## Abstract

The potential of the copula method to construct the joint probability distribution of three hydrological variables characterizing water supply and demand (WSD) is explored for the Luhun irrigation district of China. The marginal distributions of rainfall, reference crop evapotranspiration (*ET*_{0}) and irrigation water are simulated by the corresponding best-fitting cumulative distribution functions. Furthermore, the correlations between every pair of variables are quantified. On this basis, the two-dimensional joint distributions of rainfall and (*ET*_{0}) (representing natural WSD), and irrigation water and (*ET*_{0}) (representing man-made WSD), and the three-dimensional joint distribution of rainfall, irrigation water, and (*ET*_{0}) (representing natural–man-made WSD) are established. The results reveal that the best-fitting marginal distributions for rainfall and (*ET*_{0}) and irrigation water are the normal distribution and the Weibull distribution. Moreover, for rainfall and (*ET*_{0}), the Student's *t* copula is applied to obtain the joint distribution, while the corresponding copula for (*ET*_{0}) and irrigation water is the Clayton copula. Finally, the three-dimensional Student's *t* copula is selected to explore the dependence structure among rainfall, irrigation water, and (*ET*_{0}). Therefore, these joint distributions provide an efficient approach to assess water shortage risks in the irrigation district.

## INTRODUCTION

The water supply to an irrigation district comprises rainfall and irrigation water. Rainfall can be called natural water, while irrigation water may be denoted as man-made water. Rainfall and irrigation water are consumed by crops to meet their water requirement. Typically, for crops in an irrigation district, reference crop evapotranspiration indicates the crop water demand. Affected, Rainfall and are random variables as they are affected by the meteorological system and underlying surface conditions. From the perspective of hydrostatistics, irrigation water that represents an artificial water resource in an irrigation district can also be considered as a random variable. In order to describe the associated dependence structure among water supply and demand (WSD) in an irrigation district, Ding *et al.* (2011) and Zhang *et al.* (2013a, 2013b) used the Frank copula to construct the joint probability of rainfall and in an irrigation district. Zhang *et al.* (2016) indicated that it is feasible to apply the Student's *t* copula for the statistical analyses of natural–man-made water supply and water demand in an irrigation district, using the data series of rainfall, , and irrigation water.

Currently, extensively applied copula functions involve Symmetry Archimedean copula (Song *et al.* 2012) and Elliptical copula (Genest *et al.* 2007), which can reveal the dependence structure by connecting the univariate marginal distributions with the multivariate joint distribution. Notably, copula functions can be used flexibly because all variables need not obey the same marginal probability distribution or need not be transformed to follow the normal distribution (Zhang & Singh 2007).

In the context of joint probability distribution (JPD), the simplicity and capacity of the One-parameter Symmetry Archimedean copula (Song *et al.* 2012) and Elliptical copula (Genest *et al.* 2007) are evident from studies on the probabilistic characterization of drought (Guttman 1998; Dalezios *et al.* 2000; Shiau 2006; Kao & Govindaraju 2010; Xu *et al.* 2015), sea storm analysis (Michele *et al.* 2007; Corbella & Stretch 2013; Montes-Iturrizaga & Heredia-Zavoni 2015), flood risk analysis (Fu & Butler 2014; Masina *et al.* 2015; Yang *et al.* 2017), runoff and sediment (Xiong *et al.* 2014; Zhou *et al.* 2014; Huang *et al.* 2017; Wang *et al.* 2017) and streamflow simulation (Chen *et al.* 2015; Jeong & Lee 2015; Liu *et al.* 2017). Zhang *et al.* (2013a, 2013b) and Fan *et al.* (2016) provided a detailed theoretical background and methodological descriptions for applying copula to hydrology and water resources. Similarly, in this paper, the JPDs of the natural, man-made and natural–man-made water supply and demand for an irrigation district are explored using Symmetry Archimedean copula and Elliptical copula.

The main objective of this study is to explore the most appropriate copula function to establish the multivariate JPDs of WSD under different water supply conditions using the data series of rainfall, , and irrigation water from 1970 to 2013 in the Luhun irrigation district of Henan Province, China. The paper is structured as follows: the section titled ‘Methods’ introduces eight cumulative probability distribution (CPD) functions as well as the copula methods. The data series of rainfall, , and irrigation water from 1970 to 2013 in the Luhun irrigation district are presented in ‘Data series’. The section titled ‘Results and analysis’ estimates the univariate marginal distributions, and the JPDs of WSD in the irrigation district under different water supply conditions. The final section presents the conclusions.

## METHODS

### Cumulative distribution function

The most popular eight CPD functions that have been widely applied for hydrological variables include the normal, exponential, two-parameter gamma, Pearson III Type (P-III), generalized extreme value (GEV), Generalized Pareto (GP), lognormal and Weibull distributions (Song *et al.* 2012).

### Correlation of variables

Prior to introducing the copula method, the correlation of variables needs to be estimated. The three most widely used coefficients are Pearson's correlation coefficient , Kendall's correlation coefficient , and Spearman's correlation coefficient .

*et al.*2012): where is the sample, = 1,2, … ,

*n*; , are the mean values of and

*y*, respectively; and

*n*is the total number of observations.

*et al.*2012): where denotes ‘ choose ’,

*n*is the total number of observations (, ); and

*i*, = 1,2,3, … , . The denotes the symbol function, which is defined as

*et al.*2012): where

*n*is the sample size; , are the ranks of

*x*and

*y*, respectively; , are the means of and , respectively; and .

### Copula method

*H*is an -dimensional distribution function of -dimensional random variables defined in

*R*, and are their marginal distributions respectively, for any ( is a specific value of , = 1, 2, … , n) an n-dimensional copula function exists and can be expressed as follows: where

*n*is the number of variables, ( = 1, 2, … , n), is the copula function, and is the parameter of interest. If is continuous and ( = 1, 2, … , n) subjects to the uniform distribution on [0, 1],

*C*is uniquely determined.

*et al.*2012).The Gaussian and Student's

*t*copulas are commonly used Elliptical copulas in these contexts. The bivariate and trivariate Gaussian copula functions are given by Equation (6) and Equation (7), respectively: where is the marginal function of the variables; , , , and is estimated using the maximum likelihood estimation method.

*d*refers to the variable's dimensions; is the inverse function of the standard normal distribution;

*w*is the integrand variable matrix, and .

### Identification and goodness-of-fit evaluation

*et al.*2018; Mohammad & Saeed 2018):

## RESULTS AND ANALYSIS

### Data series

The Luhun irrigation district (112°5′-113°5′E, 34°10′-34°45′N) is located in the hilly area of western Henan Province, China. It comprises a total irrigation area of 1,838.48 km^{2}, as shown in Figure 1 (Zhang *et al.* 2018). Its annual average rainfall is only 611.02 mm but the annual evaporation reaches 1,034.32 mm, making the area prone to meteorological droughts. The volume of annual irrigation water for the Luhun irrigation district is 1.82 × 10^{8} m^{3}. The water is sourced from the Luhun reservoir built on the upper reaches of the Yihe River, which is the secondary tributary of the Yellow River.

The researched data series for the Luhun irrigation district comprise the daily meteorological data (including rainfall, wind speed, maximum temperature, minimum temperature, relative humidity, net radiation, etc.) and irrigation water data from 1970 to 2013. The Penman–Monteith formula (Luo *et al.* 2008) is used to calculate . The annual data series of rainfall, , and irrigation water in the Luhun irrigation district appear in Table 1, while their statistical characteristics are provided in Table 2.

Year . | Rainfall (mm) . | ET_{0} (mm)
. | Irrigation water (10^{8} m^{3})
. | Year . | Rainfall (mm) . | ET_{0} (mm)
. | Irrigation water (10^{8} m^{3})
. |
---|---|---|---|---|---|---|---|

1970 | 557.850 | 1,067.814 | 2.012 | 1992 | 679.100 | 1,009.013 | 1.699 |

1971 | 557.650 | 1,103.150 | 1.996 | 1993 | 559.000 | 972.071 | 1.225 |

1972 | 521.750 | 1,081.929 | 2.108 | 1994 | 718.900 | 1,052.835 | 1.596 |

1973 | 748.050 | 1,065.220 | 1.962 | 1995 | 548.600 | 1,072.172 | 2.489 |

1974 | 648.850 | 1,102.339 | 2.121 | 1996 | 628.300 | 981.695 | 1.375 |

1975 | 579.050 | 1,066.823 | 0.705 | 1997 | 380.600 | 1,077.800 | 2.886 |

1976 | 579.200 | 1,057.709 | 1.219 | 1998 | 781.800 | 993.259 | 1.203 |

1977 | 575.400 | 1,087.760 | 1.590 | 1999 | 616.900 | 1,043.432 | 1.679 |

1978 | 471.700 | 1,116.644 | 2.189 | 2000 | 637.100 | 1,035.435 | 1.404 |

1979 | 568.250 | 1,073.022 | 2.117 | 2001 | 401.800 | 1,109.791 | 2.430 |

1980 | 663.050 | 995.547 | 2.976 | 2002 | 599.300 | 1074.932 | 1.964 |

1981 | 456.350 | 1,073.982 | 2.900 | 2003 | 953.900 | 903.766 | 0.720 |

1982 | 611.050 | 1,013.299 | 2.821 | 2004 | 767.400 | 1,045.761 | 2.020 |

1983 | 890.000 | 990.806 | 1.000 | 2005 | 728.800 | 1,060.298 | 2.450 |

1984 | 836.900 | 942.560 | 1.102 | 2006 | 692.600 | 1,059.791 | 1.240 |

1985 | 666.950 | 971.262 | 1.710 | 2007 | 596.400 | 1,071.822 | 1.733 |

1986 | 382.750 | 1,059.945 | 2.538 | 2008 | 658.200 | 1,080.310 | 2.550 |

1987 | 571.000 | 1,008.485 | 0.702 | 2009 | 762.500 | 1,086.360 | 1.650 |

1988 | 580.400 | 989.792 | 1.239 | 2010 | 600.300 | 1,092.342 | 1.820 |

1989 | 551.400 | 920.345 | 1.669 | 2011 | 706.500 | 1,101.749 | 1.780 |

1990 | 702.550 | 911.865 | 0.603 | 2012 | 498.700 | 1,168.252 | 1.997 |

1991 | 353.750 | 944.680 | 3.290 | 2013 | 294.400 | 1,017.433 | 2.372 |

Year . | Rainfall (mm) . | ET_{0} (mm)
. | Irrigation water (10^{8} m^{3})
. | Year . | Rainfall (mm) . | ET_{0} (mm)
. | Irrigation water (10^{8} m^{3})
. |
---|---|---|---|---|---|---|---|

1970 | 557.850 | 1,067.814 | 2.012 | 1992 | 679.100 | 1,009.013 | 1.699 |

1971 | 557.650 | 1,103.150 | 1.996 | 1993 | 559.000 | 972.071 | 1.225 |

1972 | 521.750 | 1,081.929 | 2.108 | 1994 | 718.900 | 1,052.835 | 1.596 |

1973 | 748.050 | 1,065.220 | 1.962 | 1995 | 548.600 | 1,072.172 | 2.489 |

1974 | 648.850 | 1,102.339 | 2.121 | 1996 | 628.300 | 981.695 | 1.375 |

1975 | 579.050 | 1,066.823 | 0.705 | 1997 | 380.600 | 1,077.800 | 2.886 |

1976 | 579.200 | 1,057.709 | 1.219 | 1998 | 781.800 | 993.259 | 1.203 |

1977 | 575.400 | 1,087.760 | 1.590 | 1999 | 616.900 | 1,043.432 | 1.679 |

1978 | 471.700 | 1,116.644 | 2.189 | 2000 | 637.100 | 1,035.435 | 1.404 |

1979 | 568.250 | 1,073.022 | 2.117 | 2001 | 401.800 | 1,109.791 | 2.430 |

1980 | 663.050 | 995.547 | 2.976 | 2002 | 599.300 | 1074.932 | 1.964 |

1981 | 456.350 | 1,073.982 | 2.900 | 2003 | 953.900 | 903.766 | 0.720 |

1982 | 611.050 | 1,013.299 | 2.821 | 2004 | 767.400 | 1,045.761 | 2.020 |

1983 | 890.000 | 990.806 | 1.000 | 2005 | 728.800 | 1,060.298 | 2.450 |

1984 | 836.900 | 942.560 | 1.102 | 2006 | 692.600 | 1,059.791 | 1.240 |

1985 | 666.950 | 971.262 | 1.710 | 2007 | 596.400 | 1,071.822 | 1.733 |

1986 | 382.750 | 1,059.945 | 2.538 | 2008 | 658.200 | 1,080.310 | 2.550 |

1987 | 571.000 | 1,008.485 | 0.702 | 2009 | 762.500 | 1,086.360 | 1.650 |

1988 | 580.400 | 989.792 | 1.239 | 2010 | 600.300 | 1,092.342 | 1.820 |

1989 | 551.400 | 920.345 | 1.669 | 2011 | 706.500 | 1,101.749 | 1.780 |

1990 | 702.550 | 911.865 | 0.603 | 2012 | 498.700 | 1,168.252 | 1.997 |

1991 | 353.750 | 944.680 | 3.290 | 2013 | 294.400 | 1,017.433 | 2.372 |

Variable Statistical characteristics . | Rainfall (mm) . | (mm) . | Irrigation water (10^{8} m^{3})
. |
---|---|---|---|

Mean | 611.020 | 1,039.890 | 1.815 |

Maximum | 953.900 | 1,168.250 | 3.290 |

Minimum | 294.400 | 903.770 | 0.603 |

Standard deviation | 135.668 | 58.996 | 0.651 |

Skewness | 0.024 | −0.574 | 0.084 |

Kurtosis | 0.435 | −0.110 | −0.469 |

Variable Statistical characteristics . | Rainfall (mm) . | (mm) . | Irrigation water (10^{8} m^{3})
. |
---|---|---|---|

Mean | 611.020 | 1,039.890 | 1.815 |

Maximum | 953.900 | 1,168.250 | 3.290 |

Minimum | 294.400 | 903.770 | 0.603 |

Standard deviation | 135.668 | 58.996 | 0.651 |

Skewness | 0.024 | −0.574 | 0.084 |

Kurtosis | 0.435 | −0.110 | −0.469 |

It can be seen that: (1) the inter-annual variation of rainfall, , and irrigation water are all uneven; (2) the volume for exceeds that for rainfall, which means more irrigation water is needed in the irrigation district; and (3) typically, has a negative correlation with rainfall but a positive correlation exists with the irrigation water. The statistical tests on stationarity, consistency and randomness were conducted as described by Li (2016).

### Marginal distributions of rainfall, *ET*_{0}, and irrigation water

The marginal distributions of rainfall, , and irrigation water are fitted by the aforementioned eight distributions. Using the maximum likelihood method, the parameters of the eight distributions are estimated, and the results are exhibited in Table 3.

Variable Distribution and parameters . | Rainfall (mm) . | (mm) . | Irrigation water (10^{8} m^{3})
. | |
---|---|---|---|---|

Normal | 611.023 137.237 | 1,039.900 59.678 | 1.815 0.676 | |

Exponential | 611.023 | 1,039.900 | 1.815 | |

Gamma | 18.823 32.462 | 303.095 3.431 | 6.451 0.281 | |

P-III | 0.240 0.030 | 0.060 0.010 | 0.350 0.090 | |

GEV | 561.679 −0.259 | 1,021.700 −0.400 | 1.576 −0.273 | |

GP | 135.935 −1.200 | 62.500 −1.700 | 0.656 −1.023 | |

Weibull | 1,148.300 664.660 | 2,019.200 1,066.400 | 3.366 2.036 | |

Lognormal | 4.914 | 20.900 | 2.998 |

Variable Distribution and parameters . | Rainfall (mm) . | (mm) . | Irrigation water (10^{8} m^{3})
. | |
---|---|---|---|---|

Normal | 611.023 137.237 | 1,039.900 59.678 | 1.815 0.676 | |

Exponential | 611.023 | 1,039.900 | 1.815 | |

Gamma | 18.823 32.462 | 303.095 3.431 | 6.451 0.281 | |

P-III | 0.240 0.030 | 0.060 0.010 | 0.350 0.090 | |

GEV | 561.679 −0.259 | 1,021.700 −0.400 | 1.576 −0.273 | |

GP | 135.935 −1.200 | 62.500 −1.700 | 0.656 −1.023 | |

Weibull | 1,148.300 664.660 | 2,019.200 1,066.400 | 3.366 2.036 | |

Lognormal | 4.914 | 20.900 | 2.998 |

The K–S test is used to identify the eight marginal distributions. Assuming a significance level of 95%, the corresponding fractile value is 0.20056. Table 4 gives the goodness-of-fit evaluations of the eight marginal distributions for rainfall, , and irrigation water. Irrespective of the variable (rainfall, or irrigation water), the values of the K–S statistics D of the exponential and GP distributions exceed 0.20056, and hence the exponential and GP distributions are inappropriate for the situation at hand.

Distribution Variable . | Statistical evaluation . | Gamma . | Lognormal . | GP . | Exponential . | P-III . | Normal . | GEV . | Weibull . |
---|---|---|---|---|---|---|---|---|---|

Rainfall (mm) | K–S | 0.147 | 0.164 | 0.304 | 0.418 | 0.132 | 0.120 | 0.128 | 0.118 |

OLS | 0.045 | 0.053 | – | – | 0.045 | 0.039 | 0.042 | 0.045 | |

(mm) | K–-S | 0.169 | 0.171 | 0.569 | 0.581 | 0.158 | 0.163 | 0.140 | 0.115 |

OLS | 0.067 | 0.068 | - | - | 0.063 | 0.064 | 0.055 | 0.045 | |

Irrigation water (10^{8} m^{3}) | K–S | 0.107 | 0.133 | 0.201 | 0.326 | 0.088 | 0.075 | 0.076 | 0.070 |

OLS | 0.082 | 0.049 | 0.100 | - | 0.034 | 0.030 | 0.031 | 0.029 |

Distribution Variable . | Statistical evaluation . | Gamma . | Lognormal . | GP . | Exponential . | P-III . | Normal . | GEV . | Weibull . |
---|---|---|---|---|---|---|---|---|---|

Rainfall (mm) | K–S | 0.147 | 0.164 | 0.304 | 0.418 | 0.132 | 0.120 | 0.128 | 0.118 |

OLS | 0.045 | 0.053 | – | – | 0.045 | 0.039 | 0.042 | 0.045 | |

(mm) | K–-S | 0.169 | 0.171 | 0.569 | 0.581 | 0.158 | 0.163 | 0.140 | 0.115 |

OLS | 0.067 | 0.068 | - | - | 0.063 | 0.064 | 0.055 | 0.045 | |

Irrigation water (10^{8} m^{3}) | K–S | 0.107 | 0.133 | 0.201 | 0.326 | 0.088 | 0.075 | 0.076 | 0.070 |

OLS | 0.082 | 0.049 | 0.100 | - | 0.034 | 0.030 | 0.031 | 0.029 |

### Correlation between rainfall, *ET*_{0}, and irrigation water

The correlation coefficients between rainfall, , and irrigation water in the Luhun irrigation district are presented in Table 5. It can be seen that a negative correlation exists not only between rainfall and but also between and irrigation water, while a positive correlation exists only between and irrigation water. Moreover, a strong correlation exists between every pair of variables. Thus, the copula method can be applied to the given situation.

Correlation coefficient . | Rainfall and . | Rainfall and irrigation . | and irrigation water . |
---|---|---|---|

r | −0.3244 | −0.5364 | 0.3225 |

τ | −0.2389 | −0.3383 | 0.2622 |

ρ | −0.3353 | −0.4698 | 0.3875 |

Correlation coefficient . | Rainfall and . | Rainfall and irrigation . | and irrigation water . |
---|---|---|---|

r | −0.3244 | −0.5364 | 0.3225 |

τ | −0.2389 | −0.3383 | 0.2622 |

ρ | −0.3353 | −0.4698 | 0.3875 |

### The JPD of WSD under different water supply conditions

#### Under the natural water supply condition

The copula method is applied to attain the JPDs of rainfall and for the Luhun irrigation district. Because of the negative correlation between rainfall and , the Frank, Gaussian and Student's *t* copulas can be used as the copula functions. The parameter θ, Kendall's rank correlation coefficient *τ*, and the identified statistics of the Frank, Gaussian and Student's *t* copulas are estimated in Table 6. R^{2} stands for the correlation coefficients of the empirical probability distributions and calculated probability distributions.

Kendall's τ
. | Parameters or identified statistics . | Copula functions . | ||
---|---|---|---|---|

Frank . | Gaussian . | Student's t
. | ||

− 0.2389 | θ | −2.2559 | −0.3217 | = −0.3322 = 9.5928 |

D | 0.0969 | 0.1068 | 0.1042 | |

OLS | 0.0424 | 0.0425 | 0.0424 | |

BIAS | −0.0174 | −0.0168 | −0.0164 | |

AIC | −276.0700 | −275.9600 | −276.1000 | |

R ^{2} | 0.9435 | 0.9418 | 0.9428 |

Kendall's τ
. | Parameters or identified statistics . | Copula functions . | ||
---|---|---|---|---|

Frank . | Gaussian . | Student's t
. | ||

− 0.2389 | θ | −2.2559 | −0.3217 | = −0.3322 = 9.5928 |

D | 0.0969 | 0.1068 | 0.1042 | |

OLS | 0.0424 | 0.0425 | 0.0424 | |

BIAS | −0.0174 | −0.0168 | −0.0164 | |

AIC | −276.0700 | −275.9600 | −276.1000 | |

R ^{2} | 0.9435 | 0.9418 | 0.9428 |

These three copulas are identified using K–S test statistic D. Taking the significance level as , when , the corresponding fractile value of the K–S statistic is . Evidently, all three copulas pass the K–S test (Table 3). Although the values of are quite similar, the value of of the Student's *t* copula is the least, and so the Student's *t* copula is the best option to describe the dependence structure of rainfall and . Besides, Student's *t* copula also has the highest accuracy in terms of BIAS = −0.0164 compared to the other models.

Additionally, it can be seen that the Student's *t* copula has higher correlation between the computed copula and the empirical copulas compared with the other two copulas, which confirms that the Student's *t* copula is the most suitable option in the context. Using the Student's *t* copula, the JPD of rainfall and is obtained as a surface plot in Figure 3(a1) and a contour plot in Figure 3(b1).

#### Under the man-made water supply condition

In order to construct the JPD of and irrigation water, five bivariate copulas– that is, the Frank, Clayton, Gumbel, Gaussian, and Student's *t* copulas– are chosen to depict the dependence structure between the two variables.

Using the previous formulas, the parameters and K–S test statistic D are estimated, and the goodness-of-fit is assessed by using *OLS*, *BIAS* and *AIC* (shown in Table 7). It can be seen that the statistic D is lower than the fractile value D_{0} = 0.20056, which indicates that the chosen five copulas pass the K–S test and thus, they can all be accepted. However, the Clayton copula is considered as the most appropriate one because it provides the lowest values of *OLS* and *AIC*. Besides, the Clayton copula also has the highest accuracy in terms of BIAS = −0.0126 compared to other models. In addition, it can be observed that the Clayton copula performs better, as evidenced by the higher correlation coefficient (R^{2}) between the empirical copula and the calculated copula. Thus, it is reasonable that the Clayton copula be selected to describe the JPD of and irrigation water as a surface plot, displayed in Figure 3(a2), and a contour plot in Figure 3(b2).

Kendall's τ
. | Parameter . | Copula functions . | ||||
---|---|---|---|---|---|---|

Frank . | Clayton . | Gumbel . | Gaussian . | Student's t
. | ||

0.2622 | θ | 2.5015 | 0.7108 | 1.3554 | 0.3048 | = 0.3832 = 4.5056 |

D | 0.1053 | 0.0937 | 0.1129 | 0.0972 | 0.1047 | |

OLS | 0.0383 | 0.0349 | 0.0402 | 0.0413 | 0.0380 | |

BIAS | −0.0158 | −0.0126 | −0.0161 | −0.0162 | −0.0158 | |

AIC | −285.1108 | −293.2372 | −280.8594 | −278.4447 | −285.7734 | |

R ^{2} | 0.9811 | 0.9798 | 0.9798 | 0.9806 | 0.9805 |

Kendall's τ
. | Parameter . | Copula functions . | ||||
---|---|---|---|---|---|---|

Frank . | Clayton . | Gumbel . | Gaussian . | Student's t
. | ||

0.2622 | θ | 2.5015 | 0.7108 | 1.3554 | 0.3048 | = 0.3832 = 4.5056 |

D | 0.1053 | 0.0937 | 0.1129 | 0.0972 | 0.1047 | |

OLS | 0.0383 | 0.0349 | 0.0402 | 0.0413 | 0.0380 | |

BIAS | −0.0158 | −0.0126 | −0.0161 | −0.0162 | −0.0158 | |

AIC | −285.1108 | −293.2372 | −280.8594 | −278.4447 | −285.7734 | |

R ^{2} | 0.9811 | 0.9798 | 0.9798 | 0.9806 | 0.9805 |

#### Under the natural–man-made WSD condition

The JPDs of rainfall, , and irrigation water are established using the Frank, Clayton, Gumbel, Gaussian, and Student's *t* copulas. Next, the corresponding parameters are estimated using the maximum likelihood method. Then, the best-fitted joint CPDs are identified as follows. Firstly, the K–S test is used to obtain the available copula functions. Secondly, the minimum deviation squares and AIC are employed to determine the best-fitted joint distribution. Table 8 shows the results of the parameters θ, K–S test statistic *D*, *OLS* and *AIC.* The findings reveal that: (1) all the values of K–S test statistic D are less than D_{0} (0.20056), which illustrates that all the tested copula functions are appropriate to establish the dependence structure among rainfall, , and irrigation water; (2) for the three-dimensional joint distribution, the Student's *t* copula is the most suitable as it provides the lowest values of *OLS*, *AIC*, from among all copula functions. Besides, Student's *t* copula also has the highest accuracy in terms of BIAS = −0.0131 compared to other models.

Parameters . | Copula functions . | ||||
---|---|---|---|---|---|

Frank . | Gumbel . | Clayton . | Gaussian . | Student's t
. | |

−0.7929 | 0.9938 | −0.0438 | = −0.3217 = −0.5320 = 0.3048 | = −0.3404 = −0.5209 = 0.3391 = 17.9123 | |

D | 0.1361 | 0.1604 | 0.1560 | 0.1177 | 0.1137 |

OLS | 0.0452 | 0.0500 | 0.0489 | 0.0410 | 0.0404 |

BIAS | −0.0135 | −0.0130 | −0.0152 | −0.0135 | −0.0131 |

AIC | −270.4200 | −261.6200 | −263.5200 | −275.0900 | −276.3100 |

R ^{2} | 0.8668 | 0.8687 | 0.8191 | 0.8084 | 0.8108 |

Parameters . | Copula functions . | ||||
---|---|---|---|---|---|

Frank . | Gumbel . | Clayton . | Gaussian . | Student's t
. | |

−0.7929 | 0.9938 | −0.0438 | = −0.3217 = −0.5320 = 0.3048 | = −0.3404 = −0.5209 = 0.3391 = 17.9123 | |

D | 0.1361 | 0.1604 | 0.1560 | 0.1177 | 0.1137 |

OLS | 0.0452 | 0.0500 | 0.0489 | 0.0410 | 0.0404 |

BIAS | −0.0135 | −0.0130 | −0.0152 | −0.0135 | −0.0131 |

AIC | −270.4200 | −261.6200 | −263.5200 | −275.0900 | −276.3100 |

R ^{2} | 0.8668 | 0.8687 | 0.8191 | 0.8084 | 0.8108 |

Furthermore, the Student's *t* copula has a higher correlation coefficient (R^{2}), as exhibited in Table 5. Therefore, the Student's *t* copula can best characterize the dependence structure and should be used to construct the joint distribution slices plot, as exhibited in Figure 3(a3) and surface plot, as seen in Figure 3(b3).

For IR representing irrigation water, compared with the bivariate JPD of (rainfall, ) (Figure 3(a1) and 3(b1)) and (, IR) (Figure 3(a2) and 3(b2)), the trivariate JPD of (rainfall, , IR) (Figure 3(a3) and 3(b3)) is quite different. The superposition from the bivariate JPD of (rainfall, ) and (, IR) to the trivariate JPD of (rainfall, , IR), is not simple. For example, it can be seen from Figure 3(b1) that when the value of JPD is between 0.1 and 0.3, the contour plot is relatively sparse and (rainfall, ) change more obviously compared with that between 0.3 and 0.7; as for (, IR) in Figure 3(b2), the value of JPD is relatively sparse from 0.1 to 0.4 and (,IR) change more obviously compared with that between 0.4 and 0.7. For the trivariate JPD of pair (rainfall, , IR), when the value of JPD is between 0.1 and 0.2 or 0.8 and 0.9, the contour surface has larger spacing than that for JPD between 0.2 and 0.8. Thus, we conclude that considering the trivariate JPD can lead to a more accurate result.

The joint return period can be obtained with the constructed JPD of rainfall, and irrigation water. Moreover, the encountered events of (rainfall,, IR) can be obtained under the same return period. Thus, the exceeded risk of any given planning indicator in the encountered events can be calculated, and the risk scope of a planning indicator corresponding to a return period can also be provided. For example, if an appropriate distribution of (rainfall, , IR) in a typical year is determined, different groups of designed rainfall, and irrigation water surfaces can be obtained simultaneously using the same frequency amplification method. This understanding will provide an added guidance for irrigation planning.

## CONCLUSIONS

The study describes the joint behaviour of three hydrological variables (rainfall, and irrigation water) as a 44-year (1970–2013) time series of WSD in the Luhun irrigation district. In reality, the length of the time series of the three hydrological variables (rainfall, , and irrigation water) is only 44 years, and the results would be more accurate if the time series were expanded. However, the results with the 44-year time series are still feasible and can be used to explain the statistical characteristics of rainfall, , and irrigation water. The joint behaviours of these hydrological variables were investigated using the copula method, estimated marginal distributions, and their correlations. The findings imply that the trivariate JPD outperforms the bivariate distribution in terms of reflecting the water shortage risk in reality. Among the eight popular CPD functions that were tested, the marginal distribution of rainfall can be best conveyed by a normal distribution, while for and irrigation water, the Weibull distribution is the best choice. Moreover, exhibits a negative correlation with rainfall, as well as irrigation water. However, a positive correlation exists between and irrigation water.

After selection of the Symmetry Archimedean copulas and Elliptical copulas, the joint distributions of the natural WSD (involving rainfall and ) and the natural–man-made WSD (involving rainfall, irrigation water and ) can be adequately described by the Student's *t* copula. However, the dependence structure of and irrigation water, representing the man-made water demand and supply, can be well presented by the Clayton copula. This study attempted to identify the best general method for modelling the dependence structure among trivariates and showed that the copula is a robust method for water shortage risk assessment in the irrigation district.

## ACKNOWLEDGEMENTS

This research is supported by the National Key R&D Program of China (Grant No. 2018YFC0406501), Outstanding Young Talent Research Fund of Zhengzhou University (Grant No. 1521323002), Program for Innovative Talents (in Science and Technology) at University of Henan Province (Grant No. 18HASTIT014), State Key Laboratory of Hydraulic Engineering Simulation and Safety, Tianjin University (Grant No. HESS-1717), and Foundation for University Key Teacher by Henan Province of China in 2017. We are thankful for the work of editorial office and suggestions from anonymous reviewers.

## REFERENCES

*Master Thesis*

Research on the Relationship Between Water Supply and Demand of Irrigation Area with Multi-Time Scales