# Significant Extremal Dependence of a Daily North Atlantic Oscillation Index (NAOI) and Weighted Regionalised Rainfall in a Small Island Using the Extremogram

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## Abstract

**:**

## 1. Introduction

## 2. Study Area

## 3. Rainfall and NAO Data

#### 3.1. Daily Rainfall Data

#### 3.2. North Atlantic Oscillation Index (NAOI) Daily Data

## 4. Methods

#### 4.1. Regionalisation of the Daily Rainfall Series

#### 4.2. Dominant Negative and Positive NAO Phases

#### 4.3. Strictly Stationary and Regularly Varying

**X**${}_{t}$) and defining

**Y**${}_{h}$: = (${X}_{1}$, ..., ${X}_{1}$), the process (

**X**${}_{t}$) is said to be regularly varying with tail index $\alpha $ if:

**X**${}_{t}$) regularly varying, then there exists a sequence such that:

#### 4.4. Tail-Dependence

#### 4.5. Extremogram

**X**${}_{t}$. The extremogram is defined as follows:

#### 4.6. Cross-Extremogram

## 5. Results

#### 5.1. Weighted Regionalised Daily Rainfall Series

#### 5.2. Dominant Extremal NAOI Subperiods and Their Extremograms

#### 5.3. Extremal Dependence of the Regionalised Rainfall and NAOI via the Cross-Extremogram

#### 5.4. Discussion and Conclusions

**C**and

**B**in Figure 11, respectively) and a northern station (low pressure), usually making use of a southwest Icelandic station (

**F**in Figure 11). As an alternative, NAOI can be obtained from gridded climate datasets utilising orthogonal analysis or PCA. Similar methods like ensembles of simulations from a general circulation model can produce climatic datasets as the one here analysed which extends a daily NAOI back to the year 1948 and consistently updated by the NOAA PSL (Section 3.2). Because all indices from shorter to longer time scales show similar temporal evolutions and are highly correlated at interannual and interdecadal time scales, an exact NAOI definition is of less importance. However, the same time scale and length of both rainfall and NAOI have been reckoned with as determining factors in this study.

**A**in Figure 11) during the four multi-year subperiods. Thus, the fitted LOWESS curve of Figure 5 was compared to an earlier characterisation from the year 1870–2090 by [23] of a filtered NAOI DJFM single realisation obtained from ensembles means of four greenhouse gases (GSa) experiments. Only the patterns were the focus due to the resolution differences and also because absolute spectrally truncated daily NAOI and NAOI DJFM values are not comparable. Although not shown, the comparison stresses that both signals’ patterns exhibit a prevalence of negative NAOI dominance from the 1950s to 1980s, followed by a reduction of the total NAOI, or integrated area above or under the signal, of alternating sign anomalies, namely, +NAOI, −NAOI, and +NAOI. It is also apparent, that in some recent years there is an increase of more consecutive and more extremes negative NAOI observations (e.g., daily NAOI values from 2009/10 to 2011/12). Additionally, the apparent upward trend in the LOWESS curve from 1960s to 1990s is also in accordance with that based on real data by [23], and also linked to quickly climatic variables changes (e.g., the rapidly temperatures rise) in the North Atlantic Ocean region [71]. These remarks and some others through the extremogram application to the univariate daily NAOI series as mentioned in Section 5.2, strengthen the justification to follow through the extremal dependence analysis adopting subperiods, i.e., from 1948–1979 (1st subperiod), 1980–1999 (2nd subperiod), 2000–2011 (3rd subperiod), and 2012–2017 (4th subperiod).

**C**and

**F**, respectively, in Figure 11). Boxplots were constructed to represent the NAOI DJFM samples dispersion for three exceedances categories, (i) values lower than the mean, (ii) between one and two times the mean, and (iii) higher than two times the mean. Figure 13 evidences that regardless the region, most of the “weaker” extreme rainfall years, with below average number of exceedances, are associated to positive NAOI DJFM values, and the “very extreme” rainfall years, with more than twice the average number of occurrences, to negative NAOI DJFM. This is in agreement with the detected alternating character of the maximum daily rainfalls for each NAOI dominance subperiod.

**F**,

**E**,

**D**,

**C**, in Figure 11 respectively). This suggests that the −NAOI effect may shift south-eastwards from Iberian Peninsula with increased storms characterised by substantial heavy rainfalls in Madeira Island, with a somewhat symmetrical +NAOI effect but to a lesser degree. These claims are supported by the statistically significant evidence found and discussed in this extremal serial dependence analysis between the weighted regionalised rainfalls and daily NAOI compared with previous studies though focused on single rain gauges${}^{\prime}$ and on monthly and seasonal NAOI data.

## Author Contributions

## Funding

## Conflicts of Interest

## References

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**Figure 1.**Location (WGS84 coordinates) and relief of Madeira Island. Spatial distribution of the forty rain gauges considered in this study depicted by bullets and their respective identification codes from ST01–ST40.

**Figure 2.**Scree plot from the sample covariance matrix obtained via the factor analysis based on the daily rainfall from 1 January 1948 to 30 September 2017 at the 40 rain gauges (the unrotated solution of the 40-dimensions problem). Horizontal dashed line shows the 1.0 eigenvalue.

**Figure 3.**Spatial distribution of the principal factor loadings after rotation (varimax) higher than 0.60 from the factor analysis based on daily rainfall records (1948–2017) at the 40 rain gauges (WGS84). The region F1-SOU is related to the first factor, F2-CEN to the second factor, and F3-NOR to the third factor.

**Figure 4.**Weighted regionalised daily rainfall series spanning from 1 January 1948 to 30 September 2017 from the Voronoi polygons method for the identified regions via the factor analysis, F1-SOU (southern slope—12 rain gauges), F2-CEN (central region—17 rain gauges), and F3-NOR (northern slope—11 rain gauges). Daily rainfalls exceeding the 90% quantile of the upper tail 99th percentile of the corresponding time series are depicted by circles.

**Figure 5.**Daily records of NAOI from 1 January 1948 to 30 September 2017. The lower and and upper tails’ daily observations are identified by blue and red triangles, respectively; and fitted into a LOWESS curve: the −NAOI dominance subperiods from January 1947 to December 1979, and from January 2000 to December 2011 (blue vertical and horizontal bars); and the +NAOI ones from January 1980 to December 1999, and from 2012 on (red vertical and horizontal bars).

**Figure 6.**Extremograms of the univariate daily time series X (NAOI) for (

**a**) the upper tail, and (

**b**) the lower tail, for the dominance subperiods: 1st one (1948–1979), 2nd one (1980–1999), 3rd one (2000–2011), and 4th one (2012–2017). Horizontal red dashed lines indicate the upper 97.5% empirical confidence bands (for an $\alpha $ of 5%) for independent data, obtained via m = 10,000 permutations.

**Figure 7.**Cross-extremograms for the 1st subperiod (1948–1979, −NAOI) of the bivariate daily time series X (NAOI) and Y (rainfall) for (

**a**) X and Y upper tails, and for (

**b**) the lower tail in X and the upper tail in Y (vertical axes cut off at 0.60). Horizontal red dashed lines indicate the upper 97.5% empirical confidence bands ($\alpha $ of 5%).

**Figure 8.**Cross-extremograms for the 2nd subperiod (1980–1999, +NAOI) of the bivariate daily time series X (NAOI) and Y (rainfall) for (

**a**) X and Y upper tails, and for (

**b**) the lower tail in X and the upper tail in Y (vertical axes cut off at 0.60). Horizontal red dashed lines indicate the upper 97.5% empirical confidence bands ($\alpha $ of 5%).

**Figure 9.**Cross-extremograms for the 3rd subperiod (2000–2011, −NAOI) of the bivariate daily time series X (NAOI) and Y (rainfall) for (

**a**) X and Y upper tails, and for (

**b**) the lower tail in X and the upper tail in Y (vertical axes cut off at 0.60). Horizontal red dashed lines indicate the upper 97.5% empirical confidence bands ($\alpha $ of 5%).

**Figure 10.**Cross-extremograms for the 4th subperiod (2012–2017, +NAOI) of the bivariate daily time series X (NAOI) and Y (rainfall) for (

**a**) X and Y upper tails, and for (

**b**) the lower tail in X and the upper tail in Y (vertical axes cut off at 0.60). Horizontal red dashed lines indicate the upper 97.5% empirical confidence bands ($\alpha $ of 5%).

**Figure 11.**Location of: (

**A**) Madeira Archipelago, (

**B**) The Azores, (

**C**) Iberian Peninsula (mostly formed by continental Portugal and Spain), (

**D**) Newfoundland and Labrador (the easternmost province of Canada), (

**E**) The British Isles, and (

**F**) Iceland. Projection: Geographic Latitude-Longitude; datum: WGS84; planar unit: arc degrees.

**Figure 12.**Cross-extremograms for the reference period from 1948 to 2017 of the bivariate daily time series X (NAOI) and Y (rainfall) for (

**a**) X and Y upper tails, and for (

**b**) the lower tail in X and the upper tail in Y (vertical axes cut off at 0.60). Horizontal red dashed lines indicate the upper 97.5% empirical confidence bands ($\alpha $ of 5%).

**Figure 13.**Dimensionless yearly number of exceedances (horizontal axis) in each hydrological year, from 1948/49 to 2016/17, for each of the weighted regionalised daily rainfall series (F1-SOU, blue circles; F2-CEN, red squares; F3-NOR, green triangles) and their corresponding NAOI DJFM (vertical axis). Boxplots constructed based on the NAOI DJFM on an annual basis for three exceedances categories: less than one exceedance per year, from one to two exceedances per year, and more than two exceedances per year.

**Table 1.**The forty rain gauges utilised. Identification (code and name), coordinates (WGS84), elevation, areal influence according to the Voronoi polygons method, and the region to which they belong from the factor analysis.

Code | Name | Coordinate | Elevation (m.a.s.l.) | Areal Influence (km${}^{2}$) | Factor-Region | |
---|---|---|---|---|---|---|

Latitude N | Longitude W | |||||

ST01 | Areeiro | 32${}^{\circ}$43${}^{\prime}$11${}^{\u2033}$ | 16${}^{\circ}$55${}^{\prime}$01${}^{\u2033}$ | 1610.1 | 13.67 | F2-CEN |

ST02 | Bica da Cana | 32${}^{\circ}$45${}^{\prime}$22${}^{\u2033}$ | 17${}^{\circ}$03${}^{\prime}$19${}^{\u2033}$ | 1560.2 | 22.05 | F2-CEN |

ST03 | Bom Sucesso | 32${}^{\circ}$39${}^{\prime}$43${}^{\u2033}$ | 16${}^{\circ}$53${}^{\prime}$45${}^{\u2033}$ | 292.0 | 6.98 | F1-SOU |

ST04 | Cabeço do Meio-Nogueira | 32${}^{\circ}$44${}^{\prime}$08${}^{\u2033}$ | 16${}^{\circ}$53${}^{\prime}$55${}^{\u2033}$ | 995.3 | 4.07 | F2-CEN |

ST05 | Camacha-Valparaiso | 32${}^{\circ}$40${}^{\prime}$34${}^{\u2033}$ | 16${}^{\circ}$50${}^{\prime}$31${}^{\u2033}$ | 675.2 | 28.57 | F1-SOU |

ST06 | Canhas | 32${}^{\circ}$41${}^{\prime}$39${}^{\u2033}$ | 17${}^{\circ}$06${}^{\prime}$35${}^{\u2033}$ | 400.4 | 25.19 | F1-SOU |

ST07 | Caniçal | 32${}^{\circ}$44${}^{\prime}$14${}^{\u2033}$ | 16${}^{\circ}$44${}^{\prime}$19${}^{\u2033}$ | 16.2 | 11.34 | F3-NOR |

ST08 | Caramujo | 32${}^{\circ}$46${}^{\prime}$09${}^{\u2033}$ | 17${}^{\circ}$03${}^{\prime}$30${}^{\u2033}$ | 1214.5 | 30.41 | F2-CEN |

ST09 | Cascalho | 32${}^{\circ}$49${}^{\prime}$44${}^{\u2033}$ | 16${}^{\circ}$55${}^{\prime}$30${}^{\u2033}$ | 430.4 | 1.83 | F3-NOR |

ST10 | Chão dos Louros Encumeadas | 32${}^{\circ}$45${}^{\prime}$25${}^{\u2033}$ | 17${}^{\circ}$01${}^{\prime}$04" | 895.2 | 9.54 | F2-CEN |

ST11 | Covão ETA | 32${}^{\circ}$40${}^{\prime}$29${}^{\u2033}$ | 16${}^{\circ}$57${}^{\prime}$46${}^{\u2033}$ | 510.1 | 22.45 | F1-SOU |

ST12 | Curral das Freiras | 32${}^{\circ}$44${}^{\prime}$44${}^{\u2033}$ | 16${}^{\circ}$57${}^{\prime}$35${}^{\u2033}$ | 787.4 | 20.08 | F2-CEN |

ST13 | Encumeada de São Vicente | 32${}^{\circ}$45${}^{\prime}$01${}^{\u2033}$ | 17${}^{\circ}$01${}^{\prime}$00${}^{\u2033}$ | 900.2 | 1.12 | F2-CEN |

ST14 | Encumeadas Casa EEM | 32${}^{\circ}$45${}^{\prime}$14${}^{\u2033}$ | 17${}^{\circ}$01${}^{\prime}$15${}^{\u2033}$ | 1010.5 | 2.32 | F2-CEN |

ST15 | ETA São Jorge | 32${}^{\circ}$48${}^{\prime}$57${}^{\u2033}$ | 16${}^{\circ}$55${}^{\prime}$33${}^{\u2033}$ | 500.5 | 10.42 | F3-NOR |

ST16 | Fajã Penedo | 32${}^{\circ}$47${}^{\prime}$31${}^{\u2033}$ | 16${}^{\circ}$57${}^{\prime}$36${}^{\u2033}$ | 620.5 | 23.83 | F3-NOR |

ST17 | Funchal Observatório | 32${}^{\circ}$38${}^{\prime}$51${}^{\u2033}$ | 16${}^{\circ}$53${}^{\prime}$32${}^{\u2033}$ | 58.2 | 7.08 | F1-SOU |

ST18 | Lapa Branca-Curral das Freiras | 32${}^{\circ}$43${}^{\prime}$08${}^{\u2033}$ | 16${}^{\circ}$57${}^{\prime}$53" | 610.2 | 22.45 | F2-CEN |

ST19 | Lido-Cais do Carvão | 32${}^{\circ}$38${}^{\prime}$11${}^{\u2033}$ | 16${}^{\circ}$56${}^{\prime}$11${}^{\u2033}$ | 20.5 | 4.98 | F1-SOU |

ST20 | Lombo Furão | 32${}^{\circ}$44${}^{\prime}$56${}^{\u2033}$ | 16${}^{\circ}$54${}^{\prime}$39${}^{\u2033}$ | 994.5 | 13.61 | F2-CEN |

ST21 | Loural | 32${}^{\circ}$46${}^{\prime}$21${}^{\u2033}$ | 17${}^{\circ}$01${}^{\prime}$45${}^{\u2033}$ | 368.1 | 19.37 | F2-CEN |

ST22 | Lugar de Baixo | 32${}^{\circ}$40${}^{\prime}$44${}^{\u2033}$ | 17${}^{\circ}$04${}^{\prime}$59${}^{\u2033}$ | 15.1 | 10.94 | F1-SOU |

ST23 | Meia Serra | 32${}^{\circ}$42${}^{\prime}$07${}^{\u2033}$ | 16${}^{\circ}$52${}^{\prime}$12${}^{\u2033}$ | 115.3 | 12.47 | F2-CEN |

ST24 | Montado do Pereiro | 32${}^{\circ}$42${}^{\prime}$06${}^{\u2033}$ | 16${}^{\circ}$53${}^{\prime}$02${}^{\u2033}$ | 1261.0 | 6.53 | F2-CEN |

ST25 | Poiso-Posto Florestal | 32${}^{\circ}$42${}^{\prime}$46${}^{\u2033}$ | 16${}^{\circ}$53${}^{\prime}$13${}^{\u2033}$ | 1360.2 | 4.60 | F2-CEN |

ST26 | Ponta de São Jorge | 32${}^{\circ}$50${}^{\prime}$01${}^{\u2033}$ | 16${}^{\circ}$54${}^{\prime}$24${}^{\u2033}$ | 266.5 | 6.15 | F3-NOR |

ST27 | Ponta Delgada | 32${}^{\circ}$49${}^{\prime}$16${}^{\u2033}$ | 16${}^{\circ}$59${}^{\prime}$31${}^{\u2033}$ | 123.3 | 17.26 | F3-NOR |

ST28 | Porto do Moniz | 32${}^{\circ}$50${}^{\prime}$57${}^{\u2033}$ | 17${}^{\circ}$09${}^{\prime}$46${}^{\u2033}$ | 64.3 | 80.65 | F3-NOR |

ST29 | Queimadas | 32${}^{\circ}$46${}^{\prime}$59${}^{\u2033}$ | 16${}^{\circ}$54${}^{\prime}$07${}^{\u2033}$ | 881.4 | 34.66 | F3-NOR |

ST30 | Rabaçal | 32${}^{\circ}$45${}^{\prime}$30${}^{\u2033}$ | 17${}^{\circ}$07${}^{\prime}$51${}^{\u2033}$ | 1233.4 | 101.10 | F2-CEN |

ST31 | Ribeira Brava | 32${}^{\circ}$40${}^{\prime}$26${}^{\u2033}$ | 17${}^{\circ}$03${}^{\prime}$46${}^{\u2033}$ | 25.4 | 24.13 | F1-SOU |

ST32 | Ribeiro Frio | 32${}^{\circ}$43${}^{\prime}$51${}^{\u2033}$ | 16${}^{\circ}$52${}^{\prime}$58${}^{\u2033}$ | 1167.1 | 19.07 | F2-CEN |

ST33 | Sanatório | 32${}^{\circ}$40${}^{\prime}$07${}^{\u2033}$ | 16${}^{\circ}$54${}^{\prime}$02${}^{\u2033}$ | 384.1 | 11.75 | F1-SOU |

ST34 | Santa Catarina | 32${}^{\circ}$41${}^{\prime}$36${}^{\u2033}$ | 16${}^{\circ}$46${}^{\prime}$23${}^{\u2033}$ | 49.2 | 7.74 | F1-SOU |

ST35 | Santa Quitéria ETA | 32${}^{\circ}$39${}^{\prime}$39${}^{\u2033}$ | 16${}^{\circ}$57${}^{\prime}$03${}^{\u2033}$ | 321.0 | 9.20 | F1-SOU |

ST36 | Santana | 32${}^{\circ}$43${}^{\prime}$19${}^{\u2033}$ | 16${}^{\circ}$46${}^{\prime}$27${}^{\u2033}$ | 80.4 | 16.46 | F3-NOR |

ST37 | Santo António | 32${}^{\circ}$40${}^{\prime}$36${}^{\u2033}$ | 16${}^{\circ}$56${}^{\prime}$45${}^{\u2033}$ | 525.3 | 10.82 | F1-SOU |

ST38 | Santo da Serra | 32${}^{\circ}$43${}^{\prime}$33${}^{\u2033}$ | 16${}^{\circ}$49${}^{\prime}$01${}^{\u2033}$ | 660.2 | 36.08 | F3-NOR |

ST39 | Serra de Água | 32${}^{\circ}$44${}^{\prime}$31${}^{\u2033}$ | 17${}^{\circ}$01${}^{\prime}$11${}^{\u2033}$ | 573.2 | 24.34 | F2-CEN |

ST40 | Vale da Lapa | 32${}^{\circ}$49${}^{\prime}$37${}^{\u2033}$ | 16${}^{\circ}$55${}^{\prime}$40${}^{\u2033}$ | 347.0 | 5.31 | F3-NOR |

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## Share and Cite

**MDPI and ACS Style**

Espinosa, L.A.; Portela, M.M.; Rodrigues, R.
Significant Extremal Dependence of a Daily North Atlantic Oscillation Index (NAOI) and Weighted Regionalised Rainfall in a Small Island Using the Extremogram. *Water* **2020**, *12*, 2989.
https://doi.org/10.3390/w12112989

**AMA Style**

Espinosa LA, Portela MM, Rodrigues R.
Significant Extremal Dependence of a Daily North Atlantic Oscillation Index (NAOI) and Weighted Regionalised Rainfall in a Small Island Using the Extremogram. *Water*. 2020; 12(11):2989.
https://doi.org/10.3390/w12112989

**Chicago/Turabian Style**

Espinosa, Luis Angel, Maria Manuela Portela, and Rui Rodrigues.
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, no. 11: 2989.
https://doi.org/10.3390/w12112989