# A Continuous Drought Probability Monitoring System, CDPMS, Based on Copulas

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Materials and Methods

#### 2.1. CDPMS Definition

#### 2.1.1. Seasonal Threshold Definition

#### 2.1.2. Copula Fitting

_{1}, …, x

_{m}follow a marginal probability distribution function F

_{1}(x

_{1}), …, F

_{m}(x

_{m}), respectively, there exists a copula, C, that can join these marginal distribution functions in the form of a joint distribution function (Equation (3)),

_{k}(x

_{k}) = u

_{k}for k = 1, ..., m, with u

_{k}~ u(0,1) and C(u

_{1}, ..., u

_{m}) being the copula function.

#### 2.1.3. Conditional Probability

_{x}(x), ${U}_{2}$ = F

_{y}(y) and ${u}_{1}$ and ${u}_{2}$ being specific values. The conditional distribution of X given Y = y is given by:

#### 2.2. CDPMS Performance Assessment

#### 2.2.1. Brier Score (BS)

#### 2.2.2. Brier Skill Score (BSS)

_{CDPMS}) and from the BS for a reference forecast (BS

_{REF}) according to Equation (6), whose results range from −∞ to 1. BSS = 0 means no skill in comparison to the reference, and BSS = 1, perfect skill.

_{REF}was set equal to 0.20.

## 3. Precipitation Data

## 4. CDPMS for Mainland Portugal: Definition and Performance

#### 4.1. Precipitation Thresholds for Drought Recognition

#### 4.2. Copula Fitting

#### 4.3. Drought Risk Monitoring

#### 4.4. CDPMS Performance Assessment

## 5. CDRMS Applied to a Single Site

#### 5.1. CDRMS Development for Santa Marta da Montanha

#### 5.2. CDRMS Application—Drought Risk Monitoring

## 6. Discussion and Conclusion

## Author Contributions

## Funding

## Conflicts of Interest

## References

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**Figure 1.**Mainland Portugal. Surfaces of precipitation thresholds, ${R}_{N}^{*}$, for the 6-month period from October to March (SPI6), from the right to the left, for moderate (−0.84), severe (−1.28), and extreme (−1.65) droughts. Adapted from [23].

**Figure 4.**Distribution (in percentage) of the selected copulas by the 45 rain gauges as a function of the number of initial months of the rainy season.

**Figure 5.**Example of the application of the CDPMS to the continuous monitoring of the likelihood of moderate drought at the end of the six months period from October 2012 to February 2013. Drought probability from the end of October (n = 1) to the end of February (n = 5). The last map identifies the rain gauges that in fact did or did not experience drought by the end of March of 2013.

**Figure 6.**Moderate droughts. BSS values for n = 1 to n = 5. The closer to 1, the better the CDRMS performance.

**Figure 7.**Moderate droughts. BSS values for all the rain gauges as a function of the number of initial months of the rainy season with known precipitation (n).

**Figure 8.**Santa Marta da Montanha (RG07) rain gauge. Bivariate models for each coupled (${R}_{N}$, ${R}_{n}$) series along the rainy season of 2017/2018, from n = 1 to 5.

**Figure 9.**Santa Marta da Montanha (RG07) rain gauge. Probability of moderate, severe, and extreme drought events along the rainy season of 2017/2018 according to the CDPMS (dashed cells).

**Table 1.**Classes of drought intensity, associated probability, and SPI value according to [26].

Drought Class | Probability | SPI Value |
---|---|---|

Moderate Drought | 0.20 | Less than −0.84 |

Severe Drought | 0.10 | Less than −1.28 |

Extreme Drought | 0.05 | Less than −1.65 |

Class | Family | $\mathit{C}\left({\mathit{u}}_{1},{\mathit{u}}_{2}\right)$ | Parameter Range |
---|---|---|---|

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

Archimedean | Frank | $\frac{1}{\theta}\mathrm{log}\text{}\left(1+\text{}\frac{\left({e}^{\theta \text{}{u}_{1}}-1\right)\left({e}^{\theta \text{}{u}_{2}}-1\right)}{\left({e}^{\theta}-1\right)}\right)$ | $\theta \text{}\in \left(-\infty ,\text{}+\infty \right)$ |

Archimedean | Clayton | ${\left({u}_{1}{}^{-\theta}+\text{}{u}_{2}{}^{-\theta}-1\right)}^{\frac{-1}{\theta}})$ | $\theta \text{}\in \left(0,\text{}+\infty \right)$ |

Meta-Elliptical | Gaussian | ${\varphi}_{\rho}\left({\varphi}^{-1}\left({u}_{1}\right),\text{}{\varphi}^{-1}\left({u}_{2}\right)\right)$ | $\rho \text{}\in \text{}\left(-1,1\right)$ |

Meta-Elliptical | t Student | ${T}_{\rho ,v}\left({T}_{v}^{-1}\left({u}_{1}\right),\text{}{T}_{v}^{-1}\left({u}_{2}\right)\right)$ | $\rho \text{}\in \text{}\left(-1,1\right),\text{}v2\text{}$ |

**Table 3.**Name, code, identification (ID), and geographic coordinates (WGS84 system) of the 45 rain gauges of Figure 3.

Name | Code | ID | Lat (°) | Long (°) |
---|---|---|---|---|

Merufe | 01G03UG | RG01 | 42.0180 | −8.3890 |

Travancas | 03N01G | RG02 | 41.8280 | −7.3056 |

Leonte | 03I03UG | RG03 | 41.7650 | −8.1470 |

Soutelo (Chaves) | 03L02UG | RG04 | 41.7530 | −7.5348 |

Campo de Víboras | 04R03UG | RG05 | 41.5240 | −6.5580 |

Cabeceiras de Basto | 04J06UG | RG06 | 41.5127 | −7.9792 |

Santa Marta da Montanha | 04K02G | RG07 | 41.5008 | −7.7460 |

Folgares | 06N01C | RG08 | 41.3032 | −7.2828 |

Carviçais | 06P02UG | RG09 | 41.1790 | −6.8900 |

Moncorvo | 06O04UG | RG10 | 41.1650 | −7.0510 |

Adorigo | 07L01U | RG11 | 41.1460 | −7.6070 |

Pindelo dos Milagres | 09J02U | RG12 | 40.8060 | −7.9630 |

Freixedas | 09O02U | RG13 | 40.6880 | −7.1630 |

Gouveia | 11L01UG | RG14 | 40.4940 | −7.5930 |

Santo Varão | 12F02C | RG15 | 40.1840 | −8.6020 |

Góis | 13I01G | RG16 | 40.1568 | −8.1133 |

Soure | 13F01G | RG17 | 40.0521 | −8.6250 |

Penha Garcia | 13O01UG | RG18 | 40.0420 | −7.0180 |

Alvaiázere | 15G01UG | RG19 | 39.8270 | −8.3810 |

Ladoeiro | 14N02UG | RG20 | 39.8269 | −7.2660 |

Nisa | 16L03UG | RG21 | 39.5160 | −7.6690 |

Castelo de Vide | 17M01G | RG22 | 39.4116 | −7.4525 |

Pernes | 17F01UG | RG23 | 39.3910 | −8.6630 |

Bemposta | 17I02UG | RG24 | 39.3490 | −8.1410 |

Alter do Chão | 18L01UG | RG25 | 39.2182 | −7.6844 |

Pragança | 18C01G | RG26 | 39.1990 | −9.0640 |

Pavia | 20I01G | RG27 | 38.8965 | −8.0136 |

Caia (Monte Caldeiras) | 20O02UG | RG28 | 38.8873 | −7.0898 |

Santo Estevão | 20E02UG | RG29 | 38.8600 | −8.7460 |

Estremoz | 20L01G | RG30 | 38.8416 | −7.6159 |

Colares (Sarrazola) | 21A01C | RG31 | 38.8020 | −9.4570 |

Évora−Monte | 21K02UG | RG32 | 38.7690 | −7.7161 |

São Manços | 23K01UG | RG33 | 38.4605 | −7.7505 |

Barragem de Pego do Altar | 23G01C | RG34 | 38.4196 | −8.3952 |

Amieira | 24L01C | RG35 | 38.2793 | −7.5605 |

Barrancos | 25P01UG | RG36 | 38.1321 | −7.0013 |

Santa Vitória | 26I01UG | RG37 | 37.9645 | −8.0227 |

Serpa | 26L01UG | RG38 | 37.9426 | −7.6038 |

Relíquias | 27G01G | RG39 | 37.7030 | −8.4825 |

Castro Verde | 27I01G | RG40 | 37.6976 | −8.0933 |

Mértola | 28L01UG | RG41 | 37.6371 | −7.6619 |

Rosário (Almodôvar) | 28I02U | RG42 | 37.6020 | −8.0810 |

Barragem de Mira | 28G01C | RG43 | 37.5101 | −8.4433 |

Santa Catarina (Tavira) | 31K01UG | RG44 | 37.1487 | −7.7847 |

Valverde | 31E03C | RG45 | 37.0820 | −8.7180 |

**Table 4.**Precipitation thresholds, ${R}_{N}^{*}$, (mm) for the six-month period, from October to March, for the different drought intensities.

Rain Gauge ID | Drought Intensity | Rain Gauge ID | Drought Intensity | ||||
---|---|---|---|---|---|---|---|

Moderate | Severe | Extreme | Moderate | Severe | Extreme | ||

RG01 | 745.2 | 550.6 | 411.7 | RG23 | 379.6 | 288.8 | 218.2 |

RG02 | 429.9 | 351.9 | 297.6 | RG24 | 307.4 | 226.9 | 165.0 |

RG03 | 1250.0 | 933.9 | 701.4 | RG25 | 269.9 | 190.0 | 126.2 |

RG04 | 338.9 | 272.4 | 228.7 | RG26 | 442.1 | 357.1 | 294.9 |

RG05 | 252.7 | 212.2 | 189.9 | RG27 | 245.4 | 181.4 | 132.1 |

RG06 | 624.0 | 467.6 | 350.5 | RG28 | 228.6 | 169.4 | 122.5 |

RG07 | 774.4 | 614.7 | 500.8 | RG29 | 272.3 | 208.9 | 160.4 |

RG08 | 250.9 | 197.3 | 158.8 | RG30 | 281.6 | 208.1 | 151.8 |

RG09 | 290.0 | 221.5 | 172.2 | RG31 | 371.7 | 309.3 | 263.1 |

RG10 | 225.0 | 175.0 | 137.3 | RG32 | 265.6 | 201.5 | 152.4 |

RG11 | 280.2 | 223.0 | 182.2 | RG33 | 238.9 | 178.1 | 130.7 |

RG12 | 601.4 | 479.3 | 393.2 | RG34 | 269.6 | 214.9 | 174.6 |

RG13 | 311.6 | 232.8 | 170.6 | RG35 | 256.2 | 202.4 | 162.6 |

RG14 | 517.3 | 419.1 | 345.1 | RG36 | 260.6 | 204.4 | 160.0 |

RG15 | 446.4 | 361.0 | 294.2 | RG37 | 251.9 | 197.7 | 155.9 |

RG16 | 557.5 | 449.0 | 364.9 | RG38 | 237.9 | 182.4 | 140.6 |

RG17 | 415.4 | 320.1 | 246.6 | RG39 | 311.3 | 242.3 | 191.1 |

RG18 | 383.7 | 310.8 | 255.6 | RG40 | 267.9 | 215.7 | 175.7 |

RG19 | 586.6 | 468.6 | 378.5 | RG41 | 184.5 | 146.8 | 120.9 |

RG20 | 288.6 | 234.7 | 194.2 | RG42 | 284.4 | 223.2 | 176.3 |

RG21 | 331.5 | 256.7 | 199.4 | RG43 | 292.3 | 225.8 | 173.2 |

RG22 | 370.7 | 284.5 | 221.8 | RG44 | 318.5 | 239.5 | 180.3 |

RG45 | 289.5 | 229.0 | 182.3 |

Rain Gauge ID | n = 1 | n = 2 | n = 3 | n = 4 | n = 5 | Average |
---|---|---|---|---|---|---|

RG01 | 0.15 | 0.27 | 0.46 | 0.57 | 0.69 | 0.43 |

RG02 | −0.02 | 0.10 | 0.13 | 0.37 | 0.55 | 0.23 |

RG03 | 0.05 | 0.19 | 0.47 | 0.63 | 0.83 | 0.44 |

RG04 | 0.04 | 0.10 | 0.25 | 0.44 | 0.65 | 0.30 |

RG05 | 0.00 | 0.18 | 0.35 | 0.51 | 0.65 | 0.34 |

RG06 | 0.15 | 0.27 | 0.44 | 0.55 | 0.77 | 0.44 |

RG07 | 0.06 | 0.18 | 0.35 | 0.52 | 0.66 | 0.35 |

RG08 | 0.04 | 0.11 | 0.23 | 0.41 | 0.56 | 0.27 |

RG09 | −0.03 | 0.25 | 0.27 | 0.53 | 0.62 | 0.33 |

RG10 | 0.01 | 0.16 | 0.22 | 0.41 | 0.52 | 0.27 |

RG11 | 0.02 | 0.14 | 0.22 | 0.53 | 0.64 | 0.31 |

RG12 | 0.01 | 0.13 | 0.28 | 0.44 | 0.59 | 0.29 |

RG13 | −0.03 | 0.14 | 0.39 | 0.71 | 0.79 | 0.40 |

RG14 | 0.00 | 0.11 | 0.23 | 0.52 | 0.66 | 0.30 |

RG15 | 0.02 | 0.19 | 0.32 | 0.44 | 0.70 | 0.33 |

RG16 | 0.01 | 0.11 | 0.23 | 0.50 | 0.73 | 0.32 |

RG17 | 0.02 | 0.13 | 0.31 | 0.54 | 0.81 | 0.36 |

RG18 | 0.00 | 0.12 | 0.23 | 0.44 | 0.58 | 0.27 |

RG19 | −0.02 | 0.09 | 0.21 | 0.50 | 0.70 | 0.29 |

RG20 | 0.01 | 0.07 | 0.21 | 0.33 | 0.65 | 0.25 |

RG21 | 0.01 | 0.10 | 0.29 | 0.42 | 0.74 | 0.31 |

RG22 | 0.00 | 0.13 | 0.31 | 0.47 | 0.69 | 0.32 |

RG23 | 0.01 | 0.31 | 0.44 | 0.74 | 0.83 | 0.47 |

RG24 | 0.03 | 0.23 | 0.38 | 0.56 | 0.81 | 0.40 |

RG25 | 0.02 | 0.15 | 0.30 | 0.57 | 0.73 | 0.35 |

RG26 | −0.05 | 0.12 | 0.25 | 0.60 | 0.78 | 0.34 |

RG27 | 0.03 | 0.17 | 0.30 | 0.41 | 0.67 | 0.32 |

RG28 | 0.03 | 0.18 | 0.29 | 0.47 | 0.72 | 0.34 |

RG29 | 0.02 | 0.10 | 0.14 | 0.41 | 0.55 | 0.24 |

RG30 | −0.04 | 0.11 | 0.27 | 0.40 | 0.54 | 0.25 |

RG31 | 0.03 | 0.07 | 0.24 | 0.51 | 0.73 | 0.32 |

RG32 | 0.05 | 0.15 | 0.25 | 0.49 | 0.62 | 0.31 |

RG33 | 0.08 | 0.21 | 0.37 | 0.48 | 0.67 | 0.36 |

RG34 | 0.01 | 0.13 | 0.27 | 0.41 | 0.68 | 0.30 |

RG35 | 0.01 | 0.11 | 0.32 | 0.57 | 0.71 | 0.34 |

RG36 | 0.04 | 0.19 | 0.29 | 0.54 | 0.68 | 0.35 |

RG37 | −0.01 | 0.14 | 0.29 | 0.57 | 0.84 | 0.36 |

RG38 | 0.07 | 0.24 | 0.31 | 0.48 | 0.57 | 0.33 |

RG39 | 0.04 | 0.17 | 0.15 | 0.54 | 0.71 | 0.32 |

RG40 | 0.03 | 0.13 | 0.31 | 0.61 | 0.80 | 0.37 |

RG41 | 0.10 | 0.28 | 0.47 | 0.70 | 0.81 | 0.47 |

RG42 | 0.07 | 0.16 | 0.42 | 0.66 | 0.78 | 0.42 |

RG43 | 0.08 | 0.20 | 0.37 | 0.54 | 0.70 | 0.38 |

RG44 | 0.03 | 0.08 | 0.35 | 0.66 | 0.82 | 0.39 |

RG45 | 0.01 | 0.14 | 0.19 | 0.47 | 0.76 | 0.32 |

Average | 0.03 | 0.16 | 0.30 | 0.51 | 0.70 | 0.34 |

**Table 6.**Santa Marta da Montanha (RG07) rain gauge. Bivariate models for each coupled (${R}_{N}$, ${R}_{n}$) series, their parameters, Kendall Tau correlation (according to the model and empirical), AIC, and p values.

${\mathit{R}}_{\mathit{n}}$ | Family | Parameters | Kendall’s Tau | AIC | p-Value | ||
---|---|---|---|---|---|---|---|

$\mathit{\theta}$$\text{}\mathbf{or}\text{}\mathit{\rho}$ | $\mathit{v}$ | Model | Empirical | ||||

n = 1 | Frank | 1.75 | - | 0.19 | 0.19 | −6.16 | <0.05 |

n = 2 | Gaussian | 0.60 | - | 0.41 | 0.40 | −37.86 | <0.05 |

n = 3 | t Student | 0.76 | 30.00 | 0.55 | 0.53 | −75.34 | <0.05 |

n = 4 | Gaussian | 0.91 | - | 0.72 | 0.72 | −161.84 | <0.05 |

n = 5 | Gaussian | 0.96 | - | 0.83 | 0.83 | −251.14 | <0.05 |

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**MDPI and ACS Style**

Pontes Filho, J.D.; Portela, M.M.; Marinho de Carvalho Studart, T.; Souza Filho, F.d.A.
A Continuous Drought Probability Monitoring System, CDPMS, Based on Copulas. *Water* **2019**, *11*, 1925.
https://doi.org/10.3390/w11091925

**AMA Style**

Pontes Filho JD, Portela MM, Marinho de Carvalho Studart T, Souza Filho FdA.
A Continuous Drought Probability Monitoring System, CDPMS, Based on Copulas. *Water*. 2019; 11(9):1925.
https://doi.org/10.3390/w11091925

**Chicago/Turabian Style**

Pontes Filho, João Dehon, Maria Manuela Portela, Ticiana Marinho de Carvalho Studart, and Francisco de Assis Souza Filho.
2019. "A Continuous Drought Probability Monitoring System, CDPMS, Based on Copulas" *Water* 11, no. 9: 1925.
https://doi.org/10.3390/w11091925