{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,1,18]],"date-time":"2026-01-18T22:37:45Z","timestamp":1768775865419,"version":"3.49.0"},"reference-count":65,"publisher":"MDPI AG","issue":"5","license":[{"start":{"date-parts":[[2021,5,19]],"date-time":"2021-05-19T00:00:00Z","timestamp":1621382400000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"DOI":"10.13039\/501100001871","name":"Funda\u00e7\u00e3o para a Ci\u00eancia e a Tecnologia","doi-asserted-by":"publisher","award":["PD\/BD\/128509\/2017"],"award-info":[{"award-number":["PD\/BD\/128509\/2017"]}],"id":[{"id":"10.13039\/501100001871","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Climate"],"abstract":"<jats:p>This paper explores practical applications of bivariate modelling via copulas of two likely dependent random variables, i.e., of the North Atlantic Oscillation (NAO) coupled with extreme rainfall on the small island of Madeira, Portugal. Madeira, due to its small size (\u223c740 km2), very pronounced mountain landscape, and location in the North Atlantic, experiences a wide range of rainfall regimes, or microclimates, which hamper the analyses of extreme rainfall. Previous studies showed that the influence of the North Atlantic Oscillation (NAO) on extreme rainfall is at its largest in the North Atlantic sector, with the likelihood of increased rainfall events from December through February, particularly during negative NAO phases. Thus, a copula-based approach was adopted for teleconnection, aiming at assigning return periods of daily values of an NAO index (NAOI) coupled with extreme daily rainfalls\u2014for the period from December 1967 to February 2017\u2014at six representative rain gauges of the island. The results show that (i) bivariate copulas describing the dependence characteristics of the underlying joint distributions may provide useful analytical expressions of the return periods of the coupled previous NAOI and extreme rainfall and (ii) that recent years show signs of increasing climate variability with more anomalous daily negative NAOI along with higher extreme rainfall events. These findings highlight the importance of multivariate modelling for teleconnections of prominent patterns of climate variability, such as the NAO, to extreme rainfall in North Atlantic regions, especially in small islands that are highly vulnerable to the effects of abrupt climate variability.<\/jats:p>","DOI":"10.3390\/cli9050086","type":"journal-article","created":{"date-parts":[[2021,5,19]],"date-time":"2021-05-19T12:58:07Z","timestamp":1621429087000},"page":"86","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":3,"title":["Bivariate Modelling of a Teleconnection Index and Extreme Rainfall in a Small North Atlantic Island"],"prefix":"10.3390","volume":"9","author":[{"ORCID":"https:\/\/orcid.org\/0000-0003-0059-3953","authenticated-orcid":false,"given":"Luis Angel","family":"Espinosa","sequence":"first","affiliation":[{"name":"Instituto Superior T\u00e9cnico (IST), Civil Engineering Research and Innovation for Sustainability (CERIS), 1049-001 Lisbon, Portugal"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-5221-1139","authenticated-orcid":false,"given":"Maria Manuela","family":"Portela","sequence":"additional","affiliation":[{"name":"Instituto Superior T\u00e9cnico (IST), Civil Engineering Research and Innovation for Sustainability (CERIS), 1049-001 Lisbon, Portugal"}]},{"ORCID":"https:\/\/orcid.org\/0000-0003-1756-0429","authenticated-orcid":false,"given":"Jo\u00e3o Dehon","family":"Pontes Filho","sequence":"additional","affiliation":[{"name":"Hydraulic and Environmental Engineering Department (DEHA), Federal University of Cear\u00e1, Fortaleza 60020-180, Brazil"}]},{"ORCID":"https:\/\/orcid.org\/0000-0001-7502-9586","authenticated-orcid":false,"given":"Martina","family":"Zelenakova","sequence":"additional","affiliation":[{"name":"Department of Environmental Engineering, Technical University of Kosice, 042 00 Kosice, Slovakia"}]}],"member":"1968","published-online":{"date-parts":[[2021,5,19]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"269","DOI":"10.1256\/004316502320517353","article-title":"Observed climate variability and change","volume":"57","author":"Folland","year":"2002","journal-title":"Weather"},{"key":"ref_2","doi-asserted-by":"crossref","first-page":"297","DOI":"10.1016\/j.ecolind.2017.01.016","article-title":"Study of Saharan dust influence on PM10 measures in Sicily from 2013 to 2015","volume":"76","author":"Cuspilici","year":"2017","journal-title":"Ecol. Indic."},{"key":"ref_3","doi-asserted-by":"crossref","first-page":"77","DOI":"10.1002\/joc.3370130106","article-title":"Long-term variations in regional rainfall in the south-west of Western Australia and the difference between average and high intensity rainfalls","volume":"13","author":"Yu","year":"1993","journal-title":"Int. J. Climatol."},{"key":"ref_4","doi-asserted-by":"crossref","first-page":"125005","DOI":"10.1016\/j.jhydrol.2020.125005","article-title":"Projecting the future of rainfall extremes: Better classic than trendy","volume":"588","author":"Iliopoulou","year":"2020","journal-title":"J. Hydrol."},{"key":"ref_5","doi-asserted-by":"crossref","first-page":"759","DOI":"10.1002\/joc.1033","article-title":"Interannual variability of European extreme winter rainfall and links with mean large-scale circulation","volume":"24","author":"Haylock","year":"2004","journal-title":"Int. J. Climatol. A J. R. Meteorol. Soc."},{"key":"ref_6","doi-asserted-by":"crossref","first-page":"72","DOI":"10.1175\/2007JCLI1631.1","article-title":"European climate extremes and the North Atlantic Oscillation","volume":"21","author":"Scaife","year":"2008","journal-title":"J. Clim."},{"key":"ref_7","doi-asserted-by":"crossref","first-page":"345","DOI":"10.1007\/s10584-013-0705-8","article-title":"Changes in temperature and precipitation extremes in the CMIP5 ensemble","volume":"119","author":"Kharin","year":"2013","journal-title":"Clim. Chang."},{"key":"ref_8","doi-asserted-by":"crossref","first-page":"1716","DOI":"10.1002\/jgrd.50203","article-title":"Climate extremes indices in the CMIP5 multimodel ensemble: Part 1. Model evaluation in the present climate","volume":"118","author":"Sillmann","year":"2013","journal-title":"J. Geophys. Res. Atmos."},{"key":"ref_9","doi-asserted-by":"crossref","first-page":"2473","DOI":"10.1002\/jgrd.50188","article-title":"Climate extremes indices in the CMIP5 multimodel ensemble: Part 2. Future climate projections","volume":"118","author":"Sillmann","year":"2013","journal-title":"J. Geophys. Res. Atmos."},{"key":"ref_10","doi-asserted-by":"crossref","first-page":"757","DOI":"10.1175\/JCLI-D-13-00163.1","article-title":"The atmospheric energy constraint on global-mean precipitation change","volume":"27","author":"Pendergrass","year":"2014","journal-title":"J. Clim."},{"key":"ref_11","doi-asserted-by":"crossref","first-page":"12876","DOI":"10.1073\/pnas.231391598","article-title":"The North Atlantic Oscillation: Past, present, and future","volume":"98","author":"Visbeck","year":"2001","journal-title":"Proc. Natl. Acad. Sci. USA"},{"key":"ref_12","doi-asserted-by":"crossref","first-page":"479","DOI":"10.1007\/s00704-013-0856-6","article-title":"Influence of the North Atlantic Oscillation on winter rainfall in Calabria (southern Italy)","volume":"114","author":"Ferrari","year":"2013","journal-title":"Theor. Appl. Climatol."},{"key":"ref_13","doi-asserted-by":"crossref","first-page":"331","DOI":"10.1007\/s11069-005-1709-0","article-title":"The influence of the North Atlantic Oscillation on rainfall triggering of landslides near Lisbon","volume":"36","author":"Trigo","year":"2005","journal-title":"Nat. Hazards"},{"key":"ref_14","doi-asserted-by":"crossref","first-page":"819","DOI":"10.1002\/joc.1334","article-title":"Exploring the relationship between the North Atlantic oscillation and rainfall patterns in Barbados","volume":"26","author":"Charlery","year":"2006","journal-title":"Int. J. Climatol. A J. R. Meteorol. Soc."},{"key":"ref_15","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1038\/s41598-018-24069-9","article-title":"Lagged influence of Atlantic and Pacific climate patterns on European extreme precipitation","volume":"8","author":"Tabari","year":"2018","journal-title":"Sci. Rep."},{"key":"ref_16","doi-asserted-by":"crossref","first-page":"1220","DOI":"10.1175\/2007JCLI1739.1","article-title":"Positive and negative phases of the wintertime North Atlantic Oscillation and drought occurrence over Europe: A multitemporal-scale approach","volume":"21","year":"2008","journal-title":"J. Clim."},{"key":"ref_17","doi-asserted-by":"crossref","unstructured":"Hanna, E., and Cropper, T.E. (2017). North Atlantic Oscillation. Oxford Research Encyclopedia of Climate Science, Oxford Research Encyclopedias.","DOI":"10.1093\/acrefore\/9780190228620.013.22"},{"key":"ref_18","doi-asserted-by":"crossref","unstructured":"Espinosa, L.A., Portela, M.M., and Rodrigues, R. (2020). Significant Extremal Dependence of a Daily North Atlantic Oscillation Index (NAOI) and Weighted Regionalised Rainfall in a Small Island Using the Extremogram. Water, 12.","DOI":"10.3390\/w12112989"},{"key":"ref_19","doi-asserted-by":"crossref","first-page":"300","DOI":"10.1002\/wea.2155","article-title":"The weather and climate of Macaronesia: Past, present and future","volume":"68","author":"Cropper","year":"2013","journal-title":"Weather"},{"key":"ref_20","first-page":"53","article-title":"World Weather V. Memories of the Royal eteorological","volume":"44","author":"Walker","year":"1932","journal-title":"Society"},{"key":"ref_21","doi-asserted-by":"crossref","first-page":"784","DOI":"10.1175\/1520-0493(1981)109<0784:TITGHF>2.0.CO;2","article-title":"Teleconnections in the geopotential height field during the Northern Hemisphere winter","volume":"109","author":"Wallace","year":"1981","journal-title":"Mon. Weather Rev."},{"key":"ref_22","doi-asserted-by":"crossref","first-page":"715","DOI":"10.5194\/nhess-12-715-2012","article-title":"The 20 February 2010 Madeira flash-floods: Synoptic analysis and extreme rainfall assessment","volume":"12","author":"Fragoso","year":"2012","journal-title":"Nat. Hazards Earth Syst. Sci."},{"key":"ref_23","doi-asserted-by":"crossref","unstructured":"Jung, T., Vitart, F., Ferranti, L., and Morcrette, J.J. (2011). Origin and predictability of the extreme negative NAO winter of 2009\/10. Geophys. Res. Lett., 38.","DOI":"10.1029\/2011GL046786"},{"key":"ref_24","doi-asserted-by":"crossref","first-page":"266","DOI":"10.1002\/j.1477-8696.1997.tb06323.x","article-title":"The North Atlantic Oscillation and British Isles climate variability, 1865\u20131996","volume":"52","author":"Wilby","year":"1997","journal-title":"Weather"},{"key":"ref_25","doi-asserted-by":"crossref","first-page":"3235","DOI":"10.5194\/nhess-13-3235-2013","article-title":"Trends and variability in extreme precipitation indices over Maghreb countries","volume":"13","author":"Tramblay","year":"2013","journal-title":"Nat. Hazards Earth Syst. Sci."},{"key":"ref_26","doi-asserted-by":"crossref","unstructured":"Wang, C., Deser, C., Yu, J.Y., DiNezio, P., and Clement, A. (2017). El Ni\u00f1o and southern oscillation (ENSO): A review. Coral Reefs of the Eastern Tropical Pacific, Springer Nature.","DOI":"10.1007\/978-94-017-7499-4_4"},{"key":"ref_27","doi-asserted-by":"crossref","first-page":"675","DOI":"10.1016\/j.atmosres.2009.09.005","article-title":"North Atlantic Oscillation influence and weather types associated with winter total and extreme precipitation events in Spain","volume":"94","author":"Queralt","year":"2009","journal-title":"Atmos. Res."},{"key":"ref_28","unstructured":"Embrechts, P., Lindskog, F., and McNeil, A. (2001). Modelling Dependence with Copulas, D\u00e9partement de Math\u00e9matiques, Institut F\u00e9d\u00e9ral de Technologie de Zurich. Rapport Technique."},{"key":"ref_29","first-page":"C306","article-title":"Trivariate copulas for characterisation of droughts","volume":"49","author":"Wong","year":"2007","journal-title":"Anziam J."},{"key":"ref_30","doi-asserted-by":"crossref","first-page":"1725","DOI":"10.1007\/s00477-020-01866-1","article-title":"Modelling dependence between observed and simulated wind speed data using copulas","volume":"34","year":"2020","journal-title":"Stoch. Environ. Res. Risk Assess."},{"key":"ref_31","doi-asserted-by":"crossref","first-page":"85","DOI":"10.1016\/j.pce.2015.11.006","article-title":"A dependence modelling study of extreme rainfall in Madeira Island","volume":"94","author":"Lopes","year":"2016","journal-title":"Phys. Chem. Earth, Parts A\/B\/C"},{"key":"ref_32","unstructured":"Falkland, A., and Custodio, E. (1991). Hydrology and Water Resources of Small Islands: A Practical Guide, UNESCO."},{"key":"ref_33","doi-asserted-by":"crossref","first-page":"384","DOI":"10.1007\/s002540100403","article-title":"Fog precipitation on the Island of Madeira (Portugal)","volume":"41","author":"Prada","year":"2001","journal-title":"Environ. Geol."},{"key":"ref_34","first-page":"473","article-title":"Climate change scenarios in the Azores and Madeira islands","volume":"16","author":"Santos","year":"2004","journal-title":"World Resour. Rev."},{"key":"ref_35","unstructured":"Chazarra, A., Bacel\u00f3, A., Pires, V., Cunha, S., Silva, A., Marques, J., Carvalho, F., Mendes, M., Neto, J., and Mendes, L. (2011). Climate Atlas of the Archipelagos of the Canary Islands. Madeira and the Azores, Department of Meteorology and Climatology of the Institute of Meteorology."},{"key":"ref_36","doi-asserted-by":"crossref","first-page":"1709","DOI":"10.1126\/science.286.5445.1709","article-title":"Eight centuries of North Atlantic Ocean atmosphere variability","volume":"286","author":"Black","year":"1999","journal-title":"Science"},{"key":"ref_37","doi-asserted-by":"crossref","first-page":"4449","DOI":"10.1007\/s11269-020-02668-4","article-title":"Rainfall trends over a small island teleconnected to the North Atlantic oscillation-the case of Madeira Island, Portugal","volume":"34","author":"Espinosa","year":"2020","journal-title":"Water Resour. Manag."},{"key":"ref_38","first-page":"1","article-title":"Changes in daily temperature and precipitation extremes in central and south Asia","volume":"111","author":"Peterson","year":"2006","journal-title":"J. Geophys. Res. Atmos."},{"key":"ref_39","unstructured":"Hair, J., Black, W., Babin, B., Anderson, R., and Tatham, R. (1998). Multivariate Data Analysis, Prentice Hall."},{"key":"ref_40","doi-asserted-by":"crossref","first-page":"2750","DOI":"10.1109\/TPEL.2013.2273461","article-title":"Moving average filter based phase-locked loops: Performance analysis and design guidelines","volume":"29","author":"Golestan","year":"2013","journal-title":"IEEE Trans. Power Electron."},{"key":"ref_41","unstructured":"WMO (World Meteorological Organization) (2009). Guide to Hydrological Practices, Volume II Management of Water Resources and Application of Hydrological Practices, WMO (World Meteorological Organization)."},{"key":"ref_42","doi-asserted-by":"crossref","unstructured":"Espinosa, L.A., Portela, M.M., and Rodrigues, R. (2021). Rainfall trends over a North Atlantic small island in the period 1937\/1938\u20132016\/2017 and an early climate teleconnection. Theoretical Appl. Climatol., 1\u201323.","DOI":"10.1007\/s00704-021-03547-7"},{"key":"ref_43","unstructured":"NOAA (2021, January 02). Daily Climate Timeseries: NAO: NOAA PSL, Available online: https:\/\/psl.noaa.gov\/data\/timeseries\/daily\/NAO\/."},{"key":"ref_44","first-page":"229","article-title":"Fonctions de repartition an dimensions et leurs marges","volume":"8","author":"Sklar","year":"1959","journal-title":"Publ. Inst. Statist. Univ. Paris"},{"key":"ref_45","unstructured":"Schweizer, B., and Sklar, A. (2011). Probabilistic Metric Spaces, Courier Corporation."},{"key":"ref_46","unstructured":"Nelsen, R.B. (2006). Methods of constructing copulas. An Introduction to Copulas, Springer."},{"key":"ref_47","doi-asserted-by":"crossref","unstructured":"Joe, H. (1997). Multivariate Models and Multivariate Dependence Concepts, CRC Press.","DOI":"10.1201\/b13150"},{"key":"ref_48","doi-asserted-by":"crossref","first-page":"1","DOI":"10.18637\/jss.v052.i03","article-title":"Cdvine: Modeling dependence with c-and d-vine copulas in R","volume":"52","author":"Brechmann","year":"2013","journal-title":"J. Stat. Softw."},{"key":"ref_49","doi-asserted-by":"crossref","first-page":"30","DOI":"10.1061\/(ASCE)0733-9496(2001)127:1(30)","article-title":"Recurrence analysis of hydrologic droughts of differing severity","volume":"127","author":"Shiau","year":"2001","journal-title":"J. Water Resour. Plan. Manag."},{"key":"ref_50","doi-asserted-by":"crossref","first-page":"42","DOI":"10.1007\/s00477-003-0125-9","article-title":"Return period of bivariate distributed extreme hydrological events","volume":"17","author":"Shiau","year":"2003","journal-title":"Stoch. Environ. Res. Risk Assess."},{"key":"ref_51","doi-asserted-by":"crossref","first-page":"5662","DOI":"10.1002\/2016WR018959","article-title":"A copula-based nonstationary frequency analysis for the 2012\u20132015 drought in California","volume":"52","author":"Kwon","year":"2016","journal-title":"Water Resour. Res."},{"key":"ref_52","doi-asserted-by":"crossref","first-page":"795","DOI":"10.1007\/s11269-005-9008-9","article-title":"Fitting drought duration and severity with two-dimensional copulas","volume":"20","author":"Shiau","year":"2006","journal-title":"Water Resour. Manag."},{"key":"ref_53","first-page":"1","article-title":"General multivariate dependence using associated copulas","volume":"14","author":"Salazar","year":"2011","journal-title":"Stat. J."},{"key":"ref_54","first-page":"70","article-title":"Tails of Copulas","volume":"89","author":"Venter","year":"2002","journal-title":"Proc. Casualty Actuar. Soc."},{"key":"ref_55","doi-asserted-by":"crossref","first-page":"23","DOI":"10.1007\/s00376-006-0003-8","article-title":"Temporal structures of the North Atlantic Oscillation and its impact on the regional climate variability","volume":"23","author":"Huang","year":"2006","journal-title":"Adv. Atmos. Sci."},{"key":"ref_56","doi-asserted-by":"crossref","first-page":"661","DOI":"10.1007\/BF02915394","article-title":"A new North Atlantic Oscillation index and its variability","volume":"20","author":"Jianping","year":"2003","journal-title":"Adv. Atmos. Sci."},{"key":"ref_57","doi-asserted-by":"crossref","first-page":"580","DOI":"10.1002\/2013WR013763","article-title":"A copula-based precipitation forecasting model: Investigating the interdecadal modulation of ENSO\u2019s impacts on monthly precipitation","volume":"50","author":"Khedun","year":"2014","journal-title":"Water Resour. Res."},{"key":"ref_58","unstructured":"Embrechts, P., Mcneil, E., and Straumann, D. (1999). Correlation: Pitfalls and alternatives. Risk Magazine, Citeseer."},{"key":"ref_59","unstructured":"Abadi, M.S.E. (2015). Analysis of New Techniques for Risk Aggregation and Dependence Modelling. [Ph.D. Thesis, Instituto Superior de Economia e Gest\u00e3o]."},{"key":"ref_60","doi-asserted-by":"crossref","first-page":"208","DOI":"10.1287\/mnsc.45.2.208","article-title":"Correlations and copulas for decision and risk analysis","volume":"45","author":"Clemen","year":"1999","journal-title":"Manag. Sci."},{"key":"ref_61","doi-asserted-by":"crossref","first-page":"237","DOI":"10.1061\/(ASCE)1084-0699(2003)8:5(237)","article-title":"Nonparametric approach for estimating return periods of droughts in arid regions","volume":"8","author":"Kim","year":"2003","journal-title":"J. Hydrol. Eng."},{"key":"ref_62","doi-asserted-by":"crossref","first-page":"51","DOI":"10.1016\/j.jhydrol.2015.09.016","article-title":"A global analysis of the asymmetric effect of ENSO on extreme precipitation","volume":"530","author":"Sun","year":"2015","journal-title":"J. Hydrol."},{"key":"ref_63","doi-asserted-by":"crossref","first-page":"657","DOI":"10.1007\/s00704-019-02847-3","article-title":"Trivariate copula in drought analysis: A case study in peninsular Malaysia","volume":"138","author":"Yusof","year":"2019","journal-title":"Theor. Appl. Climatol."},{"key":"ref_64","doi-asserted-by":"crossref","unstructured":"Taws, S.L., Marsh, R., Wells, N.C., and Hirschi, J. (2011). Re-emerging ocean temperature anomalies in late-2010 associated with a repeat negative NAO. Geophys. Res. Lett., 38.","DOI":"10.1029\/2011GL048978"},{"key":"ref_65","doi-asserted-by":"crossref","first-page":"2540","DOI":"10.1002\/joc.4157","article-title":"Recent seasonal asymmetric changes in the NAO (a marked summer decline and increased winter variability) and associated changes in the AO and Greenland Blocking Index","volume":"35","author":"Hanna","year":"2015","journal-title":"Int. J. Climatol."}],"container-title":["Climate"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/2225-1154\/9\/5\/86\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,11]],"date-time":"2025-10-11T06:03:50Z","timestamp":1760162630000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/2225-1154\/9\/5\/86"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2021,5,19]]},"references-count":65,"journal-issue":{"issue":"5","published-online":{"date-parts":[[2021,5]]}},"alternative-id":["cli9050086"],"URL":"https:\/\/doi.org\/10.3390\/cli9050086","relation":{},"ISSN":["2225-1154"],"issn-type":[{"value":"2225-1154","type":"electronic"}],"subject":[],"published":{"date-parts":[[2021,5,19]]}}}