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Simulation and optimization for crop water allocation based on crop water production functions and climate factor under uncertainty
Applied Mathematical Modelling 37 (2013) 7708–7716
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Applied Mathematical Modelling journal homepage: www.elsevier.com/locate/apm
Simulation and optimization for crop water allocation based on crop water production functions and climate factor under uncertainty Fangfang Tong a,b, Ping Guo a,c,⇑ a
Centre for Agricultural Water Research in China, China Agricultural University, Beijing 100083, China Sinohydro Corporation Limited, Beijing 100048, China c Environmental Science and Engineering Program, University of Northern British Columbia, Prince George, Canada BC V2N 4Z9 b
a r t i c l e
i n f o
Article history: Received 1 May 2012 Received in revised form 4 November 2012 Accepted 1 March 2013 Available online 15 March 2013 Keywords: Crop water production function Meteorological factor Optimal allocation Water resources Irrigation Uncertainty
a b s t r a c t In this paper, the uncertainty methods of interval and functional interval are introduced in the research of the uncertainty of crop water production function itself and optimal allocation of water resources in the irrigation area. The crop water production functions in the whole growth period under uncertainty and the optimal allocation of water resources model in the irrigation area under uncertainty are established, and the meteorological factor is considered in the model. It can promote the practical application of the uncertain methods, reflect the complexity and uncertainty of the actual situation, and provide more reliable scientific basis for using water resources economically, fully improving irrigation efficiency, and keeping the sustainable development of the irrigated area. This approach has important value on theoretical and practical for the optimal irrigation schedule, and has very broad prospects for research and development to other related agriculture water management. Ó 2013 Elsevier Inc. All rights reserved.
1. Introduction The increased water deficits associated with overuse of surface water and the declining groundwater levels are threatening the sustainability of agricultural production in the regions of Northwest of China [1]. Therefore, the improvement of water management practices is the utmost importance to ensure crop production with existing supplies of water [2,3]. Crop water production function is the mathematical description of the influence of water supply time and quantity on crop yield during crop growth. It is the main basis of evaluating various irrigation strategies and the foundation of determining economic water quotas and optimal allocation of water. In order to use limited water resources reasonably to achieve maximum crop yield and schedule the reasonable allocation of limited water supply both in temporal and spatial scale, the research on crop water production function was rapidly developed [4–8]. For example, De Juan et al. [9] developed a model for optimal cropping patterns within the farm based on crop water production functions and irrigation uniformity. Igbadun et al. [10] presented the performance evaluation of four selected crop water production functions. Brumbelow and Georgakakos [11] presented a new algorithm type based upon differential crop yield response to irrigation that uses these crop models to determine planning-level irrigation schedules and crop water production functions.
⇑ Corresponding author at: Centre for Agricultural Water Research in China, China Agricultural University, Beijing 100083, China. Tel.: +86 10 62738496. E-mail address: [email protected] (P. Guo). 0307-904X/$ - see front matter Ó 2013 Elsevier Inc. All rights reserved. http://dx.doi.org/10.1016/j.apm.2013.03.018
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In previous research on crop water production function, the problem was simplified into the deterministic problem, and only the deterministic function of the relationship between crop yield and evapotranspiration was established, so evapotranspiration corresponding to the crop highest yield also was the deterministic value. However, there are a number of uncertain factors in the actual problems, so the actual application value of the research results is reduced, due to the uncertain factors did not be considered in reflecting the practical problems. At present, generally the deterministic optimal allocation of water resources in the irrigation area is studied [12]. Some scholars consider the random factors (e.g. rainfall, runoff) and establish some stochastic programming models [13,14]. However, still no scholar has researched the uncertainty of water production function itself and its application on the optimization of irrigation water distribution systematically. Climate change has the impacts on many fields. There are a number of studies have been undertaken in the last decades to evaluate the impacts of climate change on water resources and agriculture [15–19]. However, there are few studies to consider the meteorological factor in optimal allocation of water resources in the irrigation area, especially in optimal allocation of water resources in the irrigation area under uncertainty. In this paper, the uncertainty methods of interval and functional interval are introduced in the research of crop water production function and optimal allocation of water resources in the irrigation area to accurately reflect the real-world problem. The crop water production functions in the whole growth period under uncertainty are firstly presented. Then based on above functions, the optimal allocation of water resources model in the irrigation area under uncertainty is established, and the meteorological factor is introduced in the model. It can promote the practical application of the uncertainty methods and provide more reliable scientific basis for using water resources economically, fully improving irrigation efficiency and keeping the sustainable development of the irrigated area. This approach has important value on theoretical and practical for the optimal irrigation schedule, and has very broad prospects for research and development to other related agriculture water management. 2. Study area The research area is located in Minqin, Gansu Province, China. It belongs to Wuwei region (101°490 104°120 E, 38°030 39°280 N) and the central area is the Shiyang River of alluvial narrow and flat oasis belt. It has the characteristic of North low and South high terrain, 1200–1500 m above sea level. The natural gradient is about l per thousand. It is surrounded by high terrain with basin landform characteristics [20]. In recent decades, due to the impact of human activities and climate change, severe land desertification appeared in Minqin. The ecological environment is deteriorating and water shortage has been very prominent to the front. Hongyashan irrigation region locates in the oasis belt of Minqin County. Its effective irrigation area is about 63073.3 hectare, and it is a typical continental temperate arid climate. Annual precipitation is less than 150 mm, while annual evaporation capacity is 2000–2600 mm. Irrigation water resources available are mainly from hongyashan reservoir water and groundwater. Due to excessive use in the middle irrigation area, surface water decreased year by year, and to 2003 has been less than 100 million m3. Groundwater which was not repeated with the surface water had only 31 million m3. Water resources is very scarce, and agriculture implements deficient irrigation [21]. 3. Crop water production function in the whole growth period under uncertainty 3.1. Method of Interval regression analysis The selected research crops, spring wheat, cotton, seed melon and honey dew melon are main crops planted in Minqin. The data of crop water consumption in the whole growth period and yield came from pilot study on water production functions and optimal irrigation schedule of main crops in Shiyang river basin [22]. Due to the information used for the water production functions are not the same, the interval regression analysis is useful to address the collected information. An interval linear regression model can be written as
YðxÞ ¼ A0 þ A1 x1 þ þ An xn ¼ Ax Where x ¼ ð1; x1 ; . . . ; xn Þt is a real input vector, A ¼ ðA0 ; . . . ; An Þ is an interval coefficient vector, and YðxÞ is the corresponding estimated interval. An interval coefficient Ai is denoted as Ai ¼ ðai ; ci Þ where ai is a center and ci is a radius. There are many approaches to achieve the interval regression analysis, such as translating into quadratic optimization problems, using neural networks, using support vector machines. This research used the relatively mature interval regression analysis method based on quadratic programming, and integrates the central tendency of least squares and possibilistic property of fuzzy regression. The proposed model can be represented as follows [23]:
Min J ¼ k1 a;c
p p X X ðyj at xj Þ2 þ k2 ct jxj jjxj jt c j¼1
t
j¼1
t
Subject to a xj þ c jxj j P yj ,
at xj ct jxj j 6 yj ;
j ¼ 1; . . . ; p
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ci P 0;
i ¼ 0; . . . ; n
Where jxj j ¼ ð1; jxj1 j; . . . ; jxjn jÞt ; a ¼ ða0 ; . . . ; an Þt , c ¼ ðc0 ; . . . ; cn Þt , k1 and k2 are weight coefficients. When the value of k1 and k2 are changed, the regression results may be different.
3.2. Calculation process and result analysis In this section, the relationship between yield and water consumption in the whole growth period of the typical crops in Minqin is addressed. That means the water production function in the whole growth period. Because crop water production function model of the whole growth period presented as the quadratic model in high yield level regions [24], the interval quadratic regression analysis method is presented based on quadratic programming to analyze yield and water consumption in the whole growth period data of spring wheat, cotton, seed melon and honey dew melon in Minqin, respectively. We had
(a)
(b)
(c)
(d)
Fig. 1. Interval quadratic regression results of four kinds of crops yield and water consumption in the whole growth period. (Fig. 1(a). spring wheat; Fig. 1(b). cotton; Fig. 1(c). seed melon; Fig. 1(d). honey dew melon).
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five sets of values of k1:k2, which were equal to 1:0.0001, 1:0.5, 1:1, 0.5:1, 0.0001:1. LINGO software was used to solve above model, and the results are depicted in Fig. 1. Fig. 1 shows that when a large valued k1 compared to k2, the more central tendency would be expected, i.e., the obtained center regression line tend to be the regression line obtained by least squares regression. On the contrary, when a large valued k2 compared to k1, reducing the fuzziness of the model would be focused. The larger values of k2:k1, the less central tendency is considered. When the values of k2:k1 increase to a certain level, and then increase the values of k2:k1, the intervals obtained would not be changed. When fully considered the central tendency, the interval regression results of k1 = 1, k2 = 0.0001 are selected as uncertain crop water production functions in the whole growth period. Due to the first-order term coefficients and quadratic term coefficients of cotton and seed melon k1 = 1, k2 = 0.0001 regression results are deterministic, the water consumption corresponding to the highest yield is the certainty. For this reason, the regression values of cotton and seed melon are selected as k1 = 1, k2 = 0.5. The obtained water production functions in the whole growth period of four crops under uncertainty in this study are as follows:
Spring wheat Y 1 ðxÞ ¼ ð16419:3; 778:682Þ þ ð8:27316; 0:00942176Þx1 þ ð0:000712642; 0Þx21 ; Cotton Y 2 ðxÞ ¼ ð8010:99; 191:871Þ þ ð4:45270; 0:0103980Þx2 þ ð0:000516700; 0Þx22 Seed melon Y 3 ðxÞ ¼ ð293:266; 145:859Þ þ ð1:82795; 0Þx3 þ ð0:000247975; 3:63015E 06Þx23 ; Honey dew melon Y 4 ðxÞ ¼ ð413273; 2802:46Þ þ ð254:449; 2:17845Þx4 þ ð0:0347713; 0Þx24 Compared with determinate water production functions in the whole growth period of four crops:
R2 ¼ 0:8337
Spring wheat Y 01 ðxÞ ¼ 16001:7 þ 8:11755x1 0:000698686x21 Cotton Y 02 ðxÞ ¼ 6495:91 þ 3:72971x2 0:000434445x22
R2 ¼ 0:8351
Seed melon Y 03 ðxÞ ¼ 220:904 þ 1:83319x3 0:000267595x23
R2 ¼ 0:9926
Honey dew melon Y 04 ðxÞ ¼ 413373 þ 254:505x4 0:0347791x24
R2 ¼ 0:6130
Considering the spring wheat as an example, the measured values, predict values of interval regression model, and predict values of general regression model are compared. The results are shown in Table 1. First, whether it is interval regression model or general regression model, its predict values are very close to the measured values. A random sample of one of them, take No.1 as an example, its measured value is 4800; the center value of interval regression model is 5006.74, and relative error is 4.3%; the calculation value of general regression model is 5029.73, and relative error is 4.8%. This shows that both models can be used to describe the relationship between crop yield and water consumption in the whole growth period. Second, all prediction values of general regression model are within prediction intervals of interval regression model. Interval
Table 1 Comparison of the measured values, predict values of interval regression model Y⁄(x) and general regression model Y0 (x) (Spring wheat). No.
x (m3/hm2)
y (kg/hm2)
Y⁄(x) (kg/hm2)
Center value (kg/hm2)
Y0 (x) (kg/hm2)
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18
3900 3900 4125 4200 4500 4500 4500 4950 5000 5100 5100 5175 5250 5550 5775 5850 6600 7650
4800 4950 5550 5850 5700 6450 7200 6750 5500 7650 7200 7200 7650 7800 7950 7200 6300 5550
[4191.31, 5822.17] [4191.31, 5822.17] [4763.89, 6398.98] [4938.71, 6575.22] [5557.84, 7200.00] [5557.84, 7200.00] [5557.84, 7200.00] [6246.01, 7896.65] [6304.66, 7956.24] [6411.26, 8064.73] [6411.26, 8064.73] [6481.86, 8136.74] [6544.45, 8200.74] [6714.61, 8376.55] [6758.05, 8424.23] [6756.49, 8424.09] [6300.33, 7981.73] [4314.01, 6015.53]
5006.74 5006.74 5581.44 5756.97 6378.92 6378.92 6378.92 7071.33 7130.45 7237.99 7237.99 7309.30 7372.59 7545.58 7591.14 7590.29 7140.86 5164.77
5029.73 5029.73 5594.61 5767.19 6378.88 6378.88 6378.88 7060.62 7118.90 7224.98 7224.98 7295.37 7357.90 7529.43 7575.54 7575.19 7139.37 5208.71
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Table 2 The highest yield of each crop and water consumption of corresponding to the highest yield (interval results). Crop
The highest yield (kg/hm2)
The corresponding water consumption (m3/hm2)
Spring wheat Cotton Seed melon Honey dew melon
[6758.42, 8425.16] [1345.24, 1818.59] [2880.94, 3271.31] [41490.2, 63036.6]
[5797.96, 5811.18] [4298.72, 4318.84] [3632.56, 3740.50] [3627.57, 3690.22]
Table 3 The highest yield of each crop and water consumption of corresponding to the highest yield (determinate number results). Crop
The highest yield (kg/hm2)
The corresponding water consumption (m3/hm2)
Spring wheat Cotton Seed melon Honey dew melon
7576.35 1508.98 2918.72 52228.4
5809.15 4292.50 3425.31 3658.88
regression model provides the upper and the lower limits of predict value and center value, while predict value of general regression model is just a real number. Thus, interval regression model provides more information than general regression model does. It reflects the uncertainty and the former has more advantages than the latter. The highest yield of each crop and water consumption of corresponding to the highest yield are calculated. The results are shown in Tables 2 and 3. It is noticed in Tables 2 and 3 that all the highest yields calculated by the traditional deterministic water production functions in the whole growth period are within the highest yield intervals calculated by the uncertainty crop water production functions in the whole growth period. Furthermore, spring wheat and honey dew melon’s water consumptions corresponding to the certainty highest yields are both within the uncertain interval results. Only cotton and seed melon’s water consumptions of corresponding to the deterministic highest yields are beyond the corresponding uncertainty intervals. But the excess portions are very small, and do not affect the research results. In fact, the results of interval quadratic functions can reflect the uncertainty of actual situation, and can represent the actual situation more accurately. 4. Optimal allocation of water resources in the irrigation area under uncertainty 4.1. Optimal allocation of water resources model in the irrigation area under uncertainty 4.1.1. Model development There are two assumptions in this study: (i) the change value of soil water in the whole growth period is zero; (ii) the groundwater recharge is negligible in the irrigation area. Consider the high water management level, that is, evapotranspiration in the whole growth period is equal to effective irrigation quota plus effective rainfall. Hongyashan irrigation region in Minqin County is selected as the research object, and the spring wheat, cotton, seed melon and honey dew melon (j = 1, 2, 3, 4) are selected as study crops. Although forage acreage in this region was large, it was not selected due to lack of available data. The yields of determinate water production functions in the whole growth period are replaced by uncertain water production functions. According to the principle of maximum economic benefit, uncertain water resources optimal allocation model in irrigation area can be formulated as follows:
max f ¼
4 4 4 4 X X X X Bj Aj Y j C M j Aj ¼ Bj Aj ½k1j þ k2j ðgt M j þ pj Þ þ k3j ðgt M j þ pj Þ2 C M j Aj j¼1
j¼1
j¼1
j¼1
subject to: 4 X Mj Aj 6 gq Q j¼1
0 6 gt M j þ pj 6 ET max
j
Mj P 0 where, f±: objective function (Yuan); Bj: the price of crop j (Yuan/kg); Aj: the planting area of crop j (hm2); Yj±: the yield of crop j (kg/hm2); C: the unit cost of irrigation water (Yuan/m3); Mj±: decision variable, the irrigation quota of unit area crop j in the whole growth period (m3/hm2); gt : the field water utilization coefficient; gq : the canal system water utilization
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coefficient; pj: the effective precipitation of crop j in the whole growth period (m3/hm2); Q: the total available irrigation water (m3); ETmax j: the maximum evapotranspiration of crop j in the whole growth period (m3/hm2). 4.1.2. Solution method Assuming that 2010 was a normal flow year (P = 50%). The real data of Hongyashan irrigation region in 2010 are used to calculate the model. Transformed the original model into another formulation, which is written as follows (let x j ¼ gt M j þ pj ):
max f ¼
4 X pj CAj xj þ Bj Aj k3j ðxj Þ2 þ CAj Bj Aj k1j þ Bj Aj k2j
gt
j¼1
gt
subject to: 4 X Aj j¼1
gt
xj
p j Aj
gt
0 6 xj 6 ET max
6 gq Q
j
xj P pj The model is essentially an interval quadratic programming [25], and the calculation results are as follows: 3 2 3 2 3 2 3 2 þ þ þ f+ = 5.21502 108 Yuan, x 1 = 3719.60 m /hm , x2 = 3646.87 m /hm , x3 = 2192.70 m /hm , x4 = 3638.17 m /hm ; 8 3 2 3 2 3 2 3 þ f = 3.61456 10 Yuan, x1 = 3743.03 m /hm , x2 = 3646.87 m /hm , x3 = 2155.77 m /hm , x4 = 3576.43 m /hm2. 4.2. Optimal allocation of water resources model under meteorological factor uncertainty in the irrigation area 4.2.1. Model development When calculating the crop evapotranspiration (ET), the reference crop evapotranspiration (ET0) is the key parameter. At present, Penman–Monteith method is the most widely used method of calculating ET0. This method requires a large number of meteorological data, and the effects of various meteorological factors on ET0 are not the same, so different meteorological factors also have different effects on ET. Because the ordinary interval number cannot express the complex uncertainty relation between meteorological factors and ET, the complex functional interval is used. Specifically, from Temporal and Spatial Distribution of Crop Evapotranspiration in Shiyang River Basin [26], we know that in the 7 kinds of meteorological factors (including: average temperature, average maximum temperature, sunshine hours, average relative humidity, average wind speed, precipitation, evaporation) at Minqin, average relative humidity has the best linear relationship with reference crop evapotranspiration, so we select average relative humidity (RH) used in function interval, not other meteorological factor. We turned the certain number ET max j in constraint condition into functional interval ET max j expressed by average relative humidity, and others were unchanged. That is, only turned the constraint condition 0 6 gt M j þ pj 6 ET max j into 0 6 gt M j þ pj 6 ET max j , where, ET max j ¼ aj þ bj RH; RH is average relative humidity. Interval number is superior to determinate number, because it is considered a line of connecting interval two endpoints not a point. Functional interval is superior to interval number, because it is considered an area surrounded by connecting lines of intervals endpoints and functional equations. Therefore, functional interval can reflect the uncertainty of system more fully than interval. 4.2.2. Solution method Similarly, we solved the model after the original model transformed (let x j ¼ gt M j þ pj ). First, we did regression analysis of ETm in the whole growth period and average relative humidity of each crop in 1954–2001, and obtained the values of the coefficients aj± and bj±. The results are depicted in Fig. 2 (RH2[39, 52], unit:1%). The model is essentially an interval semi-infinite quadratic programming, and the calculation results are as follows: 3 2 3 2 3 2 3 2 þ þ f+ = 5.19327 108 Yuan, x 1 = 3788.12 m /hm , x2 = 3606.49 m /hm , x3 = 2243.41 m /hm , x4 = 3639.88 m /hm ; 8 3 2 3 2 3 2 3 þ þ f = 3.33811 10 Yuan, x1 = 4289.75 m /hm , x2 = 3326.10 m /hm , x3 = 2548.67 m /hm , x4 = 3590.04 m /hm2. 4.3. Result analysis and discussion Comparison of Tables 4 and 5 shows that all the widths of the interval solutions of introduction of meteorological factor model are larger than basic model’s except the irrigation quota of honey dew melon. The interval solution width of the spring wheat irrigation quota increases from 28.58 m3/hm2 to 611.74 m3/hm2, the interval solution width of the cotton irrigation quota increases from 0 to 341.95 m3/hm2, and the interval solution width of the seed melon irrigation quota increases from 45.04 m3/hm2 to 372.28 m3/hm2. This is because we turned the certain number ET max j in constraint condition into functional interval ET max j expressed by average relative humidity. When solved the upper and lower limits of introduction of meteorological factor model, ET max j in the corresponding constraint took the upper and lower limit functions of functional interval, respectively. So the widths of the interval solutions are enlarged. In addition, compared with the solutions of the
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(a)
(b)
(c)
(d)
Fig. 2. Interval linear regression results of four kinds of crops ETm in the whole growth period and average relative humidity. (Fig. 2(a). spring wheat; Fig. 2(b). cotton; Fig. 2(c). seed melon; Fig. 2(d). honey dew melon).
Table 4 The results of optimal allocation of water resources model in the irrigation area under uncertainty. Benefit f± (108 Yuan) The irrigation quota Mj± (m3/hm2)
Spring wheat Cotton Seed melon Honey dew melon
[3.61456, 5.21502] [3951.95, 3980.53] 3454.72 [1707.03, 1752.07] [3568.82, 3644.11]
Table 5 The results of optimal allocation of water resources model under meteorological factor uncertainty in the irrigation area. Benefit f± (108 Yuan) The irrigation quota Mj± (m3/hm2)
Spring wheat Cotton Seed melon Honey dew melon
[3.33811, 5.19327] [4035.52, 4647.26] [3063.53, 3405.48] [1813.91, 2186.19] [3585.41, 3646.19]
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first model, the spring wheat and seed melon irrigation quotas of the second model have increasing trend, the cotton irrigation quota of the second model has decreasing trend, and the honey dew melon irrigation quotas of two models are nearly equal. After comparison, it is known that the differences of the solutions of the two models, the solutions of determinate optimal allocation of water resources model, and the actual experimental results in this irrigation area are not significant. Therefore, the above two models are consistent with the actual situation, meet the requirements of optimal allocation of water resources, and can be applied in this irrigation area. IF other irrigation area meets the assumptions: (i) the change value of soil water in the whole growth period is zero; (ii) the groundwater recharge is negligible in the irrigation area, and it also has high water management level, then the models which are developed in this paper can be applied to such irrigation area according to the actual situation of irrigation area after crops data, meteorological data, and water resources development and utilization degree of irrigation area are collected. The interval results calculated by optimal allocation of water resources model in the irrigation area under uncertainty are more rich and reliable than the determinate number results calculated by traditional deterministic models, and have more practical value. Moreover, optimal allocation of water resources model under meteorological factor uncertainty in the irrigation area considers the complex uncertainty relation between meteorological factor and ET, so can reflect the impacts of climate change on the economic benefit of irrigation area and the irrigation quota of each crop in the whole growth period. These can’t be achieved by traditional deterministic models. 5. Conclusions In this paper, the interval quadratic regression analysis method is presented to analyze yield and water consumption in the whole growth period data of spring wheat, cotton, seed melon and honey dew melon in Minqin, respectively. The crop water production functions in the whole growth period under uncertainty are presented. The interval regression model provides the upper and the lower limits of predict value and center value, while predict value of general regression model is just a real number. Based on above analysis, an optimal allocation of water resources model in hongyashan irrigation area under uncertainty (model 1) is established and solved. Moreover, the certain number ET max j in constraint condition is integrated into the functional interval ET max j expressed by average relative humidity. The optimal allocation of water resources model under meteorological factor uncertainty in the irrigation area (model 2) is established. The results show that the widths of the interval solutions are enlarged, and model 2 can reflect the effect of climate change on benefit. The differences of the solutions of models 1 and 2, the solutions of determinate optimal allocation of water resources model, and the actual experimental results in this irrigation area are not significant. Therefore, the above two models are consistent with the actual situation, and can be applied in this irrigation area and promoted in other irrigation area. This paper studies on the uncertainty of water production function and its application on the optimization of irrigation water distribution. Especially the meteorological factor is introduced in optimal allocation of water resources model in irrigation area under uncertainty at the same time. This research provides more rich and reliable results than the deterministic research does, and has obvious superiority. Meanwhile, it can reflect the complexity and uncertainty of the actual situation, and has great reference value. But this study only introduces the uncertainty methods of interval and functional interval. Fuzzy, random and other uncertainties should also be considered in future studies. Acknowledgements This research was supported by the National Natural Science Foundation of China (No. 41271536, 71071154, 91125017), the National High Technology Research and Development Program of China (863 Program) (NO. 2011AA100502), the Governmental Public Research Funds for Projects of the Ministry of Agriculture (No. 201203077) and the Ministry of Water Resources (No. 200901083, 201001060, and 201001061). References [1] Q.X. Fang, L. Ma, T.R. Green, et al, Water resources and water use efficiency in the North China Plain: current status and agronomic management options, Agric. Water Manage. 97 (2010) 1102–1116. [2] M.A. Iqbal, G. Bodner, L.K. Heng, et al, Assessing yield optimization and water reduction potential for summer-sown and spring-sown maize in Pakistan, Agric. Water Manage. 97 (2010) 731–737. [3] J.W. Knox, M.G. Kay, E.K. 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[10] H.E. Igbadun, A.K.P.R. Tarimo, B.A. Salim, et al, Evaluation of selected crop water production functions for an irrigated maize crop, Agric. Water Manage. 94 (2007) 1–10. [11] K. Brumbelow, A. Georgakakos, Determining crop-water production functions using yield-irrigation gradient algorithms, Agric. Water Manage. 87 (2007) 151–161. [12] D. Zhao, D.G. Shao, B.J. Liu, Method of disposition on water resources of irrigation district and its applications, Trans. Chin. Soc. Agric. Eng. 20 (2004) 69–73 (Chinese). [13] B. Ghahraman, A.R. Sepaskhah, Optimal allocation of water from a single purpose reservoir to an irrigation project with pre-determined multiple cropping patterns, Irrigat. Sci. 21 (2002) 127–137. [14] L.N. Sethi, S.N. Panda, M.K. Nayak, Optimal crop planning and water resources allocation in a coastal groundwater basin, Orissa, India, Agric. Water Manage. 83 (2006) 209–220. [15] W. Schlenker, W.M. Hanemann, A.C. Fisher, Water availability, degree days, and the potential impact of climate change on irrigated agriculture in California, Clim. Change 81 (2007) 19–38. [16] S.L. Piao, P. Ciais, Y. Huang, et al, The impacts of climate change on water resources and agriculture in China, Nature 467 (2010) 43–51. [17] J.A. Vano, M.J. Scott, N. Voisin, et al, Climate change impacts on water management and irrigated agriculture in the Yakima river basin, Washington, USA, Clim. Change 102 (2010) 287–317. [18] G.P. Mengu, E. Akkuzu, S. Anac, et al, Impact of climate change on irrigated agriculture, Fresenius Environ. Bull. 20 (2011) 823–830. [19] B. Chen, L. Jing, B.Y. Zhang, et al, Wetland monitoring characterization and modelling under changing climate in the Canadian subarctic, J. Environ. Inf. 18 (2011) 55–64. [20] C.F. Zhang, H.S. Niu, Estimation of relative potential of three agricultural water-saving measures in Minqin Oasis, Tran. Chin. Soc. Agric. Eng. 25 (2009) 7–12 (Chinese). [21] Y. Zuo, Optimized irrigation program design of spring wheat in Hongyashan irrigation region, Gansu Water Conservancy Hydropower Technol. 46 (2010) 33–36. (Chinese). [22] T. Li, Pilot Study on water production functions and optimal irrigation schedule of main crops in Shiyang river basin, M.A.Sc. Thesis, Northwest A&F University, Yangling, Shanxi, China, 2005 (Chinese). [23] H. Tanaka, H. Lee, Interval regression analysis by quadratic programming approach, IEEE Trans. Fuzzy Syst. 6 (1998) 473–481. [24] S.Z. Kang, X.L. Su, T.S. Du, et al., Water resources transformation rule and water saving regulation and control mode of river basin scale in the northwest arid area of China – Taking Shiyang river basin in Gansu province as an example, China WaterPower Press, Beijing, China, 2009, p. 522 (Chinese). [25] M.J. Chen, G.H. Huang, A derivative algorithm for inexact quadratic program-application to environmental decision-making under uncertainty, Eur. J. Oper. Res. 128 (2001) 570–586. [26] L. Tong, Temporal and spatial distribution of crop evapotranspiration in Shiyang river basin, M.A.Sc. Thesis, Northwest A&F University, Yangling, Shanxi, China, 2004 (Chinese).
Contents lists available at SciVerse ScienceDirect
Applied Mathematical Modelling journal homepage: www.elsevier.com/locate/apm
Simulation and optimization for crop water allocation based on crop water production functions and climate factor under uncertainty Fangfang Tong a,b, Ping Guo a,c,⇑ a
Centre for Agricultural Water Research in China, China Agricultural University, Beijing 100083, China Sinohydro Corporation Limited, Beijing 100048, China c Environmental Science and Engineering Program, University of Northern British Columbia, Prince George, Canada BC V2N 4Z9 b
a r t i c l e
i n f o
Article history: Received 1 May 2012 Received in revised form 4 November 2012 Accepted 1 March 2013 Available online 15 March 2013 Keywords: Crop water production function Meteorological factor Optimal allocation Water resources Irrigation Uncertainty
a b s t r a c t In this paper, the uncertainty methods of interval and functional interval are introduced in the research of the uncertainty of crop water production function itself and optimal allocation of water resources in the irrigation area. The crop water production functions in the whole growth period under uncertainty and the optimal allocation of water resources model in the irrigation area under uncertainty are established, and the meteorological factor is considered in the model. It can promote the practical application of the uncertain methods, reflect the complexity and uncertainty of the actual situation, and provide more reliable scientific basis for using water resources economically, fully improving irrigation efficiency, and keeping the sustainable development of the irrigated area. This approach has important value on theoretical and practical for the optimal irrigation schedule, and has very broad prospects for research and development to other related agriculture water management. Ó 2013 Elsevier Inc. All rights reserved.
1. Introduction The increased water deficits associated with overuse of surface water and the declining groundwater levels are threatening the sustainability of agricultural production in the regions of Northwest of China [1]. Therefore, the improvement of water management practices is the utmost importance to ensure crop production with existing supplies of water [2,3]. Crop water production function is the mathematical description of the influence of water supply time and quantity on crop yield during crop growth. It is the main basis of evaluating various irrigation strategies and the foundation of determining economic water quotas and optimal allocation of water. In order to use limited water resources reasonably to achieve maximum crop yield and schedule the reasonable allocation of limited water supply both in temporal and spatial scale, the research on crop water production function was rapidly developed [4–8]. For example, De Juan et al. [9] developed a model for optimal cropping patterns within the farm based on crop water production functions and irrigation uniformity. Igbadun et al. [10] presented the performance evaluation of four selected crop water production functions. Brumbelow and Georgakakos [11] presented a new algorithm type based upon differential crop yield response to irrigation that uses these crop models to determine planning-level irrigation schedules and crop water production functions.
⇑ Corresponding author at: Centre for Agricultural Water Research in China, China Agricultural University, Beijing 100083, China. Tel.: +86 10 62738496. E-mail address: [email protected] (P. Guo). 0307-904X/$ - see front matter Ó 2013 Elsevier Inc. All rights reserved. http://dx.doi.org/10.1016/j.apm.2013.03.018
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In previous research on crop water production function, the problem was simplified into the deterministic problem, and only the deterministic function of the relationship between crop yield and evapotranspiration was established, so evapotranspiration corresponding to the crop highest yield also was the deterministic value. However, there are a number of uncertain factors in the actual problems, so the actual application value of the research results is reduced, due to the uncertain factors did not be considered in reflecting the practical problems. At present, generally the deterministic optimal allocation of water resources in the irrigation area is studied [12]. Some scholars consider the random factors (e.g. rainfall, runoff) and establish some stochastic programming models [13,14]. However, still no scholar has researched the uncertainty of water production function itself and its application on the optimization of irrigation water distribution systematically. Climate change has the impacts on many fields. There are a number of studies have been undertaken in the last decades to evaluate the impacts of climate change on water resources and agriculture [15–19]. However, there are few studies to consider the meteorological factor in optimal allocation of water resources in the irrigation area, especially in optimal allocation of water resources in the irrigation area under uncertainty. In this paper, the uncertainty methods of interval and functional interval are introduced in the research of crop water production function and optimal allocation of water resources in the irrigation area to accurately reflect the real-world problem. The crop water production functions in the whole growth period under uncertainty are firstly presented. Then based on above functions, the optimal allocation of water resources model in the irrigation area under uncertainty is established, and the meteorological factor is introduced in the model. It can promote the practical application of the uncertainty methods and provide more reliable scientific basis for using water resources economically, fully improving irrigation efficiency and keeping the sustainable development of the irrigated area. This approach has important value on theoretical and practical for the optimal irrigation schedule, and has very broad prospects for research and development to other related agriculture water management. 2. Study area The research area is located in Minqin, Gansu Province, China. It belongs to Wuwei region (101°490 104°120 E, 38°030 39°280 N) and the central area is the Shiyang River of alluvial narrow and flat oasis belt. It has the characteristic of North low and South high terrain, 1200–1500 m above sea level. The natural gradient is about l per thousand. It is surrounded by high terrain with basin landform characteristics [20]. In recent decades, due to the impact of human activities and climate change, severe land desertification appeared in Minqin. The ecological environment is deteriorating and water shortage has been very prominent to the front. Hongyashan irrigation region locates in the oasis belt of Minqin County. Its effective irrigation area is about 63073.3 hectare, and it is a typical continental temperate arid climate. Annual precipitation is less than 150 mm, while annual evaporation capacity is 2000–2600 mm. Irrigation water resources available are mainly from hongyashan reservoir water and groundwater. Due to excessive use in the middle irrigation area, surface water decreased year by year, and to 2003 has been less than 100 million m3. Groundwater which was not repeated with the surface water had only 31 million m3. Water resources is very scarce, and agriculture implements deficient irrigation [21]. 3. Crop water production function in the whole growth period under uncertainty 3.1. Method of Interval regression analysis The selected research crops, spring wheat, cotton, seed melon and honey dew melon are main crops planted in Minqin. The data of crop water consumption in the whole growth period and yield came from pilot study on water production functions and optimal irrigation schedule of main crops in Shiyang river basin [22]. Due to the information used for the water production functions are not the same, the interval regression analysis is useful to address the collected information. An interval linear regression model can be written as
YðxÞ ¼ A0 þ A1 x1 þ þ An xn ¼ Ax Where x ¼ ð1; x1 ; . . . ; xn Þt is a real input vector, A ¼ ðA0 ; . . . ; An Þ is an interval coefficient vector, and YðxÞ is the corresponding estimated interval. An interval coefficient Ai is denoted as Ai ¼ ðai ; ci Þ where ai is a center and ci is a radius. There are many approaches to achieve the interval regression analysis, such as translating into quadratic optimization problems, using neural networks, using support vector machines. This research used the relatively mature interval regression analysis method based on quadratic programming, and integrates the central tendency of least squares and possibilistic property of fuzzy regression. The proposed model can be represented as follows [23]:
Min J ¼ k1 a;c
p p X X ðyj at xj Þ2 þ k2 ct jxj jjxj jt c j¼1
t
j¼1
t
Subject to a xj þ c jxj j P yj ,
at xj ct jxj j 6 yj ;
j ¼ 1; . . . ; p
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ci P 0;
i ¼ 0; . . . ; n
Where jxj j ¼ ð1; jxj1 j; . . . ; jxjn jÞt ; a ¼ ða0 ; . . . ; an Þt , c ¼ ðc0 ; . . . ; cn Þt , k1 and k2 are weight coefficients. When the value of k1 and k2 are changed, the regression results may be different.
3.2. Calculation process and result analysis In this section, the relationship between yield and water consumption in the whole growth period of the typical crops in Minqin is addressed. That means the water production function in the whole growth period. Because crop water production function model of the whole growth period presented as the quadratic model in high yield level regions [24], the interval quadratic regression analysis method is presented based on quadratic programming to analyze yield and water consumption in the whole growth period data of spring wheat, cotton, seed melon and honey dew melon in Minqin, respectively. We had
(a)
(b)
(c)
(d)
Fig. 1. Interval quadratic regression results of four kinds of crops yield and water consumption in the whole growth period. (Fig. 1(a). spring wheat; Fig. 1(b). cotton; Fig. 1(c). seed melon; Fig. 1(d). honey dew melon).
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five sets of values of k1:k2, which were equal to 1:0.0001, 1:0.5, 1:1, 0.5:1, 0.0001:1. LINGO software was used to solve above model, and the results are depicted in Fig. 1. Fig. 1 shows that when a large valued k1 compared to k2, the more central tendency would be expected, i.e., the obtained center regression line tend to be the regression line obtained by least squares regression. On the contrary, when a large valued k2 compared to k1, reducing the fuzziness of the model would be focused. The larger values of k2:k1, the less central tendency is considered. When the values of k2:k1 increase to a certain level, and then increase the values of k2:k1, the intervals obtained would not be changed. When fully considered the central tendency, the interval regression results of k1 = 1, k2 = 0.0001 are selected as uncertain crop water production functions in the whole growth period. Due to the first-order term coefficients and quadratic term coefficients of cotton and seed melon k1 = 1, k2 = 0.0001 regression results are deterministic, the water consumption corresponding to the highest yield is the certainty. For this reason, the regression values of cotton and seed melon are selected as k1 = 1, k2 = 0.5. The obtained water production functions in the whole growth period of four crops under uncertainty in this study are as follows:
Spring wheat Y 1 ðxÞ ¼ ð16419:3; 778:682Þ þ ð8:27316; 0:00942176Þx1 þ ð0:000712642; 0Þx21 ; Cotton Y 2 ðxÞ ¼ ð8010:99; 191:871Þ þ ð4:45270; 0:0103980Þx2 þ ð0:000516700; 0Þx22 Seed melon Y 3 ðxÞ ¼ ð293:266; 145:859Þ þ ð1:82795; 0Þx3 þ ð0:000247975; 3:63015E 06Þx23 ; Honey dew melon Y 4 ðxÞ ¼ ð413273; 2802:46Þ þ ð254:449; 2:17845Þx4 þ ð0:0347713; 0Þx24 Compared with determinate water production functions in the whole growth period of four crops:
R2 ¼ 0:8337
Spring wheat Y 01 ðxÞ ¼ 16001:7 þ 8:11755x1 0:000698686x21 Cotton Y 02 ðxÞ ¼ 6495:91 þ 3:72971x2 0:000434445x22
R2 ¼ 0:8351
Seed melon Y 03 ðxÞ ¼ 220:904 þ 1:83319x3 0:000267595x23
R2 ¼ 0:9926
Honey dew melon Y 04 ðxÞ ¼ 413373 þ 254:505x4 0:0347791x24
R2 ¼ 0:6130
Considering the spring wheat as an example, the measured values, predict values of interval regression model, and predict values of general regression model are compared. The results are shown in Table 1. First, whether it is interval regression model or general regression model, its predict values are very close to the measured values. A random sample of one of them, take No.1 as an example, its measured value is 4800; the center value of interval regression model is 5006.74, and relative error is 4.3%; the calculation value of general regression model is 5029.73, and relative error is 4.8%. This shows that both models can be used to describe the relationship between crop yield and water consumption in the whole growth period. Second, all prediction values of general regression model are within prediction intervals of interval regression model. Interval
Table 1 Comparison of the measured values, predict values of interval regression model Y⁄(x) and general regression model Y0 (x) (Spring wheat). No.
x (m3/hm2)
y (kg/hm2)
Y⁄(x) (kg/hm2)
Center value (kg/hm2)
Y0 (x) (kg/hm2)
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18
3900 3900 4125 4200 4500 4500 4500 4950 5000 5100 5100 5175 5250 5550 5775 5850 6600 7650
4800 4950 5550 5850 5700 6450 7200 6750 5500 7650 7200 7200 7650 7800 7950 7200 6300 5550
[4191.31, 5822.17] [4191.31, 5822.17] [4763.89, 6398.98] [4938.71, 6575.22] [5557.84, 7200.00] [5557.84, 7200.00] [5557.84, 7200.00] [6246.01, 7896.65] [6304.66, 7956.24] [6411.26, 8064.73] [6411.26, 8064.73] [6481.86, 8136.74] [6544.45, 8200.74] [6714.61, 8376.55] [6758.05, 8424.23] [6756.49, 8424.09] [6300.33, 7981.73] [4314.01, 6015.53]
5006.74 5006.74 5581.44 5756.97 6378.92 6378.92 6378.92 7071.33 7130.45 7237.99 7237.99 7309.30 7372.59 7545.58 7591.14 7590.29 7140.86 5164.77
5029.73 5029.73 5594.61 5767.19 6378.88 6378.88 6378.88 7060.62 7118.90 7224.98 7224.98 7295.37 7357.90 7529.43 7575.54 7575.19 7139.37 5208.71
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Table 2 The highest yield of each crop and water consumption of corresponding to the highest yield (interval results). Crop
The highest yield (kg/hm2)
The corresponding water consumption (m3/hm2)
Spring wheat Cotton Seed melon Honey dew melon
[6758.42, 8425.16] [1345.24, 1818.59] [2880.94, 3271.31] [41490.2, 63036.6]
[5797.96, 5811.18] [4298.72, 4318.84] [3632.56, 3740.50] [3627.57, 3690.22]
Table 3 The highest yield of each crop and water consumption of corresponding to the highest yield (determinate number results). Crop
The highest yield (kg/hm2)
The corresponding water consumption (m3/hm2)
Spring wheat Cotton Seed melon Honey dew melon
7576.35 1508.98 2918.72 52228.4
5809.15 4292.50 3425.31 3658.88
regression model provides the upper and the lower limits of predict value and center value, while predict value of general regression model is just a real number. Thus, interval regression model provides more information than general regression model does. It reflects the uncertainty and the former has more advantages than the latter. The highest yield of each crop and water consumption of corresponding to the highest yield are calculated. The results are shown in Tables 2 and 3. It is noticed in Tables 2 and 3 that all the highest yields calculated by the traditional deterministic water production functions in the whole growth period are within the highest yield intervals calculated by the uncertainty crop water production functions in the whole growth period. Furthermore, spring wheat and honey dew melon’s water consumptions corresponding to the certainty highest yields are both within the uncertain interval results. Only cotton and seed melon’s water consumptions of corresponding to the deterministic highest yields are beyond the corresponding uncertainty intervals. But the excess portions are very small, and do not affect the research results. In fact, the results of interval quadratic functions can reflect the uncertainty of actual situation, and can represent the actual situation more accurately. 4. Optimal allocation of water resources in the irrigation area under uncertainty 4.1. Optimal allocation of water resources model in the irrigation area under uncertainty 4.1.1. Model development There are two assumptions in this study: (i) the change value of soil water in the whole growth period is zero; (ii) the groundwater recharge is negligible in the irrigation area. Consider the high water management level, that is, evapotranspiration in the whole growth period is equal to effective irrigation quota plus effective rainfall. Hongyashan irrigation region in Minqin County is selected as the research object, and the spring wheat, cotton, seed melon and honey dew melon (j = 1, 2, 3, 4) are selected as study crops. Although forage acreage in this region was large, it was not selected due to lack of available data. The yields of determinate water production functions in the whole growth period are replaced by uncertain water production functions. According to the principle of maximum economic benefit, uncertain water resources optimal allocation model in irrigation area can be formulated as follows:
max f ¼
4 4 4 4 X X X X Bj Aj Y j C M j Aj ¼ Bj Aj ½k1j þ k2j ðgt M j þ pj Þ þ k3j ðgt M j þ pj Þ2 C M j Aj j¼1
j¼1
j¼1
j¼1
subject to: 4 X Mj Aj 6 gq Q j¼1
0 6 gt M j þ pj 6 ET max
j
Mj P 0 where, f±: objective function (Yuan); Bj: the price of crop j (Yuan/kg); Aj: the planting area of crop j (hm2); Yj±: the yield of crop j (kg/hm2); C: the unit cost of irrigation water (Yuan/m3); Mj±: decision variable, the irrigation quota of unit area crop j in the whole growth period (m3/hm2); gt : the field water utilization coefficient; gq : the canal system water utilization
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coefficient; pj: the effective precipitation of crop j in the whole growth period (m3/hm2); Q: the total available irrigation water (m3); ETmax j: the maximum evapotranspiration of crop j in the whole growth period (m3/hm2). 4.1.2. Solution method Assuming that 2010 was a normal flow year (P = 50%). The real data of Hongyashan irrigation region in 2010 are used to calculate the model. Transformed the original model into another formulation, which is written as follows (let x j ¼ gt M j þ pj ):
max f ¼
4 X pj CAj xj þ Bj Aj k3j ðxj Þ2 þ CAj Bj Aj k1j þ Bj Aj k2j
gt
j¼1
gt
subject to: 4 X Aj j¼1
gt
xj
p j Aj
gt
0 6 xj 6 ET max
6 gq Q
j
xj P pj The model is essentially an interval quadratic programming [25], and the calculation results are as follows: 3 2 3 2 3 2 3 2 þ þ þ f+ = 5.21502 108 Yuan, x 1 = 3719.60 m /hm , x2 = 3646.87 m /hm , x3 = 2192.70 m /hm , x4 = 3638.17 m /hm ; 8 3 2 3 2 3 2 3 þ f = 3.61456 10 Yuan, x1 = 3743.03 m /hm , x2 = 3646.87 m /hm , x3 = 2155.77 m /hm , x4 = 3576.43 m /hm2. 4.2. Optimal allocation of water resources model under meteorological factor uncertainty in the irrigation area 4.2.1. Model development When calculating the crop evapotranspiration (ET), the reference crop evapotranspiration (ET0) is the key parameter. At present, Penman–Monteith method is the most widely used method of calculating ET0. This method requires a large number of meteorological data, and the effects of various meteorological factors on ET0 are not the same, so different meteorological factors also have different effects on ET. Because the ordinary interval number cannot express the complex uncertainty relation between meteorological factors and ET, the complex functional interval is used. Specifically, from Temporal and Spatial Distribution of Crop Evapotranspiration in Shiyang River Basin [26], we know that in the 7 kinds of meteorological factors (including: average temperature, average maximum temperature, sunshine hours, average relative humidity, average wind speed, precipitation, evaporation) at Minqin, average relative humidity has the best linear relationship with reference crop evapotranspiration, so we select average relative humidity (RH) used in function interval, not other meteorological factor. We turned the certain number ET max j in constraint condition into functional interval ET max j expressed by average relative humidity, and others were unchanged. That is, only turned the constraint condition 0 6 gt M j þ pj 6 ET max j into 0 6 gt M j þ pj 6 ET max j , where, ET max j ¼ aj þ bj RH; RH is average relative humidity. Interval number is superior to determinate number, because it is considered a line of connecting interval two endpoints not a point. Functional interval is superior to interval number, because it is considered an area surrounded by connecting lines of intervals endpoints and functional equations. Therefore, functional interval can reflect the uncertainty of system more fully than interval. 4.2.2. Solution method Similarly, we solved the model after the original model transformed (let x j ¼ gt M j þ pj ). First, we did regression analysis of ETm in the whole growth period and average relative humidity of each crop in 1954–2001, and obtained the values of the coefficients aj± and bj±. The results are depicted in Fig. 2 (RH2[39, 52], unit:1%). The model is essentially an interval semi-infinite quadratic programming, and the calculation results are as follows: 3 2 3 2 3 2 3 2 þ þ f+ = 5.19327 108 Yuan, x 1 = 3788.12 m /hm , x2 = 3606.49 m /hm , x3 = 2243.41 m /hm , x4 = 3639.88 m /hm ; 8 3 2 3 2 3 2 3 þ þ f = 3.33811 10 Yuan, x1 = 4289.75 m /hm , x2 = 3326.10 m /hm , x3 = 2548.67 m /hm , x4 = 3590.04 m /hm2. 4.3. Result analysis and discussion Comparison of Tables 4 and 5 shows that all the widths of the interval solutions of introduction of meteorological factor model are larger than basic model’s except the irrigation quota of honey dew melon. The interval solution width of the spring wheat irrigation quota increases from 28.58 m3/hm2 to 611.74 m3/hm2, the interval solution width of the cotton irrigation quota increases from 0 to 341.95 m3/hm2, and the interval solution width of the seed melon irrigation quota increases from 45.04 m3/hm2 to 372.28 m3/hm2. This is because we turned the certain number ET max j in constraint condition into functional interval ET max j expressed by average relative humidity. When solved the upper and lower limits of introduction of meteorological factor model, ET max j in the corresponding constraint took the upper and lower limit functions of functional interval, respectively. So the widths of the interval solutions are enlarged. In addition, compared with the solutions of the
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(a)
(b)
(c)
(d)
Fig. 2. Interval linear regression results of four kinds of crops ETm in the whole growth period and average relative humidity. (Fig. 2(a). spring wheat; Fig. 2(b). cotton; Fig. 2(c). seed melon; Fig. 2(d). honey dew melon).
Table 4 The results of optimal allocation of water resources model in the irrigation area under uncertainty. Benefit f± (108 Yuan) The irrigation quota Mj± (m3/hm2)
Spring wheat Cotton Seed melon Honey dew melon
[3.61456, 5.21502] [3951.95, 3980.53] 3454.72 [1707.03, 1752.07] [3568.82, 3644.11]
Table 5 The results of optimal allocation of water resources model under meteorological factor uncertainty in the irrigation area. Benefit f± (108 Yuan) The irrigation quota Mj± (m3/hm2)
Spring wheat Cotton Seed melon Honey dew melon
[3.33811, 5.19327] [4035.52, 4647.26] [3063.53, 3405.48] [1813.91, 2186.19] [3585.41, 3646.19]
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first model, the spring wheat and seed melon irrigation quotas of the second model have increasing trend, the cotton irrigation quota of the second model has decreasing trend, and the honey dew melon irrigation quotas of two models are nearly equal. After comparison, it is known that the differences of the solutions of the two models, the solutions of determinate optimal allocation of water resources model, and the actual experimental results in this irrigation area are not significant. Therefore, the above two models are consistent with the actual situation, meet the requirements of optimal allocation of water resources, and can be applied in this irrigation area. IF other irrigation area meets the assumptions: (i) the change value of soil water in the whole growth period is zero; (ii) the groundwater recharge is negligible in the irrigation area, and it also has high water management level, then the models which are developed in this paper can be applied to such irrigation area according to the actual situation of irrigation area after crops data, meteorological data, and water resources development and utilization degree of irrigation area are collected. The interval results calculated by optimal allocation of water resources model in the irrigation area under uncertainty are more rich and reliable than the determinate number results calculated by traditional deterministic models, and have more practical value. Moreover, optimal allocation of water resources model under meteorological factor uncertainty in the irrigation area considers the complex uncertainty relation between meteorological factor and ET, so can reflect the impacts of climate change on the economic benefit of irrigation area and the irrigation quota of each crop in the whole growth period. These can’t be achieved by traditional deterministic models. 5. Conclusions In this paper, the interval quadratic regression analysis method is presented to analyze yield and water consumption in the whole growth period data of spring wheat, cotton, seed melon and honey dew melon in Minqin, respectively. The crop water production functions in the whole growth period under uncertainty are presented. The interval regression model provides the upper and the lower limits of predict value and center value, while predict value of general regression model is just a real number. Based on above analysis, an optimal allocation of water resources model in hongyashan irrigation area under uncertainty (model 1) is established and solved. Moreover, the certain number ET max j in constraint condition is integrated into the functional interval ET max j expressed by average relative humidity. The optimal allocation of water resources model under meteorological factor uncertainty in the irrigation area (model 2) is established. The results show that the widths of the interval solutions are enlarged, and model 2 can reflect the effect of climate change on benefit. The differences of the solutions of models 1 and 2, the solutions of determinate optimal allocation of water resources model, and the actual experimental results in this irrigation area are not significant. Therefore, the above two models are consistent with the actual situation, and can be applied in this irrigation area and promoted in other irrigation area. This paper studies on the uncertainty of water production function and its application on the optimization of irrigation water distribution. Especially the meteorological factor is introduced in optimal allocation of water resources model in irrigation area under uncertainty at the same time. This research provides more rich and reliable results than the deterministic research does, and has obvious superiority. Meanwhile, it can reflect the complexity and uncertainty of the actual situation, and has great reference value. But this study only introduces the uncertainty methods of interval and functional interval. Fuzzy, random and other uncertainties should also be considered in future studies. Acknowledgements This research was supported by the National Natural Science Foundation of China (No. 41271536, 71071154, 91125017), the National High Technology Research and Development Program of China (863 Program) (NO. 2011AA100502), the Governmental Public Research Funds for Projects of the Ministry of Agriculture (No. 201203077) and the Ministry of Water Resources (No. 200901083, 201001060, and 201001061). References [1] Q.X. Fang, L. Ma, T.R. Green, et al, Water resources and water use efficiency in the North China Plain: current status and agronomic management options, Agric. Water Manage. 97 (2010) 1102–1116. [2] M.A. Iqbal, G. Bodner, L.K. Heng, et al, Assessing yield optimization and water reduction potential for summer-sown and spring-sown maize in Pakistan, Agric. Water Manage. 97 (2010) 731–737. [3] J.W. Knox, M.G. Kay, E.K. 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