# Dynamic Evolution and Copula-Based Multivariable Frequency Analysis of Meteorological Drought Considering the Spatiotemporal Variability in Northwestern China

^{1}

^{2}

^{3}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Materials and Methods

#### 2.1. Study Area

^{2}, accounting for roughly 19.46% of the total area in China. The elevation ranges from 184 to 6672 m, exhibiting significant spatial variation with a decrease from west to east. The study area has a diversified topography of mountains, plateaus, and basins, presenting as the Qinghai-Tibet Plateau, Gobi desert, desert steppe, and Loess Plateau from west to east [23].

#### 2.2. Dataset

#### 2.3. Methods

#### 2.3.1. Standardized Precipitation Evapotranspiration Index

_{n}is the net amount of radiation reaching the ground, MJ/m

^{2}d; G is the soil heat flux density, MJ/m

^{2}d; γ is the hygrometer constant; U

_{2}is the wind speed 2 m above ground level, m/s; e

_{s}is the saturated water vapor pressure of air, kPa; and e

_{a}is the actual vapor pressure, kPa.

_{n}

^{k}series. The probability distribution function F(x) is

_{0}= 2.5155, C

_{1}= 0.8028, C

_{2}= 0.0103, d

_{1}= 1.4327, d

_{2}= 0.1892, and d

_{3}= 0.0013.

#### 2.3.2. Modified Mann-Kendall Test (MMK)

#### 2.3.3. R/S Analysis

#### 2.3.4. Identification of Drought Events by a 3D Clustering Method

#### 2.3.5. Copula-Based Multivariable Frequency Analysis of Drought Events

#### Marginal Distribution of Drought Variables

#### Determination of the Optimal Copula Function

#### Copula-Based Multivariable Probability Calculation

## 3. Results

#### 3.1. Spatial-Temporal Variation of Drought at Multiple Time Scales

#### 3.1.1. Temporal Evolution Characteristics of Drought at Different Time Scales

#### 3.1.2. Spatial-Temporal Characteristics of Seasonal and Annual Drought Variation Trends

#### Temporal Characteristics of Drought Variation Trend

^{2}= 0.21).

#### Temporal Characteristics of Drought Variation Trend

#### Spatial Characteristics of Drought Variation Trend

#### 3.1.3. Spatial Variation Characteristics of Drought Intensity and Frequency

#### Spatial Characteristics of Drought Intensity

#### Spatial Characteristics of Drought Frequency

#### 3.2. Dynamic Evolution of Typical Drought Event

#### Temporal Evolution Characteristics of Drought at Different Time Scales

^{6}month·km

^{2}. The drought event initially struck an approximate area of 0.31 × 10

^{6}km

^{2}, accounting for 18.10% of the total study area. The drought center was situated in the northwest part of Haixi Mongolian-Tibetan Autonomous Prefecture, Qinghai. In January 2018, the drought area rapidly expanded to 0.63 × 10

^{6}km

^{2}, accounting for 35.90% of the total area. The drought center shifted southeastwards to the central part of Haixi Mongolian-Tibetan Autonomous Prefecture with a migration distance of 191.73 km. In February of the same year, the drought area and severity declined to 0.34 × 10

^{6}km

^{2}and 0.54 × 10

^{6}month·km

^{2}, respectively. The drought center migrated southwestwards to the northern part of the Yushu Tibetan Autonomous Prefecture at an average velocity of 255.36 km·month

^{−1}. In March, the drought magnitude continued to weaken, and the drought area reached its minimum of 0.12 × 10

^{6}km

^{2}, accounting for only 6.90% of the total study area. It was primarily concentrated in the western part of Jiuquan City, Gansu. Subsequently, the drought magnitude showed an upward trend, with an area of 0.26 × 10

^{6}km

^{2}and an intensity of 1.75, signifying the most severe level. It expanded further into central Gansu and the western part of Alxa League. In May, the drought area reduced to 0.15 × 10

^{6}km

^{2}, indicating a weakened drought situation and the termination of the drought. In summary, this drought event was mainly concentrated in the western part of the study area. The migration path of the drought center was characterized by a north-south oscillation pattern, and the drought had generally experienced five processes: occurrence, aggravation, mitigation, re-aggravation, and termination.

#### 3.3. Multivariable Frequency Analysis of Drought

#### 3.3.1. Correlation Analysis of Drought Variables

^{2}> 0.6) among the drought feature variables. Moreover, each set of drought variables had high mutual dependence and a strong correlation, with a statistically significant p-value of 0.01. Therefore, it is reasonable to use the Copula function for joint distribution modeling for the frequency analysis.

#### 3.3.2. Selection of Marginal Distributions for Drought Variables

#### 3.3.3. Selection of Optimal Copula Functions

#### 3.3.4. Joint Occurrence Probability of Drought

^{6}month·km

^{2}, respectively, the joint occurrence probability in the “or” situation is 49.4%, while it is 30.2% in the “and” situation. In both situations, the joint occurrence probability increases as the drought variables decrease. For example, when the drought duration and severity exceed 6.65 months and 2.73 × 10

^{6}month·km

^{2}, respectively, in the “or” situation, the joint occurrence probability is 20.1%; when they exceed 3.30 months and 1.97 × 10

^{6}month·km

^{2}, respectively, the probability is 49.8%.

#### 3.3.5. Conditional Probability of Drought

^{6}month·km

^{2}, the probability of drought is 93.4%. It means that there is a high likelihood of meteorological drought under such conditions. Furthermore, if the drought severity is larger than 1, 3, 5, 7, or 9 × 10

^{6}month·km

^{2}and the drought duration is greater than 5 months, the probability of meteorological drought will be 46.2%, 73.9%, 79.8%, 81.7%, and 82.6%, respectively. The distribution characteristics of probability show a dense distribution at both ends and a sparse distribution in the middle, indicating the difference in probability results at different conditional factor levels. For example, when the drought severity is high (Figure 16a), the probability results for different conditional factors have little difference. Additionally, an increase in the conditional factor (drought duration) will significantly raise the probability of meteorological drought events with moderate intensity, similar to the results of Yusof et al. [45].

^{6}month·km

^{2}and a drought duration longer than 5 months, the conditional probability of meteorological drought is 79.8%. Meanwhile, given a drought area greater than 1 × 10

^{6}km

^{2}, this figure increases to 99.8%. It can be concluded that ignoring any drought variable may significantly underestimate the occurrence probability of severe meteorological droughts, as well as the severity of drought events [17].

## 4. Discussion

## 5. Conclusions

- (1)
- Overall, the SPEI showed an upward trend in the plateau climate zone and the westerly climate zone, with rates of 0.188/10a and 0.046/10a, indicating a mitigation of drought. Conversely, the SPEI demonstrated a descending trend (−0.038/10a) in the southeast climate zone, suggesting an intensified drought situation.
- (2)
- The variation trend of drought in different seasons was mainly downward in the west part and upward in the east part of Northwestern China. From spring to winter, the humidification trend expanded towards the east, and the areas with a significant humidification trend gradually shifted towards the east.
- (3)
- The spatial distribution of drought intensity and frequency in different seasons exhibited opposite characteristics. For example, the southeastern part of the study area experienced a high drought intensity but a low drought frequency in spring. This indicated that the likelihood of high-intensity meteorological drought events occurring in the same area was relatively low, whereas low-intensity drought events were frequent.
- (4)
- The most severe drought event occurred from January 1961 to October 1962 and experienced five processes: occurrence, aggravation, mitigation, re-aggravation, and termination. The migration path was characterized by north-south oscillation.
- (5)
- The joint occurrence probabilities were consistently higher in the “or” situation than in the “and” situation for the same combination of drought variables. Furthermore, the conditional probability of drought variables, given specific conditional factors, declined as the values of these factors increased. Notably, as the conditional factors increased, there was a noticeable reduction in the drought occurrence probability for drought variables with lower values.

## Author Contributions

## Funding

## Data Availability Statement

## Conflicts of Interest

## References

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**Figure 2.**Schematic diagrams of (

**a**) spatial identification and (

**b**) temporal connection of drought patches.

**Figure 3.**Temporal evolution characteristics of SPEI series at different timescales. (

**a**) Northwest region (

**b**) Plateau climate zone (

**c**) Westerlies climate zone (

**d**) Southeast climate zone.

**Figure 4.**Temporal evolution characteristics of SPEI series at different timescales in northwestern regions between 1960 and 2018: (

**a**) northwestern China; (

**b**) Pplateau climate zone; (

**c**) westerlies climate zone; (

**d**) southeast climate zone.

**Figure 5.**Temporal change characteristics of seasonal SPEI series in northwestern regions during 1960–2018: (

**a**) spring; (

**b**) summer; (

**c**) autumn; (

**d**) winter.

**Figure 6.**Variation trend of monthly SPEI series in northwestern regions between 1960 and 2018 (“*” and “**” denote significant at 0.1 and 0.05 levels, respectively).

**Figure 7.**Spatial distributions of change trend of seasonal SPEI series in northwestern regions between 1960 and 2018: (

**a**) spring; (

**b**) summer; (

**c**) autumn; (

**d**) winter.

**Figure 8.**Spatial distributions of seasonal drought intensity in northwestern regions between 1960 and 2018: (

**a**) spring; (

**b**) summer; (

**c**) autumn; (

**d**) winter.

**Figure 9.**Spatial distribution of seasonal drought frequency in northwestern regions between 1960 and 2018: (

**a**) spring; (

**b**) summer; (

**c**) autumn; (

**d**) winter.

**Figure 10.**Spatiotemporal dynamic evolution of the No.338 meteorological drought (December 2017–May 2018).

**Figure 11.**Migration path of drought center of the No.338 meteorological drought (December 2017–May 2018). Note: the color bar denotes the cumulative value of SPEI3.

**Figure 13.**Probability-probability (PP) plot of meteorological drought duration, severity, and area.

**Figure 14.**Probability-probability (PP) plot of the optimal Copula function: (

**a**) duration-severity; (

**b**) duration-area; (

**c**) severity-area; (

**d**) duration-severity-area.

**Figure 15.**The bivariate and trivariate joint occurrence probability of drought duration (month), severity (10

^{6}month·km

^{2}), and area (10

^{6}km

^{2}): (

**a**,

**c**,

**e**,

**g**) indicate “or” situation; (

**b**,

**d**,

**f**,

**h**) indicate “and” situation.

**Figure 16.**The conditional probability of (

**a**) severity (10

^{6}month·km

^{2}) given that duration exceeds a certain value, (

**b**) duration (month) given that severity exceeds a certain value, (

**c**) area (106 km

^{2}) given that duration exceeds a certain value, (

**d**) duration (month) given that area exceeds a certain value, (

**e**) area (10

^{6}km

^{2}) given that severity exceeds a certain value, and (

**f**) severity (10

^{6}month·km

^{2}) given that area exceeds a certain value.

**Figure 17.**The conditional probability of duration (month) given that severity (10

^{6}month·km

^{2}) exceeds a certain value and area (10

^{6}km

^{2}) exceeds (

**a**) 0.6 × 10

^{6}km

^{2}and (

**b**) 1 × 10

^{6}km

^{2}.

Drought Level | SPEI | Drought Severity |
---|---|---|

I | −0.5 < SPEI | No drought |

II | −1.0 < SPEI ≤ −0.5 | Mild drought |

III | −1.5 < SPEI ≤ −1.0 | Moderate drought |

IV | −2.0 < SPEI ≤ −1.5 | Severe drought |

V | SPEI ≤ −2 | Extreme drought |

Distribution Types | Cumulative Probability Distribution | Parameters |
---|---|---|

Gam | $F(x)=\frac{{\Gamma}_{x/\beta}(\alpha )}{\Gamma (\alpha )}$ | α: shape parameter β: scale parameter |

LogL | $F(x)={\left(1+{\left(\frac{\beta}{\alpha}\right)}^{\alpha}\right)}^{-1}$ | α: shape parameter (α > 0) β: scale parameter (β > 0) |

LogN | $F(x)=\Phi \left(\frac{\mathrm{ln}x-\mu}{\sigma}\right)$ | μ: location parameter σ: scale parameter |

Wb | $F(x)=1-\mathrm{exp}\left({\left(\frac{x}{\beta}\right)}^{\alpha}\right)$ | α: shape parameter β: scale parameter |

P-III | $F(x)=\frac{{\displaystyle {\int}_{0}^{\frac{x-\mu}{\beta}}{t}^{\alpha -1}\mathrm{exp}\left(-t\right)dt}}{\Gamma \left(\alpha \right)}$ | α: shape parameter β: scale parameter μ: location parameter |

GEV | $F(x)=\{\begin{array}{l}\mathrm{exp}\left(-\left(1+k{\left(\frac{x-\mu}{\sigma}\right)}^{-\frac{1}{k}}\right)\right),k\ne 0\\ \mathrm{exp}\left(-\mathrm{exp}\left(-\left(\frac{x-\mu}{\sigma}\right)\right)\right),k=0\end{array}$ | k: shape parameter σ: scale parameter (σ>0) μ: location parameter |

GP | $F(x)=\{\begin{array}{l}1-{\left(1+k\frac{\left(x-\mu \right)}{\sigma}\right)}^{-\frac{1}{k}},k\ne 0\\ 1-\mathrm{exp}\left(-\frac{\left(x-\mu \right)}{\sigma}\right),k=0\end{array}$ | k: shape parameter σ: scale parameter (σ > 0) μ: location parameter |

Copula | Function Expression | Parameters |
---|---|---|

Frank | ${C}_{F}\left({u}_{1},{u}_{2},\cdots ,{u}_{d};\theta \right)=-\frac{1}{\theta}\mathrm{ln}\left[1+\frac{{\displaystyle \prod _{j=1}^{d}{e}^{-\theta {u}_{j}-1}}}{{\left({e}^{-\theta}-1\right)}^{d-1}}\right]$ | $\theta \in R\backslash 0$ |

Clayton | ${C}_{C}\left({u}_{1},{u}_{2},\cdots ,{u}_{d};\theta \right)=\left[\left({\displaystyle \sum _{j=1}^{d}{u}_{j}^{-\theta}}\right)-d+1\right]$ | $\theta \in \left[-1,\infty \right)\backslash 0$ |

Gumbel | ${C}_{G}\left({u}_{1},{u}_{2},\cdots ,{u}_{d};\theta \right)=\mathrm{exp}\left\{-{\left[{\displaystyle \sum _{j=1}^{d}{\left(-\mathrm{ln}{u}_{j}\right)}^{\theta}}\right]}^{\frac{1}{\theta}}\right\}$ | $\theta \in \left[1,\infty \right)$ |

Joe | ${C}_{J}\left({u}_{1},{u}_{2},\cdots ,{u}_{d};\theta \right)=1-{\left[{\displaystyle \sum _{j=1}^{d}{\left(1-{u}_{j}\right)}^{\theta}}-{\displaystyle \prod _{j=1}^{d}{\left(1-{u}_{j}\right)}^{\theta}}\right]}^{\frac{1}{\theta}}$ | $\theta \in \left[-1,\infty \right)$ |

Normal | ${C}_{N}\left({u}_{1},{u}_{2},\cdots ,{u}_{d};\sum \right)=\Phi \left({\Phi}^{-1}\left({u}_{1}\right),\cdots ,{\Phi}^{-1}\left({u}_{d}\right)\right)$ | $\sum =\left[\begin{array}{cc}1& \cdots \\ \vdots & \ddots \\ {\rho}_{1d}& \cdots \end{array}\begin{array}{c}{\rho}_{1d}\\ \vdots \\ 1\end{array}\right]$ |

Student t | ${C}_{t}\left({u}_{1},{u}_{2},\cdots ,{u}_{d};\sum ,\upsilon \right)={T}_{\sum ,\upsilon}\left({T}_{\upsilon}{}^{-1}\left({u}_{1}\right),\cdots ,{T}_{\upsilon}{}^{-1}\left({u}_{d}\right)\right)$ | $\sum =\left[\begin{array}{cc}1& \cdots \\ \vdots & \ddots \\ {\rho}_{1d}& \cdots \end{array}\begin{array}{c}{\rho}_{1d}\\ \vdots \\ 1\end{array}\right]$ |

SPEI Series | Trend Statistics | Hurst Index | Past Trend | Persistence and Future Trend | |
---|---|---|---|---|---|

Northwestern regions | Spring | 1.73 | 0.54 | Significant increase | Persistence: increase |

Summer | 1.75 | 0.52 | Significant increase | Persistence: increase | |

Autumn | 0.90 | 0.62 | Increase | Persistence: increase | |

Winter | 2.09 | 0.59 | Significant increase | Persistence: increase | |

Plateau climate zone | Spring | 2.99 | 0.63 | Significant increase | Persistence: increase |

Summer | 3.06 | 0.57 | Significant increase | Persistence: increase | |

Autumn | 1.49 | 0.60 | Increase | Persistence: increase | |

Winter | 2.19 | 0.64 | Significant increase | Persistence: increase | |

Westerlies climate zone | Spring | 0.18 | 0.48 | Increase | Unsustainability: decrease |

Summer | 0.56 | 0.53 | Increase | Persistence: increase | |

Autumn | 0.90 | 0.56 | Increase | Persistence: increase | |

Winter | 0.34 | 0.49 | Increase | Unsustainability: decrease | |

Southeast climate zone | Spring | 0.10 | 0.47 | Increase | Unsustainability: decrease |

Summer | 0.12 | 0.55 | Increase | Persistence: increase | |

Autumn | −0.86 | 0.65 | Decrease | Persistence: decrease | |

Winter | 1.77 | 0.53 | Significant increase | Persistence: increase |

Drought Variables | Pearson | Kendall | Spearman |
---|---|---|---|

Duration-Severity | 0.83 ** | 0.67 ** | 0.82 ** |

Duration-Area | 0.77 ** | 0.57 ** | 0.72 ** |

Severity-Area | 0.93 ** | 0.84 ** | 0.97 ** |

Drought Variables | K-S Test | A-D Statistics | Optimal Distribution | Parameters | ||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|

Gam | LogL | LogN | Wb | P-III | GEV | GP | Gam | LogL | LogN | Wb | P-III | GEV | GP | |||

Duration | √ | √ | √ | × | √ | √ | √ | 2.63 | 2.13 | 1.82 | 5.69 | 0.42 | 1.25 | 0.35 | GP | k = −0.077 σ = 2.438 μ = 1.581 |

Severity | √ | √ | √ | × | √ | × | √ | 2.39 | 1.08 | 0.67 | 2.92 | 5.24 | 2.19 | 1.10 | LogN | σ = 1.134 μ = −0.114 |

Area | √ | √ | √ | √ | √ | √ | √ | 0.64 | 1.50 | 1.12 | 0.81 | 0.31 | 1.13 | 0.16 | GP | k = −0.199 σ = 0.561 μ = 0.034 |

Copula Function | Gumbel | Clayton | Frank | Joe | Normal | t | Optimal Copula | Parameters | |
---|---|---|---|---|---|---|---|---|---|

D-S | AIC | −1195.90 | −1094.77 | −1235.56 | −1149.28 | −1181.84 | −1139.16 | Frank | 8.01 |

BIC | −1192.82 | −1091.70 | −1232.49 | −1146.20 | −1178.76 | −1136.09 | |||

RMSE | 0.023 | 0.032 | 0.021 | 0.027 | 0.025 | 0.028 | |||

D-A | AIC | −1226.21 | −1050.26 | −1205.33 | −1215.57 | −1174.88 | −1150.98 | Gumbel | 2.04 |

BIC | −1223.14 | −1047.19 | −1202.26 | −1212.49 | −1171.81 | −1147.90 | |||

RMSE | 0.021 | 0.037 | 0.023 | 0.022 | 0.025 | 0.027 | |||

S-A | AIC | −1231.89 | −1300.85 | −1270.51 | −1061.98 | −1266.06 | −1259.59 | Clayton | 10.63 |

BIC | −1228.82 | −1297.77 | −1267.43 | −1058.90 | −1262.99 | −1256.51 | |||

RMSE | 0.021 | 0.017 | 0.019 | 0.036 | 0.019 | 0.02 | |||

D-S-A | AIC | −1204.64 | −1056.04 | −1202.06 | −983.64 | −1147.71 | −1140.11 | Gumbel | 3.62 |

BIC | −1201.57 | −1052.97 | −1198.99 | −980.56 | −1144.63 | −1137.04 | |||

RMSE | 0.022 | 0.037 | 0.023 | 0.046 | 0.028 | 0.029 |

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## Share and Cite

**MDPI and ACS Style**

Zhang, W.; Feng, K.; Wang, F.; Wang, W.; Zhang, Z.; Wang, Y.; Huang, S.
Dynamic Evolution and Copula-Based Multivariable Frequency Analysis of Meteorological Drought Considering the Spatiotemporal Variability in Northwestern China. *Water* **2023**, *15*, 3861.
https://doi.org/10.3390/w15213861

**AMA Style**

Zhang W, Feng K, Wang F, Wang W, Zhang Z, Wang Y, Huang S.
Dynamic Evolution and Copula-Based Multivariable Frequency Analysis of Meteorological Drought Considering the Spatiotemporal Variability in Northwestern China. *Water*. 2023; 15(21):3861.
https://doi.org/10.3390/w15213861

**Chicago/Turabian Style**

Zhang, Weijie, Kai Feng, Fei Wang, Wenjun Wang, Zezhong Zhang, Yingying Wang, and Shengzhi Huang.
2023. "Dynamic Evolution and Copula-Based Multivariable Frequency Analysis of Meteorological Drought Considering the Spatiotemporal Variability in Northwestern China" *Water* 15, no. 21: 3861.
https://doi.org/10.3390/w15213861