# Copula-Based Drought Analysis Using Standardized Precipitation Evapotranspiration Index: A Case Study in the Yellow River Basin, China

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## Abstract

**:**

## 1. Introduction

## 2. Materials and Methodology

#### 2.1. Study Area

#### 2.2. Dataset

#### 2.3. Methodology

#### 2.3.1. Standardized Precipitation Evapotranspiration Index

_{i}is the monthly average temperature, I is the annual total heating index, and m is the coefficient determined by I.

_{i}is the monthly precipitation and PET

_{i}is the monthly potential evapotranspiration.

_{i}based on three-parameter Log-Logistic distribution and calculating the cumulative function:

_{0}= 2.515517, C

_{1}= 0.802853, C

_{2}= 0.010328, d

_{1}= 1.432788, d

_{2}= 0.189269, and d

_{3}= 0.001308.

#### 2.3.2. The Modified Mann-Kendall Trend Test Method

#### 2.3.3. Run Theory

_{0}(SPEI = 0), X

_{1}(SPEI = −0.3), and X

_{2}(SPEI = −0.5) were set. The identification process of drought events is as follows:

_{1}, it is preliminarily determined that drought occurs in this month.

_{2}, it is considered that there is no drought in this month.

_{0}in that month, the two adjacent drought events can be merged into one drought event. The drought duration is the sum of drought duration and one month, and the drought severity is the sum of drought severity. Otherwise, they are two independent drought events.

#### 2.3.4. Marginal Distribution and Copula-Based Models

_{s}were applied to measure the correlation between two drought characteristic variables [42].

^{2}), Akaike Information Criterion (AIC), and root mean square error (RMSE) [47]. Copula families used in this research are shown in Table 2.

#### 2.3.5. Return Period

## 3. Results

#### 3.1. Drought Characteristics in the YRB

#### 3.1.1. Temporal Evolution

#### 3.1.2. Spatial Distribution

#### 3.1.3. Trend Characteristics at the Grid Scale

_{s}values of SPEI in the YRB from 1961 to 2015. Z

_{s}> 0 indicates a downward drought trend, and Z

_{s}< 0 indicates an upward drought trend. It can be seen that the gridded drought trend of the YRB was different in each period. On the monthly scale, the average trend characteristic Z

_{s}values of all grids in the YRB from January to December were 0.08, 0.17, −0.55, −0.58, 0.01, 0.07, −0.36, −0.28, −0.21, −0.59, −0.37, and 0.12, respectively. The SPEI for seven months (March, April, July, August, September, October, and November) showed a downward trend and drought showed an upward trend, while drought showed a downward trend in the remaining months. The average Z

_{s}value was less than zero in each subzone in April, July, October, and November, indicating that drought was increasing in these months in each subzone. And the average Z

_{s}value was greater than zero in each subzone in February, indicating that drought was decreasing in February in each subzone. From January to December, the drought area percentage with an increasing trend in the YRB was 45.9%, 16.5%, 87.8%, 91.6%, 49.8%, 39.0%, 99.4%, 82.3%, 71.9%, 97.0%, 90.1%, and 35.2%, respectively. On the seasonal scale, the average trend characteristic Z

_{s}values in spring, summer, autumn, and winter were −0.53, −0.38, −0.44, and 0.99, respectively. It can be seen that drought showed an upward trend in spring, summer, and autumn, while drought showed a downward trend in winter. The seasonal drought trend characteristic was different in each subzone. The average Z

_{s}value was less than zero in each subzone in summer and autumn, which indicated that drought was increasing in both seasons in each subzone. The drought area percentage with an increasing trend in spring, summer, autumn, and winter was 89.8%, 94.8%, 95.4%, and 17.7%, respectively. The average trend characteristic Z

_{s}values failed to pass the significance test of α = 0.05, and most of Z

_{s}values were less than zero, indicating that drought generally showed a non–significant upward trend in the YRB (Figure 6).

#### 3.2. Marginal Distribution Functions and Copulas Models

#### 3.2.1. Marginal Distribution Functions

_{s}were also adopted to test the correlation between drought duration and severity. The calculated τ and ρ

_{s}can represent the correlation degree between drought duration and severity. As shown in Table 3, Kendall rank correlation coefficients of drought duration and severity were all above 0.76, and reached the maximum value (0.84) in BH. Spearman rank correlation coefficients of drought duration and severity were all above 0.89, and reached the maximum value (0.95) in BH. All correlation coefficients passed the significance test of α = 0.01, which indicated that drought duration and severity were highly correlated. Thus, Copula function can be used to establish the joint distribution function of drought duration and severity in the YRB.

#### 3.2.2. Copulas Models

^{2}, AIC, and RMSE between theoretical and empirical Copula function. Based on the principle that the smaller the values of d

^{2}, AIC, and RMSE, the higher the GOF of Copula function, the three GOF evaluation indicators were calculated respectively, and the optimal Copula functions of drought duration and severity were established in the YRB. As shown in Table 4, the selected optimal Copula functions of drought duration and severity in AL, LL, LH, IF, HL, LS, SH, BH, and YRB were Frank-copula, Normal-copula, Frank-copula, Frank-copula, Normal-copula, Frank-copula, Frank-copula, Frank-copula, and Frank-copula, respectively. All the GOF evaluation indicators of the selected optimal Copula model met the requirements. Finally, the selected optimal Copula functions were regarded as the joint distribution functions of drought duration and severity in each subzone. From the obtained optimal Copula functions, we can see that the optimal Copula functions were Frank-copula for six subzones (AL, LH, IF, LS, SH, and BH) and the whole basin. In conclusion, Frank-copula was found to be the best-fitted one in the YRB.

#### 3.3. Return Period of Droughts

## 4. Discussion

## 5. Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 2.**Temporal evolution characteristics of drought in the YRB. (

**a**–

**i**) denote above Longyangxia (AL), Longyangxia to Lanzhou (LL), Lanzhou to Hekou (LH), Inner Flow region (IF), Hekou to Longmen (HL), Longmen to Sanmenxia (LS), Sanmenxia to Huayuankou (SH), below Huayuankou (BH), and Yellow River basin.

**Figure 3.**Hovmoller-type diagrams for the temporal variability of drought at different timescales (1 to 24 months) in the YRB. (

**a**–

**i**) denote above AL, LL, LH, IF, HL, LS, SH, BH, and YRB.

**Figure 4.**Spatial distribution of drought in the YRB. (

**a**–

**l**) denote monthly trends, and (

**m**–

**p**) denote seasonal trends.

**Figure 5.**Monthly and seasonal trends of Standardized Precipitation Evapotranspiration Index (SPEI) at the grid scale in the YRB during 1961–2015. (

**a**–

**l**) denote monthly trends, and (

**m**–

**p**) denote seasonal trends.

**Figure 7.**Drought return period isolines are color coded with joint density levels with blue representing lower densities and red denoting higher densities. Blue dots show drought events. (

**a**–

**i**) denote AL (Frank-copula), LL (Normal-copula), LH (Frank-copula), IF (Frank-copula), HL (Normal-copula), LS (Frank-copula), SH (Frank-copula), BH (Frank-copula), and YRB (Frank-copula), respectively.

Name | Cumulative Distribution Function (CDF) | Parameters (Shape, Scale, Location) | Reference |
---|---|---|---|

Lognormal (Logn) | $F(x)=\mathsf{\Phi}(\frac{\mathrm{ln}(x-\gamma )-\mu}{\sigma})$ | $\mu ,\sigma ,\gamma $ | [43] |

Gen. Pareto (GP) | $F(x)=1-{[1-\frac{k}{\alpha}(x-\epsilon )]}^{1/k}$ | $k,\alpha ,\epsilon $ | [43] |

Pearson Type III (P–III) | $F(x)=\frac{1}{\alpha \Gamma (\beta )}{{\displaystyle {\int}_{\gamma}^{x}(\frac{x-\gamma}{\alpha})}}^{\beta -1}{e}^{-(\frac{x-\gamma}{\alpha})}$ | $\alpha ,\beta ,\gamma $ | [43] |

Log–Logistic (Log–L) | $F(x)={[1+{(\frac{\beta}{x-\gamma})}^{\alpha}]}^{-1}$ | $\alpha ,\beta ,\gamma $ | [44] |

Gen. Extreme Value (GEV) | $F(x)=\mathrm{exp}(-\mathrm{exp}({k}^{-1}\mathrm{ln}(1-\frac{k(x-\mu )}{\sigma})))$ | $k,\sigma ,\mu $ | [45] |

Weibull (Wbl) | $F(x)=1-\mathrm{exp}(-{(\frac{x-\gamma}{\beta})}^{\alpha})$ | $\alpha ,\beta ,\gamma $ | [46] |

Name | Mathematical Description | Parameter Range | Reference |
---|---|---|---|

Normal-copula | ${\int}_{-\infty}^{{\phi}^{-1}(u)}{\displaystyle {\int}_{-\infty}^{{\phi}^{-1}(v)}\frac{1}{2\pi \sqrt{1-{\theta}^{2}}}}}\mathrm{exp}(\frac{2\theta xy-{x}^{2}-{y}^{2}}{2(1-{\theta}^{2})})dxdy$ | $\theta \in [-1,1]$ | [48] |

t-copula | ${\int}_{-\infty}^{{t}_{{\theta}_{2}}^{-1}(u)}{\displaystyle {\int}_{-\infty}^{{t}_{{\theta}_{2}}^{-1}(v)}\frac{\Gamma (({\theta}_{2}+2)/2)}{\Gamma ({\theta}_{2}/2)\pi {\theta}_{2}\sqrt{1-{\theta}_{1}{}^{2}}}}}{(1+\frac{{x}^{2}-2{\theta}_{1}xy+{y}^{2}}{{\theta}_{2}})}^{({\theta}_{2}+2)/2}dxdy$ | ${\theta}_{1}\in [-1,1]$ and ${\theta}_{2}\in (0,\infty )$ | [48] |

Clayton-copula | $\mathrm{max}{({u}^{-\theta}+{v}^{-\theta}-1,0)}^{-1/\theta}$ | $\theta \in [-1,\infty )$\0 | [49] |

Frank-copula | $-\frac{1}{\theta}\mathrm{ln}[1+\frac{(\mathrm{exp}(-\theta u)-1)(\mathrm{exp}(-\theta v)-1)}{\mathrm{exp}(-\theta )-1}]$ | $\theta \in R\backslash 0$ | [48] |

Gumbel-copula | $\mathrm{exp}\left\{-{[{(-\mathrm{ln}(u))}^{\theta}+{(-\mathrm{ln}(v))}^{\theta}]}^{1/\theta}\right\}$ | $\theta \in [1,\infty )$ | [48] |

Zone | Drought Characteristics | Optimal Distribution | Parameters (Shape, Scale, and Location Parameter) | p | d | Kendall Rank Correlation Coefficient τ | Spearman Rank Correlation Coefficient ρ_{s} |
---|---|---|---|---|---|---|---|

AL | Duration | GEV | $k=0.17,\sigma =1.47,\mu =2.24$ | 0.09 | 0.14 | 0.82^{∗∗} | 0.94^{∗∗} |

Severity | P–III | $\alpha =1.13,\beta =2.62,\gamma =0.37$ | 0.98 | 0.05 | |||

LL | Duration | GEV | $k=0.27,\sigma =1.76,\mu =2.43$ | 0.28 | 0.12 | 0.78^{∗∗} | 0.91^{∗∗} |

Severity | GP | $k=0.10,\alpha =3.21,\epsilon =0.40$ | 0.99 | 0.05 | |||

LH | Duration | P–III | $\alpha =0.66,\beta =4.32,\gamma =1.00$ | 0.15 | 0.13 | 0.81^{∗∗} | 0.93^{∗∗} |

Severity | GEV | $k=0.39,\sigma =1.44,\mu =1.82$ | 0.96 | 0.06 | |||

IF | Duration | GP | $k=-0.10,\alpha =3.42,\epsilon =0.44$ | 0.06 | 0.16 | 0.80^{∗∗} | 0.93^{∗∗} |

Severity | P–III | $\alpha =0.72,\beta =4.00,\gamma =0.52$ | 0.70 | 0.08 | |||

HL | Duration | GEV | $k=0.29,\sigma =1.19,\mu =1.97$ | 0.06 | 0.15 | 0.76^{∗∗} | 0.89^{∗∗} |

Severity | Logn | $\mu =0.58,\sigma =0.98,\gamma =0.25$ | 0.95 | 0.06 | |||

LS | Duration | GP | $k=0.01,\alpha =3.08,\epsilon =0.60$ | 0.06 | 0.16 | 0.83^{∗∗} | 0.94^{∗∗} |

Severity | GP | $k=0.12,\alpha =2.83,\epsilon =0.33$ | 0.90 | 0.06 | |||

SH | Duration | GP | $k=-0.11,\alpha =2.98,\epsilon =0.78$ | 0.07 | 0.15 | 0.77^{∗∗} | 0.90^{∗∗} |

Severity | GP | $k=0.02,\alpha =2.83,\epsilon =0.44$ | 0.95 | 0.06 | |||

BH | Duration | GP | $k=-0.19,\alpha =3.52,\epsilon =0.58$ | 0.09 | 0.14 | 0.84^{∗∗} | 0.95^{∗∗} |

Severity | GP | $k=-0.08,\alpha =3.43,\epsilon =0.20$ | 0.56 | 0.09 | |||

YRB | Duration | P–III | $\alpha =0.69,\beta =5.09,\gamma =1.00$ | 0.13 | 0.15 | 0.80^{∗∗} | 0.92^{∗∗} |

Severity | GP | $k=0.17,\alpha =1.49,\epsilon =0.53$ | 0.96 | 0.06 |

^{∗∗}” denote that the correlation coefficients pass the significance test of α = 0.01.

Zone | Normal-Copula | t-Copula | Clayton-Copula | Frank-Copula | Gumbel-Copula | θ | ||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|

d^{2} | AIC | RMSE | d^{2} | AIC | RMSE | d^{2} | AIC | RMSE | d^{2} | AIC | RMSE | d^{2} | AIC | RMSE | ||

AL | 0.23 | −432.49 | 0.06 | 0.22 | −432.70 | 0.06 | 0.41 | −388.39 | 0.07 | 0.20 | −443.85 | 0.05 | 0.25 | −426.39 | 0.06 | 15.56 |

LL | 0.13 | −392.27 | 0.04 | 0.14 | −391.79 | 0.05 | 0.22 | −362.03 | 0.06 | 0.14 | −388.77 | 0.05 | 0.15 | −384.92 | 0.05 | 0.92 |

LH | 0.78 | −324.27 | 0.10 | 1.84 | −260.02 | 0.16 | 2.24 | −247.89 | 0.18 | 0.63 | −339.41 | 0.09 | 1.18 | −293.81 | 0.13 | 11.08 |

IF | 0.41 | −370.04 | 0.08 | 0.29 | −393.07 | 0.07 | 0.55 | −348.75 | 0.09 | 0.28 | −395.96 | 0.06 | 0.35 | −381.92 | 0.07 | 16.32 |

HL | 0.29 | −431.36 | 0.05 | 0.30 | −428.99 | 0.06 | 0.51 | −390.07 | 0.08 | 0.31 | −429.71 | 0.06 | 0.32 | −427.39 | 0.06 | 0.90 |

LS | 0.24 | −409.18 | 0.06 | 0.23 | −409.63 | 0.06 | 0.33 | −385.90 | 0.07 | 0.22 | −413.36 | 0.05 | 0.26 | −404.16 | 0.06 | 15.94 |

SH | 0.17 | −461.96 | 0.05 | 0.15 | −467.99 | 0.04 | 0.30 | −418.46 | 0.06 | 0.15 | −471.13 | 0.03 | 0.18 | −459.48 | 0.05 | 12.58 |

BH | 0.19 | −446.84 | 0.05 | 0.18 | −447.88 | 0.05 | 0.23 | −431.28 | 0.06 | 0.17 | −454.45 | 0.05 | 0.21 | −438.00 | 0.05 | 18.23 |

YRB | 0.75 | −255.79 | 0.11 | 1.56 | −210.39 | 0.16 | 2.18 | −192.55 | 0.19 | 0.47 | −283.55 | 0.09 | 0.83 | −249.32 | 0.12 | 11.41 |

Drought Characteristics | AL | LL | LH | IF | HL | LS | SH | BH | YRB |
---|---|---|---|---|---|---|---|---|---|

number of droughts | 75 | 64 | 72 | 73 | 78 | 76 | 76 | 75 | 59 |

longest drought duration (month) | 13 | 15 | 18 | 13 | 11 | 16 | 10 | 13 | 16 |

average drought duration (month) | 3.39 | 4.08 | 3.61 | 3.64 | 3.14 | 3.49 | 3.46 | 3.55 | 3.41 |

maximum drought severity | 15.75 | 16.17 | 22.83 | 12.58 | 13.16 | 15.45 | 12.15 | 14.05 | 12.44 |

average drought severity | 3.34 | 3.95 | 3.56 | 3.41 | 3.11 | 3.37 | 3.32 | 3.38 | 2.36 |

© 2019 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Wang, F.; Wang, Z.; Yang, H.; Zhao, Y.; Zhang, Z.; Li, Z.; Hussain, Z.
Copula-Based Drought Analysis Using Standardized Precipitation Evapotranspiration Index: A Case Study in the Yellow River Basin, China. *Water* **2019**, *11*, 1298.
https://doi.org/10.3390/w11061298

**AMA Style**

Wang F, Wang Z, Yang H, Zhao Y, Zhang Z, Li Z, Hussain Z.
Copula-Based Drought Analysis Using Standardized Precipitation Evapotranspiration Index: A Case Study in the Yellow River Basin, China. *Water*. 2019; 11(6):1298.
https://doi.org/10.3390/w11061298

**Chicago/Turabian Style**

Wang, Fei, Zongmin Wang, Haibo Yang, Yong Zhao, Zezhong Zhang, Zhenhong Li, and Zafar Hussain.
2019. "Copula-Based Drought Analysis Using Standardized Precipitation Evapotranspiration Index: A Case Study in the Yellow River Basin, China" *Water* 11, no. 6: 1298.
https://doi.org/10.3390/w11061298