# Copula-Based Drought Monitoring and Assessment According to Zonal and Meridional Temperature Gradients

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Materials and Methods

#### 2.1. Study Area and Data

#### 2.2. Standardized Precipitation Evapotranspiration Index

_{i}= P

_{i}− PET

_{i}

_{i}is the precipitation and PET

_{i}is the potential evaporation.

_{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. Definition of Drought Characteristics

#### 2.4. Trend Analysis

_{1}is a bilateral test. For all the distributions of i, j ≤ n and i ≠ j and the distributions of X

_{i}and X

_{j}differ from each other. The MK correlation coefficient between two variables is known as the MK statistic S. It is calculated as follows:

_{C}converges to the standard normal distribution and is calculated as shown in Equation (10) below:

_{0}will be rejected when |Z

_{C}|≥ Z

_{1–α/2}. In this case, there is a significant upward or downward trend in the current time series data α confidence level. If Z

_{C}> 0, it shows an upward trend. If Z

_{C}< 0, it shows a downward trend. ±Z

_{1−α/2}is the (1 − α/2) quantile of the standard normal distribution, and α is the test confidence level of the test. When |Z

_{C}| ≥ 1.64, 1.96, and 2.58, the α confidence values are 0.1, 0.05, and 0.01, respectively. Here, the MK test was used to analyze the drought characteristic across China, and the tendency was determined from the results of the p and z values. The trend is not significant when p > 0.05. However, the difference between the result and the zero hypothesis is evident when p < 0.05. Thus, the p-value can indicate whether the SPEI of the area has a significant change trend. The direction of trend is then determined based on the positive and negative S values.

#### 2.5. Meridional and Zonal Temperature Gradients

#### 2.6. Copula Functions

^{−1}is the inverse function of the standard normal distribution function, φ

_{ρ}is the standard multivariate normal distribution, ρ is the correlation coefficient matrix between variables and is a symmetric positive definite matrix. I is the identity matrix, and $\varsigma $n = φ

^{−1}(u

_{n}).

_{1},···,u

_{n},···,u

_{N},ρ) = T

_{ρ,v}(t

_{v}

^{−1}(u

_{1}),···,t

_{v}

^{−1}(u

_{n}),···,t

_{v}

^{−1}(u

_{N}))

_{ρ,v}is the t standard multivariate distribution, v is the degree of freedom, ρ is the correlation coefficient matrix between variables and is a symmetric positive definite matrix, | ρ| is the absolute value of the determinant of ρ, t

_{v}

^{−1}is the inverse function of the one-variable t distribution function with degrees of freedom, and $\varsigma $

_{n}= φ

^{−1}(u

_{n}).

## 3. Results

#### 3.1. Changes in Large-Scale Temperature

#### 3.2. Composite Anomalies of Drought Characteristics Associated with LOC/MTG

#### 3.3. Changes in Drought Duration and Severity in China

#### 3.4. Joint Distribution of Drought Duration and Severity

## 4. Discussion and Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## References

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**Figure 1.**Study framework for copula-based drought monitoring and assessment using zonal and meridional circulations.

**Figure 4.**Time series of global temperature gradients. (

**a**) LOC index. (

**b**) MTG index during the period from 1979 to 2019. Green dotted line represents the linear trend of the indices. Gray dashed line indicates the +1 σ and −1 σ. For each gradient, red dots indicate significant positive events while blue dots indicate significant negative events.

**Figure 6.**Long-term trend in drought components in China between 1979 and 2019. (

**a**) Drought duration. (

**b**) Drought severity.

**Figure 7.**Relationship between drought duration and severity in the (

**a**) southwest, (

**b**) northeast, and (

**c**) southwest.

**Figure 8.**Copula distributions of drought duration and severity in positive (negative) LOC/MTG years.

Category | SPEI Index |
---|---|

Normal | (−0.5, +∞) |

Slight drought | (−1.0, −0.5) |

Moderate drought | (−1.5, −1.0) |

Severe drought | (−2.0, −1.5) |

Extreme drought | (−∞, −2.0] |

Area | Category of Copula Function | Parameter Estimates | Goodness of Fit Evaluation Standard | |
---|---|---|---|---|

AIC | BIC | |||

Northwest | Gaussian copula | 0.8271 | −601.2871 | −596.9415 |

Student’s copula | 0.8271 | −599.1858 | −590.4946 | |

GH copula | 2.345 | −546.6435 | −542.2979 | |

Frank copula | 8.183 | −573.2539 | −568.9083 | |

Galambos copula | 1.645 | −554.9031 | −550.5575 | |

HuslerReiss copula | 2.252 | −565.7303 | −561.3846 | |

Northeast | Gaussian copula | 0.8319 | −820.555 | −815.905 |

Student’s copula | 0.8337 | −822.6324 | −813.3318 | |

GH copula | 2.701 | −907.3378 | −902.6875 | |

Frank copula | 8.604 | −791.2964 | −786.6461 | |

Galambos copula | 2.003 | −909.7965 | −905.1462 | |

HuslerReiss copula | 2.635 | −910.1736 | −905.5233 | |

Southwest | Gaussian copula | 0.8291 | −775.5213 | −773.5139 |

Student’s copula | 0.8291 | −773.5139 | −764.3061 | |

GH copula | 2.436 | −737.7691 | −733.1652 | |

Frank copula | 8.816 | −784.6019 | −779.998 | |

Galambos copula | 1.727 | −744.8849 | −740.2809 | |

HuslerReiss copula | 2.335 | −757.1334 | −752.5294 |

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**MDPI and ACS Style**

Otkur, A.; Wu, D.; Zheng, Y.; Kim, J.-S.; Lee, J.-H.
Copula-Based Drought Monitoring and Assessment According to Zonal and Meridional Temperature Gradients. *Atmosphere* **2021**, *12*, 1066.
https://doi.org/10.3390/atmos12081066

**AMA Style**

Otkur A, Wu D, Zheng Y, Kim J-S, Lee J-H.
Copula-Based Drought Monitoring and Assessment According to Zonal and Meridional Temperature Gradients. *Atmosphere*. 2021; 12(8):1066.
https://doi.org/10.3390/atmos12081066

**Chicago/Turabian Style**

Otkur, Abudureymjang, Dian Wu, Yin Zheng, Jong-Suk Kim, and Joo-Heon Lee.
2021. "Copula-Based Drought Monitoring and Assessment According to Zonal and Meridional Temperature Gradients" *Atmosphere* 12, no. 8: 1066.
https://doi.org/10.3390/atmos12081066