Crop Water-Saving Potential Based on the Stochastic Distance Function: The Case of Liaoning Province of China

: Scientiﬁc evaluation of crop water use efﬁciency is of great signiﬁcance for ascertaining water-saving potential and realizing efﬁcient utilization of water resources. In this paper, we calculated the water footprint of crop growth, pollution water footprint and production water footprint of 14 cities in Liaoning Province, China, by using the water footprint theory, established the crop water use efﬁciency model of stochastic frontier distance function, and analyzed the spatial-temporal variation characteristics of crop water use efﬁciency (WUE), ecological WUE and production WUE. Results show that: (1) the average water footprint of crop growth was 1.714 × 10 9 m 3 , the ecological water footprint of crop was 6.26 × 10 8 m 3 , and the water footprint of crop production was 2.34 × 10 9 m 3 from 2001 to 2017 for the whole province. (2) the WUE of crop growth was 0.821, the crop ecological WUE was 0.845 and crop production was 0.865, respectively. We concluded that Liaoning province can save 17.9% of crop consumption water, equivalent to 8.38 × 10 8 m 3 , 15.5% of ecological water, equivalent to 7.25 × 10 8 m 3 and 13.5% of production water, equivalent to 6.32 × 10 8 m 3 by strength-ening the popularization of agricultural high-efﬁciency water use technology and improving the level of policy management. This research provides a basic support for the evaluation of crop water-saving potential with the stochastic frontier approach in other regions.


Introduction
Wastewater is a link that needs to be focused on in response to the current world water shortage. Wastewaters come from the water discharged from human activities and rainwater runoff, including sewage, industrial waste water, and early rain runoff, especially from agricultural production areas [1][2][3]. The global wastewater volume is high, and the world may face its deficiency in the future [4,5]. Among them, agricultural water is the main source of water resource use, accounting for 85% of the water resource consumption [6,7]. In this field, its water-saving potential should not be underestimated. An important way of mitigating crop water resource consumption is by improving water use efficiency (WUE) in the crop production process and optimizing water resource allocation. Improving crop water use efficiency to control total water consumption is one of the main ways to alleviate the contradiction between water resource shortage and food security and determining the appropriate evaluation indexes and methods of crop water use efficiency are the precondition to accurately measure the crop water saving potential.
The evaluation of traditional agricultural WUE mostly revolves around the amount of irrigation water used. The specific process for this involves collecting irrigation water data through experiments, surveys or yearbooks, and then calculating indexes such as irrigation efficiency [8][9][10], agricultural water use intensity [11], water use efficiency [12][13][14] and agricultural water use efficiency [15,16]. In fact, the water consumption for crop growth includes irrigation water and rainwater, and more than 60% of the food production is rain-fed. Rainwater not only plays a dominant role in the water consumption for crop growth [1], but is also an important part of local water resources. The amount of irrigation water is not only related to crop types, but also forms an alternative relationship with the effective local rainfall, so it is biased to only use irrigation water to measure the agricultural water use efficiency. The introduction of the water footprint method with blue-green water (blue water-irrigation water, green water-rainwater), as an important connotation of the water footprint theory, can technically measure the amounts of irrigation water and effective rainfall required for crop growth, and overcome the calculation deviation of traditional agricultural WUE. In some studies, researchers have evaluated the utilization of water resources in agricultural production by calculating blue-green water [17][18][19][20]. These studies were focused on reducing water consumption for crop growth, and considered less the impact of agricultural production on water quality.
In the agricultural production, in addition to water consumption for crop growth, a certain amount of dilution water is needed to decrease the concentration of pollutants like chemical fertilizers and pesticides. The amount of the dilution water can be calculated from the grey water footprint. Studies show that in China nearly 60% of the grey water footprint comes from agricultural production [21], which shows that it is not a comprehensive approach to measure the true consumption of water resources in the entire agricultural production only from the perspective of blue-green water [22]. This gives new meaning to the total amount of water used in agricultural production, which is no longer limited to the water consumption required for the crop growth, but also needs to include the water amount to dilute pollutant. Some researchers considered the blue-green-grey water in the evaluation of agricultural WUE [23][24][25][26][27][28].
In view of the above researches, we found that the research methods only considered the relationship between agricultural water consumption and crop yield (or output value), which belongs to the calculation method of the single-factor agricultural WUE. This method can establish the relationship between economic benefit of agricultural production and water resource utilization by using simple and understandable calculation method, and analyze the impact of agricultural production activities on water resource ecosystem. However, this method has an implicit assumption that the output is created by water as the only input factor. This assumption ignores the contribution of other production factors [29][30][31] and the impact of the ratio between water resource and other elements on the agricultural production. Therefore, the calculation method of single-factor agricultural WUE lacks guidance for scientific and effective evaluation of agricultural WUE and rational planning of agricultural water use system. In order to make up for the deficiency of the calculation method of single-factor agricultural WUE, the total factor concept of agricultural WUE considering all inputs of agricultural production was proposed. That is, for a given level of output and other inputs, agricultural WUE is equal to the ratio of the technically feasible minimum amount of water used to the actual amount used. For example, some researchers used Stochastic Frontier Analysis (SFA) or Data Envelopment Analysis (DEA) to calculate the agricultural WUE from the perspective of total factor productivity [32][33][34][35][36][37]. Almost all of these studies take the irrigation water amount or blue-green water as the input factor of water resource in the agricultural production, ignore the grey water amount, and are thus unable to reflect the actual total water use in the whole agricultural production. However, although SFA and DEA are the most commonly used methods to calculate the total factor agricultural WUE at present, they cannot directly obtain the use efficiency of a certain input factor among multiple input factors. SFA can separate the efficiency value and error term by parameter estimation. DEA can solve the deficiency of SFA, but it cannot separate the efficiency value and error term. In order to fix these two shortcomings, the stochastic frontier distance function was proposed. This function can be used to estimate the characteristic parameters of the multi-output-input production technology. Zhou et al. [38] built an energy efficiency model based on this method and applied it in the empirical analysis of energy economy. This method can be used in the calculation of agricultural WUE to obtain more accurate results combined with the blue-green-grey water data.
We aim to obtain the total water footprint of the crop production process through water footprint theory and build a calculation model of the crop water use efficiency based on stochastic frontier distance function method, so as to more scientifically calculate and analyze the water-saving potential of the crop production. In this paper, we selected 14 cities in Liaoning Province as the research objective and calculated the structural and spatiotemporal characteristics of the crop growth water footprint, pollution water footprint, and production water footprint in agricultural regions in Liaoning from 2001 to 2017. After obtaining a more comprehensive reflection of crop type and amount of water used in the production, we respectively constructed evaluation models of the growth WUE, ecological WUE, and production WUE, and analyzed the temporal and spatial characteristics of crop WUE in each city and their improvement potential.

Overview of the Research Area
There are 14 cities in Liaoning Province, China, which are the main grain producing areas and key producing areas for high-quality fruits and various special products (in Figure 1). According to Statistical Yearbook of Liaoning Province, the area of grain crop was 34.68 × 10 5 hm 2 , the area of economic crop was 5.97 × 10 5 hm 2 , the vegetable area was 3.09 × 10 5 hm 2 , and the orchard area was 3.51 × 10 5 hm 2 in 2017. The agricultural planting structure of Liaoning Province was mainly dominated by the grain crop, accounting for 83.1% of the total planting area among which corn and rice accounted for 77.6% and 14.2% of the grain crop. solve the deficiency of SFA, but it cannot separate the efficiency value and error term. In order to fix these two shortcomings, the stochastic frontier distance function was proposed. This function can be used to estimate the characteristic parameters of the multioutput-input production technology. Zhou et al. [38] built an energy efficiency model based on this method and applied it in the empirical analysis of energy economy. This method can be used in the calculation of agricultural WUE to obtain more accurate results combined with the blue-green-grey water data. We aim to obtain the total water footprint of the crop production process through water footprint theory and build a calculation model of the crop water use efficiency based on stochastic frontier distance function method, so as to more scientifically calculate and analyze the water-saving potential of the crop production. In this paper, we selected 14 cities in Liaoning Province as the research objective and calculated the structural and spatiotemporal characteristics of the crop growth water footprint, pollution water footprint, and production water footprint in agricultural regions in Liaoning from 2001 to 2017. After obtaining a more comprehensive reflection of crop type and amount of water used in the production, we respectively constructed evaluation models of the growth WUE, ecological WUE, and production WUE, and analyzed the temporal and spatial characteristics of crop WUE in each city and their improvement potential.

Overview of the Research Area
There are 14 cities in Liaoning Province, China, which are the main grain producing areas and key producing areas for high-quality fruits and various special products (in Figure 1). According to Statistical Yearbook of Liaoning Province, the area of grain crop was 34.68 × 10 5 hm 2 , the area of economic crop was 5.97 × 10 5 hm 2 , the vegetable area was 3.09 × 10 5 hm 2 , and the orchard area was 3.51 × 10 5 hm 2 in 2017. The agricultural planting structure of Liaoning Province was mainly dominated by the grain crop, accounting for 83.1% of the total planting area among which corn and rice accounted for 77.6% and 14.2% of the grain crop. Liaoning Province belongs to the temperate continental monsoon climate zone. The rainfall and heat occur in the same season, with high cumulative temperature and four distinct seasons. The winter is long and the summer is warm, while the spring and autumn Liaoning Province belongs to the temperate continental monsoon climate zone. The rainfall and heat occur in the same season, with high cumulative temperature and four distinct seasons. The winter is long and the summer is warm, while the spring and autumn are short. The sunshine in the four seasons is uneven, being more in summer and autumn. The annual average temperature is between 7-11 • C. The temperature varies greatly from region to region due to the monsoon climate, decreasing from southwest to northeast. Annual average frost-free period is 130-200 d, gradually increasing from northwest to southeast. The rainfall in this province is uneven, with a wet area in the east and a dry one in the west. The annual precipitation in the eastern mountainous and hilly region is over 1100 mm. The annual average precipitation in the central plain is about 600 mm. The one in the western mountainous and the hilly region connected with the Inner Mongolia Plateau is about 400 mm. Liaoning is one of the provinces with severe water resources shortage in China. The spatial distribution of soil and water resources is very different and the potential of water resources development is very limited.

Methods
In this study, we obtained the total amount of crop production water consumption by the footprint theory, and then calculated the crop WUE by using the stochastic frontier distance function method. There are two advantages to obtain the amount of crop water use based on the water footprint theory: one is to calculate the total amount of the water use in the crop production; the other is to subdivide crop water consumption into crop growth and pollutant dilution consumption, so as to know the consumption direction of water resources. The crop water footprint can be named as the water footprint of crop growth, the water footprint of crop contamination and the water footprint of crop production. The crop WUE model can be further subdivided into the crop WUE, crop ecological WUE and crop production WUE.

Regional Crop Production Water Footprint Model
The regional crop production water footprint is the actual water consumption of all crop production processes in the region, including the growth water footprint and the pollution water footprint. The growth water footprint is the sum of the available precipitation (green water) and the field irrigation volume (blue water) during growth period [6,39]. Pollution water footprint refers to the water demand for diluting the pollution concentration caused by fertilizers and pesticides that cannot be fully absorbed by crops under the impact of precipitation and runoff and enter the receiving water body at a certain loss rate [1]. The calculation methods of regional crop production water footprint are shown in Equations (1)-(3): AWF = AWF grow + AWF pollution (1) AWF grow = ∑ AWF green,i + ∑ AWF blue,i AWF pollution = ∑ AWF grey,i where, AWF is the regional crop production water footprint, m 3 , AWF grow is the regional crop growth water footprint, and AWF pollution is the regional crop pollution water footprint, m 3 . AWF green,i is the regional green water footprint of the ith crop, m 3 , AWF blue,i is the regional blue water footprint, m 3 , and AWF grey,i is the regional gray water footprint, m 3 . The calculation formulas of regional green water and regional blue water are shown in Equations (4)-(7): AWF green = min (CWR, P eff ) × A (4) AWF blue = max (0, IR) × A (6) IR = CWR − P eff (7) where, CWR is the water requirement of crop per unit area, mm. P eff is the effective precipitation per unit area, mm. A is the planting area of crop, hm 2 . ET 0 is the reference evapotranspiration of crop, mm, and ET c is the water requirement of crop, mm. K c is the crop coefficient. The crop coefficients of wheat, cotton, corn, rice and other major crops referred to the "Water Demand and Irrigation for Major Crops in China", and the rest of the The grey water footprint of crop production conforms to the "short board principle", that is, the required volume of fresh water to dilute the maximum pollutant in the region is taken as the grey water footprint, while the key fertilizer pollutant is nitrogen fertilizer. Therefore, the regional grey water footprint can be calculated by Equation (8): where, AR is the yield of nitrogen fertilizer, kg/ hm 2 . α is the leaching rate, and was determined 10% based on the present research results and the situation in Liaoning [40]. C max is the maximum allowable concentration, kg/m 3 , and the mass concentration of nitrogen element in the water environmental quality standard shall not exceed 10 mg/L [13,41]. C nat is the natural background concentration of pollutant, kg/m 3 , which was assumed to be 0 in this study.

Regional Crop Water Use Efficiency Model
We took the calculation results of regional crop growth water footprint, pollution water footprint and production water footprint as the water consumption of crop production and adopted the stochastic frontier distance function (SDFA) to construct the evaluation models of crop growth WUE, ecological WUE and production WUE.
The model construction process of crop growth WUE is as follows: it was assumed that crop production takes input factors, such as capital (K), labor (L), land input (A), fertilizer (F) and water consumption (AWF), and economic income (Y) is obtained from crop production. Its economic technology (T) can be expressed in Equation (9): We referred to the method of energy efficiency to estimate energy saving potential [38], to define the Shephard directional distance function of water resource factor input, shown in Equation (10): According to the linear homogeneous characteristic of the directional distance function, Equation (10) can be converted into Equation (11): Equation (11) can then be transformed into Equation (12): Since the parameter estimation method requires the specific form of the function, we assumed that the production technique was a Cobb-Douglas function with linear homogeneous property, and the specific model form of the crop WUE is shown in Equation (13): where, i is the prefecture-level city. t is the time. v i,t is random disturbance term, satisfying the classical metering assumption. That is v i,t~i .i.d N(0,σ u 2 ). u i,t = lnD E (K i,t , L i,t , A i,t , E i,t , Y i,t ) ≥ 0, reflecting the inefficiency of water use in the crop planting process. WUE, WEI i,t = exp(−u i,t ) = ∅, u i,t follows a semi-normal distribution. Battese and Coelli set u i,t as u i,t = u i exp[−η(t − T)], where, the parameter represented by η means the influence of the time factor on the technical non-efficiency term [42]. They also set the total square deviation σ 2 = σ u 2 + σ v 2 and variance γ = σ_u 2 /σ 2 to test the proportion of technical invalid terms in the composite perturbation term, where γ is between 0 and 1. When γ approaches 1, the water use deviation is mainly determined by the term of water use inefficiency. When it approaches 0, the water use deviation is mainly determined by random errors. The construction methods of growth WUE and ecological WUE model were consistent with that of crop WUE model. The models are shown in Equations (14) and (15). (15) where AWF grow,i,t is the growth WUE of region i at time t, AWF pollution,i,t is Ecological WUE.

Data
This paper took the period from 2001 to 2017 as the research period and took 14 cities in Liaoning Province as the research units. This is because that city-level research range can reduce the impact of the economic development, natural condition, planting model and policy on the estimation result. Crop WUE analysis took agricultural gross product as output variable and took the price of 2001 as the base period for flattening. Input variables included agricultural workers, total power of machinery, crop sown area, amount of fertilizer, water footprint of regional crop growth footprint, pollution water footprint, and production water footprint, etc. All data were obtained from the Statistical Yearbook of Liaoning Province and the Statistical Yearbooks of all cities in Liaoning. We found that a total of 16 crops were planted in Liaoning Province, including soybean, sorghum, millet, potato, rice, corn, peanut, cotton, vegetable, fruit (apple, pear, grape), beet, sunflower, wheat, tobacco, sesame and hemp from 2002 to 2017. Due to the small planting area of hemp in the study period (less than 100 hm 2 ) and incoherent planting years, this study ignored hemp and considered a total of 15 crops. Meteorological data was from China Meteorological Data Network (http://data.cma.cn (accessed on 16 May 2018)), including monthly precipitation (mm), monthly average maximum temperature ( • C), monthly average minimum temperature ( • C), average wind speed (m/s), sunshine duration (h) and relative humidity (%) of 25 meteorological stations in all cities of Liaoning Province. The data of crop growth stage was from the Data Set of Crop Growth and Development status in China.

Characteristics of Regional Crop Water Footprint with Time
In order to obtain the characteristics of regional crop water footprint with time, we calculated the crop growth water footprint, pollution water footprint and production water footprint in all cities in Liaoning Province between 2001 and 2017, using a nonparametric Mann-Kendall test [43] to identify its change trends and significance, and used the variation coefficient to characterize its fluctuation degree. The results are respectively shown in Figure 2 and Table 1.

Changes of Regional Crop Growth Water Footprint
The crop growth water footprint in Liaoning Province was only 1.991 × 10 10 m 3 in 2001, followed by a wavy increase. It began to steadily rise after 2010 and reached to 2.788 × 10 10 m 3 in 2017. The increase of crop growth water footprint shows that water requirement pressure of crop increased. The M-K test value of regional crop growth water footprint for whole province was 4.24, the variation coefficient was 12.4%, and the longterm growth rate was 2.13% which was significant at the 1% level, which indicates that the crop growth water footprint of Liaoning Province presented a fluctuating increase trend during the study period. Except Shenyang, M-K test values for all cities were significantly positive, showing different degrees of growth. The long-term growth rates of eight cities were over 2%, and it was 4.52% in Fuxin. The variation coefficient of regional crop growth water footprint was between 6.73% and 24.74%, indicating that there was a great fluctuation difference of the crop growth water footprint in different cities. The most fluctuant region was Fuxin, followed by Chaoyang, where the variation coefficients were 24.74% and 20.17%, respectively. This is because these cities belong to the arid area. Only the variation coefficients in Benxi, Liaoyang and Shenyang showed a small degree of fluctuation, and they were less than 10%. The increase of the crop growth water footprint shows that crop water use pressure increased. We found that the blue-water accounted for 40.7% of the total crop growth water footprint by the analysis of the blue-green water footprint constitute, which illustrates that the rainfall could not satisfy the requirement of the crop growth, this puts forward higher requirements on water conservancy facilities, especially in Chaoyang, Fuxin and Panjin with the blue water footprint accounted for over 50%.  Note: **, * respectively mean significant at the level of 1%, 5%. Variation coefficient = standard deviation/average value * 100%. CV < 10% means weak variation and slow fluctuation. 10% < CV < 20% means medium variation and relative fluctuation.CV > 20% means strong variation and obvi-

Changes of Regional Pollution Water Footprint
Since 2000, the crop pollution water footprint in Liaoning Province steadily increased overall. It was 7.76 × 10 10 m 3 in 2001 and reached to the peak in 2013-2014 (9.76 × 10 10 m 3 , 9.71 × 10 10 m 3 ), then slowed down to 9.05 × 10 10 m 3 in 2017. This result illustrates that the policy of reducing fertilizer use in 2015 China obtained certain achievements, but the pollution water footprint degree of decline was limited. In the view of specific cities, the M-K test values of pollution water footprint in most cities were significantly positive, the long-term growth rates for 4 cities were over 2%, and the variation coefficients were below 13%, which shows a gentle rising trend. Only the M-K test value of pollution water footprint in Benxi was negative and significant at the level of 10%, and the variation coefficient was 3.22%, indicating that the pollution water footprint has shown a downward trend, but more slowly. The variation coefficient of pollution water footprint in Fuxin was more than 20%, and the long-term growth rate was 3.93%, showing fluctuation increase trend. The pollution water footprint was mainly related to nitrogen and phosphorus in fertilizer, and is closely related to crop growth requirements, farmland soil fertility and farmers' fertilization habits in different regions. Note: **, * respectively mean significant at the level of 1%, 5%. Variation coefficient = standard deviation/average value * 100%. CV < 10% means weak variation and slow fluctuation. 10% < CV < 20% means medium variation and relative fluctuation.CV > 20% means strong variation and obvious fluctuant. The long-term growth rate is defined as: (exp((ln(t + n) − ln(t))/n) − 1) * 100. If the long-term growth rate is over 2%, meaning increase fast.

Changes of Regional Crop Production Water Footprint
The crop production water footprint in Liaoning Province increased from 2.927 × 10 10 m 3 in 2001 to 3.693 × 10 10 m 3 in 2017. The M-k test value of crop production water footprint for the whole province was 4.49 and significant at 1% level, the long-term growth rate was 1.82%, and the variation coefficient was 11.07%. The M-K test values for all cities were significantly positive, the long-term growth rates for 6 cities were over 2%, and the variation coefficients were between 5.16 and 24.18%, showing fluctuation increase trend, especially in Fuxin and Chaoyang. No matter from the perspective of Liaoning province or all cities, the changes of production water footprint and growth water footprint were more similar, mainly because the proportion of growth water footprint in production water footprint was greater in all regions, and the increase speed of pollution water footprint was lower than that of growth water footprint.

Spatial Difference Analysis of Regional Crop Water Footprint
Based on K-mean-value clustering analysis, we classified the spatial distribution of crop growth water footprint, pollution water footprint and production water footprint in all cities of Liaoning Province, and the results are shown in Figure 3. Based on K-mean-value clustering analysis, we classified the spatial distribution of crop growth water footprint, pollution water footprint and production water footprint in all cities of Liaoning Province, and the results are shown in Figure 3.  showing that the differences of crop growth water footprint among cities was very obvious. The areas with higher crop growth water footprint were distributed in the north of Liaoning Province and the areas with lower crop growth water footprint was in the east of Liaoning Province, and the distribution characteristics were very similar to the distribution of crop scale. In 2001, there was only 5 cities with the crop growth water footprint less than 9 × 10 8 m 3 . By 2017, the crop growth water footprint in all cities has increased by an average of 40% compared with that in 2001, due to the increasing scale of crop irrigation. The Distribution of Regional Pollution Water Footprint The average crop pollution water footprint in Liaoning Province was 6.26 × 10 8 m 3 as shown in Figure 3b, which was 36.5% of the regional growth water footprint. The highest values were in Shenyang (1.2785 × 10 9 m 3 ) and Tieling (1.174 × 10 9 m 3 ), and the lowest ones were in Fushun (2.0001 × 10 8 m 3 ) and Benxi (9.37 × 10 8 m 3 ). The difference between the maximum and minimum value was 13.65 times, which was much greater than that the regional crop growth water footprint, and their distributions were slightly different. For example, Dalian had a high pollution water footprint, but the growing water footprint belonged to a medium area, while Fuxin and Chaoyang were on the contrary. Their distribution characteristics were not only affected by crop area, but also by soil nutrient composition and planting structure. During the study period, the regional pollution water footprint increased by 16.7% on average, and the pollution water footprint of Fuxin, Anshan, Jinzhou, Shenyang and other central and northern cities increased.
The Distribution of Regional Production Water Footprint The average value of the crop production water footprint of in Liaoning Province was 2.34 × 10 9 m 3 as shown in Figure 3c, with a difference of 11 times between the maximum value in Shenyang (4.585 × 10 9 m 3 ) and the minimum value in Benxi (4.15 × 10 8 m 3 ). Compared with 2001, the crop production water footprint increased in all cities in 2017, with an average increase of 33.46%. The number of regions with severe consumption of the crop production water footprint increased from 2 to 4, and the severity increased in three cities. The growth water footprint was the main component of the production water footprint, accounting for up to 73.2%, so its spatial distribution characteristics were very similar to the crop growth water footprint.

Analysis of Crop Water Use Efficiency
Based on the crop WUE model, we measured the WUEs of growth, ecology, production. Their differences lay in the different types of water used in crop production. The empirical results are showed in Table 2. It can be seen from the maximum likelihood value and LR value that the crop WUE model had a good fitting effect on the whole and was of value for further analysis. σ 2 was significant at the level of 5%, which indicates that the model error term had an obvious compound structure. The values of γ were all above 0.9, indicating that more than 90% of the error was caused by non-efficiency factors. η was negative, indicating that crop water use efficiency presented a downward trend over time. Note: **, * respectively mean significant at the level of 1%, 5%. The number in brackets are the sta ard errors of the coefficients.

Dynamic Analysis of Crop Water Use Efficiency
In this paper, we selected the core density curves of crop WUE in 2001, 2010 and 2 to further explore the dynamic evolution trend of the crop WUE in Liaoning Provin The results are shown in Figure 4.

Changes of Crop Growth Water Use Efficiency
The "main peak" of the crop WUE curve in Liaoning Province slightly moved to th left, reflecting that the growth WUE showed a downward trend over time. The "unimod peak" of the crop WUE curve in 2001 gradually became a relatively obvious "bimod peak" form in 2017, indicating that the growth WUE appeared a trend of two-stage diffe entiation, and most of them were concentrated in areas with low efficiency.

Changes of the Crop Ecological Water Use Efficiency
The "main" peak of crop ecological WUE curve shown in Figure 3b also showed th tendency of the left, and had more significant change than that of crop growth WUE whic illustrates that decrease amplitude of crop ecological WUE over time was higher than th of the growth WUE. The highest decrease amplitude was 0.08% during the research p riod, and the lowest one was 18.8%. The peak value of "main peak" showed a significan downward trend, and the extension area on the left side of the curve gradually decrease while the extension area on the right side gradually increased, indicating that the diffe ence of ecological WUE between different cities gradually increased over time, and th decrease amplitude with higher efficiency was significantly lower than that with low efficiency. The curve gradually turned into a "multi-peak" shape, indicating that the ec logical WUE gradually formed a certain agglomeration effect.

Changes of the Crop Production Water Use Efficiency
The "main peak" of the crop production WUE curve in Liaoning Province shown Figure 3c showed a shift to the left, and curve was relatively smooth. The decrease ampl tude was between the first two WUE, indicating that the differences of the crop produ tion WUE among different cities were gradually increasing. In 2010 and 2017, the cro production WUE curves had an obvious "bimodal" pattern, forming a relatively obviou polarization trend, and the one in 2017 was more obvious than that in 2010. It can be see

Changes of Crop Growth Water Use Efficiency
The "main peak" of the crop WUE curve shown in Figure 4a in Liaoning Province slightly moved to the left, reflecting that the growth WUE showed a downward trend over time. The "unimodal peak" of the crop WUE curve in 2001 gradually became a relatively obvious "bimodal peak" form in 2017, indicating that the growth WUE appeared a trend of two-stage differentiation, and most of them were concentrated in areas with low efficiency.

Changes of the Crop Ecological Water Use Efficiency
The "main" peak of crop ecological WUE curve shown in Figure 4b also showed the tendency of the left, and had more significant change than that of crop growth WUE which illustrates that decrease amplitude of crop ecological WUE over time was higher than that of the growth WUE. The highest decrease amplitude was 0.08% during the research period, and the lowest one was 18.8%. The peak value of "main peak" showed a significant downward trend, and the extension area on the left side of the curve gradually decreased, while the extension area on the right side gradually increased, indicating that the difference of ecological WUE between different cities gradually increased over time, and the decrease amplitude with higher efficiency was significantly lower than that with lower efficiency. The curve gradually turned into a "multi-peak" shape, indicating that the ecological WUE gradually formed a certain agglomeration effect.

Changes of the Crop Production Water Use Efficiency
The "main peak" of the crop production WUE curve in Liaoning Province shown in Figure 4c showed a shift to the left, and curve was relatively smooth. The decrease amplitude was between the first two WUE, indicating that the differences of the crop production WUE among different cities were gradually increasing. In 2010 and 2017, the crop production WUE curves had an obvious "bimodal" pattern, forming a relatively obvious polarization trend, and the one in 2017 was more obvious than that in 2010. It can be seen from its evolution that the evolution of the crop production WUE was influenced by the first two WUE, but mainly by the growth water footprint.

Spatial Analysis of Crop Water Use Efficiency
Based on K-mean-value clustering analysis, we classified the spatial distribution of multi-year average of the crop water use efficiency in all cities of Liaoning Province from 2001 to 2017, and the results are showed in Figure 5.

Distributions of Crop Growth Water Use Efficiency
The crop growth WUE in Liaoning Province shown in Figure 4a gradually decreased from east to west, which was basically consistent with rainfall distribution. The average of crop growth WUE of the whole province was 0.821, which means that the water consumption of Liaoning province should be reduced by 17.9% to eliminate the inefficiency of the crop growth water use. The highest crop growth WUE were in Dandong (0.983) and Fushun (0.975), which was closest to the water use front constructed by the model. The second higher crop growth WUE were in Benxi (0.920) and Liaoyang (0.899), and they can close to the water use front by only a small amount of adjustment. The growth WUEs in Chaoyang (0.817), Tieling (0.821) and Shenyang (0.824) constituted the slightly effective area of crop growth WUE. If attention were paid to technology introduction, optimal allocation of water resources and adjustment of planting structure in the future, these cities can be easily transformed into a highly productive area. In Jinzhou (0.745), Huludao (0.744), Dalian (0.745), Fuxin (0.734) and Yingkou (0.724), the crop growth WUE was lower than the provincial average value, which was difficult to catch up with the water use front. This means that the crop growth water footprint was too large when the output value and other input factor remained unchanged. These cities are the key areas to improve the crop growth water efficiency in the future. Planting technology and water-saving technology need to be introduced, and even the planting structure needs to be adjusted in some areas.

Distributions of the Crop Ecological Water Use Efficiency
The areas with high crop ecological WUE in Liaoning Province were concentrated in eastern and western Liaoning, as shown in Figure 4b, and showed a small range of aggregation. The mean value of crop ecological WUE in the whole province was 0.845, indicating that under the condition of constant economic output, the impact of crop on water environment can be improved to save 15.5% water consumption. The first two highest crop ecological WUE were in Benxi (0.986) and Chaoyang (0.984). The highly effective regions of crop ecological WUE included Shenyang (0.943), Fushun (0.913) and Huludao (0.891), whose crop production activities had less impact on water environment than those of other cities. The lowest crop ecological WUE was in Jinzhou (0.700) and Anshan (0.695), which illustrates that it is urgent to take relevant measures to alleviate the water environment pressure caused by crop production. After improvement, more than 30% of the pollutant dilution water consumption will be saved, showing that there is considerable room for improvement to save the water consumption.

Distributions of Crop Growth Water Use Efficiency
The crop growth WUE in Liaoning Province shown in Figure 5a gradually decreased from east to west, which was basically consistent with rainfall distribution. The average of crop growth WUE of the whole province was 0.821, which means that the water consumption of Liaoning province should be reduced by 17.9% to eliminate the inefficiency of the crop growth water use. The highest crop growth WUE were in Dandong (0.983) and Fushun (0.975), which was closest to the water use front constructed by the model. The second higher crop growth WUE were in Benxi (0.920) and Liaoyang (0.899), and they can close to the water use front by only a small amount of adjustment. The growth WUEs in Chaoyang (0.817), Tieling (0.821) and Shenyang (0.824) constituted the slightly effective area of crop growth WUE. If attention were paid to technology introduction, optimal allocation of water resources and adjustment of planting structure in the future, these cities can be easily transformed into a highly productive area. In Jinzhou (0.745), Huludao (0.744), Dalian (0.745), Fuxin (0.734) and Yingkou (0.724), the crop growth WUE was lower than the provincial average value, which was difficult to catch up with the water use front. This means that the crop growth water footprint was too large when the output value and other input factor remained unchanged. These cities are the key areas to improve the crop growth water efficiency in the future. Planting technology and water-saving technology need to be introduced, and even the planting structure needs to be adjusted in some areas.

Distributions of the Crop Ecological Water Use Efficiency
The areas with high crop ecological WUE in Liaoning Province were concentrated in eastern and western Liaoning, as shown in Figure 5b, and showed a small range of aggregation. The mean value of crop ecological WUE in the whole province was 0.845, indicating that under the condition of constant economic output, the impact of crop on water environment can be improved to save 15.5% water consumption. The first two highest crop ecological WUE were in Benxi (0.986) and Chaoyang (0.984). The highly effective regions of crop ecological WUE included Shenyang (0.943), Fushun (0.913) and Huludao (0.891), whose crop production activities had less impact on water environment than those of other cities. The lowest crop ecological WUE was in Jinzhou (0.700) and Anshan (0.695), which illustrates that it is urgent to take relevant measures to alleviate the water environment pressure caused by crop production. After improvement, more than 30% of the pollutant dilution water consumption will be saved, showing that there is considerable room for improvement to save the water consumption.

Changes of the Crop Production Water Use Efficiency
Average value of the crop production WUE was 0.865 in Liaoning Province, which means that the crop production water consumption can be saved by 13.5% considering the crop evapotranspiration and impact of the crop production on water environment at the same time, showed in Figure 5c. This saved space was less than those of the crop ecological WUE and crop growth WUE, this is because that to improve the crop production WUE, not only the factors affecting the crop growth water consumption, such as the natural conditions and planting structure, but factors influencing the pollution footprint, such as the fertilizer degree of farmland soil and farmers fertilize habits, need to be considered too. These constraints lead to the space for promoting crop production WUE being smaller. The regions with high crop production WUE included Liaoyang (0.936), Tieling (0.851), Shenyang (0.909) and Fushun (0.986), which were distributed in the east and central part of Liaoning Province. The regions with low crop production WUE included Jinzhou (0.773), Anshan (0.791), Yingkou (0.766). They should be given special attention in the formulation and implementation of water resource policy in the future.

Discussion
Our study found that the combination of the whole water footprint in crop production obtained by the water footprint theory and the stochastic frontier distance function calculation model of crop water use efficiency, can more scientifically reflect the consumption of regional scale crop water dynamic characteristics, that is, growth water use efficiency, ecological water use efficiency, and production water use efficiency revealed the water saving potential of crop in each city of Liaoning Province.
In this paper, we obtained that the average value of total water footprint of 17 years was 4.68 × 10 9 m 3 . Of this, the water footprint of crop growth was 1.714 × 10 9 m 3 , the ecological water footprint of crop was 6.26 × 10 8 m 3 , and the water footprint of crop production was 2.34 × 10 9 m 3 . We measured the real consumption of water resources in the whole process of the crop production from the perspective of blue-green-grey water. This calculation method is consistent with other methods used in agricultural water use efficiency evaluation [23][24][25][26][27][28].
Based on the water footprint calculated by the water footprint theory, we established the crop water use efficiency model by using the stochastic distance function considering the total factors of production and calculated the crop water use efficiency of all cities in Liaoning Province. The annual average crop growth WUE, ecological WUE and production WUE of all cities in Liaoning province were 0.821, 0.845 and 0.865, respectively. The results calculated by this method took into account not only the actual total water consumption in agricultural production process, but also other production factors. We compared this result with that calculated by taking the blue-green water as the water resource input of crop, showed in Figure 6. We found that crop WUEs considering the blue-green-grey water for 14 cities in Liaoning province were higher than those obtained by only considering the blue-green water. This results once again shows that in practical production, the watersaving space was smaller, conforming to the actual production conditions. In the actual crop production process, the two main reasons leading to the reduction of available water resources include crop growth water and pollutant dilution water consumption. Therefore, we have taken into account these two factors in this crop water use efficiency model, so as to reflect the real crop water consumption. The total factor crop WUE model includes the contribution of other input factors to yield and reflects the actual production situation more than that done by the single factor crop WUE model. Based on the above two points, this crop water use efficiency model can quantify the volume of water saving in technical feasibility by adjusting crop planting structure, adjusting the ratio of agricultural input factors, introducing advanced production technology and management technology, etc.
After we obtained the crop water use efficiency, we assessed that Liaoning Province can save 17.9% of crop water consumption through adjustment of crop water technology popularization and policy management, equivalent to 8.38 × 10 8 m 3 , reduce ecological water consumption by 15.5%, equivalent to 7.25 × 10 8 m 3 , and save the crop production water consumption by 13.5% when crop evapotranspiration and pollutant dilution were taken into account, equivalent to 6.32 × 10 8 m 3 .

Conclusions
Based on the panel data of 14 cities in Liaoning province from 2001 to 2017, we calculated the crop growth water footprint, crop ecological water footprint and crop production water footprint through the water footprint theory, established the crop water use efficiency model by using the stochastic distance function, and analyzed the spatial-temporal variation characteristics of crop WUE, ecological WUE and production WUE. We got the following conclusion: In this paper, we obtained that the water footprint of crop growth was 1.714 × 10 9 m 3 , the ecological water footprint of crop was 6.26 × 10 8 m 3 , and the water footprint of crop production was 2.34 × 10 9 m 3 . We also calculated that the WUE of crop growth was 0.821, the crop ecological WUE was 0.845 and crop production was 0.865, respectively. Liaoning Province can save 17.9% of crop water consumption through adjustment of crop water technology popularization and policy management, equivalent to 8.38×10 8 m 3 , reduce ecological water consumption by 15.5%, equivalent to 7.25 × 10 8 m 3 , and save the crop production water consumption by 13.5% when crop evapotranspiration and pollutant dilution were taken into account, equivalent to 6.32 × 10 8 m 3 .
The crop water use efficiency model obtained by using the stochastic distance function can be more reasonable and scientific to estimate the crop water saving potential. This method is suitable for the areas with scarce water resources and strong intention of saving water. Among them, the crop water efficiency model based on the growth water footprint can be used in agricultural areas with excessive water consumption. The crop water efficiency model based on the ecological water footprint can be used in areas with serious agricultural water pollution. The crop water efficiency model based on the production water footprint can be used in both these two areas with excessive water consumption and serious agricultural water pollution. These areas first estimate the water saving potential by our method, then realize the water saving by adjusting the planting structure, reducing fertilizer input, improving the management technology, inducing advanced planting technology. After we obtained the crop water use efficiency, we assessed that Liaoning Province can save 17.9% of crop water consumption through adjustment of crop water technology popularization and policy management, equivalent to 8.38 × 10 8 m 3 , reduce ecological water consumption by 15.5%, equivalent to 7.25 × 10 8 m 3 , and save the crop production water consumption by 13.5% when crop evapotranspiration and pollutant dilution were taken into account, equivalent to 6.32 × 10 8 m 3 .

Conclusions
Based on the panel data of 14 cities in Liaoning province from 2001 to 2017, we calculated the crop growth water footprint, crop ecological water footprint and crop production water footprint through the water footprint theory, established the crop water use efficiency model by using the stochastic distance function, and analyzed the spatial-temporal variation characteristics of crop WUE, ecological WUE and production WUE. We got the following conclusion: In this paper, we obtained that the water footprint of crop growth was 1.714 × 10 9 m 3 , the ecological water footprint of crop was 6.26 × 10 8 m 3 , and the water footprint of crop production was 2.34 × 10 9 m 3 . We also calculated that the WUE of crop growth was 0.821, the crop ecological WUE was 0.845 and crop production was 0.865, respectively. Liaoning Province can save 17.9% of crop water consumption through adjustment of crop water technology popularization and policy management, equivalent to 8.38×10 8 m 3 , reduce ecological water consumption by 15.5%, equivalent to 7.25 × 10 8 m 3 , and save the crop production water consumption by 13.5% when crop evapotranspiration and pollutant dilution were taken into account, equivalent to 6.32 × 10 8 m 3 .
The crop water use efficiency model obtained by using the stochastic distance function can be more reasonable and scientific to estimate the crop water saving potential. This method is suitable for the areas with scarce water resources and strong intention of saving water. Among them, the crop water efficiency model based on the growth water footprint can be used in agricultural areas with excessive water consumption. The crop water efficiency model based on the ecological water footprint can be used in areas with serious agricultural water pollution. The crop water efficiency model based on the production water footprint can be used in both these two areas with excessive water consumption and serious agricultural water pollution. These areas first estimate the water saving potential by our method, then realize the water saving by adjusting the planting structure, reducing fertilizer input, improving the management technology, inducing advanced planting technology.