Evaluation and Improvement of Crop Production Functions for Simulation Winter Wheat Yields with Two Types of Yield Response Factors

Water is an important item of crop production and Irrigation water is limiting for crop production in arid and semi-arid areas. On the other hand, considering the population growth and increasing the water and food needs and also limited resources of water, the optimization of water consumption, especially in section of agriculture is important. For this purpose, in first step, the production function of expected products in any region should be obtained acceptably, because in this case, the models are totally dependent on the production function. Thus, finding the optimal production function has the lowest error in the estimation of risk-taking and decision-making power in the future will be important. So this study was conducted in Esmaeil Abad in Qazvin plain in Iran in the growing season of 2009-2012. Deficit Irrigations applied on different growth stages of winter wheat. The maximum evapotranspiration 641 mm and maximum attainable yield 5847 kg/ha was determined. After that the different production functions were studied and these methods have been tried to improve a new method with the least error. On the other hand yield response factor (Ky) per month was defined as either one of the standard values by FAO and the other using correction values by Najarchi et al. (2011). The result showed that the new model in this study is normalized with yield response factors of Najarchi et al. (2011) with 5% normal root mean square (NRMSE) has the lowest error. Therefore this technique for estimating water deficits of winter wheat in the Qazvin Plain was suggested.


Introduction
The important limit for agriculture in Iran is water, especially in Qazvin plain.So paying attention to the management of water consumption in agricultural sector as the main consumer of water is necessary.The rapid increase of the population and the corresponding demand for extra water by sectors such as industries and municipals, forces the agricultural sector to use its irrigation water more efficiently.Also planning and management of available water resources in the agricultural sector are to become a national and global priority (Smith, 2000).One important thing is to discuss in agricultural land is water shortage in under irrigation lands.There are different strategies to confront with this problem on farm land which are divided into six parts including: 1) Increasing soil water storage at the time of cultivation; 2) Increase water usage by plant in soil; 3) To decrease the evaporation from soil surface; 4) optimization model for water consumption; 5) improving plant tolerance to water stress and tensions; 6) irrigation at critical growth periods (Debaeke & Aboudrare, 2004).Apart from field experiments, a robust simulation model with which specific situations can be simulated would be very useful for the proper design of deficit irrigation strategies.The mechanistic models however generally require a huge set of input data which is often not readily available outside research stations.Since they also demand an extensive site-specific calibration before they can be applied, this type of model might not be very useful for developing irrigation strategies under practical conditions (Raes et al., 2006).In this context, By means of a mechanistic crop growth model, the expected yield for several growing conditions can be estimated (Penning de Vries & Van Laar, 1982;Spitters et al., 1989;Ritchie, 1990;Goudriaan & van Laar, 1994;Bouman et al., 1996;Boote & Jones, 1998;Soltani et al., 1999 ;Robertson et al., 2001;Batchelor et al., 2002;Stockle et al., 2003;Wang et al., 2003;Ziaei & Sepaskhah, 2003;Yang et al., 2004).Also we don't have enough data to run and to calibrate these models in each region of Iran especially in Qazvin plain.So using of these models is difficult.We need sample models to estimate crop yield in practical conditions.For this purpose, we try to find monthly sample method for estimating crop yield.In this study, the sample models, Doorenbos andKassam (1979), Allen (1994), Stewart et al. (1977) and Raes (2004) are used to find best simple model.In the other hand, yield response factor (K y ) is very important to all of the models.So In the case of Qazvin Plain, we use yield response factor (K y ) that produce by Najarchi et al. (2011).Therefore, we should see that the impact of this correction factors is how much Influence on the production functions.Considering the importance of winter wheat as one of important products to supply the food needed in Iran and deficit irrigation method, the present study was to investigate the production functions with correction factor Najarchi et al. (2011) was conducted to estimate winter wheat yield to show that production function can be used and which functions will provide better answers.

Experimental Site
This study was conducted in Esmaeil Abad in Qazvin province, Iran in the growing season of 2009-2012.This experiment was performed on a land area of 600 square meters in Esmaeil Abad Research Station (49º 52' N, 36º 15' E, 1285 MSL).There was no salinity and sodium hazard by using the irrigation water.In this experiment 200 kg/ha Nitrogen and 45 kg/ha phosphate fertilizer with 5 million plants per hectare density was applied.Randomized complete block design with 5 treatments and 3 replications was used.In each crop year and after preparing the ground with dimensions of 6m long, 4m wide, for each plot (24 m) were classified.Soil physical and chemical properties are given in Tables 1 and 2 respectively.In order to cultivate the seed of Alvand was used.Deficit Irrigations applied in during the Germination, Tillering, Stem elongation, flowering, Milky and dough and ripening stages.The evapotranspiration and crop yield in each method shown in Tables 3, 4 and 5. To ensure of significant difference between treatments in per year, Duncan test was applied, and the results were added to the tables.

Measurement and Methodologies
The amount of irrigation water, in the basis of soil moisture in depths of 25, 75, 100, and 125 cm was measured, and amount of water demand in each layer in the region of plant's root relation was obtained by Equation 1.
Where d n is the net irrigation depth (m), θ fci and θ i are the volumetric soil water contents in layer i at field capacity and before irrigation, respectively (m 3 m -3 ), Δ z is the soil layer thickness (m) and n is the number of soil layers.Using Equation given by Borg and Grimes (1986) root growth during the growing season was calculated as follows: Where z r is the root depth, R is maximum rooting depth, D TM is the number of days required to reach the maximum depth, D is the number of days after they are planted, The maximum depth of the root is 1 meter for winter wheat was placed 160 days after planting (Hosseini, 2005).
After calculating the amount of irrigation water needed due to the type of treatment applied, the soil water balance Equation for each treatment was calculated as the winter wheat evapotranspiration (Jensen, 1973).
where I is the irrigation amount (mm), P is the precipitation (mm), D is the deep percolation (mm) at the bottom of the root zone, n is the number of layers, ΔS is the thickness of each soil layer (mm) and θ 1 and θ 2 are the volumetric soil water contents (cm 3 cm -3 ) before two consecutive irrigations.These values are presented in Table 3, 4, and 5.In this test, the farming field capacity and bulk density during test was assumed constant.At the time of growth season, necessary cares like weeding, spraying against pests and plant diseases were performed and harvesting was done by hand.Then the product was dried and next the seeds were separated from the Straw and yields were measured for each treatment.

Yield Production Functions
According to information obtained (Table 3, 4, and 5), different production functions used.These functions for were as follows: Doorenbos and Kassam (1979) have presented the relationship between relative yields, relative evapotranspiration, and crop coefficients were as follows: Where y a is the actual harvested yield (kg/ha), y m is the maximum attainable yield (kg/ha), K y is the yield response factor (non-dimensional), and ET a and ET m are the actual evapotranspiration (mm) and maximum evapotranspiration (mm), respectively, during the growing period.Y max and K y are crop-related coefficients, which must be known.
This Equation acts as simple and general.So it is usable for total growth period and other methods based on this method have been written.In order to use this model, using yield response factor of product recommended by FAO and once again using yield response factor of product recommended by Najarchi et al. (2012) was applied.
Where Y a is the actual harvested yield (kg/ha), Y p is the maximum attainable yield, Y ai /Y pi are expected relative yield as result of water stress in growth stage i.The expected yield is estimated by the sum of the right hand terms of Equation 1 determined for each period.
In this study, instead of taking minimum, the Average of product reduction was considered, as the Equation can be written as follows; Where Y a is the actual harvested yield (kg/ha), Y p is the maximum attainable yield, Y ai /Y pi are expected relative yield as result of water stress in growth stage i.The expected yield is estimated by the sum of the right hand terms of Equation 1 determined for each period.
Another productive method of generating functions is that Stewart et al. (1977) presented the following Equation for the stress at different growth stages: Where y a is the actual harvested yield (kg/ha), y m is the maximum attainable yield, ET ai is the actual evapotranspiration in each period (mm), ET mi is the maximum evapotranspiration in each period (mm), K yi is the yield response factor (non-dimensional) at any stage of growth, i is the stage of development, and n is the number of stages of growth.
Therefore, according to Equation 7, reduction rate in one-month period was determined.So that the final product can be gotten, as this method, one time with yield response factor of FAO and another time with yield response factor of Najarchi et al. (2011) were performed.
To improve and increase the accuracy of the Equation 7, Raes (2004) in the water and solute balance model (BUDGET), the following Equation was used where the different stages are divided into a number of smaller periods: Where ∏ stands for the product of the M functions between square brackets, M for the number of time steps with length Δt j (day), during the growth stages i, L i for the total length of the stages(day), and ET a,j and ET m,j for respectively the actual and maximum evapotranspiration during the time step j.In this study length of period (Δt j ) is 30 days, and Equation 8 was used with monthly intervals, and range of (Δt j /L i ) is given in Table 6.
So, using the table of values and evapotranspiration values in each interval, yields were estimated by this method.In this way, the yield response factors of the two types of plants were used.
To improve and increase the accuracy of the Equation 8 and to increase the accuracy of production function, it has been tried, the weighting of length of periods with yield response factor is replaced and the new Equation is as follows: Where Y a is the Actual yield (Kg/ha), Y p is the maximum attainable yield (kg/ha), ET a,j is the actual evapotranspiration (mm) in each period, and ET m,j is the maximum evapotranspiration (mm) in each period, K yi is the yield response factor (non-dimensional) at any stage of growth, i:stage development, and n is the number of stages.
The computed values for new coefficients are given in Table 6.According to the values obtained the new method was launched.Table 6.Monthly coefficients of the Raes (2004)  12 0.12 0.12 0.12 0.12 0.1 0.08

Analyses Method
Simulation yield were compared with the measurement values.The goodness of fit of the simulations was assessed with the help of three statistical estimators: Where RMSE is the Root Mean square error, n is number of data, X i is the Data Measurement and Y i is data estimated by the model.
Where NRMSE is the Normal Root Mean square error, n is the number of data, X i is the measured data, Y i is the data estimated by the model, and is the average of measured data.
Where d is Agreement index, n is the number of data, x i is the measured data, y i is the data estimated by the model, is the average of measured data, and is the average of estimated data.

Results and Discussion
According the measured values and different functions of production which were used in this study, the yield of winter wheat was estimated in any method with separation of yield response factor are given in Tables 7, 8 and 9. Eq.( 4) Eq.( 4) Eq.( 5) Eq.( 5) Eq.( 6) Eq.( 6) Eq.( 7) Eq.( 7) Eq.( 8) Eq.( 8) Eq.( 9) Eq.( 9) For better results, first, we explain the review of results of the production function of Doorenbos and Kassam (1979).For this work, the estimated parameters measured using the index of sensitive plants FAO in Figure 1a and the coefficients of yield response factor of Najarchi et al. (2011) in Figure 1b, compared to the one by one line along with the values of RMSE, NRMSE and d is presented.As, it's clear, this production function the index of sensitive plants FAO can act better and amount of normal risk is 6% and this is while with the index of sensitive plants of Najarchi et al. (2011), the amount of error reaches to 5%, thus, in the case of using this production function, coefficients of Najarchi et al. (2011) are proposed.Second function of production, which has been examined, is a minimization method, as amount of estimated values with this method is brought using FAO plant yield response factors in Figure 2a and yield response factor Najarchi et al. (2011) in Figure 2b relate to a line, one by one and together with values of RMSE, NRMSE and d is presented.Considering the results obtained, this method with two aforesaid yield response factors didn't work and only based on the extremist tension can make decision, thus, its results are not imputable and has normal error of 21-34%, as it was not suggested.The third function is the averaging method of production was investigated in this study, this method was used to correct for the minimal approach.Values estimated by this method using the FAO crop yield response factor in Figure 3a  (2011) yield response factor work better and based on the averaging of imposed tensions estimates the amount of crop.Thus, results of this method are acceptable and using Correction Coefficient (Najarchi et al., 2011) its Normal Error reduces from 8 to 6 percent.So, this method can increase the accuracy of estimate and comparing to method of minimum offers more reasonable answers.Next function of production, which has been examined, is a simple multiplication method, as in this method, water tension in each cycle, have influence on each other as multiplying and decrease the crop in this function.Amount of estimated values with this method is brought using FAO plant yield response factor in Figure 4a and yield response factor Najarchi et al. (2011) in Figure 4b relate to a line, one by one and together with values of RMSE, NRMSE and d is presented.Considering the results obtained, this method with two aforesaid yield response factors didn't work and only based on the extremist tension can make decision, thus, its results are not imputable and has normal error of 32-48%, as it was not acceptable and is not suggested.2011) Raes (2004) in order to improved simple multiplication method, the multiplying function of production with coefficients were defined to the length of each period were corrected.As in this method, water tension in each cycle, have influence on each other as multiplying and decrease the crop.Amount of estimated values with this method is brought using FAO plant yield response factorin Figure 5a   Raes ( 2004) multiplication method defined multiplication function with strength coefficients, which is dependant to the proportion of length of each time period, but, the important thing is sensitivity of plant during each period.Thus, in this study, these coefficients are distributed according to yield response factor of each period so that it can be observed that this strength proportion is much dependent to the length of period or rate of yield response factor of each period.And the severity of crop reduction in each period can be controlled with these new coefficients.Amount of estimated values with this method is brought using FAO plant yield response factor in Figure 6a

Conclusions
Considering functions of planned production in this study, it's needed that results which are obtained should be collected in a table so that we can get an appropriate conclusion.Therefore, a summary of statistical results are given in Table 10.The values obtained in the method of Doorenbos and Kassam (1979), the yield response factors of Najarchi et al. (2011) are recommended.However, this method is general and does not feature a monthly breakdown.In the Average method, the normal error rate is reduced to 8 percent.And with the use of Najarchi et al. (2011) yield response factors, this error can be reduced to 6 percent.In simple multiplication method and minimum method, we see high error as in these methods.Therefore, these methods with error of 21% and 31% are not recommended to estimate the crop of Winter wheat.The method of Raes (2004) with monthly stage cause a suitable weighting is done and the normal error rate is reduced to 8 percent.Therefore, with this method, we can reach satisfactory answers and it can act better than the average method.Also, the normal error rate is reduced to 6 percent by using corrective coefficients of Najarchi et al. (2011).In this new method that the weighing power is connected to the coefficient of sensitivity, the normal error rate is reduced to 7.5 percent and in the case of corrective coefficients of Najarchi et al. (2011), the normal error rate is reduced to 5 percent.So, what is the more effective on weighting is not length of course of plant sensitivity.Thus, the new method can provide a more acceptable solution.Therefore, among aforesaid methods, the new method works better than others and is proposed as a suitable method.Finally, in order to increase the accuracy of plant yield in Qazvin Plain response factor of Najarchi et al. (2011) is proposed.

Figure 1 .
Figure 1.Relationship between amount of measured and yield was estimated by Equation.(4) with: a) FAO plant yield response factors.b) Plant yield response factors of Najarchi et al. (2011) and Najarchi et al. (2011) plant yield response factor in Figure 3b compared to the one by one line along with the values of RMSE, NRMSE and d are presented.Considering the results obtained this method Najarchi et al.

Figure 3 .
Figure 3. Relationship between amount of measured and yield was estimated by Equation.(6) with: a) FAO plant yield response factors.b) Plant yield response factors of Najarchi et al. (2011)

Figure 4 .
Figure 4. Relationship between amount of measured and yield was estimated by Equation.7 with: a) FAO plant yield response factors.b) Plant yield response factors of Najarchi et al. (2011) and yield response factor Najarchi et al. (2011) in Figure 5b relate to a line, one by one and together with values of RMSE, NRMSE and d is presented.Considering the results obtained this method Najarchi et al. (2011) Plant yield response factor work better.Thus, results of this method are acceptable and using Correction Coefficient its Normal Error reduces from 8 to 6 percent.So, this method can accurately increase the accuracy of estimate and comparing to method of simple multiplication offers more reasonable answers.

Figure 5 .
Figure 5. Relationship between amount of measured and yield was estimated by Equation 8 with: a) FAO yield response factors.b) Plant yield response factors of Najarchi et al. (2011) and yield response factor Najarchi et al. (2011) in Figure 6b relate to a line, one by one and together with values of RMSE, NRMSE and d is presented.Considering the results obtained this method Najarchi et al. (2011) Plant yield response factor work better.Thus, results of this method are acceptable and using Correction Coefficient its Normal Error reduces from 8 to 5 percent.So, this method can accurately increase the accuracy of estimate and comparing to method ofRaes (2004) offers more reasonable answers.

Figure 6 .
Figure 6.Relationship between Amount of measured and yield was estimated by Equation 9 with: a) FAO yield response factors b) Plant yield response factors of Najarchi et al. (2011).

Table 3 .
Monthly evapotranspiration rates of winter wheat in different irrigation treatments year/month Nov. Dec.Jan.Feb.Mar.Apr.May.Jun.Means followed by the same letters in each parameter are not significantly different at 5% level of probability *

Table 4 .
Monthly evapotranspiration rates of winter wheat in different irrigation treatments year/month Nov. Dec.Jan.Feb.Mar.Apr.May.Jun.
*Means followed by the same letters in each parameter are not significantly different at 5% level of probability

Table 5 .
Monthly evapotranspiration rates of winter wheat in different irrigation treatmentsMinimum product loss is another method which was expressed by Allen (1994) and different growth period was considered: *Means followed by the same letters in each parameter are not significantly different at 5% level of probability

Table 10 .
Values of RMSE, NRMSE and d for production functions