The fluctuation and the consequent prediction of the water table is one of the key problems of hydrological environment management [1
]. Reliable prediction methods of the water table play significant roles in terms of groundwater planning and comprehensive management [2
]. Over the years, scholars have applied different methods to study the water table, including the Linear Regression Method [3
], Clustering Method [5
], ARIMA Model [6
], Genetic Programming Method, Neutral Network Method [7
], Wavelet Approach [8
], and SVM (Support Vector Machine) method [9
], as well as the joint application of several methods [10
These research methods have helped scholars to define the fluctuation of the water table as well as to predict it. However, most of these methods conduct a point prediction instead of finding the probability of an occurrence in ranges, neglecting the specific distribution of variables.
The Copula Function can be viewed as a multivariate probability distribution with uniform marginal in the interval [0,1]. In 2003, the first application of the Copula Function in the field of hydrology was undertaken [12
]. In recent years, it has been widely applied to various aspects of hydrological studies [13
]. The main applications of the Copula Function include the analysis of: precipitation characteristics [12
], the correlation between flood peak and flood volume [14
], the frequency and recurrence interval of floods [16
], characteristics of storms [19
], the frequency and recurrence intervals of droughts [21
], drought assessment [22
], risk assessment [23
], and an assessment of environmental hydrological model performance [24
]. Due to the specialty of the water table, the establishment of the Copula Function for this purpose is relatively difficult; therefore, no research that uses the Copula Function to analyze the water table has been reported. When considering the flexibility of the Copula Function establishment, this paper proposes that the Copula Function can be used to analyze the relationship between groundwater and driving factors from the perspective of probability.
The Jinghui Irrigation District, formerly known as Zhengguoqu in ancient times, is one of the four water resource projects in China. Its groundwater recharge mainly comes from surface water irrigation and precipitation. The method of surface water irrigation that was adopted in the Jinghui Irrigation District is flooding irrigation. Wheat in this area is irrigated three times a year in times of normal runoff, one time in winter, and two times in spring. In the case of low or high runoff, the irrigation frequency of the wheat can be adjusted as needed. Winter irrigation is conducted around the middle ten days of December. Spring irrigation is divided into three stages: the jointing stage, heading stage, and the filling stage. In terms of the irrigation volume, the fixed irrigation volume of winter irrigation is 9 × 104
to 10.5 × 104
, and that of the spring irrigation is 6 × 104
to 7.5 × 104
. The total annual irrigation volume is less than 18 × 104
to 22.5 × 104
. Corn in this area is irrigated one to two times in wetter years, and three to four times in dry years. The irrigation period is from July to September, and the fixed irrigation volume is 6 × 104
to 7.5 × 104
. The total annual irrigation volume of the corn is less than 19.5 × 104
to 27 × 104
. However, the actual irrigation varies according to the different situations each year. Recharging from the irrigation water infiltration is the major source of replenishment of the groundwater. Research data has indicated that, in the 1980s, the water irrigation infiltration volume utilized at least 30% of the surface water diversion; in the 1990s, the percentage dropped to about 25%; and, from 1997 to 2009, it dropped to less than 20%. After 2010, the percentage began to slowly increase again. In addition to surface water diversion, precipitation infiltration is another way to recharge groundwater. However, the effects of the latter are obviously smaller than those of the former [25
]. The Jinghui Irrigation District plays an important role in Chinese national economic life. Therefore, it has frequently been the target of research. However, this research has mainly focused on qualitative analysis and water quality. Research on the water table of the Jinghui Irrigation District is relatively rare, but it includes the research by Liu Y. (2010), who analyzed the dynamic transformation of the groundwater of the Jinghui Irrigation District from 1977 to 2005 using the Multivariate Linear Regression Method [4
]. This indicated that, ceteris puribus, if the annual precipitation of the district varies by 10 mm, the buried depth of the groundwater will vary by 3.5 cm approximately; similarly, for every million m3
of surface water irrigation volume in the district, the average buried depth of the groundwater varies 2.4 cm consequently, and for every million m3
of groundwater mining, the average buried water table in the Jinghui Irrigation District varies by ±0.53 mm. Obviously, when compared with the amount of groundwater mining, the irrigation volume of the surface water and precipitation are two significant factors that affect the buried depth of the phreatic water. Additionally, due to the data collection method, the data regarding of the amount of groundwater mining is not stable enough. Given the same conditions, the more groundwater mining that there is, the smaller the buried depth of the groundwater is, which is contrary to what occurs in reality. Although the amount of groundwater mining is a relatively important factor that affects the buried depth of the phreatic water, unstable data may lead to incorrect conclusions. Therefore, given the fixed conditions of the present amount of groundwater mining (140 ± 4 million m3
), this paper analyzes the relatively stable data of the buried depth of the phreatic water, the annual irrigation volume of the surface water, and the precipitation through the Copula Function from a statistical perspective.
In order to make explicit the probabilistic relations between the water table and the driving factors, so that these can be used to offer more reliable references for the planning and management of water resources, this paper mainly focuses on the following: (1) Estimating the marginal distribution function of each driving factor through the Kernel distribution method. (2) Establishing the two-dimensional (2D) Copula Function of each driving factor and the marginal distribution of the water table through the 2D Copula Function, and analyzing the response of each driving factor to the buried depth of groundwater. The 2D Copula Function can be built directly, and its fitting results are satisfactory. (3) Since the buried depth of the phreatic water is negatively correlated with the annual precipitation and annual irrigation volume of the surface water, the fitting results of the three-dimensional (3D) Copula Function—which is directly built through the 3D Symmetric Archimedean Copula Function—are not satisfactory. Therefore, in undertaking a real application, this paper corrects the variables first, which consequently leads to an acceptable fitting result. Again, the 3D Copula Function is applied to comprehensively analyze the response of the irrigation volume of the surface water and the annual precipitation to the buried the depth of the phreatic water when the two factors, respectively, vary. In addition, the results are compared with those of the 2D Copula Function. (4) This paper also calculates the conditional probability values. Given fixed conditions, we calculate the probability of the water table falling into different ranges, together with the volume of surface water irrigation that should lead to the expected water table.
The paper introduces a 2D and 3D Frank Copula Function to analyze the responsiveness of different driving factors to the buried depth of the phreatic water. Consequently, the fitting results are satisfactory. Furthermore, the paper compared the results of the 3D Copula Function and 2D Copula Function, jointly analyzes the responsiveness of multiple varying driving factors on the buried depth of the phreatic water, and predicts the design volume of surface water irrigation given definite precipitation. This paper adopts data from the Jinghui Irrigation District to measure and to make the relevant calculations, and draws the following conclusions:
(1) We precisely obtained the 2D and 3D joint distribution functions of the marginal distribution of the water table and the marginal distribution of each driving factor, as well as exactly calculating the probabilities of water table variation that was caused by the variation of driving factors.
(2) We utilized conditional probability equations precisely calculated the conditional probability values. Given fixed conditions, we calculated the probability of the water table falling into different ranges.
(3) The greater the level of precipitation, the shallower the buried depth of the phreatic water is. The larger the irrigation volume of surface water, the shallower the buried depth of the phreatic water. The level of precipitation and irrigation volume of the surface water can mutually promote the variation of the phreatic water buried depth.
Due to the high flexibility of the Copula Function application, it has significant advantages when establishing complicated functions of the water table and its driving factors and calculating the relevant probabilities. When compared with other methods, such as linear regression and ARIMA, the Copula Function method emphasizes the specific distribution of variables and finds the probability of an occurrence in ranges. While regulating and controlling the water level of the Jinghui Irrigation District, precipitation and the irrigation volume of the surface water are the driving factors that matter the most. This paper utilizes a multidimensional Copula Function to establish the joint distribution functions of the marginal distribution of precipitation, irrigation volume of the surface water, and the buried depth of the groundwater, and calculates the probabilities of the buried depth falling into different ranges given the different precipitation levels and irrigation volumes of the surface water. This study has provided key information regarding the relationship between the water table and its driving factors and the effective regulation and control of the water level. According to the results of this study, combined with the actual situation in the Jinghui Irrigation District, in which in recent years the groundwater buried depth has been deeper than is typical, we recommend that the administration increase the volume of surface water irrigation to adjust the groundwater level to a suitable range.
This study only focuses on the annual variation of the average groundwater level in the Jinghui Irrigation District. Studies on seasonal fluctuations and regional fluctuations of the groundwater level in the Jinghui Irrigation District will be explored in the future.