Shou-ke WEI*

1. Eawag, the Swiss Federal Institute of Aquatic Science and Technology, Dübendorf CH-8600, Switzerland

2. Department of Forest Resources Management, University of British Columbia, Vancouver BC-V6T 1Z4, Canada

3. Emodlogic Technology Inc., Vancouver BC-V5P 3R1, Canada

Estimating water deficit and its uncertainties in water-scarce area using integrated modeling approach

Shou-ke WEI*1,2,3

1. Eawag, the Swiss Federal Institute of Aquatic Science and Technology, Dübendorf CH-8600, Switzerland

2. Department of Forest Resources Management, University of British Columbia, Vancouver BC-V6T 1Z4, Canada

3. Emodlogic Technology Inc., Vancouver BC-V5P 3R1, Canada

Accurate assessment of water deficit and related uncertainties in water-scarce areas is strategically important in various fields of water resources management. This study developed a hybrid approach integrating conceptual water balance model and econometric regression to estimate water shortage and its related uncertainties in water-scarce areas. This hybrid approach was used to assess the agricultural water deficit of Beijing, an extremely water-scarce area in China. A predictive model of agricultural water demand was developed using the stepwise multiple regression method, and was validated by comparing the predicted values with observed data. Scenario analysis was employed to investigate the uncertainties of agricultural water shortage and agricultural water demand. This modeling approach can assist water administration in creating sustainable water allocation strategies in water-scarce areas.

water deficit; hybrid model; conceptual water balance model; stepwise multiple regression method; uncertainty; Beijing

1 Introduction

From an economic point of view, water resources are composite assets providing various services for the development and life of human beings and other species. However, water shortage has become a big problem in many countries, and the main reasons include unevenly distributed precipitation (Wolf 1999), increasing water consumption with population growth, degradation of water quality (UN-CSD 1994), unsustainable water resources management practices (Wang 2005), and increasing temperature (Westmacott and Burn 1997). The increasing water shortage situation urgently necessitates practical approaches for estimation of water shortage of different water users, including industry, agriculture, households, and ecology. However, studies on the methods of water shortage estimation have not appeared frequently in the literature.

This study selected Beijing as the study area, mainly because Beijing is a typical exampleof both an extremely water-scarce and an economically developed area in China. Many studies on water issues of Beijing have been conducted. For example, Cao (2003) forecasted the industrial water use in 2010 by analyzing the trend of industrial water use in urban and suburban areas of Beijing; Jia and Zhang (2003) studied the influence of water price increments on industrial water use; Wu and Zhang (2005) discussed the cause of the water crisis through analysis of water supply and consumption over the last 20 years; Li and Xu (2004) predicted the industrial water demand and water demand in other sectors using the computable general equilibrium (CGE) model; the Beijing Development and Reform Commission (BDRC) (2006) forecasted the industrial, agricultural, domestic, and ecological water consumption during the 11th Five-Year Plan; Wei and Gnauck (2007) forecasted the future water demand of industry, agriculture, households, and ecology using game-theoretic modeling approaches; and Wei et al. (2009, 2010) used statistic and econometric modeling methods to analyze and forecast domestic and industrial water demand.

The aim of this study was to develop a convenient modeling approach integrating a conceptual water balance model and econometric regression to assess the water deficit and its related uncertainties influenced by socio-economic development and environmental and ecological protection in water-scarce areas. This hybrid method was used to evaluate the agricultural water shortage of Beijing.

2 Model structure and methods

2.1 Conceptual model of water shortage estimation

Water deficit is the difference between water availability and water demand. Water availability refers to the total amount of available water under constraints of neither overexploiting groundwater nor overtaking ecological water demand. The total amount of available water is the sum of the total water resources (surface water and groundwater) and reclaimed water from wastewater and sewage in a certain area, which are influenced by climatological and hydrological factors, technology, policies, and environmental and ecological situations. Water demand (or use) includes water amounts required by industry, agriculture, and households, and ecological water use, which are usually influenced by socio-economic factors and policy changes.

Ecological water demand is the water amount that the ecology really requires, an ecological definition. The minimum ecological water demand in an area should consist of at least three parts: water required for maintaining public green areas, water required for maintaining certain water surfaces, and water required for guaranteeing the growth of trees, especially young trees. We regard ecological water use as a part of socio-economic water demand or use because ecological water use is the water amount used for the ecology in reality and it is more of a socio-economic definition. Ratios of water demand in different sectors to the socio-economic water demand are calculated to estimate the water deficits in these sectors.

Computation of total water deficits and water deficit in a certain sector can be expressed by Eqs. (1) through (13), for which the notations of the variables and symbols used in this study are summarized in Table 1.

The total water deficit of socio-economic development is calculated as follows:

The ecological water deficit is calculated as follows:

The water deficits of certain socio-economic sectors are calculated as follows:

2.2 Regression modeling methods

2.2.1 Models

We used three regression models: the linear regression model (level-level model) (Eq. (14)), the double-logarithmic model (log-log model) (Eq. (15)), and the semi-logarithmic model (log-level model) (Eq. (16)). The ordinary least squares (OLS) regression method was used to estimate the parameters of the models. The autoregressive ( AR(j)) and/or moving average ( MA(l)) terms were included in the equations of some models to remove the serial correlation, where j and l are their orders.

The level-level model assumes that a dependent variable Yiis a linear function of a set of explanatory variables Xki:

where Yiis the dependent variable with i observations, Xkiis the independent (or explanatory) variable, k is the number of independent variables, βkis the parameter of the equation or regression coefficient, and μiis the disturbance (or error) term. This modeling equation includes two components: (1) the non-random component β0+ β1X1i+ β2X2i+ …+ βkXkiand (2) the random component μi.

Table 1Notations of variables and symbols used in Eqs. (1) through (13)

The coefficientkβ in Eq. (14) can be interpreted as the marginal effect, i.e., how the dependent variable changes when the independent variable changes by an additional unit, holding all other variables in the equation constant (i.e., partial derivative).

The forms of logarithmic models can be generally expressed by the following equations:

The coefficient of the double-logarithmic model (Eq. (15)) expresses the concept of elasticity, i.e., the ratio of the percentage change in one variable to the percentage change in another variable, holding all other variables in the equation constant. In the log-level model(Eq. (16)), 100βkcan be interpreted as the percentage change in Yifor a unit increase in Xki, holding all other independent variables constant. If Xkistands for a time variable (T), 100βkexactly expresses the average growth rate of Yiduring the analysis period.

2.2.2 Modeling process

The regression modeling process can be briefly summarized as follows: (1) the preliminary variables are defined based on physical principles, (2) the input variables are selected by correlation analysis between water use and socio-economic elements, (4) the data set is divided into two parts, (5) the predictive model is formulated using part of the observed data set, (6) the best model is chosen using backward stepwise regression to determine the most influential independent variables based on the criteria of t-statistics at a significance level (p-value) less than 0.05, (7) the predictive model is validated by comparing predictions with the rest of the observed data, and (8) scenarios are designed to simulate the uncertainty of agricultural water use due to possible changes of input variables and constraint variables.

2.2.3 Model evaluation

Evaluation terms used in this study include the coefficient of determination (R2), adjusted R2( Ra2dj), residual (error), standard error, the t-statistic and associated statistical significance level (p-value), the sum of squares of errors (SSE), the standard errors of regression (SER), and the F-statistic and the associated statistical significance level (P-value). Validation methods mainly include: (1) comparing model predictions and coefficients with physical theories, (2) testing whether the model can explain the reality, (3) checking predictions with newly collected data, (4) comparing results with theoretical models and simulation data, and (5) data-splitting or cross-validation in which one part of the data set is used for model coefficient estimation and the other for evaluation of model prediction accuracy (Snee 1997). We used methods (1), (2), and (4) in this study. First, method (1) was used to check whether the model meets physical theory. If it passed the check, the data-splitting method was used to check the absolute percentage error (APE ) (Eq. (17)) and the mean absolute percentage error (MAPE ) (Eq. (18)). If it did not pass the method (1) check, we investigated whether the model could explain the real situation. If it could explain the reality, we continued to method (4). If it could not explain the reality, we judged the model to be incorrect.

where ?tY is the predicted value,tY is the observed value, and t is the size of the samples.

2.3 Scenario design

Four alternative simulation scenarios were designed to analyze the uncertainty in the case study (Table 2), and the quantified assumptions are displayed in Table 3. Business as usualdevelopment was regarded as the first scenario (S1). The other three scenarios were designed according to the possible variations of constraints and input variables in S1. The second scenario (S2) is very optimistic, in which the situation is better than that in S1 from economic and environmental perspectives. By contrast, the fourth scenario (S4) is more pessimistic. The third scenario (S3) is a situation approximately positioned between S2 and S4.

Table 2Descriptions of four scenarios

Table 3Quantified assumptions of four scenarios

3 Study area and data

3.1 Study area

Beijing is located in northeastern China, and covers an area of 16 808 km2. It has a temperate semi-humid climate dominated mainly by the Pacific monsoon with typical diversified nature. Beijing possesses five large river systems, including the Chaobai river system, Yongding river system, Juma river system, Jiyun river system, and Beiyun river system, and they are part of the Haihe River Basin (Fig. 1).

With rapid socio-economic development, water shortage has become one of the serious problems in Beijing. The current available water resources per capita are only 247 m3per year, which indicates that Beijing belongs to an area of extreme water deficit (Li and Xiu 2004; Wei 2007). Fig. 2 clearly shows that the total water resources cannot meet the total water demand in Beijing. Agriculture is the second largest water user, accounting for 34% of the total water use in Beijing. Due to water shortage, agricultural water has continuously been reallocated to meetthe water demand in domestic and industrial sectors, resulting in a decline in irrigation areas (Wei et al. 2009).

Fig. 1Beijing river systems

Fig. 2Total water resources and water uses of different sectors in Beijing from 1986 to 2007

3.2 Data source

The main types of data include the following: (1) socio-economic indicators, (2) water quantity, (3) hydrology, (4) water quality, and (5) environmental and ecological indicators. Socio-economic data, including the rural population, agricultural gross output value, rural per capita net income, irrigation area, sown area of crops, afforestation area, and consumer price index (CPI), were collected from the China Statistical Yearbook (NBSC 1996-2009) and the Statistical Yearbook of Beijing (BNBS and NBSSOB 1999-2009). Water quantity data, such as water resources, water use, and hydrological data, were collected from the China WaterResources Bulletin (CWRA 1998-2004), Beijing Water Resources Bulletin (BWB 2005), Water Resources Bulletin of the Haihe River (HRWRC 1998-2006), Statistical Yearbook of Beijing (BNBS and NBSSOB 1999-2009), and previous research by Wu and Zhang (2005). Water quality data, including industrial and urban wastewater and sewage discharge, and their relative recycling amount, were collected from the Statistical Yearbook of Beijing (BNBS and NBSSOB 1999-2009). Environmental and ecological data, including ecological water use, urban water surface areas, public green areas, and the numbers of newly planted trees, were collected from the Statistical Yearbook of Beijing (BNBS and NBSSOB 1999-2009) and the Statistical Yearbook of China (NBSC 1996-2009).

4 Results and discussion

4.1 Correlation analysis

In principle, agricultural water demand (Q) usually has correlations with rural labor forces (L), the agricultural gross output value (V), rural per capita net income (I), irrigation area (A1), sown area of crops (A2), afforestation area (A3), and time (T), and they are defined as internal variables. The agricultural gross output value and rural per capita net income are the real values or comparative values, which are calculated based on CPI. The factors of urban sewage discharge, water resources, newly planted trees, public green areas, and ecological water demand are regarded as external influencing factors.

Table 4 shows the correlation coefficient (r) between agricultural water demand and the selected variables using 22 observations. It shows that Q has a stronger linear correlation with T (r = ?0.91), V (r = ?0.83), I (r = ?0.91), A1(r = 0.88), A2(r = 0.92), and the first-order lagged values of Q (i.e., Q (? 1)) (r = 0.92). However, the coefficients (r = 0.55 and r = 0.004) show that Q has a very weak linear correlation with L and almost no linear correlation with A3.

Table 4Matrix of correlation coefficients (r)

4.2 Predictive model for agricultural water demand

All variables for model development were logarithmically transformed because a preliminary regression analysis revealed that a log-transformed model has a slightly bettergoodness of fit (R2) than a non-transformed one. Log-transformation is usually able to normalize data, reduce extreme values and heteroskedasticity, and transform a nonlinear model into a linear one, which are the main reasons that R2of a log-transformed model is better than that of a non-transformed one (Zhu and Day 2009). The results from the backward stepwise regression process show that the parameter is not significant at a p-value between 0 and 0.05 if any of the variables I, A2, and Q (? 1) are included in the model, thought the goodness of fit values are very high (R2= 0.97) in all models and the F-test for all models shows the significance at a probability level less than 10?6(Table 5). The model excluding variables I, A2, and Q (? 1) is not only statically significant (P ＜ 10?6), but also has smaller standard errors and significant t-statistics (Table 6). The model excluding the three variables is expressed by Eq. (19), and the fitted results and residuals are illustrated in Fig. 3.

Table 5Parameter significance results from backward stepwise regression models

Table 6Statistics for agricultural water demand model

This model can explain at a 97% confidence interval that a 1.00% increase of the agricultural gross output value (V) will cause an average increase of 0.56% in agricultural water demand with other factors remaining constant. With other factors fixed, the elasticity (or sensitivity) of agricultural water demand to the time span of 20 years is around 51%, which suggests that the agricultural water demand annually decreased by 2.6% on average over the 20-year period from 1986 to 2005. The 1.00% rise of irrigation area, ceteris paribus, means a0.41% average increase in water demand. Therefore, time is the most sensitive variable in this model.

Fig. 3Results of agricultural water demand predictive model versus lgV, 1lgA, and lgT

4.3 Evaluation of model prediction

Observations of agricultural water demand from 2006 to 2008 were used to check the predictions of the model, and the evaluation results are illustrated by Fig. 4, in which 2SE lines are two times of the standard error bands, which provide an approximate 95% forecast interval. The testing results show that the model has a mean absolute fitting error (Efit) of 2.7%, a mean absolute prediction error (Epre) of 6.5%, and thus a total mean absolute error of 3.2%. These results prove that the model has a very good forecasting accuracy.

Fig. 4Model evaluation results

4.4 Models of input variables of agricultural water demand

The models for predicting agricultural gross output value (V), irrigation area (A1), urban sewage discharge (W), minimum urban ecological water demand (edQ ), and the ratio of agricultural water demand to total water demand (Ka) are summarized in Table 7. Besides these variables, a dummy variable (also known as an indicator variable) (Dt) was included in models to indicate the absence or presence of some categorical effects, which usually change the outcomes of the models.

Table 7Models and statistics used to calculate input variables in agricultural water demand model and its related constraint variables

4.5 Prediction results

Table 8 provides the modeling results of agricultural water availability (aaQ), water demand (daQ ), and water shortage (saQ) from 2007 to 2018 in scenario 1 (S1). The results reveal that agricultural water demand will continuously decrease from 12.4 × 108m3to 10.3 × 108m3from 2009 to 2018, and this will mainly be due to declines of water availability and irrigation areas, as well as some of the agricultural water supply being appropriated by industry and households. It also shows that the agricultural sector will face the problem of water shortages of 6.5 × 108m3to 7.1 × 108m3during this period, indicated by the gap between the estimated available water for agriculture and the agricultural water demand.

Table 8Modeling results of aaQ , daQ , and saQ from 2007 to 2018 (108m3)

4.6 Scenario results

The comparison results of the four scenarios are shown in Fig. 5. The results reveal thatagricultural water demand shows a deceasing trend from 12.99 × 108m3to 10.80 × 108m3in the optimistic scenario (S2) and from 12.05 × 108m3to 10.02 × 108m3in the pessimistic scenario (S4) from 2009 to 2018 (Fig. 5(a)). In the meantime, the ratio of the agricultural water demand to the total water demand (Fig. 5(c)) and agricultural water availability (Fig. 5(b)) also show a decreasing trend in these four scenarios. Comparing the agricultural water demand with the water availability (Fig. 5(b)), it was found that during the ten years beginning from 2009 Beijing’s agriculture will still face water shortages in all four scenarios (Fig. 5(d)).

Fig. 5Comparison results of four scenarios in Beijing from 2009 to 2018

The results of the four scenarios indicate that agriculture will demand more water in the faster-developing scenario (S2) than the more slowly-developing scenario (S4). In each scenario, however, agricultural water demand is decreasing with agricultural growth, and this is mainly because a developed economy usually includes a high efficiency of water consumption and a high volume of wastewater reclamation. In addition, the results reveal that Beijing’s agriculture will face water deficits in all the four scenarios, mainly due to increasing ecological water demand and a decreasing agricultural water demand ratio in total water demand in the future. Furthermore, the results also suggest that ecological water demand will be the main driver to decrease the agricultural water demand ratio in total water demand,though high water use efficiency might also change this ratio. From the results of the four scenarios, it is clear that Beijing’s agriculture will face severe water shortages even under the optimistic scenario (S2).

5 Conclusions

This study developed a hybrid method integrating a conceptual water balance model and econometric regression analysis for estimating water shortage, water demand, and relative uncertainties in water-scarce areas. This method was used to assess the agricultural water shortage in Beijing during the period from 2009 to 2018. It is found that agricultural water demand has strong correlations with the agricultural gross output value, irrigation area, and time in this study area. The modeling results show that the agricultural water demand will decrease due to efficient water use, water-saving policies, decreasing irrigation areas, and the allocation of more water to meet ecological water demand during the forecast period. However, due to insufficient water availability, the agriculture of Beijing will face a severe water deficit problem even under the most optimistic scenario. The results suggest that besides developing technologies to reclaim wastewater, to consume water efficiently, to sufficiently use precipitation water, and to desalinate sea water, Beijing may have to transfer water from other places to meet its water demand. The analysis methods and results of this study can support the water administration in creating water policies and plans. The limitation of this study, however, is that incomplete climate data are not sufficient for us to analyze the influence of climate change on agricultural water use, so this will be listed as one of future study topics.

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(Edited by Yun-li YU)

This work was supported by the Sino-Swiss Science and Technology Cooperation Program of Switzerland, and the Ministry of Science and Technology of China (Grant No. 2009DFA22980).

*Corresponding author (e-mail: shouke.wei@gmail.com)

Received Sep. 2, 2011; accepted Oct. 12, 2012