# Uncertainty of Hydrological Drought Characteristics with Copula Functions and Probability Distributions: A Case Study of Weihe River, China

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Materials and Methods

#### 2.1. Weihe River Basin and Data

^{2}, is the largest tributary of the Yellow River in North China (Figure 1) (between 104°–110° E and 34°–38° N). This basin originates from the north of Niaoshu Mountain with an altitude of 3485 m above sea level. The most important topographic feature of the Weihe River Basin is the Loess Plateau in the north, which is the main source of sediments in the river [21]. The annual average temperature ranges between 9.3 °C and 14.4 °C, the annual mean precipitation amounts are in the range of 558–750 mm with a general increasing trend from north to south, and the annual mean runoff amounts are 10.37 billion m

^{3}. The runoff from July to September accounts for about 60–70% of the annual discharge [22]. Agricultural losses due to local drought disasters occupy over 50% of the total losses [23].

#### 2.2. Defining Drought Duration and Severity

^{3}/s) is converted into runoff depth (mm) by:

^{3}/s, $t$ is the monthly time in seconds, and $A$ is the drainage area in km

^{2}.

#### 2.3. Marginal Distribution Model and Copula-Based Models

#### 2.4. Return Period of Droughts

## 3. Results

#### 3.1. Drought Duration and Severity Characteristics

#### 3.2. Marginal Distributions and Copula Functions

#### 3.2.1. Selected Marginal Distributions

#### 3.2.2. Selected Copula Functions

#### 3.3. Return Period of Droughts

## 4. Sensitivity and Uncertainty of the Drought Frequency

#### 4.1. Effects of the Selection of Margin Distributions to the Return Period

#### 4.2. Effects of the Selection of Copula Functions to the Return Period

#### 4.3. Effects of Human Activities on Drought Frequency

## 5. Conclusions

- (1)
- There are more drought events at Huaxian station (lower basin) than at Linjiacun station (upper basin), but there are longer drought durations and greater severity at Linjiacun station.
- (2)
- Based on the AD test, five models (EXP, GAM, LOGN, GEV, WBL) are acceptable for the fit of the drought duration and drought severity at LJC and XY stations. The GP model is not acceptable for the goodness-of-fit of drought duration and drought severity at three stations, the EXP model is rejected for the fit of drought severity at Huaxian station.
- (3)
- Based on ordinary least squares (OLS) and Akaike information criterion (AIC), the Frank copula is the best joint distribution function at Linjiacun and Huaxian stations, while the Clayton copula is the best-fitted model at Huaxian station.
- (4)
- The co-occurrence return period is greater than both the return periods defined by drought duration and drought severity separately, while the joint return period is shorter than both of the return periods. This shows that these two kinds of combination return periods can be regarded as two extreme conditions of the marginal distribution return period. It is possible to estimate the interval of the actual return period according to the co-occurrence return period and the joint return period.
- (5)
- The drought return period is sensitive to the selected marginal distribution and different copula functions. Therefore, it is important to select proper marginal distributions and copula functions, and the sensitivity and uncertainty of hydrological droughts should be paid more attention on the modeling and designing of drought models with consideration to the condition of water resources and the requirement of water management.

## Acknowledgments

## Author Contributions

## Conflicts of Interest

## References

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**Figure 2.**Threshold level method with a variable threshold to define the drought duration and drought severity.

**Figure 4.**The number events of different drought (

**a**) durations and (

**b**) severities at three stations.

**Figure 5.**Marginal distribution modeling of drought duration and drought severity. (

**a**): LJC drought duration; (

**b**): XY drought duration; (

**c**): HX drought duration; (

**d**): LJC drought severity; (

**e**): XY drought severity; (

**f**): HX drought severity.

**Figure 6.**Probability-probability (PP) plot of the joint distributions of duration-severity at three stations. (

**a**): LJC; (

**b**): XY; (

**c**): HX.

**Figure 7.**Co-occurrence return period and joint return period of the best-selected copula. (

**a**): LJC drought duration; (

**b**): XY drought duration; (

**c**): HX drought duration; (

**d**): LJC drought severity; (

**e**): XY drought severity; (

**f**): HX drought severity.

**Figure 8.**The return period of drought duration and severity based on different marginal distributions at three stations. (

**a**): LJC drought duration; (

**b**): XY drought duration; (

**c**): HX drought duration; (

**d**): LJC drought severity; (

**e**): XY drought severity; (

**f**): HX drought severity.

**Figure 9.**Sensitivity of the co-occurrence return period and joint return period of three drought events at Huaxian station and the selection of the univariate distribution and the copulas. (

**a**): co-occurrence return period of drought event 1; (

**b**): co-occurrence return period of drought event 2; (

**c**): co-occurrence return period of drought event 3; (

**d**): joint return period of drought event 1; (

**e**): joint return period of drought event 2; (

**f**): joint return period of drought event 3.

Distribution | CDF | Parameters |
---|---|---|

Exponential (EXP) | $F\left(x\right)=1-{e}^{-x/\mu}$ | $\mu $: scale |

Gamma (GAM) | $F\left(x\right)=\frac{{\beta}^{-\alpha}}{\mathrm{\Gamma}\left(\alpha \right)}{\displaystyle {\int}_{0}^{x}{t}^{\alpha -1}{e}^{-t/\beta}dt}$ | $\alpha $: shape |

$\beta $: scale | ||

Log-normal (LOGN) | $F\left(x\right)=\frac{1}{\sigma \sqrt{2\pi}}{\displaystyle {\int}_{0}^{x}\frac{e-{\left(\mathrm{ln}\left(t\right)-\mu \right)}^{2}/2{\sigma}^{2}}{t}dt}$ | $\mu $: mean |

$\sigma $: standard deviation | ||

Generalized Pareto (GP) | $F\left(x\right)=1-\mathrm{exp}\left({\kappa}^{-1}\mathrm{ln}\left(1-\frac{\kappa \left(x-\xi \right)}{\alpha}\right)\right)$ | $\kappa $: shape |

$\alpha $: scale | ||

$\xi $: location | ||

Generalized extreme value (GEV) | $F\left(x\right)=\mathrm{exp}\left(-\mathrm{exp}\left({\kappa}^{-1}\mathrm{ln}\left(1-\frac{\kappa \left(x-\xi \right)}{\alpha}\right)\right)\right)$ | $\kappa $: shape |

$\alpha $: scale | ||

$\xi $: location | ||

Weibull (WBL) | $F\left(x\right)=1-{e}^{-{\left(x/a\right)}^{b}}{\mathrm{{\rm I}}}_{\left(0,\infty \right)}\left(x\right)$ | $a$: scale |

$b$: shape |

Copulas | CDF | Parameters |
---|---|---|

Clayton | $C\left(u,v\right)={\left({u}^{-\theta}+{v}^{-\theta}-1\right)}^{-\frac{1}{\theta}}$ | $\theta \ge 0$ |

Frank | $C\left(u,v\right)=-\frac{1}{\theta}\mathrm{ln}\left[1+\left(\frac{\left({e}^{-\theta u}-1\right)\left({e}^{-\theta v}-1\right)}{{e}^{-\theta}-1}\right)\right]$ | $\theta \ne 0$ |

Gumel | $C\left(u,v\right)=\mathrm{exp}\left\{-{\left[{\left(-\mathrm{ln}u\right)}^{\theta}+{\left(-\mathrm{ln}v\right)}^{\theta}\right]}^{\frac{1}{2}}\right\}$ | $\theta \ge 1$ |

Mean | Std.dev. | Max | Min | Cv | SK | ||
---|---|---|---|---|---|---|---|

Drought duration (month) | LJC | 3.391 | 3.363 | 17 | 1 | 0.992 | 2.172 |

XY | 3.059 | 3.343 | 17 | 1 | 1.093 | 2.316 | |

HX | 2.382 | 2.273 | 11 | 1 | 0.954 | 1.879 | |

Drought Severity (mm) | LJC | 3.009 | 5.081 | 30.436 | 0.019 | 1.689 | 3.964 |

XY | 3.726 | 5.387 | 23.946 | 0.002 | 1.446 | 2.513 | |

HX | 2.175 | 2.959 | 15.169 | 0.001 | 1.360 | 2.074 |

Correlation Coefficient | LJC ($\mathit{D}~\mathit{S}$) | XY ($\mathit{D}~\mathit{S}$) | HX ($\mathit{D}~\mathit{S}$) |
---|---|---|---|

$\rho $ | 0.883 | 0.916 | 0.885 |

$\tau $ | 0.705 | 0.644 | 0.639 |

${\rho}_{s}$ | 0.827 | 0.803 | 0.776 |

Margin | Parameter | LJC | XY | HX | |
---|---|---|---|---|---|

Duration | EXP | $\mu $ | 3.391 | 3.059 | 2.382 |

GAM | $\alpha $ | 1.612 | 1.399 | 1.743 | |

$\beta $ | 2.103 | 2.186 | 1.366 | ||

LOGN | $\mu $ | 0.880 | 0.720 | 0.555 | |

$\sigma $ | 0.789 | 0.838 | 0.731 | ||

GP | $\kappa $ | −0.017 | 0.072 | −0.056 | |

$\alpha $ | 3.449 | 2.838 | 2.515 | ||

GEV | $\kappa $ | 5.062 | 3.783 | 4.958 | |

$\alpha $ | 0.330 | 0.011 | 0.159 | ||

$\xi $ | 1.065 | 1.003 | 1.032 | ||

WBL | $a$ | 3.637 | 3.197 | 2.580 | |

$b$ | 1.196 | 1.105 | 1.234 | ||

Severity | EXP | $\mu $ | 35.600 | 67.316 | 89.357 |

GAM | $\alpha $ | 0.691 | 0.594 | 0.287 | |

$\beta $ | 51.513 | 113.247 | 311.259 | ||

LOGN | $\mu $ | 2.696 | 3.169 | 2.073 | |

$\sigma $ | 1.510 | 1.805 | 7.138 | ||

GP | $\kappa $ | 0.381 | 0.529 | 0.747 | |

$\alpha $ | 21.939 | 36.178 | 38.080 | ||

GEV | $\kappa $ | 0.706 | 1.042 | 1.239 | |

$\alpha $ | 12.958 | 21.278 | 25.334 | ||

$\xi $ | 11.151 | 15.605 | 16.587 | ||

WBL | $a$ | 29.791 | 53.411 | 51.179 | |

$b$ | 0.768 | 0.706 | 0.417 |

LJC | XY | HX | |||||
---|---|---|---|---|---|---|---|

AIC | AD | AIC | AD | AIC | AD | ||

p-Value | p-Value | p-Value | |||||

Duration | EXP | −76.856 | 0.243 | −76.856 | 0.243 | −39.335 | 0.393 |

GAM | −98.198 | 0.945 | −98.198 | 0.945 | −50.448 | 0.962 | |

LOGN | −88.137 | 0.635 | −88.137 | 0.635 | −46.052 | 0.857 | |

GP | −90.31 | 0.000 | −90.31 | 0.000 | −52.657 | 0.000 | |

GEV | −93.289 | 0.873 | −93.289 | 0.873 | −53.595 | 0.982 | |

WBL | −100.749 | 0.975 | −100.749 | 0.975 | −59.937 | 0.986 | |

Severity | EXP | −207.934 | 0.0295 | −230.471 | 0.021 | −307.066 | 0.0068 |

GAM | −263.939 | 0.3543 | −291.92 | 0.3002 | −425.32 | 0.479 | |

LOGN | −232.344 | 0.1144 | −256.857 | 0.0866 | −339.99 | 0.0348 | |

GP | −427.811 | 0 | −475.892 | 0 | −676.391 | 0 | |

GEV | −282.934 | 0.6941 | −312.324 | 0.4499 | −386.761 | 0.174 | |

WBL | −281.932 | 0.5106 | −315.351 | 0.6396 | −415.183 | 0.2938 |

Copula | Parameters and Goodness of Fit Index | LJC | XY | HX |
---|---|---|---|---|

Clayton | $\theta $ | 2.098 | 7.132 | 0.050 |

OLS | 0.071 | 0.057 | 0.181 | |

AIC | −86.783 | −94.304 | −34.542 | |

Frank | $\theta $ | 9.829 | 40.229 | 28.017 |

OLS | 0.054 | 0.047 | 0.077 | |

AIC | −96.422 | −100.643 | −53.330 | |

Gumbel | $\theta $ | 2.887 | 4.299 | 2.687 |

OLS | 0.050 | 0.063 | 0.104 | |

AIC | −98.522 | −90.740 | −46.728 | |

Best Function | Gumbel | Frank | Frank |

Drought Duration (month) | Drought Severity (mm) | ||||||
---|---|---|---|---|---|---|---|

LJC | XY | HX | LJC | XY | HX | ||

Return period (year) | 2 | 7.452 | 8.152 | 6.551 | 1.156 | 1.690 | 1.041 |

5 | 12.500 | 12.962 | 9.272 | 3.449 | 5.702 | 4.357 | |

10 | 15.370 | 15.759 | 10.903 | 6.327 | 9.520 | 7.711 | |

20 | 17.841 | 18.184 | 12.329 | 10.954 | 13.859 | 11.483 | |

50 | 20.718 | 21.021 | 14.001 | 21.752 | 20.269 | 16.877 | |

100 | 22.685 | 22.966 | 15.149 | 36.024 | 25.564 | 21.168 |

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**MDPI and ACS Style**

Zhao, P.; Lü, H.; Fu, G.; Zhu, Y.; Su, J.; Wang, J.
Uncertainty of Hydrological Drought Characteristics with Copula Functions and Probability Distributions: A Case Study of Weihe River, China. *Water* **2017**, *9*, 334.
https://doi.org/10.3390/w9050334

**AMA Style**

Zhao P, Lü H, Fu G, Zhu Y, Su J, Wang J.
Uncertainty of Hydrological Drought Characteristics with Copula Functions and Probability Distributions: A Case Study of Weihe River, China. *Water*. 2017; 9(5):334.
https://doi.org/10.3390/w9050334

**Chicago/Turabian Style**

Zhao, Panpan, Haishen Lü, Guobin Fu, Yonghua Zhu, Jianbin Su, and Jianqun Wang.
2017. "Uncertainty of Hydrological Drought Characteristics with Copula Functions and Probability Distributions: A Case Study of Weihe River, China" *Water* 9, no. 5: 334.
https://doi.org/10.3390/w9050334