The mathematical formulation was presented by Zand-Parsa & Sepaskhah (2001) to determine the farm net income in case that the crop yield and price are a function of applied water and N as follows:

where A is the irrigated area (ha), W is the applied water (m ³ /ha), R is the rainfall (m ³ /ha), N is the N application rate (kg/ha), Nr is the soil residual mineral N (kg/ha), y (W+R, N+Nr) is the crop yield (kg/ha), c (W, N) is the production costs (Rls ¹ /ha), P _c (W+R, N+Nr) is the crop price (Rls/kg), i _l (W+R, N+Nr) is the net income per unit area (Rls/ha), and I _f (W+R, N+Nr) is the total net income (Rls) from all irrigated area. Contrary to W and N that are management parameters and can be selected, R and Nr are not easily available under field conditions; however, R can be predicted. Although the values of R and Nr affect the crop yield and price, they do not influence the crop production costs.

The irrigated area (A) can be a function of used water and is not dependent on N fertilizer because of no limitation for N use. The irrigated area can be determined as follows:

where W _T is the total available water supply (m ³ ).

Under land limited conditions (when the arable land area is limited), the farmer put all available arable land under irrigation and cannot increase the land area. Therefore, net income per unit area gets a maximum value since the partial derivative of farm net income equation [Eqn (1)] with respect to W and N was set to zero as follows:

The data used in this investigation were obtained from Karimi & Naderi (2008) , who conducted an experiment at Karaj Research Center, Karaj, I. R. of Iran in 2003. Table 1 shows the soil physical and chemical properties ( Karimi & Naderi, 2008 ). The experiment was a split-plot design (1 plot = 3×6 m ² ) with randomized complete blocks arrangement with three replications. Irrigation treatments were the main plots and N fertilizer rates were subplots. Four levels of irrigation water (40%, 80%, 120% and 160% of evaporation from the surface of class A evaporation pan) and four of N fertilizer (urea) (0, 90, 180 and 270 kg N/ha) were applied. Irrigation frequency was the same for all treatments (every 7 days). Sum of the applied water and seasonal rainfall for different irrigation treatments were 9500, 11850, 14500 and 16450 m ³ /ha, respectively. Seasonal rainfall was 207 m ³ /ha. MSC2 cultivar of sugar beet was planted. Seeds were planted on rows with spacing between rows of 0.6 m and distance between seeds on rows of 0.2 m. After harvest, the sugar concentration was determined by standard procedures by the sugar refining factory. The soil residual mineralized nitrogen (NO ₃ and NH ₄ ) of the root zone was considered as 255.15 kg/ha for two soil layers (5% of the soil total N (Table 1), as reported by Dhanke & Vass, 1973 ).

Using the above-mentioned analysis to plan the deficit irrigation is contingent on the prediction of seasonal rainfall amount before the start of the growing season or on the amount of optimum applied water that should be determined by using occurrence probability analysis for a given rainfall ( Sepaskhah et al. , 2008 ).

Rainfall amounts during the growing season for 21 years in the study area are available ( http://www.irimo.ir ). A frequency analysis was applied for the amounts of optimum applied water at different N levels obtai­ned in different years. The occurrence probability was estima­ted by Weibull equation ( Chow et al. , 1988 ) as follows:

Based on the sugar beet root yield, sugar content, sum of irrigation water and rainfall and sum of N application rate and residual N, reported by Karimi & Naderi (2008) , crop yield and sugar concentration functions were determined by multiple regression analysis as follows:

Solving Eqns (11a) and (11b) based on the coe­fficients of Eqn (13), the amount of applied water (W _m ) and nitrogen (N _m ) that resulted in maximum yield was obtained as follows:

Under land limiting conditions, saved water com­pared to W _m cannot be used to irrigate extra land area. This amount of water maximized the benefit for each unit of land. By solving Eqns (7a) and (7b) based on yield [Eqn (13)], crop price [Eqn (15)] and cost [Eqn (16)] functions for R=207 m ³ /ha, Nr=255.15 kg/ha, P _c16 =290 Rls/kg, P _n =913 Rls/kg and P _w =120 Rls/m ³ ,

the calculated optimum water (W _l ) and N (N _l ) under land limiting conditions were 12174.2 m ³ /ha and

262.6 kg/ha, respectively. For more assessments, W _l and N _l were calculated at different rainfalls, water costs, N cost, base crop prices and soil residual N. Results are shown in Figs. 1 and 2 which indicate that as rainfall increased, W _l decreased, whereas N _l did not vary (Fig. 1a). Therefore, rainfall had not any effect on the optimal amount of N _l . For constant N cost (P _n =913 Rls/kg) and base crop price (P _c16 =290 Rls/kg), by increasing water cost (P _w ), optimum amounts of water and N decreased and increased, respectively (Fig. 1a). For constant water cost (P _w =120 Rls/m ³ ) and base crop price (P _c16 =290 Rls/kg),

by increasing N cost (P _n ), the optimum amount of N decreased; however, the optimum amount of water increased (Fig. 1b). For a given rainfall, the optimum level of water increased by an increase in base crop price (Fig. 1c). In other words, maximum net income was found at a higher depth of water by an increase in base crop price. For a given rainfall, N _l variation was very low by a change in base crop price (Fig. 1c) and was higher for a higher water and N cost (Figs. 1a and 1b). The value of W _l was not affected by soil residual N content, and the value of N _l decreased by an increase in Nr (Fig. 2).

Net income was calculated based on different irri­gation water and N applications at different water and N costs and base crop price for constant rainfall and soil residual N. Relationships between net income and irrigation water at different water costs and base crop prices for R=207 m ³ /ha and N+Nr=517 kg/ha are shown in Fig. 3. As mentioned above, the maximum net income per unit area (the maximum points) occurred at optimum irrigation water for different water costs (Fig. 3a). An increase in water cost decreased net income. Also, maximum net income was obtained at lower irrigation water depth by the increase in water cost compared with P _w =0 (arrow in Fig. 3a). Maximum net income increased by increase in the base crop price. The maximum net income was obtained at higher irrigation water by an increase in base crop price (arrow in Fig. 3b). The net income under land limiting conditions at different N costs for P _c16 =290 Rls/kg and W+R=

12381 m ³ /ha are shown in Fig. 4. The maximum net income per unit area (maximum point) occurred at optimum N application. The maximum net income decreased and occurred at a lower N level by an increase in N cost (arrow in Fig. 4).

For an optimum amount of water under water limiting conditions, the saved water with respect to W _m can be used to increase the A (more planting area). Therefore, net income gained from total A is increased. By solving Eqns (10a) and (10b) based on yield [Eqn (13)], crop price [Eqn (15)] and cost [Eqn (16)] functions, for R=207 m ³ /ha, Nr=255.15 kg/ha, P _c16 =290 Rls/kg, P _n =913 Rls/kg and P _w =120 Rls/m ³ , the calculated optimum water (W _w ) and N (N _w ) under water limiting conditions were 8410.6 m ³ /ha and

300.7 kg/ha, respectively.

For more assessments, the values of W _w and N _w were calculated at different rainfall amounts, water costs, N costs, base crop prices and soil residual N contents. The results are shown in Figs. 5 and 6. By an increase in rainfall (Fig. 5), residual N (Fig. 6), water cost (Fig. 5a) and base crop price (Fig. 5c), the value of applied water decreased. The value of W _w increased by an increase in the N cost (Fig. 5b). For N _w , with exception of base crop price (Fig. 5c), the value of N _w decreased by increase in rainfall (Fig. 5), soil residual N content (Fig. 6), water cost (Fig. 5a), and N cost (Fig. 5b). For a constant water cost and base crop price, by increasing N cost, the optimum N decreased, and optimum water increased (Fig. 5b). Net income decreased by an increase in N cost. Therefore, the maximum net income per unit area occurred at a higher water level in which yield was higher. For a given rainfall, the optimum level of applied water decreased by an increase in base crop price. In other words, maximum net income occurred at lower applied water by an increase in base crop price (Fig. 5c).

The net income under conditions of water limiting for seasonal rainfall of 20.7 mm and N+Nr=517 kg/ha at different water costs and base crop prices are shown in Fig. 7.

Optimum water, N, grain yield, net income and land increase for Nr=255 kg/ha, P _w =120 Rls/m ³ , P _n =

913 Rls/kg and P _c16 =290 Rls/kg at different rainfall depths were calculated and presented in Table 2. Rainfall occurrence resulted in increase in the amount of total net income due to a decrease in water cost. In Fig. 7, the maximum points occurred at irrigation depth that is equal to W _w . The maximum point was shifted to the left and optimum irrigation water decreased by an increase in the water cost and base crop price (arrows in Fig. 7). The net income under water limiting conditions at different N costs for P _c16 =290 Rls/kg and W+R=12381 m ³ /ha are shown in Fig. 8. Maximum net income in the total irrigated area (maximum point) occurred at optimum N application. The maximum net income decreased and occurred at a lower N level by increase in N cost due to increase in production costs (arrow in Fig. 8).

When the crop price is fix (it does not change with crop quality), its value will be determined based on supply and demand system or government pricing. For fixed crop price, term pc ( W + R , N + Nr ) in Eq. (2) changed to a constant value (P _c ). Anyway, the fixed crop price may be either higher than the variable crop price or lower than it. Therefore, the results of these two situations are similar to those obtained by changing the base crop price in Eqn. (15). For example, as presented in Table 2, optimum water, optimum N, grain yield, net income and land increase for

Nr=255 kg/ha, P _w =120 Rls/m and P _n =913 Rls/kg in case of the fixed crop price and equal to the base crop price (P _c = P _c16 =290 Rls/kg) at different rainfall depths were calculated. In other words, crop price did not change with crop quality and it was assumed that the value of SC in Eqn. (15) was 16 and

P _c = P _c16 =290 Rls/kg. In these conditions, crop price was higher than the varied crop price due to root sugar concentration lower than 16% (mean sugar content is 13% in Iran; Tavakoli & Fardad, 1999 ). As mentioned above, increasing of P _c16 , to higher crop price under constant crop price conditions resulted in higher W _l , W _w , N _w and total net income and lower N _l compared with the crop price under varied crop prices (Table 2).

For planning deficit irrigation, the amounts of optimum applied water at different N levels were obtained for 21 years of seasonal rainfall values in the study area. A frequency analysis was applied for the amounts of optimum applied water at different N levels obtained in different years. The occurrence probability for the values of optimum applied water was estimated by the Weibull equation [Eqn (12)]. The suitable distribution function was Gumbel (Fig. 9) based on the Kolmogorov-Smirnov index of goodness of fit. Results of distribution function showed that under water limiting conditions, the probability of occurrence for W _w as 50% and 80% that is corresponding to 3001.9 and 3400.3 m ³ /ha of water for N _w =0.0 kg/ha, 7135.6 and 7430.3 m ³ /ha of water for N _w =100.0 kg/ha and 7996.0 and 8278.9 m ³ /ha of water for N _w =200.0 kg/ha, respectively, could be used in irrigation water resources planning, which is in accordance with the probability of occurrence of 50% and 80% for seasonal rainfall.