# Construction of a Time-Variant Integrated Drought Index Based on the GAMLSS Approach and Copula Function

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Study Area and Research Framework

#### 2.1. Study Area and Data Sources

^{2}, which includes cultivated land area of 21,400 km

^{2}accounting for more than 50% to the total area of Anhui Province [22]. For the impacts of subhumid monsoon climate, the annual average precipitation of Huaibei Plain is approximately 869.6 mm but differs significantly within and between different years [2,22]. The annual precipitation of a wet year is about three to four times compared to dry years, and the precipitation during flood season (from June to September) accounts for more than 75% of the total [2]. In addition, the annual average runoff of Huaibei Plain is about 7772 million m

^{3}, while the annual variation difference is more obvious compared to precipitation [15,22]. For a long time, because of inappropriate land resources development, a sharp decrease in vegetation and excessive deforestation, etc., water and soil resources erosion became serious. The groundwater levels dropped prominently as well in Huaibei Plain, which resulted in frequent flooding and drought disasters. Drought disasters have occurred in 48 of the past 60 years, and drought duration, severity and losses all presented an aggravating trend especially after the 1990s. For instance, the drought disaster situation of Anhui Province in 2019 was the most serious during the past 40 years, and drought variation in nearly 51 counties reached severe grade. Therefore, a great variety of statistics and existing research findings indicate that it is urgent to conduct integrated drought index construction and practical application research in Huaibei Plain, Anhui Province.

#### 2.2. Research Framework

## 3. Methodologies

#### 3.1. Generalized Additive Models for Location, Scale and Shape (GAMLSS) Method

#### 3.1.1. Introduction to GAMLSS

_{t}(t = 1, 2, …, n) of stochastic variable Y is denoted as f(y

_{t}|θ

_{t}), and its distribution parameter series varying with time parameter t is denoted as θ

_{t}= (θ

_{t}

_{1}, θ

_{t}

_{2}, θ

_{t}

_{3}, θ

_{t}

_{4}) = (μ

_{t}, σ

_{t}, ν

_{t}, τ

_{t}), in which, variables μ and σ are defined as position parameter and scale parameter, corresponding to average vector and mean square deviation vector of stochastic variable Y, respectively. Variables ν and τ are uniformly defined as shape parameter, corresponding to the skewness vector and kurtosis vector of stochastic variable Y separately [25,26]. If denoting g

_{m}(∙) as the monotonic function between parameter θ

_{m}and its corresponding explanatory variables X

_{m}as well as random effect items, then it becomes

_{m})

_{n}

_{×Jm}is the explanatory variables matrix, β

_{m}is the regression coefficient vector with the length equaling J

_{m}, J

_{m}denotes the number of random effect variables of the m

_{th}parameter, (D

_{jm})

_{n}

_{×qjm}is the random effect variable matrix, γ

_{jm}is normally distributed random variable vector with the length equaling q

_{jm}, and q

_{jm}is the number of random factors corresponding to j

_{th}random effect variable [26]. Equation (1) is primarily utilized to describe the linear or nonlinear relationship of explanatory variables as well as linear relationship of random effect items for different distribution parameters, if neglecting the influences of random effect items, the original monotonic function of GAMLSS model can be simplified as follows

_{t}and σ

_{t}varying with time parameter t in this study, and the distribution parameter estimation method of GAMLSS model can be further simplified as a linear function mode varying with time parameter t [28,29], as follows

_{0}and a

_{1}are combined coefficients of position parameter μ

_{t}, parameters b

_{0}and b

_{1}are combined coefficients of scale parameter σ

_{t}, and parameter t is time concomitant variable.

#### 3.1.2. Parameter Optimization of GAMLSS

_{m}(∙) between explanatory variable X and its distribution parameters including time-variant position variable μ

_{t}and scale variable σ

_{t}[28]. Actually, the entire function derivation and simulation analysis of GAMLSS method is frequently accomplished by the program package of R language software, which can provide a variety of distribution functions for users. Combining previous application study findings of GAMLSS model in related meteorological and hydrological fields, we selected five types of two-parametric distribution functions including standard normal distribution (NO), log normal distribution (LOGNO), Weibull distribution (WEI), gamma distribution (GA) and Gumbel distribution (GU), to derive the time-variant PDF of drought indicators precipitation and soil moisture of different cities in Huaibei Plain [26,28], which is shown in Table 1.

_{t}and scale variable σ

_{t}were all constant, revealing that the variation of distribution parameters μ

_{t}and σ

_{t}all satisfied stationary features. Scenario 2, position variable μ

_{t}varied with time variable t while scale variable σ

_{t}was constant. Scenario 3, position variable μ

_{t}and scale variable σ

_{t}were all varying with time variable t [26,29]. Furthermore, the fitting performance of different distribution functions was evaluated by a generalized Akaike information criterion (AIC) [26,27], and the distribution function with minimum AIC value was recognized as the optimal fitting function of distribution parameters. In other words, if the AIC value of distribution functions of Scenario 1 was minimum, the variation of distribution parameters μ

_{t}and σ

_{t}did not present nonstationary characteristics.

#### 3.2. Copula Function

#### 3.3. Calculation Procedures of Integrated Index CTVDI

_{t}and scale variable σ

_{t}varying with time variable, and then the optimal time-variant and monthly scale PDF. The u(x) and v(y) of precipitation and soil moisture were eventually derived under AIC principle for different cities in Huaibei Plain in this study.

_{1}, R

_{2}and R

_{3}are crucial to accurately derive the starting and end time of drought process [31,35]. In this study, according to Standard for Hydrological Information and Hydrological Forecasting (GB/T22482-2008), the threshold values of parameters R

_{1}, R

_{2}and R

_{3}are 0, −0.5 and −1, respectively, and then the historical drought event samples could be obtained based on the proposed CTVDI. Finally, drought characteristic variables, including drought duration and severity corresponding to different drought events could be obtained to describe drought variation conditions [35].

_{D}(d) and F

_{S}(s), respectively, then the drought frequency, i.e., the occurrence probability of future drought events with drought characteristic variables satisfying D > d and S > s simultaneously, could be represented as follows [19,35]

_{D}

_{, S}(d, s) is the combined probability distribution function of drought duration and severity derived by copula function as introduced in Section 3.2. In addition, the return period of extreme hydrological events is another important concept in the field of hydraulic engineering designing processes. The return period of drought events with drought characteristic variables satisfying D > d and S > s, simultaneously, could be denoted as follows [19,36]

## 4. Results and Discussion

#### 4.1. Performance Analysis of GAMLSS Model

#### 4.2. Derivation of CTVDI Series and Its Application in Drought Process Recognition

#### 4.3. Drought Frequency and Return Period Determination Analysis

## 5. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

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**Figure 2.**Normal QQ figure of residual distribution of optimal distribution pattern in Huaibei Plain: (

**A**) precipitation, (

**B**) soil moisture.

**Figure 3.**Normal worm figure of residual distribution of optimal distribution pattern in Huaibei Plain: (

**A**) precipitation, (

**B**) soil moisture.

**Figure 5.**Variation of PDF of drought duration, severity and joint probability distribution in Huaibei Plain: (

**A**) PDF of drought duration, (

**B**) PDF of drought severity, (

**C**) joint probability distribution of drought duration and severity.

Name | Determination of Distribution Parameters | |
---|---|---|

NO | ${f}_{X}\left(x|\mu ,\sigma \right)=\frac{1}{\sqrt{2\pi \sigma}}\cdot \mathrm{exp}\left[-\frac{{\left(x-\mu \right)}^{2}}{2{\sigma}^{2}}\right]$ | $E\left(X\right)=\mu $ $Var\left(X\right)={\sigma}^{2}$ |

LOGNO | ${f}_{X}\left(x|\mu ,\sigma \right)=\frac{1}{\sqrt{2\pi \sigma}}\cdot \frac{1}{x}\cdot \mathrm{exp}\left[-\frac{\mathrm{log}\left(x\right)-\mu )}{2{\sigma}^{2}}\right]$ | $E\left(X\right)=\mathrm{exp}{\left({\sigma}^{2}\right)}^{1/2}\cdot {e}^{\mu}$ $Var\left(X\right)=\mathrm{exp}\left({\sigma}^{2}\right)\cdot \left[\mathrm{exp}\left({\sigma}^{2}\right)-1\right]$ |

WEI | ${f}_{X}\left(x|\mu ,\sigma \right)=\frac{\sigma {x}^{\sigma -1}}{{\mu}^{\sigma}}\cdot \mathrm{exp}\left[-{\left(\frac{x}{\mu}\right)}^{2}\right]$ | $E\left(X\right)=\mu \xb7G\left(1/\sigma \right)$ $Var\left(X\right)={\mu}^{2}\xb7\left\{G\left(\frac{2}{\sigma +1}\right)-{\left[G\left(\frac{1}{\sigma +1}\right)\right]}^{2}\right\}$ |

GA | ${f}_{X}\left(x|\mu ,\sigma \right)=\frac{1}{{\left({\sigma}^{2}\mu \right)}^{1/{\sigma}^{2}}}\cdot \frac{{x}^{1/\left({\sigma}^{2}-1\right)}\cdot {e}^{-x/\left({\sigma}^{2}\mu \right)}}{G\left(1/{\sigma}^{2}\right)}$ | $E\left(X\right)=\mu $ $Var\left(X\right)={\mu}^{2}\xb7{\sigma}^{2}$ |

GU | ${f}_{X}\left(x|\mu ,\sigma \right)=\frac{1}{\sigma}\cdot \mathrm{exp}\left[\left(\frac{x-\mu}{\sigma}\right)-\mathrm{exp}\left(\frac{x-\mu}{\sigma}\right)\right]$ | $E\left(X\right)=\mu -\mu \sigma \approx \mu -0.57722\sigma $ $Var\left(X\right)={\pi}^{2}\xb7{\sigma}^{2}/6$ |

Name | Type | Huaibei | Suzhou | Bozhou | Bengbu | Fuyang | Huainan | ||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|

P | SM | P | SM | P | SM | P | SM | P | SM | P | SM | ||

Scenario 1:constant μ and σ | NO | 601 | 266 | 593 | 276 | 602 | 252 | 600 | 259 | 602 | 261 | 592 | 275 |

LOGNO | 602 | 268 | 598 | 279 | 596 | 255 | 603 | 264 | 589 | 266 | 604 | 280 | |

WEI | 598 | 262 | 591 | 273 | 596 | 247 | 596 | 247 | 595 | 251 | 589 | 259 | |

GA | 597 | 267 | 592 | 278 | 594 | 254 | 597 | 262 | 590 | 265 | 592 | 278 | |

GU | 614 | 263 | 606 | 274 | 625 | 248 | 613 | 246 | 625 | 251 | 604 | 257 | |

Scenario 2:μ = f(t) and constant σ | NO | 602 | 257 | 595 | 263 | 603 | 246 | 601 | 259 | 603 | 256 | 593 | 271 |

LOGNO | 603 | 259 | 599 | 266 | 596 | 249 | 604 | 264 | 589 | 261 | 603 | 276 | |

WEI | 598 | 253 | 593 | 259 | 598 | 241 | 598 | 243 | 596 | 246 | 591 | 259 | |

GA | 599 | 258 | 594 | 265 | 595 | 248 | 599 | 262 | 591 | 259 | 593 | 275 | |

GU | 615 | 254 | 607 | 258 | 627 | 240 | 615 | 241 | 626 | 245 | 606 | 256 | |

Scenario 3:μ = f(t) and σ = s(t) | NO | 604 | 259 | 597 | 264 | 604 | 248 | 601 | 257 | 605 | 257 | 592 | 271 |

LOGNO | 602 | 261 | 595 | 268 | 597 | 251 | 597 | 262 | 590 | 262 | 601 | 277 | |

WEI | 599 | 254 | 594 | 259 | 598 | 242 | 596 | 245 | 597 | 247 | 586 | 256 | |

GA | 599 | 260 | 593 | 267 | 595 | 250 | 595 | 261 | 592 | 260 | 585 | 275 | |

GU | 617 | 254 | 609 | 260 | 624 | 242 | 613 | 243 | 627 | 246 | 603 | 254 |

**Table 3.**Variation of residual distribution parameters of optimal distribution pattern in Huaibei Plain.

City | Name | Type | Distribution Parameter | Residual Distribution Parameter | ||||
---|---|---|---|---|---|---|---|---|

AV | VA | SC | KC | FC | ||||

Huaibei | P | GA | μ = exp(4.7862) | 0.0018 | 1.0192 | −0.2744 | 2.5630 | 0.9929 |

σ = exp(−0.7287) | ||||||||

SM | WEI | μ = exp(3.6082 − 0.0018·t) | −0.0019 | 1.0158 | 0.0465 | 2.4647 | 0.9922 | |

σ = exp(2.8581) | ||||||||

Suzhou | P | WEI | μ = exp(4.9376) | 0.0003 | 1.0013 | 0.1000 | 2.5842 | 0.9938 |

σ = exp(0.9520) | ||||||||

SM | GU | μ = (36.8974 − 0.0835·t) | 0.0005 | 0.9956 | 0.1559 | 2.4970 | 0.9908 | |

σ = exp(0.7262) | ||||||||

Bozhou | P | GA | μ = exp(4.7933) | 0.0000 | 1.0185 | −0.0009 | 2.8585 | 0.9966 |

σ = exp(−0.7785) | ||||||||

SM | GU | μ = (34.7393 − 0.0484·t) | 0.0007 | 0.9987 | 0.1415 | 2.6827 | 0.9953 | |

σ = exp(−0.5639) | ||||||||

Bengbu | P | GA | μ = (4.6641 − 0.0053·t) | 0.0054 | 1.0252 | −0.0231 | 2.0978 | 0.9879 |

σ = exp(−0.4236 − 0.0146·t) | ||||||||

SM | GU | μ = (38.3270 − 0.0394·t) | −0.0063 | 1.0873 | −0.3833 | 2.7571 | 0.9889 | |

σ = exp(0.5237) | ||||||||

Fuyang | P | LOGNO | μ = 4.6452 | 0.0000 | 1.0185 | −0.1165 | 2.7462 | 0.9946 |

σ = exp(− 0.7432) | ||||||||

SM | GU | μ = 36.1412 − 0.0461·t | −0.0005 | 1.0271 | −0.0491 | 2.8125 | 0.9968 | |

σ = exp(0.5921) | ||||||||

Huainan | P | GA | μ = exp(4.5209 + 0.0065·t) | 0.0061 | 1.0253 | −0.1981 | 2.3835 | 0.9901 |

σ = exp(−0.3317 − 0.0162·t) | ||||||||

SM | GU | μ = 37.1387 − 0.0348·t | −0.0109 | 1.1009 | −0.4277 | 2.5492 | 0.9872 | |

σ = exp(0.2777 + 0.0122·t ) |

Name | Type | Huaibei | Suzhou | Bozhou | Bengbu | Fuyang | Huainan | Average |
---|---|---|---|---|---|---|---|---|

P and CTVDI | LCC | 0.68 | 0.72 | 0.73 | 0.77 | 0.75 | 0.74 | 0.73 |

KCC | 0.68 | 0.67 | 0.71 | 0.79 | 0.68 | 0.73 | 0.71 | |

SM and CTVDI | LCC | 0.83 | 0.82 | 0.92 | 0.91 | 0.88 | 0.85 | 0.87 |

KCC | 0.73 | 0.71 | 0.87 | 0.82 | 0.76 | 0.77 | 0.78 |

**Table 5.**Statistic result of historical drought characteristic variable, 1960–2014 in Huaibei Plain.

Name | Huaibei | Suzhou | Bozhou | Bengbu | Fuyang | Huainan |
---|---|---|---|---|---|---|

Drought event amount | 91 | 82 | 80 | 80 | 74 | 76 |

Average of drought duration (month) | 3.1 | 3.04 | 3.19 | 2.53 | 3.23 | 2.88 |

Average of non-drought duration (month) | 4.13 | 4.37 | 4.93 | 5.86 | 5.04 | 6.13 |

Maximum of drought duration (month) | 16 | 12 | 14 | 13 | 14 | 9 |

Average of drought severity | 4.22 | 4.05 | 4.26 | 3.57 | 4.4 | 3.98 |

Maximum of drought severity | 20.89 | 15.57 | 18.61 | 18.95 | 18.43 | 15.94 |

Kendall correlation coefficient of drought duration and severity | 0.78 | 0.72 | 0.77 | 0.7 | 0.73 | 0.74 |

City | T ≤ 2a (Light Drought) | 2a < T ≤ 6a (Moderate Drought) | 6a < T ≤ 20a (Severe Drought) | T > 20a (Extreme Drought) | Total Amount |
---|---|---|---|---|---|

Huaibei | 60 | 18 | 8 | 5 | 91 |

Suzhou | 52 | 18 | 7 | 5 | 82 |

Bozhou | 51 | 16 | 9 | 4 | 80 |

Bengbu | 50 | 17 | 9 | 4 | 80 |

Fuyang | 46 | 16 | 11 | 6 | 79 |

Huainan | 45 | 15 | 8 | 3 | 71 |

City | No. | Time of Drought Event | Drought Duration /Month | Drought Severity | Joint Frequency | Return Period /Year |
---|---|---|---|---|---|---|

Huaibei | 1 | August 1966–June 1967 | 11 | 17.56 | 0.0094 | 64.2 |

2 | November 1967–July 1968 | 9 | 14.19 | 0.0239 | 25.2 | |

3 | October 1991–January 1993 | 16 | 14.08 | 0.0038 | 160.1 | |

4 | September 1998–September 1999 | 12 | 13.23 | 0.0152 | 39.6 | |

5 | March 2001–November 2001 | 9 | 18.43 | 0.0074 | 81.2 | |

Suzhou | 1 | May 1966–January 1967 | 9 | 13.78 | 0.0183 | 34.2 |

2 | September 1967–July 1968 | 11 | 15.57 | 0.0088 | 70.3 | |

3 | January 1978–December 1978 | 12 | 10.46 | 0.0074 | 87.9 | |

4 | March 1981–April 1982 | 12 | 13.54 | 0.0215 | 28.7 | |

5 | March 2001–November 2001 | 9 | 14.95 | 0.0133 | 47.5 | |

Bozhou | 1 | May 1966–June 1967 | 14 | 18.62 | 0.0056 | 121.6 |

2 | March 1978–January 1979 | 11 | 12.17 | 0.0202 | 33.5 | |

3 | October 1991–August 1992 | 11 | 11.51 | 0.0203 | 33.3 | |

4 | March 2001–November 2001 | 9 | 18.26 | 0.0084 | 80.3 | |

Bengbu | 1 | May 1966–June 1967 | 13 | 18.95 | 0.0044 | 157.9 |

2 | November 1967–June 1968 | 8 | 13.62 | 0.0206 | 34.3 | |

3 | March 1978–March 1979 | 13 | 15.55 | 0.0064 | 109.9 | |

4 | October 2010–June 2011 | 9 | 11.25 | 0.0253 | 27.6 | |

Fuyang | 1 | December 1960–July 1961 | 8 | 11.05 | 0.0275 | 27.3 |

2 | October 1967–May 1968 | 8 | 15.94 | 0.0072 | 104.3 | |

3 | August 1978–March 1979 | 8 | 7.62 | 0.0304 | 24.7 | |

4 | November 1983–May 1984 | 7 | 11.11 | 0.0363 | 20.7 | |

5 | October 1991–June 1992 | 9 | 11.04 | 0.0168 | 44.7 | |

6 | March 2001–November 2001 | 9 | 12.98 | 0.0153 | 50.4 | |

Huainan | 1 | May 1966–June 1967 | 14 | 20.89 | 0.0051 | 136.1 |

2 | February 1978–March 1979 | 13 | 15.6 | 0.0087 | 79.1 | |

3 | March 2001–November 2001 | 9 | 20.49 | 0.0077 | 89.1 |

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**MDPI and ACS Style**

Bai, X.; Jin, J.; Wu, C.; Zhou, Y.; Zhang, L.; Cui, Y.; Tong, F.
Construction of a Time-Variant Integrated Drought Index Based on the GAMLSS Approach and Copula Function. *Water* **2023**, *15*, 1653.
https://doi.org/10.3390/w15091653

**AMA Style**

Bai X, Jin J, Wu C, Zhou Y, Zhang L, Cui Y, Tong F.
Construction of a Time-Variant Integrated Drought Index Based on the GAMLSS Approach and Copula Function. *Water*. 2023; 15(9):1653.
https://doi.org/10.3390/w15091653

**Chicago/Turabian Style**

Bai, Xia, Juliang Jin, Chengguo Wu, Yuliang Zhou, Libing Zhang, Yi Cui, and Fang Tong.
2023. "Construction of a Time-Variant Integrated Drought Index Based on the GAMLSS Approach and Copula Function" *Water* 15, no. 9: 1653.
https://doi.org/10.3390/w15091653