Optimizing Water Consumption Using Crop Water Production Functions

There are two options for increasing crop production: Increasing acreage and improving land utilization. Food supply in the early period of agriculture was obtained only by expanding the physical area of land. Gradually, improving land productivity using different agronomy and management techniques for food production has had a more prominent role than increase in physical land area. As today, the acreage increase does not play key role in the crop production. Historical samples are proof of this claim. Historical trends indicate that, although application of fertilizers, pesticides and plant breeding could improve yield per unit area, but gradually due to improper management of resources, pollution caused by chemical fertilizer and pesticides has led to agricultural sustainability is threatened. While in terms of plant breeding, in many cases the plants have reached their biological production limit. But irrigation management has still key role to increase land productivity and food production. Currently, 2.5 billion people in terms of employment, food and income are dependent on irrigated agriculture. It is estimated that within the next 30 years, 80 percent of the additional food needed for world population is rely on irrigation. Although the increase in irrigated agriculture can play a significant role in providing world food needs, but water resources for such development is limited. Therefore, more effective use of current resources in irrigation is emphasized. Groundwater extraction in the short term is effective in producing food, but its continuance would not be sustainable in the long run. Agricultural is the largest consumer of water in the world. About 80 to 90 percent of exploited water resources are used in the agricultural section. In this section, it is possible to increase water productivity and use of saved water for the new land could increase crop production. Generally, traditional management to reduce risk from water shortages in the agricultural sector was demand management, mainly to develop water storage using dams or water captivates had been focused. Of course, it is led to various problems of environmental because of changing in water hydrologic cycle. Today, most countries have reached the peak utilization of their water resources, extracting the main accessible water. It is natural that in these circumstances water extraction costs more and more expensive every day, and in practical point of view, agricultural practices will be non-economic. In these conditions, the agricultural sector had to compete with other sectors to use of each unit of


Introduction
There are two options for increasing crop production: Increasing acreage and improving land utilization.Food supply in the early period of agriculture was obtained only by expanding the physical area of land.Gradually, improving land productivity using different agronomy and management techniques for food production has had a more prominent role than increase in physical land area.As today, the acreage increase does not play key role in the crop production.Historical samples are proof of this claim.Historical trends indicate that, although application of fertilizers, pesticides and plant breeding could improve yield per unit area, but gradually due to improper management of resources, pollution caused by chemical fertilizer and pesticides has led to agricultural sustainability is threatened.While in terms of plant breeding, in many cases the plants have reached their biological production limit.But irrigation management has still key role to increase land productivity and food production.Currently, 2.5 billion people in terms of employment, food and income are dependent on irrigated agriculture.It is estimated that within the next 30 years, 80 percent of the additional food needed for world population is rely on irrigation.Although the increase in irrigated agriculture can play a significant role in providing world food needs, but water resources for such development is limited.Therefore, more effective use of current resources in irrigation is emphasized.Groundwater extraction in the short term is effective in producing food, but its continuance would not be sustainable in the long run.Agricultural is the largest consumer of water in the world.About 80 to 90 percent of exploited water resources are used in the agricultural section.In this section, it is possible to increase water productivity and use of saved water for the new land could increase crop production.Generally, traditional management to reduce risk from water shortages in the agricultural sector was demand management, mainly to develop water storage using dams or water captivates had been focused.Of course, it is led to various problems of environmental because of changing in water hydrologic cycle.Today, most countries have reached the peak utilization of their water resources, extracting the main accessible water.It is natural that in these circumstances water extraction costs more and more expensive every day, and in practical point of view, agricultural practices will be non-economic.In these conditions, the agricultural sector had to compete with other sectors to use of each unit of water.In other words, a serious question arises whether in the long term water extraction in large scale to crop production is economy?or use of water in other sectors like industry has a higher comparative advantage?Rapid population growth and the need for more production has caused the agricultural sector would have more demand for water than any other water consuming sectors (industry, domestic, drinking).So, the main challenge of the agricultural sector is to produce more crops from less water.In this context, increasing of water use efficiency is one of the important strategies for more production.One of the important solutions to reduce hunger, poverty and maintaining the previous level of production will be increasing water productivity proportional to increase in demand for food.Increasing water use efficiency while maintaining stable production and reducing water consumption, a significant volume of water can store without a new structure to be constructed.There are two basic solutions for increasing water productivity: These are reducing water consumption with proven production (irrigation management) and increasing production with a fixed amount of water (agronomy management) that eventually will lead to more production with less water.Reducing water consumption while maintaining the previous production, includes all activities that are leading to improve irrigation management.Ways to increase water productivity from the perspective of improving irrigation management can be summarized as follows: -improving irrigation efficiency (improving conventional irrigation systems and change to modern irrigation systems) -optimal allocation of water resources deficit or alternate irrigation optimize water use by determination of water -yield relations (production functions) -the use of uncommon water reduce evaporation Increasing production per unit water consumption is achievable by improving the production management.The production management refers to all agricultural activities that are leading to increase production.The following can be cited In this context: -breading activities to make drought resistant cultivars reduce the period of plant growth increasing the depth and root density in soils change and reform cropping pattern on the basis of the highest plant productivity crop rotation application of conservation tillage techniques for more efficient use of available moisture in the soil, irrigation water and rain controlling pests and diseases improving soil fertility Irrigation strategy management to increase production would not be the same in all areas.In areas where enough water is available, full irrigation strategy could be a suitable option, while in areas where water is limited, deficit irrigation is appropriate strategy and finally in areas where water resources are saline, use of management strategies for achieving sustainable production as well as economic yield is suitable management.However, any optional strategies are needed to specific management, and they are a function of time and space.But in all cases, knowledge of water-yield relation (production function) is necessary in order to achieve optimal amounts of irrigation water and sustainable production.The purpose of this section is to optimize water consumption using production functions in terms of saline and non-saline water application.

Crop production functions
To express the relationship between inputs and crop yield, production functions are used.Crop production functions are mathematical relationship between yield and inputs used in the production process.In other words, the crop production function identifies the conversion rate of input to output.The statistical data obtained from field observations or controlled experimental design can be used to estimate of production functions.Overall form of the crop production function can be written as follows: This equation shows the amount of production determined by different amounts of inputs (n).Production factors can be classified in different ways.Some factors are variables and some others are fixed.Some of the factors are very important and some others not significant.The crop production function is usually estimated based on a few variable factors under controlled.Using the estimated production function can be defined different scenarios based on user-defined.The amount of yield in different levels of inputs used to crop production function, marginal production, the final value of each of the factors of production and marginal rate of technical substitution factors could be calculated.

Optimizing water consumption
Optimizing water consumption is a type of management options that may establish relationship between land and water under limitation of water/land conditions.So that crop production is economically affordable and technically possible.Generally, Figure 1 shows relationship between the applied water (AW) as a function of gross income and cost (English, 1990).As shown, there is curvilinear relation between gross income and applied water and a linear relation with cost function.In fact, the gross income increases with water and reaches the maximum point, then decreases mainly due to increasing water cost and also decreasing in production.But cost function rises linearly with increasing water amount.Net income is equal to the difference between gross income and cost (distance between two curves in Figure 1).Selection of appropriate strategy depends on the presence or absence of limiting factors previously mentioned.If land is limiting factor, using the amount of water maximizing net benefit per unit of land, is the best strategy (AW L in Figure 1).Although with increasing amount of water, gross income could be increased so that maximum crop production is obtained with full irrigation (AW F on the right AW L in Figure 1), but net income resulting from application of AW F is less than AW L .Reducing the amount of water applied from AW L and move to the left curve gradually decreases the net income, so that net income in AW d per unit of land is equal to net income from full irrigation.It is clear that full irrigation needed to achieve maximum production, but to get the highest net profit not need to full irrigation.In practical point of view, in water scarcity areas economic benefit could be achieved using deficit irrigation scheduling.In areas where water is a limiting factor, the costs of supply, transmission and distribution of water are the most serious challenges.In these areas, due to water shortages a large part of the farms are not cultivated or remained as rainfed.According to presented analysis, in terms of water consumption there is a point that irrigation depth is less than the amount of crop water requirement and full irrigation depth, however its net profit per unit of land is equal to net income from full irrigation (AW d ).So with the saved water for irrigating new lands can produce more revenue achieved without paying additional cost for new water supply.To achieve optimal levels of water consumption and choosing appropriate management options, it is necessary to estimate variable amounts of derived functions such as AW L , AW F and AW d .

Determination of threshold values
For optimum irrigation scheduling, driving water-yield relation (crop water production function) is required for determination of optimum amount of water.Generally, the quadratic form of crop water production is used to describe relationship between irrigation water and crop yield: where, Y is crop yield (t ha -1 ), I is irrigation water (mm) and a 0 , a 1 and a 2 are constants.
The cost equation is used as: where, C is production cost per hectare and b 0 and b 1 are constants.Constant coefficients in above equations can be derived using various levels of water against their corresponding yields as follows (English, 1990): (5) where, I F , I L and I d are full irrigation, optimum amount of irrigation where land is limiting and water is limiting, respectively (mm) and Pc is the crop price per kg.Above equations, were modified by Sepaskha & Akbari (2005) for case having rainfall (P) as follows: where, , , and are the similar constants as equation 2 when total applied water (AW) is replaced with I.The following sections present how to determine the optimal amount of water consumption by using analysis of production functions in different conditions.

Wheat production function under supplementary irrigation
Supplementary irrigation plays a key role in crop production in many countries in the world.In most rainfed areas, the amount of rainfall and its distribution is not suitable throughout growing season of winter crops.Therefore, increasing crop production is necessary to timing irrigation with supplementary irrigation.At present, supplementary irrigation covers 80 percent of cultivated areas over the world, producing 60 percent of the global production (Harris, 1991).Wheat yield under rainfed conditions in arid and semiarid regions of the world, including West Asia and North Africa, varies between 0.6 to 1.5 t/ha.In these areas, supplemental irrigation significantly increases yield and water productivity.The experimental results on wheat show that water productivity is increased with deficit irrigation scheduling (Sun et al., 2006;Zhang et al., 2005;Zhang et al., 2006).Supplemental irrigation potentially increases wheat yield and water productivity.Irrigation water for achieving optimum water productivity is not the same in different areas of the world, mainly due to different climate.For example, the highest water productivity of wheat in northern Syria (Zhang and Oweis, 1999) was corresponded with 440-500 mm of water application (140-180 mm irrigation water), in northern China ( Zhang et al., 1999), with 400 mm (120-160 mm irrigation water) and in Oregon America (English and Nakamora, 1989) with 750-850 mm of water applied (350-450 mm irrigation water).A field study was conducted to compare various genotypes of wheat to water using line source sprinkler irrigation during growing season of wheat (2005)(2006).The geographical location of the farm was 36 o 54' N, 54 o 25' E and 155 m above mean sea level.This area represents medium annual rainfall, mostly falls in winter (November-April).Seasonal rainfall during the wheat growth stages was about 250 mm.Four irrigation treatments were provided by the decline with distance from the line source, during the growing season to meet 100 %(W 1 ), 76 %(W 2 ), 52 %(W 3 ) and 39% (W 4 ) crop water requirement.Optimal irrigation water application was considered by generated crop water production functions for six cultivars of wheat under supplementary irrigation.The experiment used a strip plot design to examine the effect of the fixed irrigation rates on six cultivars treatments (C 1 =TAJAN, C 2 = N-80-6, C 3 = N-80-7, C 4 = N-80-19, C 5 = N-81-18, C 6 = Desconcido) with four replications.The soil texture of the experimental plots was silty clay loam in the surface layer (0-30 cm) and silty loam in the deeper layers (30-90 cm), having soil bulk density of 1.3 and 1.4 gr /cm 3 , respectively.Wheat yield variation of different cultivars as a function of applied water (AW) illustrated in figure 1, separately.A quadratic form of production function was used to describe the relationship between applied water and yield.The general trend is similar to all the cultivars, so that with increasing applied water, yield increases and at a certain point that yield reaches the peak value, more increasing water will result in decreasing the yield.Estimated wheat production function for any cultivars showed that they have different constants as well as different response to water.As a result, appropriate cultivar could be selected due to provide different production from a certain amount of water.For example, the cultivars C 1 -C 6 produced 4.2, 4.8, 4.5, 4.4, 4.6 and 4.2 t/ha against 350 mm applied water, respectively (Fig. 2).Crop water production and cost functions were used to describe optimum scheduling of irrigation water.Due to the presence of effective rainfall in the region, optimum irrigation water under full irrigation (I f ) and deficit irrigation (I d ) were estimated as function of seasonal rainfall (with the assumption that wheat price equal to 1700 Rials/kg and production cost without irrigation costs are 2.5 million Rials/ha, 10000Rials=US$1).The results revealed that, optimum irrigation water depth will be decreased when seasonal rainfall increased.Required water under full irrigation strategy for maximizing production in cultivars C 1 to C 6 was 362, 370, 335, 342, 340 and 345 mm.Optimum irrigation water under deficit irrigation for all cultivars as a function of seasonal rainfall was presented in Table 1.It is observed that under deficit irrigation, whenever during the wheat growing season 250 mm of rainfall occurs with a similar distribution, all cultivars will not require to irrigate.If rain does not happen (in terms of deficit irrigation) C 2 cultivar will need more water than the others (331 mm).So, if assumes that the average value of 250 mm rainfall occurs during the wheat growing season, in case of full supplemental irrigation two or three irrigation event will be needed to all the cultivars.However, in case of deficit irrigation cultivars will not require irrigation water (Table 2

Comparison of deficit and full irrigation strategies
Generally, when farmers face water scarcity, as water resources are not enough to follow the full irrigation, they have two options (Oweis and Hachum, 2003): in the first case may be, they apply the available water for irrigating a part of the farm and leave the rest as rainfed.
In the second case may be, they apply less water to irrigate the whole farm (deficit irrigation).The quantitative comparison of deficit and full irrigation strategies are showed in Table 2

Soybean production function
Soybean is one of the most important crops for oil and protein production in the world.Generally, soybean is planted in warm and semi-warm climate and relatively resistance to low and very high temperatures, but its growth rate is reduced at temperatures higher than 35 ˚C and less than 18 ˚C (FAOSTAT, 2001).Because soybean having a high concentration of protein (36%), oil (18%) and carbohydrate (20%) planted in almost all parts of the world for human consumption, livestock and plants (Boydak et al., 2002).3.
As shown in Table 4, the determination of coefficient (R 2 ) of the quadratic function for all cultivars are more than 84% and it is found that the effects of water levels on soybean grain yield is significant.This indicates that 84% of the yield variability of soybean is explained by the variations of applied water (AW).A quadratic form has been reported between yield and applied water by many researchers (Stewart and Hagan, 1983;Zhang and Oweis, 1999;Sepaskhah and Akbari, 2005).The grain yield of soybean cultivars as a function of AW in 2005AW in , 2006AW in and 2005AW in -2006 are illustrated in Fig. 4. Figure 4 states that the yield of soybean cultivars is increased as a decline slope with increasing water amount.In other words, for Crop Production Technologies 82 every certain amount of water increase, soybean yield will not increase proportionally.After a certain level of water, with increasing water, the yield starts to decrease (Fig. 4).The equation coefficients on three cultivars are different.It is found that the yields have a different response to given water.For example, the most productive use of water in the Sahar (2.5 t/ha), G 3 (3 t/ha) and DPX (3.7 t/ha) were reached with about 490, 490 and 510 mm of AW, respectively.For a certain amount of AW, the DPX cultivar produced more grain yield than the 2 other cultivars (Fig. 4).   4 shows that in deficit irrigation condition with increasing effective rainfall during the soybean growing season amount of optimum irrigation water decreases to achieve optimal production.In full irrigation strategy to obtain maximum grain yield the cultivars Sahar, G 3 , and DPX are required to 550, 580 and 640 mm of AW, respectively.Water requirement of soybean is 450-600 mm to produce maximum depending on climatic conditions and variety (FAO, 2002).But, in deficit irrigation strategy, if no rainfall occurs during the growing season, Sahar, G 3 and DPX need 425, 435 and 435 mm of irrigation water, and if 200 mm effective rainfall occurs the cultivars will not need to irrigation water.

Factors
Under no irrigation water and with 200 mm effective rainfall the grain yield of Sahar, G 3 and DPX were obtained at 1, 1.2 and 1.5 t/ha, respectively.However, with increasing the amounts of AW from optimal level increasing yield is possible, but this strategy for areas that facing with water shortages are not appropriate.that for all cultivars total production under deficit irrigation (W 4 ) was more than full irrigation (W 1 ) for a given levels of farm.For example, in Sahar cultivar, if full irrigation (345 mm) is applied to irrigate one hectare and left a 3.85 hectare as rainfed, totally 4.5 ton grain yield can be obtained.Instead of full irrigation of one hectare, if deficit irrigation (71mm) is applied for all the 4.85 hectare, totally 5.8 ton grain yield is achievable.This trend with more appropriate results for G 3 and DPX cultivars are presented in Table 5.Table 5. Quantitative comparison of full (W 1 ) and deficit irrigation (W 4 ) in three soybean cultivars Two-year average has shown that there was a tendency for more production in DPX cultivar than other cultivars, mainly due to appropriate response to water.As a result, using deficit irrigation and allocation of saved water to other lands where facing with water scarcity is the optimal strategy to maximize production as well as farmer's income.It is clear that water stress does not have similar effect on the whole crop growth.The advantage of deficit irrigation can be achieved by saving water during those periods when the crop is less sensitive to water stress.

Water-salinity production function
Water scarcity and salinity are major problems in reducing crop production in the arid and semi-arid regions of the word.Due to scarcity of fresh water, use of saline water is becoming inevitable to meet agricultural crop water requirement.In some water shortage areas of the world, farmers usually apply saline water especially to winter cereals in the pre-sowing and early stages of the crop growth.Farmers are faced with the challenges of optimal water applications for given saline water and economical effect of using saline water.Therefore, it is necessary to estimate crop yield in response to joint salinity and water stress conditions resulting from the use of any given set of water quantity and quality.
Several studies have explored the response of crops to salinity and water stress, i.e., use of saline water for irrigation purposes (Ayers and Westcot, 1985;Tanji, 1990;Rhoades et al., 1992).Blending non-saline and saline water, or cyclic use of these types of water, have been developed to use saline drainage water for irrigation.Water-salinity production functions are estimated by two different approaches.The first one uses mathematical models to describe the combined effects of the amount of applied water and its salinity on crop yields (Letey et al., 1985;Letey and Dinar, 1986).The second approach experimentally estimates production functions by statistical analysis from a wide range of water qualities and quantities (Dinar et al., 1985;Dinar and Knapp, 1986;Russo and Bakker, 1986;Datta et al., 1998;Datta and Dayal, 2000;Kiani & Abbasi, 2009).The first approach is valuable.Because it www.intechopen.comallows exploration of different management strategies by using appropriate water-salinity production functions saving time and costs.But, due to the implicit assumptions, the practical application is restricted.However, several studies have been published where production functions have been verified by field data, or have been used to assess the dual effects of salinity and the amount of applied water on crop yields (Letey et al., 1985;Shani and Dudley, 2001;Oster et al., 2007).An experiment was conducted in the northern region of Golestan province (37°07′ N, 54°07′ E) in Iran, for two growing seasons (2001-2002 and 2002-2003).The mean annual rainfall in this region is 330 mm of which more than 80% occurs in autumn and winter (November to April).The average annual temperature and relative humidity are 17.8 ˚C and 75%, respectively.The experimental design was a randomised complete block design with split plot layout considering water quantity as the main plot and water quality as subplot with three replications.The treatments consisted of four levels of irrigation water providing 50(W 1 ), 75(W 2 ), 100(W 3 ) and 125(W 4 ) percent of the crop water requirement and four levels of irrigation water salinity of 1.5(S 1 ), 8.5(S 2 ), 11.5(S 3 ) and 14.2(S 4 ) dS/m.Saline waters for different irrigation salinity levels were obtained by mixing various ratios of well (1 dS/m) and drainage channel (10-30 dS/m) waters.
In the first year, a pre-sowing of 26 mm non-saline water was applied to all treatments using a sprinkler irrigation system, and three irrigations were subsequently applied using surface irrigation.In the second year, due to suitable rainfall at sowing date, water was applied in four irrigations after the vegetative stage.The amount of irrigation water during the growing season, for W 1 , W 2 , W 3 and W 4 treatments was equal to 118, 163, 209 and 246 mm in the first year and 104, 160, 212 and 264 mm in the second year, respectively.Effective rainfall in the growing season was 163 and 184 mm in 2001-02 and 2002 -2003, respectively.Data obtained from the experimental plots were used to investigate crop response to both electrical conductivity of soil water (EC sw ) and soil water content ( θ ), assuming that the crop responded only to these two factors.Other possible factors affecting the yield were assumed to be constant according the following relationship: Where, X is the constant vector for considering other factors affecting the yield.
The following various production functions were explored using the collected experimental data: Linear: Cobb-Douglas: Quadratic: Transcendental: The optimal production function was selected based upon statistical analysis.The coefficients of various production functions were estimated using the Ordinary Least Square (OLS) technique (SPSS, version 11.5).T-statistic of model's coefficients for determination of significantly different, F-value, R 2 , Standard Error (SE) and Relative Error (RE) were estimated for comparison of various production functions.The RE was computed by: where, Y m and Y p are measured and predicted yield.
The marginal production of water content (MP θ ) and salinity (MP ECsw ), the marginal value product of θ (VMP θ ) and EC sw (VMP ECsw ) and the marginal rate of technical substitution of θ and EC sw (MRTS ECsw θ ) were determined using the selected production function as: = . (18) = .
where, P y is wheat price.
Table 6 shows the amount of matric potential (h), osmotic potential (h o ) and wheat grain yield in different treatments.Generally, a rather consistent decrease in yield with increasing salinity application for a given irrigation treatment and a rather constant increase in yield with increasing water application for a given saline water.In the W 1 S 1 treatment, the total water potential was -6678 cm, which is somewhat lower than the total potential of -6,403 cm in the W 3 S 3 treatment.The yield reduction for the W 1 S 1 treatment compared to W 3 S 1 was 17%, and for the W 3 S 3 treatment it was 7%.For the W 1 S 1 treatment, the matric potential contributed 38% of the total water potential as compared to 26% for the W 3 S 3 treatment.The results suggest that the reduction of the yield due to joint salinity and water stress are not confirmed by the previously proposed simple linear additive and multiplicative concepts.For example, in the first year, yield reduction for the W 1 S 1 and W 3 S 4 treatments were, respectively, 0.17 and 0.12 compared to W 3 S 1 , while the yield reduction for the combined stress W 1 S 4 was 0.26.The same results were obtained in the second year.Matric and osmotic potentials were additive in their effect on yield, but the amount of yield reduction due to unit increase of matric potential was more than that due to unit increase of osmotic potential.

Treatments Matric Potential
Osmotic Potential Yield h (cm) h o (cm) (kg ha -1 ) Irrigation Salinity 2001Salinity -2002Salinity 2002Salinity -2003Salinity 2001Salinity -2002Salinity 2002Salinity -2003Salinity 2001Salinity -2002Salinity 2002Salinity -2003 -2,565 -2,421 -2,595 -3,391 -1,528 -1,489 -1,492 -1,427 -4,113 -4,303 -5,267 -5,922 -3,615 -4,212 -5,111 -5,330 3,213 3,216 3 -2,555 -4,248 -4,531 -5,383 -2,904 -3,833 -4,880 -5,402 3,825 3,783 3,773 3,467 510 4,839 4,690 4,405 4,331 474 Table 6.Average seasonal matric and osmotic potential and grain yield per treatment combination The estimated coefficients and the statistical analysis of various production functions are shown in Table 8.The results indicate that transcendental and quadratic production functions provide a better fit to the data.The determination coefficient (R 2 ) of the transcendental function is about 0.94 (two years average) and larger than the other functional forms.The R 2 values in the transcendental and quadratic production functions suggest that 94% of the yield variability is explained by the variations of EC sw and θ .It is also found that the transcendental model provides a higher F-value and lower standard error (SE) compared to the quadratic model (Table 7).Furthermore, RE of the various production functions suggests that the transcendental model is the best.In addition, the transcendental form of the production function is found useful in describing input-output data including all three stages of the production curve with increasing positive, decreasing positive and negative marginal products (Sankhayan, 1988).As a result, in this study the transcendental model is found as a suitable wheat production function under salinity-water stress conditions.The isoquant curves of wheat relative yield obtained using the transcendental production function are shown in Figure 5.These curves provide the different combinations of θ and EC sw that result in the same yield.For a given θ , increase of EC sw results in the decrease of relative yield and for a given EC sw relative yield increases with increasing θ .In general, the relative yield is strongly affected by both θ and EC sw .Isoquant curves (Figure 5) indicate that each one of the two factors (EC sw and θ) can be substituted by the other one for a wide range in order to achieve equal amount of yield.The higher level of EC sw can be used with increasing soil water content, without yield reduction.For a given relative yield, results show that the optimal combination of EC sw and θ is located where the slope of the isoquant lines is parallel to the θ axis.From this point forward, an increase of θ will not result in increasing the yield.For example, a relative yield of 85% can be obtained with water content of 25% and EC sw of 8 dS/m.If EC sw increases to 12.2 dS/m, previous relative yield (85%) can be achieved, but the water content must be increased to 27%.The direction of the isoquant curves changes for EC sw less than 6.5 dS/m.The slope of the isoquant curves becomes parallel to EC sw axis.This means that by increasing EC sw from 0 to 6.5 dS/m the relative yield will increase slightly with low water contents, but sharply with relatively high water contents.Maas and Hoffman (1977) did not consider soil water contents.They obtained and reported yield as a function of the ECe.In fact, soil water content was not a limiting factor in their study.Consequently, their function applies only to well watered conditions and inherently cannot be construed to have implications to situations where soil water content is limiting.Letey et al. (1985) assumed that the relative yield becomes maximal at EC sw equal to zero.However, it is clear that the relative yield does not become maximal at EC sw equal to zero because EC sw = 0 implies that there is not any mass (i.e., fertilizer) in the soil, which it can provide minimum relative yield.Of course, Letey et al. (1985) did not intend an extrapolation to a situation where all nutrients are removed from the soil.The MRTS θECsw for the two growing seasons were 0.96 and 0.67, respectively.This means that one unit increase in EC sw requires the soil water content to be increased by a factor of 0.96 and 0.67 to prevent the yield reduction in two studied years.The marginal production of θ (MP θ ) and EC sw (MP ECsw ) were 125 and -121 kg/ha, respectively in the first year, and 232 and -158 kg/ha, respectively in the second year.It means that the marginal value of income per hectare due to one unit increase in water content is US$50 (two years average) and the marginal value of damage per hectare due to an increase in EC sw is US$38 (Py = US$0.28).Using the Cobb-Douglas model, Datta and Dayal (2000) reported that the marginal value product of water quantity of irrigation water and the marginal value of damage of water quality were US$1.2/ha and US$4.2/ha, respectively.

90
The iso-( Y θ ∂∂) and iso-( YE C s w ∂ ∂ ) lines determined using the transcendental model are shown in Figs.6a and 6b, respectively.In the dry soil (for θ less than 21%), the rate of change of the yield with θ ( Y θ ∂ ∂ ) increase as θ increase and EC sw decrease.On the contrary, in the wet soil (for θ more than 22%) Y θ ∂ ∂ decreases as both θ and EC sw increase (Figure 6a).The rate of change of yield with θ becomes negative at water content of over 32% (data was not showed in Figure 6a).Figure 5a revealed that Y θ ∂ ∂ is influenced by θ changes, but slightly changes with EC sw variations.Findings of Letey et al. (1985) for a given EC e showed that YW ∂∂ was affected by applied water.The decreasing yield per unit increase in soil water content is consistent with the findings of Letey et al. (1985).The YE C s w ∂ ∂ increases (with negative slope) as both θ and EC sw increase (Figure 6b).In the dry soil, the EC sw variation has a little effect on YE C s w ∂ ∂ and θ variation has a more effect on YE C s w ∂∂ .In contrary, in the wet soil, the EC sw variation has a more effect on YE C s w ∂ ∂ and θ variation has a little effect on YE C s w ∂ ∂ .

Conclusion
Knowledge of water-yield relation (production function) is necessary in order to achieve optimal amounts of irrigation water and sustainable production.Optimizing water use is a type of management options that may establish relationship between land and water under limitation of water/ land conditions so that the crop production is economically affordable and technically also be possible.Optimal irrigation water application was considered by generated crop water production functions for some cultivars of wheat and soybean in different amount of water under non-saline and also combined salinity and water stress conditions.Estimated water production function for any cultivars showed that they have different constants as well as different response to water, resulting the appropriate varieties for higher productivity of water can be selected.Quantitative comparison of two strategies a b of full irrigation and deficit irrigation has shown that if producers instead of full irrigation the part of farm, use deficit irrigation to increase irrigated area, total production and total revenue will increase.To minimize risk with water stress on crop yield reduction, it is needed to know sensitivity of different growth stages of crop to water stress.Considering the trend of crop yield under different amounts of irrigation water (steep slope in the deficit irrigation and less slope in full irrigation) priority allocation water resource to little water areas is a very effective strategy for optimal use of water resources, increasing production and farmers income.The reduction in yield due to joint salinity and water stress was not confirmed by the simple linear additive and multiplicative concepts.It was found that transcendental and quadratic production functions reasonably well predicted the yield under salinity and water stress conditions.The results revealed that yield reduction due to a unit increase of matric potential (i.e., water content) was not the same as that due to a unit increase of osmotic potential (i.e., salinity).The effect of soil water content on the rate of change of wheat yield was more pronounced than the effect of salinity.Iso-quant curves indicated that soil water content and salinity can be substituted by one another for a wide range to achieve equal amount of yield.Crop-water productions derived with the experimental approaches are site-and year-specific.Therefore, generalization of the production functions would not be so easy.

Fig. 1 .
Fig. 1.Income and cost curves as a function of applied water (AW)

Fig. 2 .
Fig. 2. Wheat grain yield as a function of applied water in different cultivars A field study was conducted to consider water-yield relation in three genotypes of soybeans, optimum irrigation depth, comparison of deficit and full irrigation at Gorgan Research Station in two lasted growing seasons(2005 & 2006).In this experiment, four irrigation treatments (W 1 , W 2 , W 3 and W 4 ) were provided by the decline in irrigation with distance from a line source.The experiment was based on a strip plot design to examine the effect of the fixed irrigation rates on three cultivars treatments (SAHAR, G 3 & DPX) with four replications.Generally, obtained highest soybean yield is consisted with treatment that has received the highest water (W 1 ).In W 4 treatment, grain yield of soybean were 47% and 40% of the W 1 treatment, for 2005 and 2006, respectively.The average values of 2-years irrigation water in W 2 , W 3 and W 4 were 80, 49 and 24% of W 1 .Irrigation water application in both years with distance from the line source was reduced for all three cultivars (Fig.3).The largest amount of irrigation was W 1 treatment being 360 mm in 2005 and 342 mm in 2006.In the first year in terms of irrigation water, W 4 treatment in Sahar, G 3 and DPX has received 17, 19 and 24% as compared with W 1 treatment, respectively (the same trend also observed in the second year).

Fig. 3 .
Fig. 3. Applied water (AP) in different treatments (2-year average) The estimated coefficient using equation 2 and their statistical analysis in different cultivars were presented for 2005, 2006 and average 2-years in Table3.As shown in Table4, the determination of coefficient (R 2 ) of the quadratic function for all cultivars are more than 84% and it is found that the effects of water levels on soybean grain yield is significant.This indicates that 84% of the yield variability of soybean is explained by the variations of applied water (AW).A quadratic form has been reported between yield and applied water by many researchers(Stewart and Hagan, 1983;Zhang and Oweis, 1999;Sepaskhah and Akbari, 2005).The grain yield of soybean cultivars as a function ofAW in  2005AW in   , 2006AW in    and 2005AW in   -2006   are illustrated in Fig.4.Figure4states that the yield of soybean cultivars is increased as a decline slope with increasing water amount.In other words, for

Fig. 4 .
Fig. 4. Soybean grain yield as a function of applied water (AW)

Fig. 5 .
Fig. 5. Calculated isoquant curves of wheat relative yield (Yr) as a function of the 2001-2002 seasonal average θ and ECsw

Fig. 6 .
Fig. 6.Calculated isoquant curves representing the variation of δY/δθ (a) and δY/δEC sw as a function of the 2001-02 seasonal average θ and EC sw

Table 2 .
). Optimum irrigation water as a function of seasonal rainfall in different cultivars of wheat

Table 2 .
that deficit irrigation is more useful strategy for obtaining more production as compared to full irrigation.For example, in C 1 cultivar, if full irrigation (99 mm) is applied to irrigate one hectare and left a 1.54 hectare as a rainfed, totally it can be obtained 7.27 ton grain yield.But, if instead of full irrigation of one hectare, it is applied deficit irrigation (39 mm) for all the 2.54 hectare, totally 9.02 ton grain yield can be obtained.This trend with more appropriate result for other cultivars is presented in Table2.It is observed that in W 4 treatment total yield in C 1 to C 6 cultivars were increased 24%, 40%, 47%, 40%, 48% and 32%, respectively, when compared to the W 1 treatment.To minimize risk with water stress on crop yield reduction, it is needed to know sensitivity of different growth stages of wheat to water stress.Comparison of total production in different cultivars of wheat at full (W 1 ) and deficit (W 4 ) irrigation

Table 3 .
Equation constants and their statistical analysis in three soybean cultivars

Table 7 .
Values in the parenthesis show t statistic values and standard error (SE), respectively.**significant at level P <0.01 by LSD range.*significant at level P <0.05 by LSD range.ns= not significant Estimated coefficients for each of the examined wheat water-salinity production functions.