# Analysis of Characteristics of Dry–Wet Events Abrupt Alternation in Northern Shaanxi, China

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## Abstract

**:**

## 1. Introduction

## 2. Study Area and Data

## 3. Methodology

#### 3.1. Daily Dry–Wet Events Abrupt Alternation Index DWAAI

#### 3.2. Identification of Characteristic Variables of Dry–Wet Events Abrupt Alternation

#### 3.3. Construct the Marginal Distribution Function of Characteristic Variables

#### 3.4. Determination of Copula Function

_{1}, X

_{2},…, X

_{N}with an edge distribution of FX

_{1}(x

_{1}), FX

_{2}(x

_{2}),…, FX

_{N}(x

_{N}). The two-dimensional copula function can combine DWu and DWa, and the formula is as follows:

#### 3.5. Calculation of Return Periods

## 4. Results

#### 4.1. DWAAI Index Applicability Verification

#### 4.2. Spatial Distribution Characteristics of Dry–Wet Events Abrupt Alternation

#### 4.3. Determination of Copula

#### 4.3.1. Correlation Analysis of Dry–Wet Events Abrupt Alternation Characteristic Variables

#### 4.3.2. Determine the Appropriate Marginal Distribution Function

#### 4.3.3. Determine the Appropriate Copula

#### 4.4. Joint Probability Distribution

#### 4.5. Joint Return Period

## 5. Discussion and Conclusions

## Author Contributions

## Funding

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 2.**Changes of rainfall and DWAAI before and after four typical dry–wet events abrupt alternation at Yulin Station.

**Figure 3.**The frequency distribution of dry–wet events abrupt alternation of different grades in northern Shaanxi during 1960–2019: (

**a**) light; (

**b**) moderate; (

**c**) severe.

**Figure 4.**The marginal distribution fitting and quantile–quantile(Q–Q) plots of 10 stations dry–wet events abrupt alternation variables from 1960 to 2019.

**Figure 5.**The joint probability Jingbian (

**a**) and Baota (

**b**) dry–wet events abrupt alternation variables from 1960 to 2019.

**Figure 6.**Joint return period contour line of dry–wet events abrupt alternation in Jingbian (

**a**) and Baota (

**b**).

DWAAI | DWAA Level |
---|---|

>0~10 | None |

>10~16 | Light |

>16~24 | Moderate |

>24 | Severe |

Copula Type | Copula Formula | Parameter Range |
---|---|---|

Joe | $1-{\left[{\left(1-u\right)}^{\theta}+{\left(1-v\right)}^{\theta}-{\left(1-u\right)}^{\theta}{\left(1-v\right)}^{\theta}\right]}^{1/\theta}$ | $\theta \in \left[1,\infty \right]$ |

Farlie-Gumbel-Morgenstern (FGM) | $uv\left[1+\theta \left(1-u\right)\left(1-v\right)\right]$ | $\theta \in \left[-1,1\right]$ |

Burr | $u+v-1+{\left[{\left(1-u\right)}^{-\frac{1}{\theta}}+{\left(1-v\right)}^{-\frac{1}{\theta}}-1\right]}^{-\theta}$ | $\theta \in \left(0,\infty \right)$ |

Marshall-Olkin | $\mathrm{min}\left[{u}^{\left(1-{\theta}_{1}\right)}v,u{v}^{\left(1-{\theta}_{2}\right)}\right]$ | ${\theta}_{1},{\theta}_{2}\in \left[0,\infty \right)$ |

Fischer-Hinzmann | ${\left\{{\theta}_{1}{\left[\mathrm{min}\left(u,v\right)\right]}^{{\theta}_{2}}+\left(1-{\theta}_{1}\right){\left[uv\right]}^{{\theta}_{2}}\right\}}^{1/{\theta}_{2}}$ | ${\theta}_{1}\in \left[0,1\right],{\theta}_{2}\in R$ |

Roch-Alegre | $\mathrm{exp}\left\{1-{\left[{\left({\left({\left(1-\mathrm{ln}\left(u\right)\right)}^{{\theta}_{1}}-1\right)}^{{\theta}_{2}}+{\left({\left(1-\mathrm{ln}\left(v\right)\right)}^{{\theta}_{1}}-1\right)}^{{\theta}_{2}}\right)}^{\frac{1}{{\theta}_{2}}}+1\right]}^{\frac{1}{{\theta}_{1}}}\right\}$ | ${\theta}_{1}\in \left(0,\infty \right),{\theta}_{2}\in \left(1,\infty \right)$ |

Tawn | $\mathrm{exp}\left\{\mathrm{ln}\left({u}^{\left(1-{\theta}_{1}\right)}\right)+\mathrm{ln}\left({v}^{\left(1-{\theta}_{2}\right)}\right)-{\left[{\left(-{\theta}_{1}\mathrm{ln}\left(u\right)\right)}^{{\theta}_{3}}+{\left(-{\theta}_{2}\mathrm{ln}\left(v\right)\right)}^{{\theta}_{3}}\right]}^{\frac{1}{{\theta}_{3}}}\right\}$ | ${\theta}_{1},{\theta}_{2}\in \left[0,1\right],{\theta}_{3}\in \left[1,\infty \right)$ |

Station | Pearson | p Value |
---|---|---|

Yulin | 0.4858 | 0.0000 |

Shenmu | 0.5310 | 0.0000 |

Dingbian | 0.2501 | 0.0226 |

Jingbian | 0.4803 | 0.0000 |

Wuqi | 0.2751 | 0.0043 |

Hengshan | 0.3570 | 0.0028 |

Suide | 0.3509 | 0.0052 |

Baota | 0.2696 | 0.0202 |

Yanchang | 0.4046 | 0.0005 |

Luochuan | 0.3280 | 0.0034 |

**Table 4.**The marginal distribution function and parameter values corresponding to dry–wet events abrupt alternation variables of meteorological stations in northern Shaanxi.

Station | Variable | Function | Parameter | Value |
---|---|---|---|---|

Yulin | DWu | Generalized Pareto | k | −1.0184 |

sigma | 5.8354 | |||

theta | 1.7711 | |||

DWa | Generalized Extreme Value | k | 0.3638 | |

sigma | 2.8110 | |||

mu | 8.3674 | |||

Shenmu | DWu | Inverse Gaussian | mu | 4.7247 |

lambda | 36.1879 | |||

DWa | Generalized Extreme Value | k | 0.3392 | |

sigma | 2.8688 | |||

mu | 8.5185 | |||

Dingbian | DWu | Gamma | a | 9.5575 |

b | 0.4582 | |||

DWa | Loglogistic | mu | 2.0924 | |

sigma | 0.2440 | |||

Jingbian | DWu | Gamma | a | 10.0449 |

b | 0.4859 | |||

DWa | Generalized Extreme Value | k | 0.3192 | |

sigma | 2.9900 | |||

mu | 8.3685 | |||

Wuqi | DWu | Inverse Gussian | mu | 4.3756 |

lambda | 39.5976 | |||

DWa | Inverse Gussian | mu | 9.6819 | |

lambda | 97.3290 | |||

Hengshan | DWu | Loglogistic | mu | 1.5660 |

sigma | 0.1611 | |||

DWa | Generalized Pareto | k | −0.2260 | |

sigma | 6.8166 | |||

theta | 5.2102 | |||

Suide | DWu | Gamma | a | 8.2977 |

b | 0.5568 | |||

DWa | Generalized Pareto | k | −0.4034 | |

sigma | 8.8193 | |||

theta | 4.6478 | |||

Baota | DWu | Gamma | a | 11.1731 |

b | 0.3811 | |||

DWa | Generalized Pareto | k | 0.0697 | |

sigma | 3.7401 | |||

theta | 6.2085 | |||

Yanchang | DWu | Nakagami | mu | 2.3438 |

omega | 21.1845 | |||

DWa | Generalized Extreme Value | k | 0.2636 | |

sigma | 2.7252 | |||

mu | 8.3915 | |||

Luochuan | DWu | Loglogistic | mu | 1.4604 |

sigma | 0.1986 | |||

DWa | Inverse Gussian | mu | 10.8882 | |

lambda | 80.5783 |

Station | Copula | RMSE | NSE | SD |
---|---|---|---|---|

Yulin | Fischer-Hinzmann | 0.3212 | 0.9820 | 4.7266 |

Roch-Alegre | 0.3515 | 0.9784 | ||

Joe | 0.3677 | 0.9764 | ||

Shenmu | Fischer-Hinzmann | 0.2658 | 0.9858 | 4.5622 |

Joe | 0.2920 | 0.9829 | ||

Burr | 0.2966 | 0.9824 | ||

Dingbian | Roch-Alegre | 0.2741 | 0.9879 | 3.3515 |

Joe | 0.3042 | 0.9851 | ||

Burr | 0.3071 | 0.9848 | ||

Jingbian | Joe | 0.2911 | 0.9863 | 4.3874 |

Fischer-Hinzmann | 0.2833 | 0.9871 | ||

Roch-Alegre | 0.2843 | 0.9870 | ||

Wuqi | Burr | 0.2123 | 0.9913 | 2.4379 |

Joe | 0.2145 | 0.9911 | ||

Roch-Alegre | 0.2139 | 0.9912 | ||

Hengshan | Joe | 0.2356 | 0.9884 | 3.4591 |

Burr | 0.2372 | 0.9883 | ||

Fischer-Hinzmann | 0.2316 | 0.9888 | ||

Suide | Roch-Alegre | 0.1879 | 0.9931 | 3.3441 |

Fischer-Hinzmann | 0.2033 | 0.9919 | ||

Joe | 0.2327 | 0.9894 | ||

Baota | Tawn | 0.1906 | 0.9927 | 3.2504 |

Marshal-Olkin | 0.1969 | 0.9922 | ||

FGM | 0.2208 | 0.9902 | ||

Yanchang | Fischer-Hinzmann | 0.3342 | 0.9816 | 4.0443 |

Joe | 0.3565 | 0.9790 | ||

Burr | 0.3605 | 0.9786 | ||

Luochuan | Roch-Alegre | 0.2293 | 0.9907 | 3.2095 |

Joe | 0.2511 | 0.9889 | ||

Burr | 0.2552 | 0.9885 |

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**MDPI and ACS Style**

Wang, J.; Rong, G.; Li, K.; Zhang, J. Analysis of Characteristics of Dry–Wet Events Abrupt Alternation in Northern Shaanxi, China. *Water* **2021**, *13*, 2384.
https://doi.org/10.3390/w13172384

**AMA Style**

Wang J, Rong G, Li K, Zhang J. Analysis of Characteristics of Dry–Wet Events Abrupt Alternation in Northern Shaanxi, China. *Water*. 2021; 13(17):2384.
https://doi.org/10.3390/w13172384

**Chicago/Turabian Style**

Wang, Junhui, Guangzhi Rong, Kaiwei Li, and Jiquan Zhang. 2021. "Analysis of Characteristics of Dry–Wet Events Abrupt Alternation in Northern Shaanxi, China" *Water* 13, no. 17: 2384.
https://doi.org/10.3390/w13172384