge during extremes - a case study for the German Bight. Nat. Hazards Earth Sys. Sci.
**2016**, 16, 2373–2389. [Google Scholar] [CrossRef] [Green Version] - Sánchez-Arcilla, A.; García-León, M.; Gracia, V.; Devoy, R.; Stanica, A.; Gault, J. Managing coastal environments under climate change: Pathways to adaptation. Sci. Total Environ.
**2016**, 572, 1336–1352. [Google Scholar] [CrossRef] [Green Version] - Krien, Y.; Testut, L.; Islam, A.K.M.S.; Bertin, X.; Durand, F.; Mayet, C.; Tazkia, A.R.; Becker, M.; Calmant, S.; Papa, F.; et al. Towards improved storm surge models in the northern Bay of Bengal. Cont. Shelf Res.
**2017**, 135, 58–73. [Google Scholar] [CrossRef] - Grases, A.; Gracia, V.; García-León, M.; Lin-Ye, J.; Sierra, J.P. Coastal Flooding and Erosion under a Changing Climate: Implications at a Low-Lying Coast (Ebro Delta). Water
**2020**, 12, 346. [Google Scholar] [CrossRef] [Green Version] - Silva, A.T.; Naghettini, M.; Portela, M.M. On some aspects of peaks-over-threshold modeling of floods under nonstationarity using climate covariates. Stoch. Environ. Res. Risk Assess.
**2016**, 30, 207–224. [Google Scholar] [CrossRef] - Du, T.; Xiong, L.H.; Xu, C.Y.; Gippel, C.J.; Guo, S.; Liu, P. Return period and risk analysis of nonstationary low-flow series under climate change. J. Hydrol.
**2015**, 527, 234–250. [Google Scholar] [CrossRef] [Green Version] - Vousdoukas, M.I.; Voukouvalas, E.; Annunziato, A.; Giardino, A.; Feyen, L. Projections of extreme storm surge levels along Europe. Clim. Dyn.
**2016**, 1–20. [Google Scholar] [CrossRef] [Green Version] - Conte, D.; Lionello, P. Characteristics of large positive and negative surges in the Mediterranean Sea and their attenuation in future climate scenarios. Glob. Planet. Chang.
**2013**, 111, 159–173. [Google Scholar] [CrossRef] - Marcos, M.; Jordà, G.; Gomis, D.; Pérez, B. Changes in storm surges in southern Europe from a regional model under climate change scenarios. Glob. Planet. Chang.
**2011**, 77, 116–128. [Google Scholar] [CrossRef] - Barnston, A.G.; Livezey, R.E. Classification, Seasonality and Persistence of Low-Frequency Atmospheric Circulation Patterns. Mon. Weather Rev.
**1987**, 115, 1083–1126. [Google Scholar] [CrossRef] - Hurrell, J.W.; Deser, C. North Atlantic climate variability: The role of the North Atlantic Oscillation. J. Marine Syst.
**2009**, 78, 28–41. [Google Scholar] [CrossRef] - Lionello, P.; Malanotte-Rizzoli, P.; Boscolo, R. Mediterranean Climate Variability; Elsevier: Amsterdam, The Netherlands, 2006; Volume 4. [Google Scholar]
- Lionello, P.; Sanna, A. Mediterranean wave climate variability and its links with NAO and Indian Monsoon. Clim. Dyn.
**2005**, 25, 611–623. [Google Scholar] [CrossRef] - Lionello, P.; Cogo, S.; Galati, M.; Sanna, A. The Mediterranean surface wave climate inferred from future scenario simulations. Glob. Planet. Change
**2008**, 63, 152–162. [Google Scholar] [CrossRef] - Lin-Ye, J.; García-León, M.; Gràcia, V.; Ortego, M.I.; Lionello, P.; Sánchez-Arcilla, A. Multivariate statistical modeling of future marine storms. Appl. Ocean Res.
**2017**, 65, 192–205. [Google Scholar] [CrossRef] [Green Version] - Trigo, R.M.; Osborn, T.J.; Corte-Real, J.M. The North Atlantic Oscillation influence on Europe: Climate impacts and associated physical mechanisms. Clim. Res.
**2002**, 20, 9–17. [Google Scholar] [CrossRef] - Ulbrich, U.; Lionello, P.; Belusic, D.; Jacobeit, J.; Knippertz, P.; Kuglitsch, F.G.; Leckebusch, G.C.; Luterbacher, J.; Maugeri, M.; Maheras, P.; et al. 5—Climate of the Mediterranean: Synoptic Patterns, Temperature, Precipitation, Winds, and Their Extremes. In The Climate of the Mediterranean Region; Lionello, P., Ed.; Elsevier: Oxford, UK, 2012; pp. 301–346. [Google Scholar] [CrossRef]
- Lionello, P.; Galati, M.B. Links of the significant wave height distribution in the Mediterranean sea with the Northern Hemisphere teleconnection patterns. Adv. Geosci.
**2008**, 17, 13–18. [Google Scholar] [CrossRef] [Green Version] - Tsimplis, M.N.; Shaw, A.G.P.; Flather, R.A.; Woolf, D.K. The influence of the North Atlantic Oscillation on the sea-level around the northern European coasts reconsidered: The thermosteric effects. Philos. Trans. A Math. Phys. Eng. Sci.
**2006**, 364, 845–856. [Google Scholar] [CrossRef] [PubMed] - Woodworth, P.; Flather, R.A.; Williams, J.A.; Wakelin, S.L.; Jevrejeva, S. The dependence of U.K. extreme sea levels and storm surges on the North Atlantic Oscillation. Cont. Shelf Res.
**2007**, 27, 935–946. [Google Scholar] [CrossRef] - Cid, A.; Menéndez, M.; Castanedo, S.; Abascal, A.J.; Méndez, F.J.; Medina, R. Long-term changes in the frequency, intensity and duration of extreme storm surge events in southern Europe. Clim. Dyn.
**2016**, 46, 1503–1516. [Google Scholar] [CrossRef] - Lionello, P.; Conte, D.; Marzo, L.; Scarascia, L. The contrasting effect of increasing mean sea level and decreasing storminess on the maximum water level during storms along the coast of the Mediterranean Sea in the mid 21st century. Glob. Planet. Chang.
**2016**, 151, 80–91. [Google Scholar] [CrossRef] - Wadey, M.P.; Haigh, I.D.; Brown, J.M. A century of sea level data and the UK’s 2013/14 storm surges: An assessment of extremes and clustering using the Newlyn tide gauge record. Ocean Sci.
**2014**, 10, 1031–1045. [Google Scholar] [CrossRef] [Green Version] - Wakelin, S.L.; Woodworth, P.L.; Flather, R.A.; Williams, J.A. Sea-level dependence on the NAO over the NW European Continental Shelf. Geophy. Res. Lett.
**2003**, 30. [Google Scholar] [CrossRef] - Hemer, M.A.; Trenham, C.E. Evaluation of a CMIP5 derived dynamical global wind wave climate model ensemble. Ocean Model.
**2016**, 103, 190–203. [Google Scholar] [CrossRef] [Green Version] - Trenberth, K.; Fasullo, J.; Shepherd, T. Attribution of climate extreme events. Nat. Clim. Chang.
**2015**, 5, 725–730. [Google Scholar] [CrossRef] - Campos, Á.; García-Valdecasas, J.M.; Molina, R.; Castillo, C.; Álvarez-Fanjul, E.; Staneva, J. Addressing Long-Term Operational Risk Management in Port Docks under Climate Change Scenarios—A Spanish Case Study. Water
**2019**, 11, 2153. [Google Scholar] [CrossRef] [Green Version] - Marcos, M.; Tsimplis, M.N.; Shaw, A.G.P. Sea level extremes in southern Europe. J. Geophys. Res. Oceans
**2009**, 114. [Google Scholar] [CrossRef] [Green Version] - Lionello, P.; Conte, D.; Reale, M. The effect of cyclones crossing the Mediterranean region on sea level anomalies on the Mediterranean Sea coast. Nat. Hazards Earth Syst. Sci.
**2019**, 19, 1541–1564. [Google Scholar] [CrossRef] [Green Version] - De Michele, C.; Salvadori, G.; Passoni, G.; Vezzoli, R. A multivariate model of sea storms using copulas. Coast. Eng.
**2007**, 54, 734–751. [Google Scholar] [CrossRef] - Wahl, T.; Mudersbach, C.; Jensen, J. Assessing the hydrodynamic boundary conditions for risk analyses in coastal areas: A multivariate statistical approach based on Copula functions. Nat. Hazards Earth Syst. Sci.
**2012**, 12, 495–510. [Google Scholar] [CrossRef] [Green Version] - Corbella, S.; Stretch, D.D. Simulating a multivariate sea storm using Archimedean copulas. Coast. Eng.
**2013**, 76, 68–78. [Google Scholar] [CrossRef] - Masina, M.; Lamberti, A.; Archetti, R. Coastal flooding: A copula based approach for estimating the joint probability of water levels and waves. Coast. Eng.
**2015**, 97, 37–52. [Google Scholar] [CrossRef] - Woollings, T.; Blackburn, M. The North Atlantic jet stream under climate change and its relation to the NAO and EA patterns. J. Clim.
**2012**, 25, 886–902. [Google Scholar] [CrossRef] [Green Version] - Marra, G.; Radice, R. Bivariate copula additive models for location, scale and shape. Comp. Stat. Data Anal.
**2017**, 112, 99–113. [Google Scholar] [CrossRef] - Stocker, T.; Qin, D.; Plattner, G.K.; Tignor, M.; Allen, S.; Boschung, J.; Nauels, A.; Xia, Y.; Bex, V.; Midgley, P. IPCC, 2013: Summary for Policymakers. In Climate Change 2013: The Physical Science Basis. Contribution of Working Group I to the Fifth Assessment Report of the Intergovernmental Panel on Climate Change; Cambridge University Press: Cambridge, UK; New York, NY, USA, 2013. [Google Scholar]
- Scoccimarro, E.; Gualdi, S.; Bellucci, A.; Sanna, A.; Fogli, P.G.; Manzini, E.; Vichi, M.; Oddo, P.; Navarra, A. Effects of Tropical Cyclones on Ocean Heat Transport in a High-Resolution Coupled General Circulation Model. J. Clim.
**2011**, 24, 4368–4384. [Google Scholar] [CrossRef] [Green Version] - Butterworth, S. On the theory of filter amplifiers. Wireless Eng.
**1930**, 7, 536–541. [Google Scholar] - Rockel, B.; Will, A.; Hense, A. The Regional Climate Model COSMO-CLM (CCLM). Meteorol. J.
**2008**, 17, 347–348. [Google Scholar] [CrossRef] - Lionello, P.; Mufato, R.; Tomasin, A. Sensitivity of free and forced oscillations of the Adriatic Sea to sea level rise. Clim. Res.
**2005**, 29, 23–29. [Google Scholar] [CrossRef] [Green Version] - Egozcue, J.J.; Pawlowsky-Glahn, V.; Ortego, M.I.; Tolosana-Delgado, R. The effect of scale in daily precipitation hazard assessment. Nat. Hazards Earth Sys. Sci.
**2006**, 6, 459–470. [Google Scholar] [CrossRef] [Green Version] - Yee, T.W.; Wild, C.J. Vector Generalized Additive Models. J. R. Stat. Soc. Series B. Methodol.
**1996**, 58, 481–493. [Google Scholar] [CrossRef] - Fessler, J. Nonparametric fixed-interval smoothing with vector splines. Signal Process. IEEE Trans.
**1991**, 39, 852–859. [Google Scholar] [CrossRef] [Green Version] - Hastie, T.J.; Tibshirani, R.J. Generalized Additive Models; CRC Press: Boca Raton, FL, USA, 1990; Volume 43. [Google Scholar]
- Rigby, R.A.; Stasinopoulos, D.M. Generalized additive models for location, scale and shape. J. R. Stat. Soc. Series C Appl. Stat.
**2005**, 54, 507–554. [Google Scholar] [CrossRef] [Green Version] - Yee, T.W.; Stephenson, A.G. Vector generalized linear and additive extreme value models. Extremes
**2007**, 10, 1–19. [Google Scholar] [CrossRef] - Coles, S. An introduction to Statistical Modeling of Extreme Values; Springer: London, UK, 2001; pp. 801–804. [Google Scholar]
- Koenker, R. Quantile Regression; Econometric Society Monographs; Cambridge University Press: Cambridge, UK, 2005. [Google Scholar]
- Northrop, P.J.; Jonathan, P. Threshold modeling of spatially dependent non-stationary extremes with application to hurricane-induced wave heights. Environmetrics
**2011**, 22, 799–809. [Google Scholar] [CrossRef] - Jonathan, P.; Ewans, K.; Randell, D. Joint modeling of extreme ocean environments incorporating covariate effects. Coast. Eng.
**2013**, 79, 22–31. [Google Scholar] [CrossRef] [Green Version] - Akaike, H. Factor analysis and AIC. Psychometrika
**1987**, 52, 317–332. [Google Scholar] [CrossRef] - Tamura, Y.; Sato, T.; Ooe, M.; Ishiguro, M. A procedure for tidal analysis with a Bayesian information criterion. Geophys. J. Int.
**1991**, 104, 507–516. [Google Scholar] [CrossRef] [Green Version] - Sklar, A. Fonctions dé Repartition à n Dimension et Leurs Marges; Université Paris 8: Paris, France, 1959. [Google Scholar]
- Nelsen, R. An Introduction to Copulas; Springer Science & Business Media: New York, NY, USA, 2007. [Google Scholar]
- Okhrin, O.; Okhrin, Y.; Schmid, W. On the structure and estimation of hierarchical Archimedean copulas. J. Econom.
**2013**, 173, 189–204. [Google Scholar] [CrossRef] - Lin-Ye, J.; García-León, M.; Gràcia, V.; Sánchez-Arcilla, A. A multivariate statistical model of extreme events: An application to the Catalan coast. Coast. Eng.
**2016**, 117, 138–156. [Google Scholar] [CrossRef] [Green Version] - Kendall, M. A new measure of rank correlation. Biometrika
**1937**, 6, 83–93. [Google Scholar] - Salvadori, G.; De Michele, C.; Durante, F. On the return period and design in a multivariate framework. Hydrol. Earth Syst. Sci.
**2011**, 15, 3293–3305. [Google Scholar] [CrossRef] [Green Version] - Joe, H. Dependence Modeling with Copulas; Chapman and Hall/CRC: London, UK, 2014. [Google Scholar]
- Salvadori, G.; De Michele, C.; Kottegoda, N.T.; Rosso, R. Extremes in Nature: An Approach Using Copulas; Springer Science & Business Media: Amsterdam, The Netherlands, 2007; Volume 56. [Google Scholar]
- Kullback, S. Information Theory and Statistics; Courier Corporation: Chelmsford, MA, USA, 1997. [Google Scholar]
- Aitchison, J. On criteria for measures of compositional difference. Math. Geol.
**1992**, 24, 365–379. [Google Scholar] [CrossRef] - Pawlowsky-Glahn, V.; Egozcue, J.J.; Tolosana-Delgado, R. Modeling and Analysis of Compositional Data; John Wiley & Sons: Hoboken, NJ, USA, 2015. [Google Scholar]
- Aitchison, J. The statistical analysis of compositional data. J. R. Stat. Soc. Ser. B Methodol.
**1982**, 44, 139–177. [Google Scholar] [CrossRef] - Egozcue, J.J.; Pawlowsky-Glahn, V.; Mateu-Figueras, G.; Barceló-Vidal, C. Isometric logratio transformations for compositional data analysis. Math. Geol.
**2003**, 35, 279–300. [Google Scholar] [CrossRef] - Pawlowsky-Glahn, V.; Egozcue, J.J. Geometric approach to statistical analysis on the simplex. Stoch. Environ. Res. Risk Assess.
**2001**, 15, 384–398. [Google Scholar] [CrossRef] - Obermann-Hellhund, A.; Conte, D.; Somot, S.; Torma, C.Z.; Ahrens, B. Mistral and Tramontane wind systems in climate simulations from 1950 to 2100. Clim. Dyn.
**2018**, 50, 693–703. [Google Scholar] [CrossRef] [Green Version] - Ullmann, A.; Moron, V. Weather regimes and sea surge variations over the Gulf of Lions (French Mediterranean coast) during the 20th century. Int. J. Clim.
**2008**, 28, 159–171. [Google Scholar] [CrossRef] - Bertin, X.; Li, K.; Roland, A.; Bidlot, J.R. The contribution of short-waves in storm surges: Two case studies in the Bay of Biscay. Cont. Shelf Res.
**2015**, 96, 1–15. [Google Scholar] [CrossRef] - Brown, J.M.; Wolf, J. Coupled wave and surge modeling for the eastern Irish Sea and implications for model wind-stress. Cont. Shelf Res.
**2009**, 29, 1329–1342. [Google Scholar] [CrossRef] [Green Version] - Marcos, M.; Rohmer, J.; Vousdoukas, M.I.; Mentaschi, L.; Le Cozannet, G.; Amores, A. Increased Extreme Coastal Water Levels Due to the Combined Action of Storm Surges and Wind Waves. Geophys. Res. Lett.
**2019**, 46, 4356–4364. [Google Scholar] [CrossRef] [Green Version] - Mase, H.; Tsujio, D.; Yasuda, T.; Mori, N. Stability analysis of composite breakwater with wave-dissipating blocks considering increase in sea levels, surges and waves due to climate change. Ocean Eng.
**2013**, 71, 58–65. [Google Scholar] [CrossRef] [Green Version] - Tsimplis, M.N.; Álvarez-Fanjul, E.; Gomis, D.; Fenoglio-Marc, L.; Pérez, B. Mediterranean Sea level trends: Atmospheric pressure and wind contribution. Geophys. Res. Lett.
**2005**, 32, L20602. [Google Scholar] [CrossRef] - Ullmann, A.; Pirazzoli, P.; Moron, V. Sea surges around the Gulf of Lions and atmospheric conditions. Glob. Planet. Chang.
**2008**, 63, 203–214. [Google Scholar] [CrossRef] - Vanem, E. Joint statistical models for significant wave height and wave period in a changing climate. Mar. Struct.
**2016**, 49, 180–205. [Google Scholar] [CrossRef]

**Figure 1.**(

**a**) North-Western Mediterranean Sea. The red line rectangle encloses the study area; (

**b**) Zoom at the Catalan coast. The red dots are the available stations from the barotropic model, whereas the yellow rhombus is the Barcelona tidal gauge (TG-BCN; REDMARvalidation station). PROF are the isobaths (in meters). Water depth in meters.

**Figure 2.**Flowchart of the proposed hybrid methodology to model the extreme surge events. The steps of the hybrid methodology are: (1) a process-based model, (2) a statistical model, and (3) validation of the statistical model. The statistical model (a bivariate copula model) is fed by the output data from the process-based model. The statistical model can adopt two different strategies, with either a pseudo-non-stationary and a non-stationary computation of the distribution parameters. Rectangles refer to data and rhombs to processes. CMCC-CM5, Centro Euro-Mediterraneo sui Cambiamenti Climatici Climate Model; GPD, Generalized Pareto Distribution; VGAM, Vectorial Generalized Additive Model; VGLM, Vectorial Generalized Linear Model.

**Figure 3.**Definition of variables for a single extreme surge event. The red solid time series represent the surge. The discontinuous red line is the threshold. The blue dot shows the maximum level in the event (${S}_{p}$). The event has an intermediate calm period of less than 72 h.

**Figure 4.**Descriptive plots of the two surge components at the N-2 Tossa model point (period from 1950–2100). Data extracted from the barotropic model. (

**a**) Continuous red line: surge-level at the peak of the extreme surge event (${S}_{p}$, in meters); (

**b**) continuous cyan line: duration of the extreme surge event (D, in hours).

**Figure 5.**Descriptive plots of the two surge components at the N-5 Badalona model point (period from 1950–2100). Data extracted from the barotropic model. (

**a**) Continuous red line: surge-level at the peak of the extreme surge event (${S}_{p}$, in meters); (

**b**) continuous cyan line: Duration of the extreme surge event (D, in hours).

**Figure 6.**Descriptive plots of the two surge components at the N-1 Llanca model point (period from 1950–2100). Data extracted from the barotropic model. (

**a**) Continuous red line: surge level at the peak of the extreme surge event (${S}_{p}$, in meters); (

**b**) continuous cyan line: duration of the extreme surge event (D, in hours).

**Figure 7.**Descriptive plots of the two surge components at the N-7 Marquesa model point (period from 1950–2100). Data extracted from the barotropic model. (

**a**) Continuous red line: surge-level at the peak of the extreme surge event (${S}_{p}$, in meters); (

**b**) continuous cyan line: duration of the extreme surge event (D, in hours).

**Figure 8.**Akaike Information Criterion (AIC) indices common to both strategies, since they share the same structure of threshold and frequency of extreme surge events. They show the goodness-of-fit of the different combinations of climate indices: North Atlantic oscillation ($NAO$), East Atlantic-West Russian pattern ($EAWR$), and Scandinavian pattern ($SC$). (

**a**) Threshold of the extreme surge event (${h}_{0}$) and (

**b**) frequency of extreme surge events ($\lambda $). The prefix “d-” means “first time-derivative of”, and the prefix “d2-” means “second time-derivative of”. The discontinuous red line shows the selected combination of climate patterns in (

**a**) ${h}_{0}$ and (

**b**) $\lambda $.

**Figure 9.**Akaike Information Criterion (AIC) indices common to both strategies, since they share the same structure of the location parameters. They compare the effect of different climate indices: North Atlantic oscillation ($NAO$), East Atlantic-West Russian pattern ($EAWR$), and Scandinavian pattern ($SC$) on the location parameter ${x}_{0}$ of a generalized Pareto distribution fit to (

**a**) the surge level at the peak of the extreme surge event (${S}_{p}$) and (

**b**) the duration of the extreme surge event (D). The prefix “d-” means “first time-derivative of”, and the prefix “d2-” means “second time-derivative of”. The discontinuous red line shows the selected combination of climate patterns on ${x}_{0}$.

**Figure 10.**Strategy #1. Akaike Information Criterion (AIC) indices. They compare the effect of different climate indices: North Atlantic oscillation ($NAO$), East Atlantic-West Russian pattern ($EAWR$), and Scandinavian pattern ($SC$) on the scale parameter $\beta $ of a generalized Pareto distribution fit to (

**a**) the surge level at the peak of the extreme surge event (${S}_{p}$) and (

**b**) the duration of the extreme surge event (D). The prefix “d-” means “first time-derivative of”, and the prefix “d2-” means “second time-derivative of”. The discontinuous red line shows the selected combination of climate patterns on $\beta $.

**Figure 11.**Strategy #1. Kendall’s $\tau $s of the dependence parameter of a Gumbel Archimedean copula at the different model points. Each color line represents a node from Figure 1. This plot presents discontinuities in time-slices that do not have enough extreme surge events to be fit by a stationary copula.

**Figure 12.**Strategy #2. Akaike Information Criterion (AIC) indices. They compare the effect of different climate indices: North Atlantic oscillation ($NAO$), East Atlantic-West Russian pattern ($EAWR$), and Scandinavian pattern ($SC$) on the scale parameter $\beta $ of a generalized Pareto distribution fit to (

**a**) the surge level at the peak of the extreme surge event (${S}_{p}$) and (

**b**) the duration of the extreme surge event (D). The prefix “d-” means “first time-derivative of”, and the prefix “d2-” means “second time-derivative of”. The discontinuous red line shows the selected combination of climate patterns on $\beta $.

**Figure 13.**Strategy #2. Akaike Information Criterion (AIC) indices. They compare the effect of different climate indices: North Atlantic oscillation ($NAO$), East Atlantic-West Russian pattern ($EAWR$), and Scandinavian pattern ($SC$) on Kendall’s $\tau $ of the dependence parameter of a Gumbel Archimedean copula. The discontinuous red line shows the selected combination of climate patterns on $\tau $.

**Figure 14.**Strategy #2. Dependence parameter $\tau $ of the Gumbel Archimedean copula at different model points, under the climate change scenario RCP8.5. Each color line represents a node from Figure 1.

**Table 1.**Most influential climate indices on parameters. The variables are: surge level at the peak of the extreme surge event (${S}_{p}$) and duration of the extreme surge event (D). The climate indices are: North Atlantic oscillation ($NAO$), East Atlantic-West Russian pattern ($EAWR$), and Scandinavian pattern ($SC$). The prefix “d-” means “first time-derivative of”, and the prefix “d2-” means “second time-derivative of”. ${C}_{r}$ is a coefficient from the vectorial generalized additive model and can be compared to the relative order of magnitude of the variable to the climate index.

${\mathit{S}}_{\mathit{p}}$ | D | |||
---|---|---|---|---|

Both strategies | Storm surge threshold | $SC$ (${C}_{r}\phantom{\rule{3.33333pt}{0ex}}=\phantom{\rule{3.33333pt}{0ex}}0.134$) | ||

Frequency $\lambda $ | $SC$ (${C}_{r}\phantom{\rule{3.33333pt}{0ex}}=\phantom{\rule{3.33333pt}{0ex}}0.156$) | |||

Location parameter (${x}_{0}$) of the generalized Pareto distribution | – | – | ||

Strategy #1: Lin-Ye | Scale parameter $\beta $ of the generalized Pareto distribution | $dSC+{d}^{2}SC$ (${C}_{r}=-0.85$) | $NAO$ (${C}_{r}=0.16$) | |

Strategy #2: Marra-Radice | $\beta $ | $EAWR$ (${C}_{r}=0.120$) | – | |

Kendall’s $\tau $ of the copula | $NAO$ (${C}_{r}=-4.104$) |

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## Share and Cite

**MDPI and ACS Style**

Lin-Ye, J.; García-León, M.; Gràcia, V.; Ortego, M.I.; Lionello, P.; Conte, D.; Pérez-Gómez, B.; Sánchez-Arcilla, A.
Modeling of Future Extreme Storm Surges at the NW Mediterranean Coast (Spain). *Water* **2020**, *12*, 472.
https://doi.org/10.3390/w12020472

**AMA Style**

Lin-Ye J, García-León M, Gràcia V, Ortego MI, Lionello P, Conte D, Pérez-Gómez B, Sánchez-Arcilla A.
Modeling of Future Extreme Storm Surges at the NW Mediterranean Coast (Spain). *Water*. 2020; 12(2):472.
https://doi.org/10.3390/w12020472

**Chicago/Turabian Style**

Lin-Ye, Jue, Manuel García-León, Vicente Gràcia, María Isabel Ortego, Piero Lionello, Dario Conte, Begoña Pérez-Gómez, and Agustín Sánchez-Arcilla.
2020. "Modeling of Future Extreme Storm Surges at the NW Mediterranean Coast (Spain)" *Water* 12, no. 2: 472.
https://doi.org/10.3390/w12020472