## Abstract

For the irrigation district, irrigation water is the manual water supply for the farmland while reference crop evapotranspiration (ET_{0}) reflects water demand. Thus, the joint distribution of irrigation water and ET_{0} can reveal water shortage risk under the condition of the manual water supply. In order to understand their relationships and overcome the drawbacks of different marginal distributions of hydrological variables, Archimedean copulas are introduced. Based on the data series of ET_{0} and irrigation water in the Luhun irrigation district of China, the univariate marginal distributions of ET_{0} and irrigation water are first selected. Then, with the Gumbel–Hougaard copula in the Archimedean copulas, the joint distribution of ET_{0} and irrigation water is proposed. The results show that the best-fitting marginal distributions of ET_{0} and irrigation water are generalized extreme values and normal distributions, respectively, but for their joint distribution, the Gumbel–Hougaard copula is the best-fitting one. The water shortage risks with different encounter situations of ET_{0} and irrigation water are better revealed using the proposed copula-based joint distribution.

## INTRODUCTION

The features of the manual water supply and water demand in the irrigation district can be revealed based on the analysis of irrigation water and reference crop evapotranspiration (ET_{0}). ET_{0} is a basic and key parameter used to estimate the crop water requirement (Kong *et al.* 2013). If rainfall in the irrigation district cannot meet the crop water requirement, irrigation water is needed. The crop water requirement occupies the majority of irrigation water and exhibits a positive correlation with it. Thus, there is some relevance between ET_{0} and irrigation water (Yamauchi 2014). Usually, irrigation water represents the manual water supply, while ET_{0} denotes water demand in the irrigation district.

Both ET_{0} and irrigation water can be influenced by a number of factors, such as precipitation, temperature, wind speed, relative humidity, solar radiation, etc. (Yu *et al.* 2002; Mojid *et al.* 2015), and can display complex spatial–temporal variations. These indicate that the necessity of studying the probability distributions of ET_{0} and irrigation water are based on their long-term data series. Now, research on the spatial–temporal variation characteristics of ET_{0} and irrigation water has found that the changing characteristics of ET_{0} and irrigation water are complicated and uncertain (Mo *et al.* 2004; Liu *et al.* 2007; Dinpashoh *et al.* 2011; Wang *et al.* 2014). In order to reveal these changing characteristics more effectively, the probability distributions of ET_{0} and irrigation water are studied based on their long time data series (Khanjani & Busch 1982; Zhao *et al.* 2015).

However, the analyses of the univariate distribution of ET_{0} and irrigation water without considering their dependence structure are inadequate (Mishra *et al.* 2013; Oh *et al.* 2013; Feng *et al.* 2014; Zhao *et al.* 2014; Riediger *et al.* 2016), which may lead to incomplete or even erroneous conclusions. Therefore, it is necessary to apply a mathematical method that can conserve the dependence structure to capture the joint probability behaviors of ET_{0} and irrigation water. Ding *et al.* (2011) proposed that the joint distribution based on the dependence among multivariate distributions extracts more information.

The copula is a useful method for assessing multivariate distributions (Kao & Govindaraju 2010), which makes it the ideal method to study the joint probability of ET_{0} and irrigation water. Currently, the copula is widely used in hydrology and water resources studies, but there are only few applications relevant to irrigation water. Copulas are functions that can combine univariate cumulative probability distributions to construct a multivariate joint distribution, as well as fully express the dependence structure among random variables (Joe 1997; Genest & Favre 2007), so they are flexible whatever the univariate marginal distribution is. In the field of hydrology and water resources, the copula method is widely used to describe the dependence structure among random variables. Its intelligibility and flexibility are evident from the related research on drought (Serinaldi *et al.* 2009; Kao & Govindaraju 2010; Yoo *et al.* 2014; Salvadori & De Michele 2015; Zhang *et al.* 2015), sea storm analysis (De Michele *et al.* 2007; Zheng *et al.* 2013), flood risk analysis (Svensson & Jones 2004; Grimaldi & Serinaldi 2006; Pinya *et al.* 2009; Ghizzoni *et al.* 2012; Zheng *et al.* 2014, 2015), tail dependence (Di Bernardino & Rullière 2016; Wang & Xie 2016), runoff and sediment (Zhang *et al.* 2014a, 2014b), and streamflow simulation (Chen *et al.* 2015; Jeong & Lee 2015). The detailed theoretical background and methodological descriptions of copula application in hydrology can be found in Klüppelberg & Rootzén (2006) and Jaworski *et al.* (2009). Recently, except for the research by Ding *et al.* (2011) and Zhang *et al.* (2014a, 2014b), who analyzed the probabilistic behaviors of rainfall and ET_{0}, few studies have been concerned with manual water supply and water demand in the irrigation region.

Therefore, this study aims to construct the joint distribution of ET_{0} and irrigation water with the copula method based on the annual observed data series of ET_{0} and irrigation water from 1970 to 2013 of the Luhun irrigation district in China. The results from this study allow us to explore the statistical changing characteristics of the manual water supply and water demand in the irrigation district, and then to capture their joint probability behaviors.

The subsequent sections of this paper are organized as follows: A brief introduction of copula is described in ‘Methodology,’ including the identification and goodness-of-fit evaluation of a univariate marginal distribution and copula-based joint distribution. The used ET_{0} and irrigation water data series of the Luhun irrigation district in China are introduced in the ‘Case study’. The results and applications of statistical analysis on the marginal distributions and the joint distribution of ET_{0} and irrigation water are provided in ‘Results and applications’. Finally, conclusions are stated in ‘Conclusions’.

## METHODOLOGY

An overview of the methodology in this study is: (1) select the appropriate univariate marginal distributions of ET_{0} and irrigation water according to their statistical characteristics; (2) estimate the parameter of the copula function; (3) determine the best-fitting copula function with the identification and goodness-of-fit evaluation method; (4) construct the joint distribution of ET_{0} and irrigation water.

### Selection of univariate marginal distribution

Hydrological variables have been generally assumed to be subject to a P-III-type distribution in China (Huang 2003). However, in order to better reflect the statistical probability characteristics of ET_{0} and irrigation water, seven commonly used marginal distributions in hydrology and water resources (Chen 2013) were applied to select the most appropriate one, including two-parameter gamma, two-parameter lognormal, generalized Pareto, exponential, the classic P-III-type, normal and generalized extreme value (GEV) distributions.

_{0}and irrigation water, the parameters of these marginal distributions were estimated by the maximum likelihood estimation (MLE) method. The identifications of the marginal distribution were derived by the Kolmogorov–Smirnov (KS) test with a 5% significant level. Then, the root mean square error (RMSE) was calculated to select the best-fitting marginal distribution. RMSE is given by: where and are, respectively, the theoretical and empirical frequency of the marginal distribution, and

*n*is the number of observations.

### Copula method

*X*and

*Y*, assume that the univariate marginal distribution functions are and , respectively. Meanwhile, let and . According to Sklar's theorem (Sklar 1959), there exists a unique copula

*C*to link these two marginal distributions. The copula function is generally expressed as follows: where is the joint distribution of pairs with its marginal distribution submitted to uniform distributions on [0, 1]. is the parameter of copula

*C*.

*C*is unique with the following joint probability density function : where

*c*is the joint probability density function of copula

*C*expressed as: where

*u*and

*v*denote, respectively, a specific value of and .

### Bivariate Archimedean copula functions

Bivariate Archimedean copula functions with only one parameter are extensively applied in hydrology and water resources (Cherubini *et al.* 2004; Grimaldi *et al.* 2005). In this study, three one-parameter Archimedean copulas are employed (as shown in Table 1) (Genest & Mackay 1986).

Archimedean copulas . | . | Relation between and . | Range of . |
---|---|---|---|

Clayton | |||

Gumbel–Hougaard | |||

Frank |

Archimedean copulas . | . | Relation between and . | Range of . |
---|---|---|---|

Clayton | |||

Gumbel–Hougaard | |||

Frank |

#### Parameter estimation

*u*and

*v*, and it can be estimated by the relations between and Kendall's rank coefficient . can measure the nonlinear dependence between random variables and is calculated as: where are observations of the random variables

*X*and

*Y*for :

As shown in Table 1, it is obviously seen that the three Archimedean copulas are suitable to describe the positive dependence, but for the negative dependence, only the Frank copula can be used.

#### Identification and goodness-of-fit evaluation of copula function

*D*is written as: where represents the value of observation of the copula function; is the number of such that and , and

*n*is the number of observations. When the statistical magnitude

*D*is less than its critical value, the assumed distribution passes the KS test.

*X*and

*Y*are exhibited as and , respectively. Thus, the empirical copula is defined as: where

*n*is the sample size, is the indicator function of set

*A*satisfying if

*A*is true, otherwise , and

*u*and

*v*are two variables between 0 and 1.

*n*is the sample size. If the value of OLS is close to zero, the copula function performs more efficiently. If few differences exist in the values of OLS, the Akaike information criterion (AIC) is applied for the selection of the best-fitting copula function. AIC can be given by: where

*k*is the number of parameters of the copula function, and

*n*is the sample size.

## CASE STUDY

The Luhun irrigation district (34°10′28″–34°45′34″N, 112°5′06″–112°5′24″E) is located in the west of Henan province in China (Figure 1). As a larger irrigation district, it crosses the Yellow River basin and Huai River basin with a total area of 1,838.48 km^{2}. Dominated by the Siberia winter monsoon and the East Asian summer monsoon, the climate in the Luhun irrigation district is typically characterized by hot and rainy summers, and cold and dry winters. The average annual rainfall is about 600 mm, but 60% of it occurs from June to September. The average annual evaporation is above 1,400 mm, and the average annual drought duration is 112 days. It is evident that the Luhun irrigation district is prone to meteorological drought. The average annual irrigation water is 1.81 × 10^{8} m^{3}, and it is mainly from the Luhun reservoir located on the Yi River (a tributary of the Yellow River) with a total storage capacity of 13.20 × 10^{8} m^{3}.

The ET_{0} is estimated by the Penman–Monteith formula recommended by the Food and Agriculture Organization (FAO) in 1998 (Allen *et al.* 1998), and the FAO Penman–Monteith model requires the input of meteorological variables. Thus, it is necessary to obtain data for these meteorological variables. The data used in this study from 1970 to 2013 were obtained from the Irrigation Administration Bureau of the Luhun irrigation district, and included irrigation water, daily rainfall, average wind velocity, mean relative humidity, maximum temperature, minimum temperature, and sunshine duration. Figure 2 shows the data series of annual ET_{0} and irrigation water from 1970 to 2013. Table 2 represents the statistical characteristics of annual ET_{0} and irrigation water.

Variable . | Minimum . | Maximum . | Mean . |
---|---|---|---|

ET_{0} (mm) | 724.59 | 1,168.25 | 1,034.89 |

Irrigation water (10^{8} m^{3}) | 0.60 | 3.29 | 1.81 |

Variable . | Minimum . | Maximum . | Mean . |
---|---|---|---|

ET_{0} (mm) | 724.59 | 1,168.25 | 1,034.89 |

Irrigation water (10^{8} m^{3}) | 0.60 | 3.29 | 1.81 |

## RESULTS AND APPLICATIONS

### Marginal distributions of ET_{0} and irrigation water

_{0}and irrigation water were fitted by two-parameter gamma, two-parameter lognormal, generalized Pareto, exponential, classic P-III-type, normal and GEV distributions, respectively. The parameters of marginal distributions were estimated by the MLE method. According to Massey (1951), when the sample size is 44, the KS statistic

*D*with the 5% significance level is 0.201. If

*D*of a marginal distribution is less than 0.201, it passes the KS test. Table 3 shows the statistic

*D*of these seven marginal distributions. The results show that for ET

_{0}, the marginal distributions of gamma, lognormal, P-III, normal, and GEV all pass the KS test, while for irrigation water, the marginal distributions of gamma, lognormal, Pareto, P-III, normal, and GEV all pass. Table 4 shows the RMSE of these marginal distributions. It can be seen that due to the smallest value of RSME, the normal distribution is the preferred marginal distribution for irrigation water and the GEV distribution for ET

_{0}. Thus, the cumulative distribution function (CDF) of the GEV distribution is applied to ET

_{0}, and it can be presented as:

. | Two-parameter gamma . | Two-parameter lognormal . | Pareto . | Exponential . | P-III . | Normal . | GEV . |
---|---|---|---|---|---|---|---|

ET_{0} (mm) | 0.169 | 0.169 | 0.528 | 0.560 | 0.170 | 0.165 | 0.125 |

Irrigation water (10^{8} m^{3}) | 0.107 | 0.133 | 0.200 | 0.326 | 0.088 | 0.075 | 0.076 |

. | Two-parameter gamma . | Two-parameter lognormal . | Pareto . | Exponential . | P-III . | Normal . | GEV . |
---|---|---|---|---|---|---|---|

ET_{0} (mm) | 0.169 | 0.169 | 0.528 | 0.560 | 0.170 | 0.165 | 0.125 |

Irrigation water (10^{8} m^{3}) | 0.107 | 0.133 | 0.200 | 0.326 | 0.088 | 0.075 | 0.076 |

RMSE . | Two-parameter gamma . | Two-parameter lognormal . | Pareto . | Exponential . | P-III . | Normal . | GEV . |
---|---|---|---|---|---|---|---|

ET_{0} (mm) | 0.082 | 0.086 | – | – | 0.075 | 0.076 | 0.065 |

Irrigation water (10^{8} m^{3}) | 0.082 | 0.049 | 0.100 | – | 0.034 | 0.030 | 0.030 |

RMSE . | Two-parameter gamma . | Two-parameter lognormal . | Pareto . | Exponential . | P-III . | Normal . | GEV . |
---|---|---|---|---|---|---|---|

ET_{0} (mm) | 0.082 | 0.086 | – | – | 0.075 | 0.076 | 0.065 |

Irrigation water (10^{8} m^{3}) | 0.082 | 0.049 | 0.100 | – | 0.034 | 0.030 | 0.030 |

### Joint distribution of ET_{0} and irrigation water

Based on the Archimedean copula functions, the joint distribution construction procedure of ET_{0} and irrigation water involves: (1) calculation of the association between ET_{0} and irrigation water with Kendall's ; (2) estimation of the parameter according to the relations between and Kendall's ; (3) identification of the Archimedean copula function by KS test; and (4) determination of the best-fitted copula by employing OLS and AIC.

As noted in Table 5, the proposed Archimedean copulas all pass the KS test (for their < 0.201) obviously. But the Gumbel–Hougaard copula function gives a better fit than the other two copulas because of its smallest OLS value as well as AIC value.

Kendall's . | Parameter . | Copulas . | ||
---|---|---|---|---|

Clayton . | Frank . | Gumbel–Hougaard . | ||

0.254 | 0.680 | 2.411 | 1.340 | |

0.113 | 0.104 | 0.112 | ||

OLS | 0.043 | 0.042 | 0.042 | |

AIC | −274.752 | −277.957 | −278.030 |

Kendall's . | Parameter . | Copulas . | ||
---|---|---|---|---|

Clayton . | Frank . | Gumbel–Hougaard . | ||

0.254 | 0.680 | 2.411 | 1.340 | |

0.113 | 0.104 | 0.112 | ||

OLS | 0.043 | 0.042 | 0.042 | |

AIC | −274.752 | −277.957 | −278.030 |

_{0}and irrigation water. Let

*u*and

*v*denote the marginal distribution of annual ET

_{0}and irrigation water of the Luhun irrigation district. Their joint distribution is expressed as: where

*x*and

*y*denote ET

_{0}and irrigation water, respectively.

Figure 3 gives the goodness-of-fit evaluation of , which indicates that the correlation coefficient between the theoretical copula and the empirical copula is 0.97. It means the selected Gumbel–Hougaard copula is reasonable. Also, the joint distribution is plotted in Figure 4.

### Analysis of the joint distribution

_{0}and IR, two different joint probability distributions can be presented as follows to reveal the possible encounter situations of the manual water supply and water demand: where

*X*and

*Y*denote ET

_{0}and IR,

*x*and

*y*are their specific values, and are the marginal distributions of ET

_{0}and IR, respectively. is the copula-based joint distribution of ET

_{0}and IR. is joint probability distribution I given or . indicates joint probability distribution II given and . The contour plots of and are described in Figures 5 and 6. Additionally, the exceeding probability of ET

_{0}and marginal probability distribution of IR are presented in Figures 7 and 8, respectively.

From Figures 5 and 6, it can be observed that the counter plot of shows the inverse direction to that of . If , . Meanwhile, the contour plots of and exhibit various encounter situations of ET_{0} and irrigation water, so the joint probability with given combinations of pairs (ET_{0}, IR) can be achieved. Take for example that the probability is 0.89 for encounter situation pairs (ET_{0} ≥ 1,000 mm or IR ≤ 1.8 × 10^{8}m^{3}) and 0.29 for pairs (ET_{0} ≥ 1,000 mm and IR ≤ 1.8 × 10^{8} m^{3}). Similarly, the different pairs (ET_{0}, IR) with a given joint probability are also obtained from the contour plots of and . Typically, the encounter situations with the probability of 0.7 include pairs (ET_{0} ≥ 1,100 mm or IR ≤ 1.9 × 10^{8} m^{3}), (ET_{0} ≥ 1,050 mm or IR ≤ 1.55 × 10^{8}m^{3}), (ET_{0} ≥ 850 mm and IR ≤ 2.25 × 10^{8}m^{3}) and (ET_{0} ≥ 950 mm and IR≦2.45 × 10^{8}m^{3}), etc. Moreover, this bivariate joint probability of pairs (ET_{0}, IR) is obviously less than the exceeding probability of ET_{0} and marginal probability of IR. For instance, the exceeding probability of ET_{0} ≥ 1,000 mm is 0.66 (noted in Figure 7), and the marginal probability of IR ≤ 2.0 × 10^{8}m^{3} is 0.61 (shown in Figure 8). They are all larger than the bivariate joint probability of pairs (ET_{0} ≥ 1,000 mm and IR ≤ 2.0 × 10^{8}m^{3}) with the value of 0.38. Thus, if the unvariate marginal distribution of ET_{0} and IR is considered only in irrigation activities, the statistical characteristics of the manual water supply and water demand can be explained incorrectly. Especially for irrigation planning and drought resistance, ET_{0} and IR need to be involved simultaneously to reflect the actual water shortage risk with the manual water supply and water demand.

In practice, the larger ET_{0} and the less IR are of more concern in irrigation planning and management. According to Table 2, the encounter situations of the maximum of ET_{0} and the minimum of IR from 1970 to 2013 in the Luhun irrigation district are considered as pairs (ET_{0} ≥ 1,034 mm or IR ≤ 1.8 × 10^{8} m^{3}) and pairs (ET_{0} ≥ 1,034 mm and IR ≤ 1.8 × 10^{8}m^{3}). This reveals the water shortage risk under the condition of manual water supply and water demand. From Figures 5 and 6, it can be seen that and , which shows the irrigation water in the Luhun irrigation district can usually satisfy water demand, but the water shortage risk still exists and should be paid more attention.

## CONCLUSIONS

The study employed seven CDFs to describe the marginal probabilistic behaviors of ET_{0} and irrigation water and found that the GEV and normal distributions are the best-fitted ones for them respectively. The Gumbel–Hougaard copula among the Archimedean copulas was employed to construct the bivariate joint distribution of ET_{0} and irrigation water. The empirical distribution was in close agreement with the theoretical distribution by which the statistical characteristics of the manual water supply and water demand in Luhun irrigation district were explored.

With the copula-based joint distribution of ET_{0} and irrigation water, two different probabilities of pairs (ET_{0}, IR) could be estimated effectively to reflect water shortage risk with the manual water supply in the Luhun irrigation district. In fact, the water shortage risk of extreme encountered situations of the maximum of ET_{0} and the minimum of IR in the Luhun irrigation district are more concerned. Compared with the univariate marginal distribution, the bivariate joint distribution of ET_{0} and irrigation water involves these two variables simultaneously, so the water shortage risk can be revealed more comprehensively, and the results are naturally more reasonable. It can be concluded that the irrigation water in the Luhun irrigation district can basically satisfy water demand, but water shortage risk still exists.

Moreover, the bivariate joint distribution of ET_{0} and irrigation water has a broader application. In practice, if an appropriate distribution process of ET_{0} and irrigation water of a typical year is known, using the same frequency amplification method, the several groups of ET_{0} and irrigation water with the same occurrence probability can be obtained. Thus, if a certain occurrence probability is given in the future, the corresponding encounter situation of ET_{0} and irrigation water can be found. This will provide technology support for irrigation management and drought resistance in the Luhun irrigation district.

## ACKNOWLEDGEMENTS

This research is supported by the National Natural Sciences Foundation of China (Grant No. 51309202), National Key R&D Program of China (No. 2016YFC0401407), and the Key Scientific Research Project in the University of Henan Province (No. 15A570011), and Outstanding Young Talent Research Fund of Zhengzhou University (Grant No. 1521323002).

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