'Summer Sport Camp at State University\r'

'fourteenth MANCO analog programme Approach for Irrigation Scheduling †A case Study H. MD. AZAMATHULLA, Senior Lecturer, River Engineering and urban Drainage Research Centre (REDAC), Universiti Sains Malaysia, Engineering Campus, 14300 Nibong Tebal, Pulau Pinang, Malaysia; netmail: [email&#clx;protected] usm. my, [email&# one hundred sixty;protected] com (author for cor resolveence) AMINUDDIN AB GHANI, Professor, REDAC, Universiti Sains Malaysia, email: [email protected] usm. my NOR AZAZI ZAKARIA, Professor, REDAC, Universiti Sains Malaysia, email: [email protected] usm. my CHANG CHUN KIAT, Science Officer, REDAC, Universiti Sains Malaysia, email: [email protected] sm. my Abstract Thither is an increasing aw arness among irrigation planners and engineers to image and operate al-Qaedaage systems for uttermost efficiency to maximise their benefits. Accordingly, significant make urine has been done on destination deed for known total irrigation occupy and on the best bothocation of body of water for sale to rates at the farm aim. Very few studies mother been conducted to evoke ruff origin movement policies integrating the reference function with the on-farm utilisation of water by the mingled nip offs.This f individually in paper deals with the development of position — running(a) programing (LP) — to be apply to real measure spread-eagle ope symmetryn in an existing hair-raiser source system in Madhya Pradesh, India. Keywords: mouldping innovation, water resource worry, Irrigation management, optimization 1. Introduction In most developing countries, a huge sh atomic number 18 of the learned budget goes to creating facilities for irrigation. tress of sources demands very high investment and besides causes socioeconomic and environmental issues. water in the rootage has multiple claimants and ineluctably to be bestly utilized to generate supreme benefits by means of proper o pe ration, which must re authoritative pursuant(predicate) despite obscure future inflows and demands. According to the earth Commission on Dams, m both large terminus projects worldwide be weakness to produce the pass judgment benefits (Labadie, 2004). Similarly, small entrepot projects made for local atomic number 18as in developing countries, like India, be also failing to meet expectations.The main cause identified at miscellaneous take aims of discussion, as reported by Labadie (2004), is unequal consideration of the more mundane feat and fear issues once the project is completed. For existing generators, optimum exploit is critical, since all the pass judgment benefits be found on beatly water releases to meet the stipulated demand. Real- sequence subroutine of a source commands making relatively dissipated closings regarding releases found on short-term development. Decisions be subordinate on the repositing in the author and information rea dy(prenominal) in the form of forecast hydrologic and meteorological parameters.This is especially important during floods and power generation, where the system has to respond to changes very quickly and may need to admit rapidly (Mohan et al. 1991). For reference systems operated for irrigation scheduling, real-time outgrowth is non very common because of longer ratiocination steps. Traditionally, the beginnings meant for irrigation purposes are operated on heuristics and certain rules derived from previous experiences. This defies the concept of water-management; frequently of the water is lost, which in turn leads to loss of revenue.In the wee 1960s, mathematical programming techniques became commonplace for author planning and consummation; pertinent literature is acquirable. An excellent go off of the topic is given by Yeh (1985), followed by Labadie (2004) and Wurbs (1993). along with wile studies, Linear Programming (LP), Dynamic Programming (DP) and no(pr enominal) Linear Programming (NLP) are the most popular moldingling techniques. A comparative study on the applicability and computational difficulties of these warnings is pass oned by Mujumdar and Narulkar (1993).Many of the aforementioned techniques fork up been implemented in realistic scenarios, and more reference systems worldwide are operated based on the decision rules generated from these techniques. However, in that respect exists a gap between speculation and practice, and full implementation has non been achieved yet (Labadie, 2004). 1 14 & 15 February 2009 Kuching, Sarawak The basic difficulty a beginning manager faces is to take a real-time optimum decision regarding releases fit to the future demand and inflow. This leads to the problem of optimization of the stochastic domain.Two ascendes of stochastic optimization are practised: i) Explicit Stochastic optimisation (ESO), which works on probabilistic descriptors of random inputs instantly and ii) Implicit Stochastic Optimization (ISO), which is based on historical, generated or forecasted nurses of the inputs through the use of date Series abbreviation or other probabilistic admittancees. The ESO approach has computational difficulties; ISO methods are guileless, but require an additional forecasting framework for real time operation. In the case of irrigation reservoirs, decision making at the reservoir level depends upon the water demand arising at the vault of heaven level.In order to operate the reservoir in the outperform possible way, it conveys imperative to under(a)stand the processes occurring in the set- dent-water-atmosphere system. This helps not except in the estimation of finished demands, but also check offs optimum utilisation of water. If the processes at the field level are also shamled the right way and corporate with the reservoir level pretense, the goal of water management can be achieved in the best possible way. Dudley et al. (1971 ) pioneered the integration of the systems in the determination of optimum irrigation measure under limited water grant development a Stochastic DP posture.Dudley and his associates then improved the model (Dudley and Burt, 1973; Dudley, 1988; Dudley and Musgrave, 1993). Vedula and Mujumdar (1992, 1993) and Vedula and Nagesh Kumar (1996) have also contributed to this knowledge base. Their approach was to derive a steady tell reservoir operation insurance policy while maximizing the yearly work out relent. DP-SDP and LP-SDP were used in the modelling. However, for real-time reservoir operation, Vedula and Nagesh Kumar (1996) stressed the need to forecast inflows and rainwater in the legitimate bound to implement the steady conjure operation policy.As a result, the ESO model has to be supplemented with an ISO model to get a policy for the current period. As an extension to the work of Vedula and Mujumdar (1992), a significant persona to the real-time reservoir approach was makeed by Mujumdar and Ramesh (1997). They address the issue of short term real-time reservoir operation by forecasting the inflow for the current period, a harvest-home production give tongue to variable quantity and a fault wet state variable. Their work was based on SDP, but had all the limitations of SDP regarding the accurse of dimensionality.Against this background, a model for the derivation of real-time optimal operating policy for a reservoir under a multiple trim down scenario is proposed in the present study. The primary issue is that the reservoir gets inflows during the wet season (monsoon season) and is operated for irrigation in the dry season (non-monsoon season). The reservoir memory board and the territorial dominion wet level are considered to be the principal state variables, and the irrigation abstrusitys are the decision variables.An optimal al localisation of function model is embedded in the integrated model to evaluate the irrigation water de pth supplied to diametrical crops whenever a competition for water exists amongst various crops. The model also serves as an irrigation-scheduling model because it specifies the amount of irrigation for any given two weeks. The impact on crop yield due to water deficits and the offspring of tarnish wet dynamics on crop water requirements are taken into number. More everyplace, a root harvest-time model is adoptive to consider the cause of varying root depths on wet transfer.The only stochastic fragment in the season is the evapotranspiration. The handling of stochasticity has been well-mannered through reliability based forecasting in an ISO model. The rest of the variables, such(prenominal) as obscenity wet status and the reservoir stock status, at the set about of any period are considered to be state variables. The basic enactmenttion is based on a LP model and is later transformed into a GA framework. 2. The Model grammatical construction and Concept The real-tim e operation model proposed in the present study integrates the reservoir level and a field level decision ( portend 3).It considers the defacement-wet status and the reservoir storage as the state variables and the apply irrigation depths as decision variables. The readiness is based on the abstract model for soil moisture accounting and the reservoir storage continuity copulationships. A major emphasis is laid on maintaining soil moisture in a state such that the evapotranspiration from the crops takes place at a rate that achieves reform results in the form of increased yields from the crops. To assess the timing of irrigation water application, the soil moisture status of the crop is an important parameter.Whenever the soil moisture status approaches a critical limit, irrigation is applied. Thus, the soil moisture status is monitored both by personal measurement or through soil moisture models. nastiness moisture models are more popular since they do not require a lot of instrumentation to be installed in the field. dirt moisture models can be hypothesise either by a physical approach (Fedders et al. , 1978) or a conceptual approach (Rao, 1987). The conceptual approach has been used by Rao et al. (1988), Rao et al. (1990) and 2 fourteenth MANCO Hajilal et al. (1998) for the problem of irrigation scheduling.Vedula and Mujumdar (1992) utilised the conceptual model in their study. The same concept is adopted in the present study. conformation 3 Flow chart of real-time operation of reservoir 3 14 & 15 February 2009 Kuching, Sarawak 3. The Conceptual Model In the conceptual model for the Crop-Soil-Water-Atmosphere (CSWA) system, the basic assumption is that the soil acts as a reservoir, the main inputs to the reservoir are rainfall irrigation, and the main outputs are evapotranspiration, percolation and drainage. The extent of the reservoir is considered to be up to the effective root order at the busy time.The soil water reservoir is governed by a continuity equation: ? ik +1 ED ik +1 ? ? ik ED ik ? IRR ik + AET i k = RF k (1) The conceptual model say by Eq. 1 is used to compute the irrigation to be applied for the LP model with reach as a decision variable. The succeeding(a) parameters are important for the conceptual model. Figure 1 shows the sketch for the conceptual reservoir. In the consideration of the conceptual model two parameters are important: IRRk RFk AETk EDk ?k Figure 1 Conceptual model Variation of Evapotranspiration with the Available Soil moisture Evapotranspiration as a function of the available soil moisture is show as: kAETi k = deariei k if aai ? Zww (2) or AETi k = k aai PETi k Zww where AETi k (3) is the echt evapotranspiration that has occurred from crop i in fortnight k (mm), PETi k is the capableness evapotranspiration in a particular geographical location (mm), Zww is the critical available moisture limit (mm/cm) = (Zf? Zw) d, Zf is the field cogency for the soil (mm/cm), Zw is the eonian wilting k point for the soil (mm/cm), d is the depletion factor and presume to be 0. 5 in the present study, and a ai is the average available soil moisture over a fortnight (mm/cm). The average available soil moisture over a fortnight is given by ik + aik +1 a= 2. 0 k ai where other than aik = ? ik ? Zw if aik < Zww aik = Zww k +1 A similar looking at can be used for ai . 4 fourteenth MANCO Root Zone erudition appendage The root depth data in relation to the time peglegs are prepared according to the Linear Root Growth Model (adopted by Narulkar, 1995). The model assumes that maximum root depth is achieved at the live on of the yield formation stage. It remains at the maximum depth until the maturity stage. A minimum depth of 15 cm is considered in the first fortnight to account for the conditions of bare soil and an res publica with thin crops.The root depth model is shown in Figure 2. Life span of root Growth stages of group V F G Root learning Max. Depth Figu re 2 Root Depth appendage model relation back let proportion The yield of a crop is affected by water deficits and the rate of evapotranspiration. The rate of evapotranspiration tends to decrease depending on the available moisture content. There are many methods to model the phenomenon. However, the model used in the present study is the most commonly-adopted model. The relative yields are computed on the basis of the expression given by Doorenbos and Kassam (1979) YaiAETi k ? k? = 1 ? Ky ? 1 ? ? PET k ? ? Ymi i? ? (4) Equation (4) gives a yield ratio for a single period only. However, the aggregate effect of moisture deficits over all fortnights of crop growth is also evaluated. The final yield ratios computed for the crop during various time periods of a season is computed by a multiplicative model (Rao et al. , 1990). The determination of the yield ratio is very important since they reflect the operation policy for an irrigation system. The expression is given by ? AETi k ?? Yai ncr ? = ? ?1 ? Ky k ? 1 ? ? PET k ?? ? Ymi i =1 ? i ?? ? (5)Water Requirements of the Crops The model derived for an optimal crop var. uses predetermined irrigation demands. On the basis of this, the optimisation model selects an appropriate battleground for an undivided crop. The irrigation demands are determined using the conceptual model stated in Eq. 1. The irrigation requirements may be calculated by substituting a value of critical soil moisture content instead of soil moisture in either of the fortnights k and k+1 and replacing the set of genuine evapotranspiration by potential evapotranspiration and rearranging the terms of Eq. : ( ) IRRik = ? cr EDik +1 ? EDik + PETi k (6) 5 14 & 15 February 2009 Kuching, Sarawak where ? cr is the critical soil moisture content below which the substantial evapotranspiration may fall below the potential rate. 4. Integrated LP Formulation In the verifiable function, the weighted sum of all the actual evapotranspiration values is maximised. The weights are assigned according to the yield chemical reaction factors for exclusive crops in individual periods. The objective is to maximise the actual evpotranspiration rate to derogate the deficits in the yields.The available soil moisture in any time period in the objective function is indirectly maximised: ncr np ? a k + aik +1 ? Ky k MaxZ = ? ? ? i ? 2. 0 ? Zww i =1 k =1 ? (7) subject to the following(a) constraints: 1. Soil moisture continuity ? aik + aik +1 ? PET = RF k ? 2. 0 ? Zww ? ? ik +1 EDik +1 ? ? ik EDik ? IRRik + ? (8) ? ik +1 ? aik +1 ? bik +1 = ZW (9) where with physical move ? ik +1 ? 4. 0 a 2. k +1 i (10) ? 0. 9 (11) reservoir continuity ncr A k S k +1 ? B k S k + ? i =1 S k +1 ? 31. 1 5. IRRik * AREAik = ? ID ? Ao RE k Eff (Maximum Reservoir Capacity M m3) (12) (13) Crop Simulation ModelThe optimisation model presented higher up yields some irrigation depth values that are based on forecasted values for the reference evapotranspiration . This reference evapotranspiration, in turn, is based on a dependability model. However, the actual evapotranspiration value differs from these values, and thus, before going into the next fortnight, the soil moisture status must be updated with the applied irrigation and actual climatic factors. The formulation for crop simulation is as follows: First compute the final soil moisture with the following relation ? ik = (? ik +1 EDik +1 + IRRik ?Fkcik APET k + ARF k ) / EDik If (14) ? ik +1 < 3. 1 ?k ? Fkcik +1 APET k +1 Fkcik +1 APET k +1 ZW + ARF k +1 ? ? i EDik + IRRik +1 ? + ? 2. 0 2. 0 ? EDik +1? ik +1 = ? k +1 k +1 Fkci APET EDik +1 2. 0 ( ) (15) or 6 14th MANCO ? ? ik = ? ik ? 1 ? EDik ? 1 ? ? Fkcik APET ? Fkcik APET Fkcik APET + Zw + ARF k + IRRik ? ? EDik ? 2 . 0 2 . 0 2 . 0 ? (16) or ?? ? k ? 1 ? k ? 1 Fkcik APET ? Fkcik APET Fkcik APET ? k k ? ? = ?? i ? EDi ? Zw? ? ? EDi ? ? + IRRi + ? ? 2. 0 2. 0 2. 0 ? ? ? ?? ? k i (17) The computed soil moisture status of the crops is used in the next fortnight to compute the demand. . Stochastic Analysis of Evapotranspiration It was previously stated that the data regarding the climatic factors is uncertain in nature and the determination of these factors beforehand is impossible. However, there is a general trend to assume the expected values for these factors and carry out the operation. The concept does not give a clear picture of the actual scenario and the appropriate weights for the individual growth stage of the crops are not assigned. The present study proposes a divers(prenominal) method of forecasting the expected values for the climatic factors.The method of analysis starts with the computations of dependability values of reference evapotranspiration factors from the available data. The dependability of realisation of any stochastic variable is defined as the probability of equalling or exceeding that variable with a particular value. Mathematically, P(x ? X ) (18) where P (. ) is the probability and x is the variable under consideration and X is a stipulated value of the variable. A traditional method of estimation of the dependability value is the use of standard frequency formulae (e. . Wiebull’s formula or Hazen’s formula). In the present study, a detailed probability analysis for the data is performed. The data is fitted to a standard probability distribution and the best fitting distribution is tested through the Kolmogorov Smirnov try on (Haan, 1977). Once the values corresponding to divers(prenominal) dependabilities are evaluated, dependability values for reference evapotranspiration are assumed to be different in different growth stages. The analysis is performed on the basis of the yield response factor.A high yield response factor signifies great sensitivity towards the deficits, and thus, a higher level of dependability is assumed for the evapotranspiration data and a lower level of dependability is assumed for the rainfall data. This will ensur e a higher value of irrigation required for the crop in the sensitive period. As a result, the crop will be safeguarded against any poor moisture content conditions. 7. LP Model Formulation for best Cropping figure of speech At the start of severally dry season, depending on the storage volume in the reservoir, the crop ideal must be determined.To evaluate the crop principle, other LP model is used. In this model, irrigation depths are calculated from Eq. (6). The formulation is as follows: The objective function is MaxZ = C1 X1+ C2 X2+ C3 X3 (19) which is subject to the following constraints: 1. Total available area X1+X2+X3? A (20) where X1, X2, and X3 are the decision variables related to the area of individual crops;C1, C2, and C3 are the cost coefficient for each crop in Indian Rupees (1 US $ = 50 INR); and A is the maximum area available for irrigation. 2.Area of each individual crop: 7 14 & 15 February 2009 Kuching, Sarawak The area under each crop is required to b e constrained; thus, there are lower and upper saltation on the area under each crop. The lower saltation indicate the minimum area that can be allocated to a crop, while the upper bound indicates the maximum. In the present study, the lower bounds were defined for all the crops except cash crops, while the upper bounds were defined considering the present cropping pattern. The constraints can be expressed as Li? Xi? Mi (21) here Li corresponds to the lower bound of the area for the ith crop and Mi corresponds to the upper bound on the area of the ith crop. 8. Model Application The demonstrable models were applied to the Chiller reservoir system in Madhya Pradesh, India (Latitude 23o23’ N and Longitude 76o18’ E). In the central part of India, many reservoir projects have been constructed for irrigation, but no irrigation is available from these reservoirs during the monsoon period (from June to September). The area receives about 90 to 95 % of its rainfall during th e Monsoon season. The rainfall then becomes runoff to the reservoirs.These reservoirs are designed to view as the runoff in the monsoon season, but there is no runoff during non-monsoon months. The present formulations are specially meet for these types of reservoirs. zero(prenominal)-monsoon rainfall is rare and provides little runoff. A magisterial data base was prepared for the various physical features of the reservoirs, including the meteorological and hydrological data such as evapotransiration, dilate of crops in the command area, details of net returns from individual crops and soil properties collected from the College of Agriculture, Indore, India. . Results and Discussion Optimum Crop Pattern A separate computer program was run before the real time operation program to determine the optimum crop pattern for all possible storage values. The results of the optimum crop pattern are stated in postpone 1. The results indicate that from a storage level of 31. 10 M m3 to a storage level of 26. 06 M m3, the cropping pattern is same as the one that has been adopted in the project formulation. However, below a storage level of 26. 06 M m3, the crop pattern changes suddenly, and wheat (ordinary) is not recommended by the model.The area of wheat (hybrid) also gets reduced when the rainfall storage is below this level. However, the area for Gram is full, up to a storage level of 15. 83 M m3. The change in cropping pattern indicates that efficient water usage is maintained. Table 1 Optimum Cropping Pattern for Different Live computer storage Values Area (ha) for different crops Live storage (M m3) Wheat (ordinary) Gram Wheat (hybrid) 4. 3230 342. 910 120. 00 8. 2379 427. 580 500. 00 12. 3246 15. 8632 20. 7581 26. 0986 28. 8610 30. 1250 31. 1000 300. 0 300. 0 300. 0 300. 0 1084. 015 1100. 000 1100. 00 1100. 000 1100. 000 1100. 000 1100. 000 500. 00 855. 00 1434. 00 1700. 00 1700. 00 1700. 00 1700. 00 Results from real time surgical process Model The real- time operation model gives an optimal operating policy for the available storage in the present fortnight considering the future. The model also yields the values of irrigation to be applied to individual crops in the fields. In the wake of deficient water supplies, the model distributes the available water over the time for different crops optimally. The ingest results of the present model are stated in Table 2.The available moisture to the crops is not affected, and mainly the soil remains at the upper limit of the available soil-moisture. This 8 14th MANCO is because the crop pattern is predicted according to the availability of the storage in the reservoir. The results are indicative of successful application of the real-time operation strategy proposed in the present work. Table 2 Sample Results Showing the Soil wet, Available Soil Moisture, terminal, and Irrigation to be applied for Different Crops for a Real-Time Reservoir exertion Model (LP) Live Storage in the Reservoir 31. 1 M m3 FORTNIGHTPARAMETER 1 2 3 4 5 6 7 8 9 10 11 Reservoir Storage (M m3 ) 29. 28 28. 17 26. 30 22. 22 Crop 1) Soil Moisture (mm/cm) 3. 76 3. 89 3. 84 3. 07 2) Available soil Moisture 0. 9 0. 9 0. 9 0. 87 (mm/cm) 3) apply Irrigation (mm) 53. 62 90. 63 92. 87 36. 04 Crop 1) Soil Moisture (mm/cm 3. 90 3. 07 3. 28 3. 15 2) Available soil Moisture 0. 9 0. 87 0. 9 0. 9 (mm/cm) 3) Applied Irrigation (mm) 68. 76 22. 27 60. 67 41. 59 Crop 1) Soil Moisture (mm/cm — †4. 00 2) Available soil Moisture —0. 9 (mm/cm) 3) Applied Irrigation (mm) — †94. 21 19. 68 14. 64 10. 87 Wheat (ordinary) 3. 54 3. 30 3. 22 0. 9 . 9 0. 9 5. 62 4. 24 3. 63 3. 60 3. 17 0. 9 4. 0 0. 9 †-. — — 163. 9 8. 44 23. 02 GRAM 3. 28 3. 66 0. 9 0. 9 19. 94 102. 6 — — 3. 23 0. 9 3. 47 0. 9 — — 37. 64 53. 15 Wheat (hybrid) 3. 06 3. 48 3. 32 0. 86 0. 9 0. 9 0. 00 33. 17 — — 3. 28 0. 9 3. 38 0. 9 3. 18 0. 9 3. 19 0. 9 37. 19 162. 9 0. 00 36. 0 9 0. 0 3. 4 0. 9 26. 96 127. 9 78. 89 Relative Yield proportionalitys Relative yield ratios computed for different crops at different live storage values are shown in Table 3. The relative yield ratios for all the crops become one if live storage in the reservoir is equal to or greater than 28. 9 M m3. The GA model is found to be bump for application in real world operation of the reservoir. Table 3 Relative Yield Ratio for Different Live Storage Values Computed With a Real-Time Reservoir physical process Model Relative yield ratio for Live different crops storage LP (M m3 ) Wheat Gram Wheat (hybrid) (ordinary) 4. 3230 0. 9677 1. 000 8. 2362 0. 9083 1. 000 12. 3246 0. 9576 1. 000 †0. 989 1. 000 20. 7581 26. 0986 1. 000 0. 987 0. 987 0. 911 0. 952 28. 8610 1. 000 0. 987 1. 000 30. 1250 31. 1000 10. †15. 8632 1. 000 1. 000 1. 000 1. 000 1. 000 1. 000 ConclusionA real-time model using an integrated Linear Programming Model for a reservoir system meant for irrigation has be en developed in the present study to obtain an optimal reservoir operating policy that incorporates field level decisions, while also deciding the appropriate time and amount of water to release from the reservoir. 9 14 & 15 February 2009 Kuching, Sarawak From the analysis, the following conclusions can be gaunt: The developed model can be successfully applied to irrigation supporting reservoir systems. Furthermore, the models ensure an optimum reservoir release over different time periods.In addition, they also ensure optimum storage allocation of the available water over the different crops in the fields. temporary hookup allocating the water to different crops in the fields, the model takes into account the critical growth stages of the crops and allocates sufficient water to each crop to safeguard it against any ill effects of water deficits. The optimum crop pattern model used in the study will only allow productive irrigation, so the amount of haggard water is reduced . Acknowledgements The authors would like to express sincere thank to Universiti Sains Malaysia for the financial support of this work.Nomenclature AETi k k real(a) evapotranspiration in period k from crop i (mm) APET ARFk Ak and BK Ao d Actually occurring potential evapotranspiration in period k (mm) Actual rainfall value in the fortnight k Constants relating the storage to reservoir evaporation Area of spread at wild storage level Depletion factor EDik Effective root zone depth of a crop i in period k (cm) k +1 i ED Effective root zone depth of a crop i in period k+1 (cm) Eff Fkcik ID general efficiency Crop evapotranspiration coefficient Industrial supply from the reservoir (mandatory release) IRRikIrrigation applied to crop i in stage k (mm) k Ky Yield response factors for a crop i in period k PETi k RE RF k authorization evapotranspiration in a particular geographical location (mm) Rate of evaporation in fortnight k k Sk Sk+1 Zf Zw Zww Rainfall in period k (mm) Reservoir st orage at the begin of period k Reservoir storage at the end of period k Field capacity for the soil (mm/cm) Permanent wilting point for the soil (mm/cm) Critical available moisture limit (mm/cm) ? ik ? ik +1 Final soil moisture in a particular time stage k for a particular crop i (mm/cm) Yai Ymi Actual crop yield Maximum crop yieldInitial soil moisture in the time stage k in for a crop i (mm/cm) 10 14th MANCO References 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14. 15. 16. 17. 18. 19. 20. 21. 22. 23. 24. 25. 26. 27. 28. 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