# Bivariate Modelling of a Teleconnection Index and Extreme Rainfall in a Small North Atlantic Island

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## Abstract

**:**

## 1. Introduction

## 2. Physical Features of the Study Area

## 3. Materials and Methods

#### 3.1. Rain Gauge Data

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

#### 3.3. Bivariate Copula

**C**—introduced by Sklar [44]—completely defined through the functional identity ${F}_{XY}=\mathbf{C}({F}_{X},{F}_{Y})$.

**C**is a bivariate copula, hereafter copula. It is precisely the copula that captures the features of a joint distribution. Moreover, copulas measure the association and dependence structure properties connecting random variables. In this research work, a possible way of analysing bivariate data (NAOI, X, and extreme rainfall, Y) consisted of investigating the dependence function and the marginals separately. This approach was convenient for the climate variability assessment, as this allowed studying the dependence structure independently of any marginal effect. Regardless of the marginal laws involved, the analysis of this research work was focused on practical applications of modelling copulas for teleconnection and not on the equations involved. Nevertheless, the main mathematical descriptions related to the copula concept [44] are presented next.

**Definition**

**1.**

**Theorem**

**1.**

**C**and $\phi [0,1]\to [0,\infty ]$ is a continuous and factually reducing function. The Meta-elliptic Gaussian and Student’s t, and the Archimedean Clayton, Frank, and Gumbel, were tested to verify best fit. Table 2 presents the formulations of the candidate copula families. When the Archimedean copula is rotated 180°, it is called a survival copula and can invert the predefined tail dependence to best fit the data.

#### 3.3.1. Copula Parameter Estimation

#### 3.3.2. Best-Fitted Copula

**C**with $\theta $ parameter(s) is defined by Equation (5):

#### 3.3.3. Bivariate Return Periods

## 4. Results

#### 4.1. Extreme Daily Rainfall Distribution Analysis

#### 4.2. Alignment of the NAOI and Extreme Rainfall Series

#### 4.3. Bivariate Joint Distributions and Constructed Copulas

#### 4.4. Bivariate Return Periods of the Previous NAOI and Extreme Rainfall Events

## 5. Discussion and Conclusions

#### 5.1. Negative NAO Persistence and Climate Variability

#### 5.2. Copula-Based Modelling of NAO Teleconnection in Extreme Rainfall

#### 5.3. Climate Variability Assessment Based on the Bivariate Return Periods

#### 5.4. Challenges and Advances Related to Extreme Rainfall Analyses

## Author Contributions

## Funding

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 1.**Coordinates WGS84 (UTM zone 28N) and relief of Madeira Island, Portugal. Spatial distribution of the six rain gauges used in this study are depicted by bullets and their identification codes (description in Table 1).

**Figure 2.**The winter (DJF) extreme daily rainfalls, i.e., the highest 45 rainfalls or 1% of the retrieved 4500 daily data at each rain gauge from 1967/1968 through 2016/2017, in absolute (

**a**) and dimensionless (

**b**) terms.

**Figure 3.**Example of the non-smoothed (1d) and smoothed daily NAOI series in a period prior to the 20 February 2010 flash floods, landslides and debris. The smoothed NAOI values were computed by a low-pass filter, or moving average, from the previous 2 days (2d, $m=2$) to previous 30 days (30d, $m=30$) and assigned to the most recent record date.

**Figure 4.**Scatter plot of previous smoothed NAOI and extreme rainfalls, both at daily scale, for $m=14$ and $l=39$ days. The 45 bivariate observations (1967/1968–2016/2017) at the AR-C rain gauge are depicted by bullets and red diamonds, the latter being the events that occurred between December 2000 and the end of the reference period. The correlation (r) between previous NAOI and extreme rainfalls is −0.84.

**Figure 5.**Normalised contour plots of the selected bivariate copulas—namely, Survival Gumbel (AR-C and SC-E), Frank (BC-C), Student’s t (FO-S), Gumbel (PD-N), and Gaussian (PP-W).

**Figure 6.**Contour lines of the joint return periods ${\mathrm{T}}_{NI\phantom{\rule{0.166667em}{0ex}}\mathrm{a}nd\phantom{\rule{0.166667em}{0ex}}RN}$ of the 45 bivariate events in each rain gauge. The letters, from A to J, stand for the different return periods between 2 and 50,000 years. The events occurring from 1967/1968 to 1999/2000 are depicted by grey bullets, and those from 2000/2001 and 2016/2017 by red diamonds.

**Figure 7.**Conditional return periods ${\mathrm{T}}_{RN\mid NI\phantom{\rule{0.166667em}{0ex}}\ge ni}$ for different previous NAOI values. The bivariate observations from 1967/1968 to 1999/2000 are depicted by grey bullets, and from 2000/2001 and 2016/2017 by red diamonds.

**Table 1.**The six rain gauges adopted in the study. Identification (code and name), coordinates WGS84 (UTM zone 28N), and elevation. In the code, the character after the hyphen indicates the location of the gauge, i.e., C for centre and S, N, W, and E for southern, northern, western, and eastern coastal areas, respectively.

Code | Name | UTM-X Easting (m) | UTM-Y Northing (m) | Elevation (m.a.s.l.) |
---|---|---|---|---|

AR-C | Areeiro | 320,746.000 | 3,621,552.000 | 1610.0 |

BC-C | Bica da Cana | 307,604.000 | 3,625,815.000 | 1560.0 |

FO-S | Funchal Observatório | 322,831.000 | 3,613,854.000 | 58.0 |

PD-N | Ponta Delgada | 313,525.801 | 3,633,232.797 | 123.0 |

PP-W | Ponta do Pargo | 288,512.995 | 3,632,569.609 | 339.0 |

SC-E | Santa Catarina | 333,770.000 | 3,618,611.000 | 49.0 |

Copula Class | Copula Family | Mathematical Formulation |
---|---|---|

Meta-elliptic | Gaussian | ${\varphi}_{\rho}({\varphi}^{-1}(u),{\varphi}^{-1}(v))$ |

Student’s t | ${\mathrm{T}}_{\rho ,}({\mathrm{T}}_{}^{-1}(u),{\mathrm{T}}_{}^{-1}(v))$ | |

Clayton | ${({u}^{-\theta}+{v}^{-\theta}-1)}^{\frac{-1}{\theta}}$ | |

Archimedean | Frank | $-{\theta}^{-1}\mathrm{log}\left[1+\frac{({e}^{\phantom{\rule{0.166667em}{0ex}}\theta \phantom{\rule{0.166667em}{0ex}}u}-1)({e}^{\phantom{\rule{0.166667em}{0ex}}\theta \phantom{\rule{0.166667em}{0ex}}v}-1)}{({e}^{\phantom{\rule{0.166667em}{0ex}}\theta}-1)}\right]$ |

Gumbel | $\mathrm{exp}\left\{-{[{(-\mathrm{ln}u)}^{-\theta}+{(-\mathrm{ln}v)}^{\theta}]}^{\frac{1}{\theta}}\right\}$ |

**Table 3.**Statistics of the bivariate observations (previous NAOI and extreme rainfall) at each rain gauge from 1967/1968–2016/2017. E(L) is the expected interarrival time, and Kendall tau ($\tau $) correlates the two variables.

Rain Gauge | Number of Observations | E(L) (year) | Kendall tau ($\mathit{\tau}$) | Previous NAOI, $-\mathit{NI}$ | Extreme Rainfall (mm), $\mathit{RN}$ | |||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|

Mean | Min | SD | CV | Mean | Max | SD | CV | |||||

AR-C | 45 | 0.90 | −0.71 | −183.31 | −366.03 | 69.42 | −0.38 | 165.00 | 257.60 | 29.64 | 0.18 | |

BC-C | 45 | 0.90 | −0.51 | −185.08 | −366.03 | 58.52 | −0.32 | 159.03 | 327.30 | 39.19 | 0.25 | |

FO-S | 45 | 0.90 | −0.50 | −174.85 | −366.03 | 78.25 | −0.45 | 58.94 | 111.00 | 17.65 | 0.30 | |

PD-N | 45 | 0.90 | −0.29 | −177.44 | −366.03 | 70.70 | −0.40 | 84.03 | 172.10 | 30.37 | 0.36 | |

PP-W | 45 | 0.90 | −0.27 | −173.34 | −366.03 | 80.92 | −0.47 | 57.53 | 133.30 | 22.20 | 0.39 | |

SC-E | 45 | 0.90 | −0.42 | −180.94 | −366.03 | 72.98 | −0.40 | 57.16 | 101.90 | 14.21 | 0.25 |

**Table 4.**Selected copula families, Kendall tau ($\tau $), univariate distributions used for candidate of marginal distributions of $NI$ and $RN$, and parameters of the models at each rain gauge. Par stands for the copula parameter, and Par1 and Par2 for the marginal distribution parameters.

Rain Gauge | Copula | Opposite Previous NAOI, $\mathit{NI}$ | Extreme Rainfall, $\mathit{RN}$ | |||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|

Family | Par | $\mathit{\tau}$ | AIC | Univariate Dist. | Par1 | Par2 | AIC | Univariate Dist. | Par1 | Par2 | AIC | |||

AR-C | Survival Gumbel | 3.28 | 0.70 | −66.61 | log-normal | 5.15 | 0.36 | 502.09 | log-normal | 5.09 | 0.17 | 429.61 | ||

BC-C | Frank | 4.90 | 0.45 | −18.91 | log-normal | 5.17 | 0.30 | 490.18 | log-normal | 5.05 | 0.21 | 445.14 | ||

FO-S | Student’s t | 0.63 | 0.44 | −27.18 | log-normal | 5.07 | 0.42 | 509.67 | log-normal | 4.04 | 0.26 | 374.56 | ||

PD-N | Gumbel | 1.38 | 0.28 | −9.53 | log-normal | 5.10 | 0.39 | 506.36 | log-normal | 4.38 | 0.32 | 423.23 | ||

PP-W | Gaussian | 0.50 | 0.33 | −10.79 | Weibull | 2.32 | 196.04 | 522.39 | log-normal | 4.00 | 0.31 | 387.31 | ||

SC-E | Survival Gumbel | 1.69 | 0.41 | −16.43 | log-normal | 5.12 | 0.39 | 508.15 | log-normal | 4.02 | 0.22 | 357.72 |

**Table 5.**At each of the six rain gauges, dates and characteristics of the three bivariate observations (previous NAOI and extreme daily rainfalls) with the highest return periods ${\mathrm{T}}_{NI\phantom{\rule{0.166667em}{0ex}}\mathrm{a}nd\phantom{\rule{0.166667em}{0ex}}RN}$ and their corresponding univariate, bivariate, and conditional return periods rounded to the nearest integer in years.

Rain Gauge | Date ${\mathbf{T}}_{\mathit{NI}\phantom{\rule{0.166667em}{0ex}}\&\phantom{\rule{0.166667em}{0ex}}\mathit{RN}}$ | Previous NAOI, $-\mathit{NI}$ | Extreme Rainfall (mm), $\mathit{RN}$ | ${\mathbf{T}}_{\mathit{NI}}$ (year) | ${\mathbf{T}}_{\mathit{RN}}$ (year) | ${\mathbf{T}}_{\mathit{NI}\phantom{\rule{0.166667em}{0ex}}\&\phantom{\rule{0.166667em}{0ex}}\mathit{RN}}$ (year) | ${\mathbf{T}}_{\mathit{RN}\mid \mathit{NI}\phantom{\rule{0.166667em}{0ex}}\ge \mathit{ni}}$ (year) |
---|---|---|---|---|---|---|---|

AR-C | 21/02/2010 | −366.0 | 257.6 | 491 | 2646 | 4407 | 259,844 |

26/01/2011 | −313.0 | 230.0 | 180 | 422 | 674 | 14,560 | |

05/12/1991 | −309.4 | 210.0 | 169 | 129 | 293 | 5922 | |

BC-C | 26/01/2011 | −313.0 | 327.3 | 278 | 44,618 | 324,643 | 10,843,673 |

15/12/1976 | −260.2 | 259.6 | 82 | 1175 | 3010 | 29,683 | |

21/02/2010 | −366.0 | 150.4 | 1009 | 15 | 1072 | 129,742 | |

FO-S | 02/02/2010 | −307.4 | 111.0 | 143 | 1600 | 1991 | 34,109 |

26/01/2011 | −313.0 | 103.4 | 156 | 758 | 1011 | 18,887 | |

21/02/2010 | −366.0 | 96.8 | 355 | 401 | 787 | 33,537 | |

PD-N | 02/02/2010 | −307.4 | 172.1 | 152 | 1034 | 1525 | 27,882 |

01/01/1969 | −293.4 | 155.2 | 120 | 449 | 742 | 10,721 | |

26/01/2011 | −313.0 | 145.5 | 167 | 279 | 598 | 12,010 | |

PP-W | 26/01/2011 | −313.0 | 133.3 | 160 | 3794 | 7975 | 153,443 |

27/02/2009 | −290.3 | 126.6 | 100 | 2295 | 4157 | 49,821 | |

21/02/2010 | −366.0 | 85.0 | 585 | 107 | 1348 | 94,663 | |

SC-E | 10/02/1969 | −209.4 | 101.9 | 29 | 2642 | 3420 | 12,027 |

26/01/2011 | −313.0 | 88.0 | 151 | 432 | 1517 | 27,517 | |

21/02/2010 | −366.0 | 76.0 | 364 | 104 | 1031 | 45,060 |

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**MDPI and ACS Style**

Espinosa, L.A.; Portela, M.M.; Pontes Filho, J.D.; Zelenakova, M.
Bivariate Modelling of a Teleconnection Index and Extreme Rainfall in a Small North Atlantic Island. *Climate* **2021**, *9*, 86.
https://doi.org/10.3390/cli9050086

**AMA Style**

Espinosa LA, Portela MM, Pontes Filho JD, Zelenakova M.
Bivariate Modelling of a Teleconnection Index and Extreme Rainfall in a Small North Atlantic Island. *Climate*. 2021; 9(5):86.
https://doi.org/10.3390/cli9050086

**Chicago/Turabian Style**

Espinosa, Luis Angel, Maria Manuela Portela, João Dehon Pontes Filho, and Martina Zelenakova.
2021. "Bivariate Modelling of a Teleconnection Index and Extreme Rainfall in a Small North Atlantic Island" *Climate* 9, no. 5: 86.
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