# Multivariate Drought Frequency Analysis using Four-Variate Symmetric and Asymmetric Archimedean Copula Functions

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## Abstract

In drought frequency analysis, as the number of drought variables increases, the joint behavior between these variables needs to be studied. Therefore, this study aims to develop a flexible four-variate joint distribution function of the regional stochastic nature of drought. Using run theory, drought duration, severity, peak, and inter-arrival time were abstracted from the Standardized Precipitation Evapotranspiration Index (SPEI) aggregated at six months, observed in mainland China between 1961 and 2013. As these drought variables showed significant dependence properties and followed different marginal distributions, we employed and compared six four-variate symmetric and asymmetric Archimedean copulas (i.e., Frank, Clayton, Gumbel–Hougaard). The best-fitting model for each region was carefully selected using RMSE, AIC, and BIAS goodness-of-fit tests. Results revealed that the empirical and theoretical probabilities of the symmetric Clayton in regions NE (Northeast), CS (Central and Southern China), EMC (Entire China), and symmetric Frank in regions NC (North China), SC (South China), IM (Inner Mongolia), NW (Northwest), TP (Tibet Plateau) agreed well. Symmetric Frank copula was considered the best-fit for station-based drought analysis in EMC. Based on these copulas, the drought probabilities and return periods for the occurrence of drought events over the next 5, 10, 20, 50, and 100 years in each region were hereby comprehensively explained, and the results shown here could be helpful in the appraisal of the adequacies of water supply systems under drought conditions in all regions. This study showed that a four-variate copula approach is a vital tool for probabilistic interpretation of hydrological and meteorological data in the different climatic region of mainland China.

## Keywords

Symmetric and asymmetric copula Four-variate drought analysis Probabilistic analysis Return period Water management SPEI## 1 Introduction

Drought-affected areas have remarkably increased over the past 50 years in China as a result of variation in precipitation (*P*) and temperature (Yang et al. 2013; Leng et al. 2015; Chang et al. 2016). In fact, a noticeable extreme winter-spring droughts events occurred in southwest China from August 2009 to May 2010 (Lu et al. 2011; Lu et al*.* 2012; Zhang et al. 2012; Yang et al. 2013), while in 2011 the middle and lower reaches of the Yellow River were impacted by the spring-summer drought (Lu et al. 2013; Zhao et al. 2013; Chang et al. 2016). Droughts have caused greatest damages in China during 1949–1995, and many damages have resulted to economic losses more than the US $12 billion (Xu et al., 2015a; Qin et al. 2015; Chang et al. 2016). Consequently, droughts are of prominent concern in the outlining and control of water resources. Studying meteorological droughts is necessary by itself, and also because they act as antecedents to the longer-lasting and more significant agricultural and hydrological droughts (Haslinger et al., 2014, Wilhite et al., 2014).

The popular method of assessing drought risk requires the calculation of the probability that a specific value of drought variable will be exceeded, which is equal to the evaluation of the recurrence interval (Prohaska et al. 2008; Chen et al. 2012). This procedure is normally focused on the univariate frequency analysis. In this analysis, the authors have concentrated on some characteristic of drought and assume that the other variables are constant. A direct attempt to simplify the multi-dimensional character of droughts has been investigated by several authors (Shiau, 2006; Ganguli and Reddy, 2014; Huang et al., 2014a). Although the simplistic analysis produces practical results, it does not strictly represent the actual mathematical interactions between the various dimensions of the phenomenon and obviously cannot give accurate results. Since drought variables are usually reliant on one other, a proper frequency analysis of droughts should consider such dependencies within a suitable multivariate framework.

It should be mentioned that the attention of researchers in multivariate modeling has increased considerably in the last decades, due to the application of copulas. The comprehensive theoretical structure proposed by Sklar (1959) can be gathered from Nelsen (2006), Joe (1997) and Salvadori et al. (2007). There have been diverse cases of copula applications in the setting of drought management (Serinaldi et al., 2009; Shiau and Modarres, 2009; Kao and Govindaraju, 2010; Wong et al., 2010; Song and Singh, 2010; Mishra and Singh, 2010; Mirabbasi et al., 2012). This includes return-period estimation (Salvadori and De Michele, 2010), multivariate simulation (AghaKouchak et al. 2010), propose a new drought indicator (Kao and Govindaraju, 2010), and many other theoretical studies of multivariate severe issues (Salvadori and De Michele, 2015; Zhang et al. 2015a; She et al. 2016). There are several types of copula functions, which have been described (e.g. Nelsen, 1999). The regularly applied copula for hydrological analysis belongs to four classes which includes: Archimedean class (AMH, Frank, Clayton, Gumbel, and Joe), elliptical class (student t and normal), extreme value class (Gumbel, Galambos, Tawn, Husler-Reiss, and t-EV), and miscellaneous class (Farlie–Gumbel–Morgenstern and Plackett) (Shiau and Modarres, 2009; Mirabbasi et al. 2012; Lee et al. 2013; Xu et al., 2015b). The Archimedean class of copulas are characterized by their simple structure and strong representativeness (Huang et al., 2014a; Tsakiris et al., 2016).

There are many applications of two-variate copula-based drought studies in many countries. For instance in China, Shiau (2006) built a joint distribution between drought severity and duration in Southern Taiwan while Shiau and Modarres (2009) employed Clayton copula to model drought severity and duration frequency curves for two climatic regions in Iran. Tsakiris et al. (2016) analyzed drought severity and areal extent utilizing Gumbel-Hougaard copula in Greece. The Frank and Gumbel–Hougaard have been considered the best copulas for illustrating drought occurrence frequency in Canada and Iran, respectively (Lee et al. 2013). Meta-Gaussian copula was applied for the joint modeling of drought variables within Texas (Song and Singh, 2010) while drought duration and severity of Sharafkhaneh in the northwest of Iran was modeled using Galambos copula (Mirakbari et al., 2010). In each of the cases, some bivariate probabilistic properties of droughts were investigated.

Three-variate Archimedean copulas have been employed for the joint distribution of rainfall, drought and flood events (Grimaldi and Serinaldi, 2006; Zhang and Singh 2007). Using asymmetric Archimedean copulas, Serinaldi and Grimaldi (2007) described an inference procedure to carry out a three-variate frequency analysis. Ma et al. (2013) created a joint distribution of drought duration, severity, and peak using elliptical, asymmetric and symmetric Archimedean copulas. Because of the flexibility of meta-elliptical copula (Fan et al. 2016), many meta-elliptical, Clayton, Frank, Gumbel-Hougaard, and Ali-MikhailHaq copulas have been employed to create a joint distribution of drought severity, duration, and interval time, and the best copula was chosen (Song and Singh, 2010).Three-variate Plackett copulas have been adopted for the investigations of extreme rainfall cases (Kao and Govindaraju, 2008) and for modeling a joint distribution of drought duration, severity and inter-arrival time (Song and Singh, 2010). Student’s *t* copula has been utilized to identify three-variate drought events under the control of La-Nina, EL-Nino, and natural climate state in New South Wales, Australia (Wong et al. 2010). Other studies on drought frequency analysis using copula includes Huang et al. (2014a, b), Zhao et al. (2015), Zhang et al. (2015b), Zhang et al. (2017) among others.

Crucial hurdles remain not only in drought characterization but also in the interpretation of drought variables to information relevant for regional monitoring, early warning, and water resources planning. Assuming that the territorial unit affected by drought is the entire region, there are two variables which together characterize each drought event; the severity and the duration. If the peak and interarrival time of drought are considered as an additional variable, a four-dimensional copulas approach may be used in which duration, severity, peak and interarrival time of droughts can be jointly analyzed. Recently, the applications of four-variate copula have been reported. De Michele et al. (2007) proposed a procedure towards building four-variate distributions, stating two copulas for a separate two-variate candidate case and utilizing the approach to give a four-variate nature of sea state.

Employing a four-variate student copula, Serinaldi et al. (2009) built the corresponding joint distributions of drought length, mean, minimum SPI values, and drought mean areal extent to investigate their joint probabilistic characteristics. So far, most of the research focused on parametric two-variate and symmetric three-variate copulas (Ganguli and Reddy, 2014), but four-variate copulas are somewhat limited because their constructions are very complicated. Although there have been some attempts to construct multivariate extensions of two-variate Archimedean copula (Embrechts et al., 2003; Whelan, 2004; Savu and Trede, 2010), however such studies seldom compares the performances of four-variate symmetric and asymmetric copulas in different climatic regions of China, considering probability and return periods of drought events. Further, knowing that droughts are multiplex phenomena, two- and three-variate analysis cannot give an exhaustive appraisal of droughts because the investigation could lead to a deficient drought probabilistic estimation (De Michele et al. 2005; Grimaldi and Serinaldi, 2006; Chebana and Ouarada, 2011; Ganguli and Reddy, 2014; Xu et al., 2015a; Tsakiris et al. 2016).

Given the challenges as mentioned above in drought characterization, a focused research is required to comprehensively analyze and delineate drought trends and spatiotemporal patterns more flexibly and intuitively. It is, therefore, the objective of this paper to produce a methodology using the four-variate copulas approach for analyzing the multivariate drought frequency of droughts. In particular, we built a joint distribution utilizing drought duration (D*d*), severity (D*s*), peak (D*p*) and inter-arrival time (D*i*), based on four-variate symmetric (Clayton, Frank, and Gumbel) and asymmetric (Clayton, Frank, and Gumbel) Archimedean Copulas. We showed how it could be used (i) to observe the temporal evolution of a drought event, and (ii) to perform a real-time appraisal of drought in seven climatic regions of mainland China over 1961–2013. The risks of drought events have been described based on joint probability and return periods, which has become a conventional method for a risk-based plan of water resources structures.

The paper is prepared as follows. Section 2 describes details about the study area and data, Standardized Precipitation Evapotranspiration Index (SPEI) calculation and estimation of *Dd*, *Ds*, *Dp,* and D*i*. Section 3 introduces the four-variate symmetric and asymmetric Archimedean copula. Section 4 compares the performance of the copulas. Section 5 selects the best-fitted copula to build a joint distribution of *Dd*, *Ds*, *Dp,* and D*i* in seven climatic regions of China. Section 6 contains some concluding remarks.

## 2 Study Region and Data

Geographically, China lies between 15^{0}N - 60^{0}N and 75°E - 135°E. By its geographical extent, China is endowed with diverse landforms that include hills, mountains, high plateaus and deserts in the western reaches, while in the central and east areas, the land slopes into broad plains and deltas. From the higher elevations in the west, thousands of rivers drain the country eastwards; the most significant are the Yangtze (i.e., the longest river in China and third longest in the world), Heilong (Amur), Mekong, Pearl, and Yellow river. The climate of China fluctuates significantly from one region to another because of its tremendous space, complex topography as well as variation in monthly *P* and temperature (Zhai *et al.* 2010; Wu *et al.* 2011). The northern China and western are influenced by dry climate while humid and semi-humid predominates over eastern China (Wu *et al.* 2011).

*et al.*2017). Northeast humid/semi-humid warm region (NE, 72 stations), North China humid/semi-humid temperate zone (NC, 104 stations), Central and Southern China humid subtropical zone (CS, 165 stations), South China humid tropical zone (SC, 57 stations), Inner Mongolia steppe zone (IM, 44 stations), Northwest desert areas (NW, 61 stations), and Qinghai-Tibet Plateau (TP, 49 stations) (Fig. 1). Regions NE, NC, CS, SC are associated with Eastern Monsoon. NW is an arid region, and the replenishment of water in this region is mostly from melting glacial and perennial frozen soil, not from

*P*.

The dataset employed in this research to appraise drought events are daily and monthly weather data from 552 national basic meteorological stations in mainland China from 1961 to 2013. These 53-year records were obtained from the China Meteorological Data Sharing Network. From Fig. 1, the meteorological stations are not uniformly spread as more are situated within the east than western regions particularly the Qinghai-Tibet Plateau. The non-parametric tests were used to cross-check the reliability and quality of the climatic data. According to Helsel and Hirsch (1992), Kendall autocorrelation test, Mann-Kendall trend test, and Mann-Whitney homogeneity tests for mean and variance were employed to test randomness, homogeneity, and absence of trends. The results showed that randomness and stationarity of the weather data were fixed between the critical points (5% statistical significance level).

## 3 Methods

### 3.1 Drought Indices and Drought Identification

*P*and potential evapotranspiration (PET) for i

^{th}month as follows:

_{i}values are synthesized in each time scale as:

Next, D_{i} is fitted with the three-parameter log-logistic distribution. Finally, the SPEI is obtained as standardized values, and details of the estimation can be found in Vicente-Serrano et al. (2010). The mean value of the SPEI is 0, and the standard deviation is 1. In this study, SPEI at the 6-month timescale, denoted by SPEI-6 is estimated for each station and seven regions. SPEI-6 is most useful for describing the shallow soil moisture available to crops (Reddy and Singh, 2014; Abdi et al. 2016).

*d*, D

*s*, D

*p,*and D

*i*, are extracted. The definitions of D

*d*, D

*s*, and D

*p*, can be found in Shiau (2006), Mishra and Singh (2010) and Ayantobo et al. (2017) while D

*i*is described as the interval between the commencement of a drought to the start of the subsequent drought (Song and Singh, 2010). Water manager needs to be aware of the risk of having drought scenarios of different D

*d*, D

*s*, D

*p*, and D

*i*during drought scenarios. Therefore, a four-variate distribution must be constructed using copula functions.

### 3.2 Marginal Distribution of Drought Variable

The univariate distribution forms the framework of four-variate analysis using copulas. The marginal distributions for D*d*, D*s*, and D*p* for different climatic regions across mainland China have been documented (Ayantobo et al. 2017). In the case of D*i*, ten marginal distributions which include exponential, two-parameter exponential, Weibull, three-parameter Weibull, generalized extreme value, inversion Gaussian, three-parameter inversion Gaussian, generalized Pareto, gamma, and three-parameter gamma are compared to choose the best distribution to fit D*i*.

*i*are compared against the empirical non-exceedance probabilities estimated utilizing Gringorten’s plotting position function as follows (Gringorten, 1963; Cunnane, 1978; Song and Singh, 2010):

Here, *n* is sample size, N_{m} is the amount of *x*_{i} regarded as *x*_{j}≤*x*_{i}, *i* = 1, …, *n*, 1≤*j*≤*i*. Utilizing the observed data (*x*), the marginal parameters are calculated employing the maximum likelihood estimation.

### 3.3 Empirical Four-Variate Distribution of Drought Variables

*Dd*,

*Ds*,

*Dp*, and

*Di*can be modified from Song and Singh (2010):

Here, n is size of the sample. *N*_{mnlp} is amount of (*x*_{i}, *y*_{i}, *z*_{i}, *k*_{i}) regarded as *x*_{j} ≤ *x*_{i}, *y*_{j} ≤ *y*_{i}, *z*_{j} ≤ *z*_{i}, and *k*_{j} ≤ *k*_{i}, *i* = 1, …, *n*, 1≤*j*≤*i*.

### 3.4 Joint Cumulative Probability Distribution of Drought Variables

*Dd*,

*Ds*,

*Dp*, and

*Di*are represented using X

_{1}, X

_{2}, X

_{3}, and X

_{4}; and \( {u}_1={F}_{x_1}\left({x}_1\right) \), \( {u}_2={F}_{x_2}\left({x}_2\right) \), \( {u}_3={F}_{x_3}\left({x}_3\right) \), and \( {u}_4={F}_{x_4}\left({x}_4\right) \) denote their cumulative distribution function (CDF), respectively. Therefore, their joint CDF could be represented as (Song and Singh, 2010):

### 3.5 Modeling Four-Variate Drought Variables Using Copulas

*F*

_{1, 2, …. , n}(

*x*

_{1},

*x*

_{2}, ….,

*x*

_{n}) is many variables CDF of

*n*associated arbitrary variables of X

_{1}, X

_{2}, . . . ., X

_{n}with the corresponding candidate CDF

*F*

_{1}(

*x*

_{1}),

*F*

_{2}(

*x*

_{2}), …. ,

*F*

_{n}(

*x*

_{n}), next the n-dimensional CDF including univariate distributions of

*F*

_{1}(

*x*

_{1}),

*F*

_{2}(

*x*

_{2}), …. ,

*F*

_{n}(

*x*

_{n}) could be written as shown below (Sklar 1959; Nelsen, 2006; Ganguli and Reddy 2014):

Here, *C* is a *d*-dimensional copula in the form: [0, 1] d → [0, 1], with association parameter * θ*;

*F*

_{k}(

*x*

_{k}) =

*u*

_{k}for

*k*= 1, . . .,

*n*;

*U*

_{k}~

*U*(0, 1).

*n*-dimensional Archimedean copula could be represented as (Chen et al. 2012):

*n*-copula and can be rewritten as (Chen et al. 2012):

Description of the four-variate symmetric and asymmetric Archimedean copula

Copula Family | C(u |
---|---|

CLAYTON (Symmetric) | \( {\left({\mathrm{u}}_1^{-\uptheta}+{\mathrm{u}}_2^{-\uptheta}+{\mathrm{u}}_3^{-\uptheta}+{\mathrm{u}}_4^{-\uptheta}-3\right)}^{\frac{-1}{\uptheta}} \) |

CLAYTON (Asymmetric) | \( {\left({\mathrm{u}}_4^{-{\uptheta}_1}+{\left({\left({\mathrm{u}}_1^{-{\uptheta}_3}+{\mathrm{u}}_2^{-{\uptheta}_3}-1\right)}^{\frac{\uptheta_2}{\uptheta_3}}+{\mathrm{u}}_3^{-{\uptheta}_2}-1\right)}^{\frac{\uptheta_2}{\uptheta_3}}-1\right)}^{\frac{\uptheta_2}{\uptheta_3}} \) |

FRANK (Symmetric) | \( \frac{-1}{\theta_1}\ln \left(1+\frac{-1}{\theta_1}\right) \) |

FRANK (Asymmetric) | \( \frac{-1}{\theta_1}\ln \left(1+\frac{-1}{\theta_1}\left({e}^{-{\uptheta}_1{\mathrm{u}}_4}-1\right)\bullet \left({\left(1-\frac{-1}{\theta_1}\bullet \left(1-{\left(1-\frac{-1}{\theta_1}\bullet \left(1-{e}^{-{\uptheta}_3{\mathrm{u}}_1}\right)\left(1-{e}^{-{\uptheta}_3{\mathrm{u}}_2}\right)\right)}^{\frac{-1}{\theta_1}}\right)\bullet \left(1-{e}^{-{\uptheta}_2{\mathrm{u}}_3}\right)\right)}^{\frac{-1}{\theta_1}}-1\right)\right) \) |

GUMBEL (Symmetric) | \( \exp \left(-{\left[{\left(-\ln\ {\mathrm{u}}_1\right)}^{\uptheta}+{\left(-\ln\ {\mathrm{u}}_2\right)}^{\uptheta}+{\left(-\ln\ {\mathrm{u}}_3\right)}^{\uptheta}+{\left(-\ln\ {\mathrm{u}}_4\right)}^{\uptheta}\right]}^{\frac{1}{\uptheta}}\right) \) |

GUMBEL (Asymmetric) | \( \exp \left\{-{\left[{\left(-\ln\ {u}_4\right)}^{\uptheta_1}+{\left({\left({\left(-\ln {u}_1\right)}^{\uptheta_3}+{\left(-\ln {u}_2\right)}^{\uptheta_3}\right)}^{\frac{\uptheta_2}{\uptheta_3}}+{\left(-\ln {u}_3\right)}^{\uptheta_2}\right)}^{\frac{\uptheta_2}{\uptheta_3}}\right]}^{\frac{\uptheta_2}{\uptheta_3}}\right\} \) |

### 3.6 Copula Parameters Estimation

Where *n,* x_{c}(*i*)*,* x_{0}(*i*)are sample size, i^{th} calculated value, i^{th} observed value respectively.

### 3.7 Selection of Appropriate Copula Function

Where E [* ∘*] = Mean square error (MSE) is the expectation operator;

*n,*x

_{c}(

*i*)

*,*x

_{0}(

*i*)are sample size, i

^{th}calculated value, i

^{th}observed value respectively and k is the number of parameters utilized in getting the computed value. The copula with least values of AIC and RMSE is selected as the best-fitted fuction.

*u*), the empirical copula,

*C*

_{n}(

*u*), and fitted parametric copula,

*C*

_{p}(

*u*), are compared to select the best-fitted function. Constructing

*C*

_{n}(

*u*) requires: (1) ranking

*Dd*,

*Ds*, D

*p*and

*Di*variables considering their pseudo-observations (

*U*

_{i,1},

*U*

_{i,2},

*U*

_{i,3},

*U*

_{i,4}), (2) the empirical CDFs are constructed based on the ranks and (3) the empirical four-variate copula are calculated using the empirical CDFs (Nelsen, 2006; Genest and Favre, 2007):

Where I (A) is the pointer variable of function A which gives a number of 1 if A is valid and 0 if A is invalid. Ranks of i^{th} *Dd*, *Ds*, D*p* and *Di* value are given as *R*_{i, 1}, *R*_{i, 2}, *R*_{i, 3} and *R*_{i, 4} respectively (Mirabbasi et al., 2012). The Quantile-Quantile (Q-Q) plot is then used to compare the closeness between *C*_{n}(*u*) and *C*_{p}(*u*). graphically.

### 3.8 Four-Variate Joint Drought Frequency Analysis

_{1}, X

_{2},. . .,X

_{d}exceeds their corresponding thresholds (

*X*

_{1}

*> x*

_{1},

*. . .*,

*X*

_{d}

*> x*

_{d}), the joint probability and return period of all drought events can be computed. In this study, the four-variate joint probabilities,

*P*

_{DSPI}, of (

*Dd*≥

*d*,

*Ds*≥

*s*,

*Dp*≥

*p*,

*Di*≥

*i*) can be calculated utilizing the copula-based procedure modified from Shiau (2006), Ganguli and Reddy (2014):

*T*

_{DSPI}, of (

*Dd*≥

*d*,

*Ds*≥

*s*,

*Dp*≥

*p*,

*Di*≥

*i*) can be estimated as follows:

In the equations above, *ζ* = *N/q*, *N* = period of SPEI time-series (years), *q* = amount of drought events in *N* years. *C*_{DSPI}(*d*, *s*, *p*, *i*)= four-variate joint distributions, *P*_{DSPI} and *T*_{DSPI}, expresses the joint probability and return periods of occurrence of *Dd* or *Ds* or *Dp* or *Di* exceeding a particular value of *d*, *s*, *p* and *i*, respectively.

## 4 Results and Discussions

### 4.1 Drought Properties and Joint Dependence

*i*extracted from SPEI-6 time series are shown in Table 2. The average regional values of D

*i*across NE, NC, CS, SC, IM, NW, TP, and EMC during 1961–2013 were 8.9, 11.7, 12.4, 10.1, 10.1, 14.7, 9.9, and 12.1 months, respectively. Further, the qualitative joint dependence of

*Di*against D

*i*,

*Ds*, and

*Dp*was examined by employing a graphical tool in the form of a scatter plot matrix (Salvadori et al. 2007; Genest and Favre, 2007). According to Serinaldi et al. (2009), the scatter plot helps to synthesize information on dependence structure and marginals. The joint dependence between D

*i*, D

*s*and D

*p*have been studied (Ayantobo et al. 2017) and would not repeat again here. The joint dependence of

*Di*against D

*i*,

*Ds*, and

*Dp*in regions NE, NC, CS, SC, IM, NW, TP, and EMC are shown in Fig. 3. Figure 3 gave a visual judgment on the pair-wise dependence between drought variables (Genest and Favre, 2007). Although some regions showed the collection of points in the lower left corner, an uphill pattern relationship was observed from left to right. For each pair of variables, a growing positive linear relationship existed.

Statistical description of drought inter-arrival time across different regions during 1961–2013

Statistic | Climatic region | EMC | ||||||
---|---|---|---|---|---|---|---|---|

NE | NC | CS | SC | IM | NW | TP | ||

Event | 68 | 52 | 51 | 63 | 63 | 43 | 64 | 53 |

Range | 22 | 34 | 32 | 30 | 38 | 60 | 35 | 25 |

Mean | 8.97 | 11.71 | 12.42 | 10.07 | 10.05 | 14.74 | 9.92 | 12.10 |

Variance | 27.24 | 67.21 | 65.06 | 51.21 | 56.51 | 164.54 | 61.66 | 57.70 |

Std. Deviation | 5.22 | 8.19 | 8.07 | 7.16 | 7.52 | 12.83 | 7.85 | 7.60 |

Coef. Variation | 0.58 | 0.70 | 0.65 | 0.71 | 0.75 | 0.87 | 0.79 | 0.63 |

Std. Error | 0.64 | 1.15 | 1.14 | 0.92 | 0.95 | 1.98 | 0.99 | 1.05 |

Skewness | 0.82 | 1.09 | 0.58 | 1.10 | 1.51 | 1.64 | 1.28 | 0.52 |

Kurtosis | 0.98 | 1.17 | −0.23 | 0.78 | 3.23 | 3.25 | 1.29 | −0.82 |

*Dd*and

*Ds*as well as

*Ds*and

*Dp*were high while the correlations between

*Dd*and

*Dp*were small. The dependence analysis demonstrated it in Fig. 3 and Table 3 that the mutual dependence of drought variables are tremendous and hence suitable for building a regional joint distribution using copula functions.

Values of classical, Kendall and Spearman’s correlation coefficients for drought variables in different regions between 1961 and 2013

Region | Variable | Correlation Coefficient | ||||||||
---|---|---|---|---|---|---|---|---|---|---|

Classical | Kendall | Spearman | ||||||||

r | t (obs) | t (crit) | t | t (obs) | t (crit) | p | t (obs) | t (crit) | ||

NE |
| 0.67 | 7.27 | 1.997 | 0.69 | 7.70 | 1.997 | 0.67 | 7.35 | 1.997 |

| 0.63 | 6.59 | 0.50 | 4.69 | 0.59 | 5.92 | ||||

| 0.51 | 4.82 | 0.45 | 4.13 | 0.53 | 5.01 | ||||

NC |
| 0.78 | 8.93 | 2.009 | 0.70 | 6.87 | 2.009 | 0.78 | 8.73 | 2.009 |

| 0.75 | 7.99 | 0.61 | 5.43 | 0.77 | 8.40 | ||||

| 0.62 | 5.62 | 0.53 | 4.43 | 0.68 | 6.59 | ||||

CS |
| 0.72 | 7.33 | 2.010 | 0.62 | 5.49 | 2.010 | 0.65 | 5.96 | 2.010 |

| 0.65 | 5.92 | 0.53 | 4.32 | 0.64 | 5.80 | ||||

| 0.51 | 4.17 | 0.42 | 3.25 | 0.51 | 4.12 | ||||

SC |
| 0.76 | 9.05 | 2.000 | 0.72 | 8.15 | 2.000 | 0.73 | 8.32 | 2.000 |

| 0.72 | 8.10 | 0.57 | 5.43 | 0.68 | 7.24 | ||||

| 0.61 | 6.01 | 0.51 | 4.57 | 0.62 | 6.19 | ||||

IM |
| 0.83 | 11.76 | 2.000 | 0.72 | 8.13 | 2.000 | 0.74 | 8.67 | 2.000 |

| 0.80 | 10.52 | 0.58 | 5.59 | 0.71 | 7.83 | ||||

| 0.66 | 6.84 | 0.50 | 4.46 | 0.63 | 6.35 | ||||

NW |
| 0.53 | 3.97 | 2.020 | 0.58 | 4.57 | 2.020 | 0.65 | 5.40 | 2.020 |

| 0.49 | 3.61 | 0.54 | 4.13 | 0.67 | 5.72 | ||||

| 0.45 | 3.23 | 0.46 | 3.29 | 0.60 | 4.75 | ||||

TP |
| 0.67 | 7.15 | 1.999 | 0.69 | 7.49 | 1.999 | 0.70 | 7.72 | 1.999 |

| 0.62 | 6.24 | 0.53 | 4.97 | 0.63 | 6.32 | ||||

| 0.54 | 5.08 | 0.46 | 4.11 | 0.55 | 5.13 | ||||

EMC |
| 0.73 | 7.52 | 2.008 | 0.68 | 6.64 | 2.008 | 0.73 | 7.61 | 2.008 |

| 0.74 | 7.81 | 0.64 | 5.89 | 0.77 | 8.54 | ||||

| 0.66 | 6.21 | 0.56 | 4.79 | 0.69 | 6.75 |

### 4.2 Marginal Distribution for Inter-Arrival Time

*Dd*,

*Ds*, and

*Dp*(Shiau and Modarres, 2009; Song and Singh, 2010; Wong et al. 2010; Lee et al. 2013). The marginal distributions and associated parameters for

*Dd*,

*Ds*, and

*Dp*in different regions of mainland China have been documented (Ayantobo et al. 2017). These distributions showed that the empirical and theoretical distributions had an excellent agreement. In the case of D

*i*, many marginal distributions were also compared and the goodness of fit evaluated. According to Table 4, the KS and AD values were compared for each distribution, and their respective ranks in each region are shown. It was demonstrated that the KS was better than the AD test. Therefore, based on the KS test, the generalized Pareto was the best for regions NE, NC, CS, SC, NW, TP and EMC while gamma was the best for region IM because they had the smallest values. As given in Table 5, the parameters of the distributions were calculated using the MLE and after that used to fit the drought data.

Comparison of selected marginal distributions for drought inter-arrival time across various regions between 1961 and 2013

Region | Statistic | Marginal Distributions | |||||||||
---|---|---|---|---|---|---|---|---|---|---|---|

Exponential | Exponential (2P) | Gamma | Gamma (3P) | Gen. Extreme Value | Gen. Pareto | Inv. Gaussian | Inv. Gaussian (3P) | Weibull | Weibull (3P) | ||

NE |
| 0.262 | 0.261 | 0.206 | 0.293 | 0.174 | 0.171 | 0.237 | 0.199 | 0.193 | 0.281 |

| 5.964 | 32.273 | 2.326 | 23.388 | 1.630 | 13.303 | 5.753 | 1.908 | 1.689 | 32.72 | |

NC |
| 0.157 | 0.110 | 0.082 | 0.178 | 0.081 | 0.069 | 0.131 | 0.104 | 0.083 | 0.266 |

| 2.150 | 15.614 | 0.389 | 20.448 | 0.423 | 0.327 | 2.189 | 0.549 | 0.365 | 23.549 | |

CS |
| 0.156 | 0.164 | 0.138 | 0.219 | 0.119 | 0.089 | 0.188 | 0.150 | 0.129 | 0.125 |

| 2.393 | 10.904 | 1.202 | 18.117 | 0.669 | 4.198 | 4.468 | 1.013 | 0.626 | 9.105 | |

SC |
| 0.180 | 0.107 | 0.091 | 0.199 | 0.101 | 0.077 | 0.137 | 0.106 | 0.093 | 0.169 |

| 2.638 | 18.195 | 0.438 | 8.137 | 0.533 | 0.340 | 1.943 | 0.715 | 0.589 | 11.73 | |

IM |
| 0.180 | 0.138 | 0.101 | 0.250 | 0.114 | 0.109 | 0.172 | 0.157 | 0.105 | 0.195 |

| 2.609 | 18.641 | 0.713 | 14.031 | 0.873 | 0.651 | 2.354 | 1.291 | 0.832 | 23.380 | |

NW |
| 0.127 | 0.071 | 0.081 | 0.178 | 0.082 | 0.063 | 0.108 | 0.087 | 0.083 | 0.084 |

| 0.793 | 6.764 | 0.309 | 5.177 | 0.365 | 0.188 | 1.003 | 0.300 | 0.516 | 11.189 | |

TP |
| 0.183 | 0.118 | 0.119 | 0.153 | 0.134 | 0.092 | 0.152 | 0.114 | 0.138 | 0.133 |

| 2.095 | 12.806 | 0.864 | 7.275 | 1.054 | 0.593 | 1.824 | 0.957 | 1.309 | 19.441 | |

EMC |
| 0.166 | 0.135 | 0.112 | 0.301 | 0.092 | 0.061 | 0.159 | 0.092 | 0.077 | 0.089 |

| 2.751 g | 10.317i | 0.755f | 23.964j | 0.541c | 0.277a | 2.965 h | 0.603d | 0.398b | 0.750e |

The best-fitted marginal distributions with estimated parameters for different regions across mainland China between 1961 and 2013

Regions | Estimated Parameters for Marginal Distributions | |||
---|---|---|---|---|

β | α | λ | ||

NE | Generalized Pareto | −0.609 | 11.90 | 1.576 |

NC | Generalized Pareto | −0.305 | 13.493 | 1.363 |

CS | Generalized Pareto | −0.554 | 18.168 | 0.732 |

SC | Generalized Pareto | −0.205 | 10.351 | 1.473 |

IM | Gamma | 1.787 | 5.623 | |

NW | Generalized Pareto | −0.007 | 13.490 | 1.349 |

TP | Generalized Pareto | −0.063 | 9.183 | 1.286 |

EMC | Generalized Pareto | −0.499 | 16.267 | 1.249 |

### 4.3 Copula-Based Four-Variate Joint Distributions

*Dd*,

*Ds*,

*Dp*, and D

*i*. Generally, it was shown that symmetric copulas fit better than the asymmetric copulas because they had the lowest RMSE and AIC values in most regions. In the symmetric class, the RMSE and AIC values of Clayton and Frank were more or less the same but gave different fittings in the different region. Clayton copula was best for regions NE, CS and EMC while Frank copula was the best for NC, SC, IM, NW, and TP because they had the smallest values of RMSE and AIC.

Copula parameter, RMSE, AIC and BIAS of four-variate symmetric and asymmetric Archimedean copulas for different regions between 1961 and 2013

Copula | Statistic | Climatic region | EMC | ||||||
---|---|---|---|---|---|---|---|---|---|

NE | NC | CS | SC | IM | NW | TP | |||

CLAYTON (Symmetric) | θ | 3.790 | 4.940 | 3.680 | 5.300 | 6.280 | 5.940 | 5.490 | 7.340 |

RMSE | 0.063 | 0.033 | 0.027 | 0.044 | 0.042 | 0.040 | 0.037 | 0.039 | |

AIC | −161.214 | −152.218 | −157.561 | −168.865 | −171.183 | −118.484 | −180.949 | −147.231 | |

BIAS | −9.478 | −0.001 | −10.130 | −13.091 | −9.750 | −7.791 | −12.854 | 0.719 | |

CLAYTON (Asymmetric) | θ | 6.950 | 8.760 | 7.740 | 16.620 | 12.480 | 21.460 | 16.670 | 156.070 |

θ | 1.840 | 5.340 | 2.280 | 3.850 | 4.480 | 4.240 | 4.070 | 3.810 | |

θ | 0.530 | 0.710 | 0.620 | 0.900 | 0.890 | 0.670 | 0.740 | 0.900 | |

RMSE | 0.065 | 0.086 | 0.088 | 0.062 | 0.068 | 0.069 | 0.076 | 0.061 | |

AIC | −130.961 | −110.744 | −106.368 | −150.084 | −146.655 | −98.276 | −142.033 | −126.961 | |

BIAS | −0.009 | 1.671 | −1.672 | −4.602 | −10.242 | −1.817 | −4.679 | 4.024 | |

FRANK (Symmetric) | θ | 9.420 | 10.730 | 9.400 | 12.700 | 13.710 | 13.290 | 13.090 | 17.510 |

RMSE | 0.069 | 0.033 | 0.030 | 0.041 | 0.042 | 0.039 | 0.033 | 0.041 | |

AIC | −156.164 | −152.441 | −153.424 | −172.250 | −171.224 | −118.767 | −187.520 | −144.509 | |

BIAS | 7.382 | 7.027 | 4.839 | 1.522 | 4.878 | 2.392 | −0.474 | 6.744 | |

FRANK (Asymmetric) | θ | 6.480 | 5.810 | 6.920 | 4.650 | 7.560 | 5.380 | 4.440 | 5.920 |

θ | 2.160 | 1.937 | 2.307 | 1.550 | 2.520 | 1.793 | 1.480 | 1.973 | |

θ | 0.040 | 0.010 | 0.050 | 0.010 | 0.050 | 0.030 | 0.010 | 0.020 | |

RMSE | 0.069 | 0.088 | 0.067 | 0.063 | 0.074 | 0.064 | 0.069 | 0.066 | |

AIC | −114.474 | −81.448 | −86.125 | −108.598 | −104.214 | −73.069 | −107.608 | −89.746 | |

BIAS | 3.494 | −9.436 | 4.851 | −25.608 | −7.349 | −6.830 | −28.379 | −12.356 | |

GUMBEL (Asymmetric) | θ | 3.060 | 3.510 | 3.170 | 3.910 | 4.420 | 4.100 | 4.210 | 5.600 |

RMSE | 0.071 | 0.034 | 0.033 | 0.042 | 0.046 | 0.041 | 0.037 | 0.042 | |

AIC | −154.504 | −150.787 | −149.110 | −171.622 | −166.834 | −117.616 | −181.509 | −143.919 | |

BIAS | 3.752 | 5.474 | 1.436 | −1.373 | −0.750 | −0.186 | −3.009 | 4.119 | |

GUMBEL (Asymmetric) | θ | 25.750 | 10.150 | 21.730 | 10.330 | 7.880 | 15.080 | 12.830 | 16.010 |

θ | 12.840 | 0.860 | 1.030 | 0.690 | 11.870 | 0.820 | 0.800 | 0.850 | |

θ | 0.170 | 0.150 | 0.210 | 0.150 | 0.150 | 0.190 | 0.160 | 0.190 | |

RMSE | 0.332 | 0.348 | 0.295 | 0.356 | 0.351 | 0.331 | 0.342 | 0.347 | |

AIC | −86.922 | −68.057 | −84.822 | −75.979 | −76.923 | −66.505 | −80.373 | −73.817 | |

BIAS | 13.294 | 5.844 | −8.053 | 15.578 | 14.691 | −8.184 | 14.230 | 1.133 |

^{0}diagonal line uniquely. The maximum deviations from the diagonal line were observed for Gumbel plots, showing that the functions might not be suitable for building joint dependence. The symmetric Clayton in regions NE, CS and EMC, and symmetric Frank in regions NC, SC, IM, NW, and TP exhibited good agreement between the theoretical and empirical probabilities, and seemed very satisfying in modeling drought variables. Therefore, these copulas were thereafter selected for regional drought frequency analysis.

### 4.4 Four-Variate Joint Drought Frequency Analysis

#### 4.4.1 Regional Four-Variate Probabilities of Drought Events

*d*,

*s*,

*p*and

*i*were estimated and presented in Fig. 7. Next, the

*P*

_{DSPI}in each of these regions were obtained using Eq. (14). Fig. 7 shows the

*P*

_{DSPI}, representing the historical, 5-, 10-, 20-, 50- and 100-year return periods to provide adequate information about the potential drought risk associated with D

*d*, D

*s*, D

*p*and D

*i*across different regions. Overall, as the return period increased, the trend of the joint probability decreased. Hence, the simultaneous existence of higher degree drought events in different regions was less frequent as the year increases. For example, in sub-region I, considering 20- years univariate return period, the values of

*d*,

*s*,

*p*, and

*i*, were 13.56 months, 17.46, 2.44 and 20.73 months, respectively. The

*P*

_{DSPI}(

*Dd*≥ 13.56,

*Ds*≥ 17.46,

*Dp*≥ 2.44,

*Di*≥ 20.73)was 0.124.

These results could provide useful hints in appraising the drought risk in different regions. For instance, regions CS and NE had the highest probability of about 0.57, indicating that drought management and planning are needed within these regions. It was shown that regions SC and IM had a low probability of 0.48, which meant a low combination risk. The joint probability not only confirmed the occurrence of regional drought events but also gave a quantitative approach to analyze the probability of drought under varying *d*, *s*, *p* and *i* situations.

#### 4.4.2 Regional Four-Variate Return Period of Drought Events

The *ζ* values were 0.78, 1.02, 1.04, 0.84, 0.84, 1.23, 0.83 and 1.00 months for regions NE, NC, CS, SC, IM, NW, TP and EMC, respectively. Using the average historical and univariate return periods of *d*, *s*, *p* and *i* estimated for 5-, 10-, 20-, 50- and 100-year, the *T*_{DSPI} were obtained for each region. Figure 7 shows the *T*_{DSPI} for the historical, 5-, 10-, 20-, 50- and 100-year return periods. The graph showed that the *T*_{DSPI} trends increased with the year.

For example, in region NE, considering 20- years univariate return period, the values of *d*, *s*, *p*, and *i*, were 13.56 months, 17.46, 2.44 and 20.73 months, respectively. The *T*_{DSPI}, (*Dd* ≥ 13.56, *Ds* ≥ 17.46, *Dp* ≥ 2.44, *Di* ≥ 20.73)was 6.27 years. The*T*_{DSPI}were lowest in regions NE and CS. This implied that drought frequency would be higher in these regions compared to the other regions. The return period of drought variables obtained from univariate frequency analysis is higher than those by joint distribution for the AND case (*T*_{DSPI}). This showed that the univariate analysis does not furnish satisfactory knowledge about the drought risks associated with the four-variates. If engineers design hydraulic structures based on the results from univariate frequency analysis, the drought variables may be overestimated and this will lead to an increased cost of the structure. These results could be valuable in risk evaluation of water resources operations under severe and extreme drought situations.

The *P*_{DSPI} and *T*_{DSPI} showed that the result computed agreed with the actual data and confirmed that the selected copula functions for each region fitted the data well. Figure 7 further explained that for a particular recurrence interval, the *P*_{DSPI} showed a decreased trend while the *T*_{DSPI} showed an increased trend. For example, given the occurrence of *d*, *s*, *p*, and *i* in region NE, the estimated *P*_{DSPI} of 5- and 100-year plan drought are 0.332 and 0.029, respectively. Considering these illustrations, the calculated results seemed feasible. Therefore, from Fig. 7, the *P*_{DSPI} and *T*_{DSPI} of any other drought events in regions NE, NC, CS, SC, IM, NW, TP, and EMC can be obtained directly through interpolation.

#### 4.4.3 Spatial Distribution of Four-Variate Probability and Return Periods

*d*,

*s*,

*p*, and

*i*for EMC were 5.81 months, 4.96, 1.15 and 11.1 months respectively. Similar to the regional drought analysis, the station-based

*P*

_{DSPI}and

*T*

_{DSPI}of drought events exceeding these specific values, (

*i*.

*e*.,

*Dd*≥ 5.81,

*Ds*≥ 4.96,

*Dp*≥ 1.15,

*Di*≥ 11.1)were estimated for each station using the symmetric Frank copula. The spatial distribution of the

*P*

_{DSPI}and

*T*

_{DSPI}are mapped in Fig. 8.

The *P*_{DSPI}varied from 0.34 to 0.75. In this case, most of northern China experienced higher probability spreading from the northwest to northeast China. Notably, it seems that the *P*_{DSPI} were very high in NW and some stations in NC and TP. Based on the spatial pattern of drought events, the drought risk resulted from these regions are consistent. On the other hand, the results for southern China including SC and CS were slightly different as relatively lower *P*_{DSPI} was observed. Accordingly, counter measures of drought hazards need to be set for proactive actions especially in regions that had greater drought risks.

The *T*_{DSPI} ranged from 1.02 to 6.79 years. Long *T*_{DSPI}were found around NC while short *T*_{DSPI} dominated parts of CS, NW, and TP. Medium*T*_{DSPI}were found in eastern China, most especially NE, NC, CS. The spatial pattern of *P*_{DSPI} and *T*_{DSPI} suggested tremendous variations within different regions. In most cases, regions with high *P*_{DSPI}are often associated with short *T*_{DSPI} and vice versa. Severe droughts are most liable to eventuate in most of the northwestern and southwestern regions because of short *T*_{DSPI}. Also, because these areas are commercially advanced with huge populations, severe drought events could place grave danger on water resources in these regions. This requisite information is needed by water managers and other government agencies for adequate planning as well as management of water resources under severe drought situations. In this study, four-variate distribution was determined and successfully used.

## 5 Summary and Conclusions

Most meteorological phenomena are multiplicative, and the advantages of using a multivariate assessment strategy are evident in this study. In particular, drought events are characterized by *Dd*, *Ds*, *Dp,* and D*i* and the joint drought risk, which reveals their probability at the same time, is vital for management and planning of water resources under drought situations. Using the symmetric and asymmetric Archimedean copula functions, annual droughts were studied as four-dimensional phenomena incorporating these variables to build the joint four-variate distributions. Next, the *P*_{DSPI} for different *T*_{DSPI} in each region and EMC were then estimated.

The foremost outcomes of this research were summed as follows: (1) For D*i*, the generalized Pareto was the best distribution for regions NE, NC, CS, SC, NW, TP, and EMC while gamma was the best distribution for region IM. (2) The RMSE and AIC values were utilized to choose the suitable copula. We employed Clayton copula in regions NE, CS, and EMC, and Frank copula in regions NC, SC, IM, NW, and TP. Symmetric Frank copula was used for the station-based drought analysis. The selected copula functions provided the best fit for the dependence structures of *Dd*, *Ds*, *Dp,* and D*i*, and consequently, it was adopted for the joint risk drought analysis. (3) The risk of drought event is determined based on *P*_{DSPI} and *T*_{DSPI}, which furnished vital information for drought analysis. For instance, by using only the univariate information provided by either *Dd*, *Ds*, *Dp,* and D*i*, may yield under or overestimates of the actual drought state, and, in turn, of the corresponding risk. (4) Analyzing the probabilities and return periods of four-variate drought events, the trends or changes in drought events were then assessed over time. The concept of return period used in this study is a familiar concept in the community of water resources professionals. However, for a deep understanding of drought event, the ultimate evaluation was based on the hazard which the event possesses, and the vulnerability of this region to the drought event. Therefore, based on the results of our analysis, it is reasonable to increase drought control concerns especially in regions with very high *P*_{DSPI} and low *T*_{DSPI}. As a recommendation, a drought return period of below ten years seems to indicate that the drought is entering an alert or a dangerous state.

This study added to a better valuable knowledge in the sphere of disaster management, especially concerning the appraisal of drought issues and the performance of full drought-risk investigations. Our insights on the real-time evaluation of meteorological droughts of different regions may provide valuable information about the possible evolution of drought episode and also help the water manager to plan effective mitigation strategies. As a conclusion, in this paper, a multivariate frequency analysis of meteorological droughts in seven climatic regions is addressed using Copulas. Such an approach is flexible, comprehensive and offered other several advantages over previous definitions of multivariate frequency analysis (Salvadori et al., 2015). The copula-derived drought analysis considers complete interdependencies between the drought variables over a geographical region. Finally, in this paper, no consideration has been included for the climate change influence on the above analysis although climate change will affect the estimation of the return period of droughts. For this aspect, rigorous and systematic attempts should be made for estimating the anticipated changes in the meteorological parameters which directly or indirectly affect the *Dd*, *Ds*, *Dp,* and D*i* and ultimately the frequency of drought events.

## Notes

### Acknowledgments

This work was supported by the National Key Research and Development Program of China (grant number 2017YFC0403303), China Natural Science Foundation (No. U1203182), and the China 111 project (B12007).

### Compliance with Ethical Standards

### Conflict of Interest

The authors declare that they have no conflict of interest.

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