# Copula-Based Multivariate Frequency Analysis of the 2012–2018 Drought in Northeast Brazil

^{1}

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## Abstract

**:**

## 1. Introduction

## 2. Study Area

## 3. The 2012–2018 Drought

^{6}hm³ in December 2011 to 13 × 10

^{6}hm³ by the end of 2016, with some hydrographic regions with total collapse. This issue was more accentuated and prolonged in central and southern regions, i.e., HR8–HR12, which represent 74% of the state’s total accumulation capacity. With the prolongation of the drought, the small and medium reservoirs (with storage capacity below 75 hm³ according to Ceará State’s declaration no. 23.068/1994 [62]) started to collapse, both in terms of quantity and quality, enhancing the costs of capturing and distributing water at longer distances.

## 4. Data and Methods

#### 4.1. Data

#### 4.2. Drought Analysis

#### 4.3. Statistical Inference

Clayton | ${\left({\mathrm{u}}_{1}{}^{-\theta}+{\mathrm{u}}_{2}{}^{-\theta}-1\right)}^{\frac{-1}{\theta}})$ | (3) |

Frank | $-\frac{1}{\theta}\mathrm{log}\left(1+\frac{\left({\mathrm{e}}^{-\theta {\mathrm{u}}_{1}}-1\right)\left({\mathrm{e}}^{-\theta {\mathrm{u}}_{2}}-1\right)}{\left({\mathrm{e}}^{-\theta}-1\right)}\right)$ | (4) |

Gumbel | $\mathrm{exp}\{-{\left[{\left(-\mathrm{ln}{\mathrm{u}}_{1}\right)}^{\theta}+{\left(-\mathrm{ln}{\mathrm{u}}_{2}\right)}^{\theta}\right]}^{\frac{1}{\theta}}$ | (5) |

Gaussian | ${\mathsf{\varphi}}_{\mathsf{\rho}}\left({\mathsf{\varphi}}^{-1}\left({\mathrm{u}}_{1}\right),{\mathsf{\varphi}}^{-1}\left({\mathrm{u}}_{2}\right)\right)$ | (6) |

t-Student | ${\mathrm{T}}_{\mathsf{\rho},\mathrm{v}}\left({\mathrm{T}}_{\mathrm{v}}^{-1}\left({\mathrm{u}}_{1}\right),{\mathrm{T}}_{\mathrm{v}}^{-1}\left({\mathrm{u}}_{2}\right)\right)$ | (7) |

#### 4.4. Frequency Analysis

#### 4.4.1. Univariate Return Period

#### 4.4.2. Bivariate Return Period

## 5. Results

#### 5.1. Drought Analysis

#### 5.2. Frequency Analysis

## 6. Discussions and Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

## References

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**Figure 1.**Location of Ceará State, its hydrographic regions and main reservoirs (storage capacity higher than 100 hm³).

**Figure 2.**Anomalies of cumulative rainfall (mm), and oceanic index: Interhemispheric Tropical Atlantic Gradient (IHTAG), Nino 3.4, Atlantic Multidecadal Oscillation (AMO) and Pacific Decadal Oscillation (PDO). In red are the accumulated rainfall deficit and oceanic index associated with dry years in the region. In blue, the exceeding rainfall and the oceanic index associated with wet years.

**Figure 5.**Standardized Precipitation Index ($SPI$) for the aggregated period of 1 to 36 months, from 1973 to 2019. Warm colors represent periods of drought in Ceará State.

**Figure 6.**$SPI$ values for the 12 hydrographic regions of Ceará State organized from northern to southernmost position, HR01–HR12. Warm colors represent dry years and cold colors represent wet years. Spatial coverage of extreme events can be visually detected.

**Figure 7.**Scatterplot of duration $D$ and severity $S$ of the recorded droughts by hydrographic region. In solid red, the drought with onset in 2012 and highlighted are the periods of the most extreme events.

**Figure 8.**The differences between autocorrelation of time series of $SPI12$ and the discretized $SPI{12}_{DEC}$ for HR04. The $SPI12$ time series presented strong autocorrelation due to the moving window used to compute its values. In order to provide frequency analysis, independence of the time series was archived by performing a discretization by using $SPI{12}_{DEC}$

**Figure 9.**The drought return period of the "or" and "and" cases, i.e., ${T}_{D\mathrm{or}S}$ and ${T}_{DS}$, for each hydrograph region, HR01–HR12. Contour lines correspond to the return period (years); blue dots are the drought events that occurred over the time series, and in red is the drought started in 2012. The “and” cases have higher return periods than the “or” ones, as the first is more restrictive than the second. The return period of any given drought can be found by providing the associated duration and severity.

**Figure 10.**Exposure to drought hazard of an event with average characteristics of the 2012–2018 event. The lower the return period, the higher the exposure to drought hazard.

**Table 1.**Descriptive statistics of drought events and the variables duration and severity by river basin in the period 1911–2018. In bold, when the current drought event initiated in 2012 is equal to the maximum event in time series.

Region | No. Drought Events | Inter- Arrival Time | Duration (Years) | Severity | ||||||
---|---|---|---|---|---|---|---|---|---|---|

Max | 2012–2018 Drought | Mean | CV | Max | 2012–2018 Drought | Mean | CV | |||

HR01 | 25 | 4.32 | 6 | 5 | 2.08 | 0.71 | 5.10 | 5.10 | 1.69 | 0.91 |

HR02 | 26 | 4.15 | 6 | 5 | 2.00 | 0.85 | 5.06 | 4.27 | 1.65 | 0.93 |

HR03 | 26 | 4.15 | 6 | 5 | 2.00 | 0.82 | 5.11 | 5.11 | 1.64 | 0.91 |

HR04 | 25 | 4.32 | 6 | 6 | 2.04 | 0.74 | 6.01 | 6.01 | 1.73 | 0.95 |

HR05 | 24 | 4.5 | 6 | 6 | 2.04 | 0.73 | 7.47 | 7.47 | 1.85 | 0.95 |

HR06 | 26 | 4.15 | 6 | 5 | 2.00 | 0.85 | 4.85 | 4.85 | 1.69 | 0.93 |

HR07 | 26 | 4.15 | 7 | 7 | 1.96 | 0.82 | 5.38 | 5.38 | 1.62 | 0.98 |

HR08 | 22 | 4.91 | 10 | 7 | 2.64 | 0.90 | 6.76 | 6.76 | 1.92 | 1.01 |

HR09 | 22 | 4.91 | 6 | 6 | 2.36 | 0.77 | 7.07 | 7.07 | 1.99 | 0.91 |

HR010 | 23 | 4.7 | 7 | 7 | 2.39 | 0.76 | 6.24 | 6.24 | 1.85 | 0.89 |

HR011 | 23 | 4.7 | 7 | 7 | 2.43 | 0.71 | 7.54 | 7.54 | 1.85 | 0.97 |

HR012 | 23 | 4.7 | 7 | 7 | 2.43 | 0.79 | 5.88 | 5.88 | 1.83 | 0.94 |

Hydrographic Region | Duration | Severity | Copula |
---|---|---|---|

HR01 | Log-normal | Exponential | Gumbel |

(µ = 0.53, σ = 0.61) | (λ = 0.59) | (θ = 2.26, τ = 0.56) | |

HR02 | Log-normal | Exponential | Survival Clayton |

(µ = 0.43, σ = 0.67) | (λ = 0.61) | (θ = 1.78, τ = 0.47) | |

HR03 | Log-normal | Exponential | Gumbel |

(µ = 0.44, σ = 0.65) | (λ = 0.61) | (θ = 2.38, τ = 0.58) | |

HR04 | Log-normal | Exponential | Survival Clayton |

(µ = 0.49, σ = 0.64) | (λ = 0.58) | (θ = 2.56, τ = 0.56) | |

HR05 | Log-normal | Exponential | Survival Clayton |

(µ = 0.50, σ = 0.63) | (λ = 0.54) | (θ = 2.28, τ = 0.56) | |

HR06 | Log-normal | Exponential | Survival Clayton |

(µ = 0.43, σ = 0.67) | (λ = 0.59) | (θ = 1.70, τ = 0.46) | |

HR07 | Log-normal | Exponential | Survival Clayton |

(µ = 0.43, σ = 0.64) | (λ = 0.62) | (θ = 1.56, τ = 0.44) | |

HR08 | Log-normal | Exponential | Survival Clayton |

(µ = 0.65, σ = 0.77) | (λ = 0.52) | (θ = 2.32, τ = 0.54) | |

HR09 | Log-normal | Exponential | Survival Clayton |

(µ = 0.60, σ = 0.7) | (λ = 0.50) | (θ = 2.33, τ = 0.54) | |

HR010 | Log-normal | Exponential | Survival Clayton |

(µ = 0.62, σ = 0.70) | (λ = 0.54) | (θ = 1.90, τ = 0.49) | |

HR011 | Log-normal | Exponential | Survival Clayton |

(µ = 0.66, σ = 0.68) | (λ = 0.54) | (θ = 2.23, τ = 0.53) | |

HR012 | Log-normal | Exponential | Survival Clayton |

(µ = 0.62, σ = 0.71) | (λ = 0.55) | (θ = 3.20, τ = 0.62) |

**Table 3.**Description of the 2012 onset drought event for each hydrographic region. The univariate return period (years) of drought duration (${T}_{D}$) and severity (${T}_{S}$ ), and the bivariate ${T}_{DorS}$ and ${T}_{D\&S}$ return periods (years).

Hydrographic Region | Drought Period | ${\mathit{T}}_{\mathit{D}}$ | ${\mathit{T}}_{\mathit{S}}$ | ${\mathit{T}}_{\mathit{D}\mathit{o}\mathit{r}\mathit{S}}$ | ${\mathit{T}}_{\mathit{D}\&\mathit{S}}$ | Rank in the Set of Events | ||
---|---|---|---|---|---|---|---|---|

Duration | Severity | Joint | ||||||

HR01 | 2012–2016 | 113 | 88 | 72 | 155 | 2 | 1 | 2 |

HR02 | 2012–2016 | 106 | 56 | 52 | 124 | 3 | 3 | 3 |

HR03 | 2012–2016 | 115 | 94 | 77 | 157 | 3 | 1 | 3 |

HR04 | 2012–2017 | 206 | 141 | 131 | 234 | 1 | 1 | 1 |

HR05 | 2012–2017 | 223 | 254 | 191 | 313 | 1 | 1 | 1 |

HR06 | 2012–2016 | 106 | 73 | 63 | 136 | 3 | 1 | 3 |

HR07 | 2012–2018 | 465 | 117 | 115 | 499 | 1 | 1 | 1 |

HR08 | 2012–2018 | 106 | 165 | 98 | 188 | 2 | 1 | 2 |

HR09 | 2012–2017 | 111 | 168 | 102 | 193 | 1 | 1 | 1 |

HR010 | 2012–2018 | 161 | 136 | 112 | 215 | 1 | 1 | 1 |

HR011 | 2012–2018 | 160 | 275 | 150 | 309 | 1 | 1 | 1 |

HR012 | 2012–2018 | 152 | 119 | 110 | 171 | 1 | 1 | 1 |

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## Share and Cite

**MDPI and ACS Style**

Pontes Filho, J.D.; Souza Filho, F.d.A.; Martins, E.S.P.R.; Studart, T.M.d.C. Copula-Based Multivariate Frequency Analysis of the 2012–2018 Drought in Northeast Brazil. *Water* **2020**, *12*, 834.
https://doi.org/10.3390/w12030834

**AMA Style**

Pontes Filho JD, Souza Filho FdA, Martins ESPR, Studart TMdC. Copula-Based Multivariate Frequency Analysis of the 2012–2018 Drought in Northeast Brazil. *Water*. 2020; 12(3):834.
https://doi.org/10.3390/w12030834

**Chicago/Turabian Style**

Pontes Filho, João Dehon, Francisco de Assis Souza Filho, Eduardo Sávio Passos Rodrigues Martins, and Ticiana Marinho de Carvalho Studart. 2020. "Copula-Based Multivariate Frequency Analysis of the 2012–2018 Drought in Northeast Brazil" *Water* 12, no. 3: 834.
https://doi.org/10.3390/w12030834