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Downscaling Crop Water Sensitivity Index Using Monotone Piecewise Cubic Interpolation
Pedosphere 23(5): 662–667, 2013 ISSN 1002-0160/CN 32-1315/P c 2013 Soil Science Society of China Published by Elsevier B.V. and Science Press
Downscaling Crop Water Sensitivity Index Using Monotone Piecewise Cubic Interpolation∗1 SHANG Song-Hao∗2 State Key Laboratory of Hydroscience and Engineering, Tsinghua University, Beijing 100084 (China) (Received April 18, 2013; revised July 18, 2013)
ABSTRACT Crop-water production functions quantitatively describe the relationship between crop yield and field evapotranspiration. The crop water sensitivity indexes of crop-water production functions, a key factor for optimizing irrigation scheduling in case of water scarcity, are usually obtained from field experiments or other sources for crop growth stages, while their values in shorter intervals are preferred for practical irrigation scheduling. We proposed a method to downscale the sensitivity index from growth stages to shorter intervals by monotone piecewise cubic interpolation of the cumulative sensitivity index curve. This method was used to estimate sensitivity indexes in irrigation intervals of about 10 d for corn and wheat in central Shanxi Province of China. Results showed that the downscaled sensitivity index could reflect the impact of water stress on crop growth both at different growth stages and within each stage. Scenario analysis of water stress at a single growth stage of wheat showed the rationality of downscaling water sensitivity index from growth stages to shorter intervals through interpolation of cumulative sensitivity index, and this proposed downscaling method was superior to the traditional linear downscaling method. Key Words:
corn, crop-water production function, growth stage, water stress, wheat
Citation: Shang, S. H. 2013. Downscaling crop water sensitivity index using monotone piecewise cubic interpolation. Pedosphere. 23(5): 662–667.
INTRODUCTION Crop-water production functions quantitatively describe the relationship between crop yield and field evapotranspiration, which is useful for calculating crop yield under water stress conditions (Jensen, 1968). Meanwhile, crop yield or economic profit dependent on the yield is usually the objective function for the optimization model of deficit irrigation that determines the optimal irrigation date with limited water available for irrigation (Georgiou and Papamichail, 2008; Ganji and Shekarriz fard, 2010). Therefore, sensitivity index for the crop-water production functions is important for the optimization of irrigation scheduling in areas suffering from water scarcity. Besides, crop-water production functions can also be used in assessing the impact of climate change on crop yield (Lhomme et al., 2009; Ferreira and Rao, 2011). Crop-water production functions are usually expressed as empirical formulae relating crop yield to evapotranspiration in the whole growth season (Stewart and Hagan, 1973), at a specified growth stage ∗1 Supported
(Doorenbos and Kassam, 1979), or at different growth stages (Jensen, 1968; Howell and Hiler, 1975). The last type usually uses multiplicative (Jensen, 1968; Rao et al., 1988) or additive (Howell and Hiler, 1975) forms to express the combined effect of water stress at several growth stages. Sensitivity analysis of several cropwater production functions indicated that the multiplicative form is preferable to the additive form (Kaboosi and Kaveh, 2012). Among these crop-water production functions, the Jensen (1968) model of multiplicative form is often used in the optimization of irrigation scheduling with limited water supply (e.g., Shang and Mao, 2006; Wang et al., 2007; Georgiou and Papamichail, 2008; Ganji and Shekarriz fard, 2010). Sensitivity indexes of the Jensen (1968) model or other parameters of crop-water production functions can be determined from irrigation experiments in local environment (e.g., Wang and Sun, 2003; Igbadun et al., 2007), which is usually time-consuming and costly. Doorenbos and Kassam (1979) analyzed available field experimental data and proposed yield response factors for water stress at a particular growth stage for main
by the National Natural Science Foundation of China (No. 51279077) and the National Key Technology R&D Program of China (No. 2013BAB05B03). ∗2 Corresponding author. E-mail: [email protected].
DOWNSCALING CROP WATER SENSITIVITY INDEX
crops. Kipkorir and Raes (2002) proposed a function to transform the readily available yield response factors of Doorenbos and Kassam (1979) to the sensitivity indexes of the Jensen (1968) model. However, both yield response factors and sensitivity indexes are usually available in the time scale of growth stages. In irrigation scheduling, their values in shorter irrigation intervals are preferred rather than their values during a particular growth stage (Kipkorir and Raes, 2002; Georgiou and Papamichail, 2008). Therefore, there is a need to downscale the sensitivity index from growth stages to shorter irrigation intervals. To solve this problem, Tsakiris (1982) proposed a procedure for estimating sensitivity indexes in given time intervals using the cumulative values of the sensitivity index (cumulative sensitivity index) of the Jensen (1968) model. In this procedure, the cumulative sensitivity index is plotted against growth time with piecewise straight lines, and then sensitivity index in a given time interval can be estimated from the difference between the cumulative sensitivity index at the end and at the beginning of the considered time interval. This procedure has been used in several irrigation optimization models (e.g., Georgiou and Papamichail, 2008; Ganji and Shekarriz fard, 2010). The procedure is simple, but it does not consider the variation of sensitivity index within a growth stage. As a result, sensitivity index is evened in each stage, and irrigation intervals of the same duration within a growth stage have the same sensitivity index (Tsakiris, 1982), which is not sound for describing crop sensitivity to soil water stress. Besides, the rationality of downscaling sensitivity index from cumulative sensitivity index has not been fully investigated. To reflect the variation of the sensitivity index both at different stages and within each stage, some empirical formulae have been used to depict the cumulative sensitivity index curves, such as the logistic curve (Wang and Sun, 2003; Shang and Mao, 2006) and the power function-based sigmoid curve (Han et al., 2010). The sensitivity index estimated from the sigmoid curves is smaller in the initial and final growth stages and larger in the middle growth stages, which accords with the general variation pattern of crop sensitivity in response to water stress. However, the sigmoid curves are not flexible enough to depict the complex variation of cumulative sensitivity index in the whole growth season, especially for winter wheat with a long frost period and very low water sensitivity index in this period. The main objective of this study was to propose a sound method to downscale sensitivity index from
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growth stages to shorter irrigation intervals, based on monotone piecewise cubic interpolation of cumulative sensitivity index. The proposed method was used to estimate sensitivity indexes in time intervals of about 10 d for corn and wheat in central Shanxi Province of China, and scenarios of water stress at a single growth stage of wheat were analyzed to demonstrate the rationality of the proposed downscaling method. DOWNSCALING METHOD FOR CROP SENSITIVITY INDEX The Jensen (1968) model for crop-water production function is as follows: ETi λi Y = Ym ETm,i i=1 n
y=
(1)
where y, Y , and Ym are the relative, actual, and maximum crop yields, respectively; ETi and ETm,i are the actual and maximum evapotranspiration during the growth stage i, respectively; n is the number of crop growth stages; and λi is the crop water sensitivity index at the stage i. Crop water sensitivity indexes of the above model are usually available for crop growth stages, while the cumulative sensitivity index curve (Tsakiris, 1982) provides the possibility to estimate the sensitivity index in an arbitrary time interval from available sensitivity indexes for growth stages. Since sensitivity indexes are greater than 0, the cumulative sensitivity index is a monotonic increasing function of growth time. If this function is known, the sensitivity index in an arbitrary time interval can be calculated from the difference between the final and initial values of the cumulative sensitivity index in the time interval. Moreover, the first derivative of the cumulative sensitivity index is the value of sensitivity index per unit time. Therefore, the main task to estimate sensitivity index in preferred irrigation intervals is to find an appropriate monotonic function to depict the cumulative sensitivity index curve from limited points in the curve. For a crop under consideration, the whole growth season is divided into n stages, and the duration and the sensitivity index of the ith stage are Δti and λi (i = 1, 2, · · · , n), respectively. The cumulative sensitivity index at time ti (i = 0, 1, · · · , n), Zi , can be calculated from: Z0 = 0, Zi =
i j=1
and
λj = Zi−1 + λi ,
i = 1, 2, · · · , n (2)
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t0 = 0,
S. H. SHANG
ti =
i
Δtj = ti−1 + Δti ,
j=1
i = 1, 2, · · · , n
(3)
Now, we have n + 1 points, (ti , Zi ), i = 0, 1, · · · , n, in the cumulative sensitivity index curve. Values of the cumulative sensitivity index at any other time may be estimated with appropriate interpolation methods. The procedure proposed by Tsakiris (1982) approximated the cumulative sensitivity index with the following piecewise linear interpolation function: Zl (t) = Zi
t − ti+1 t − ti + Zi+1 , ti − ti+1 ti+1 − ti
ti ≤ t ≤ ti+1 ,
i = 0, 1, · · · , n − 1
(4)
where Zl (t) is the piecewise linear interpolation function. The piecewise linear interpolation is simple and can preserve the monotonicity of the cumulative sensitivity index curve, but the first derivative of this interpolation function is piecewise constant. Consequently, the sensitivity index is evened over each growth stage, and the variation of crop sensitivity in response to water stress within the stage is not considered. To overcome the evenness of the sensitivity index in each growth stage by piecewise linear interpolation, higher order polynomial or piecewise polynomial interpolation is an alternative. However, commonly used higher order interpolation can not preserve the monotonicity of the cumulative sensitivity index curve. Here we used a monotone piecewise cubic interpolation (Fritsch and Carlson, 1980), which is characterized by passing through all n + 1 points, preserving the monotonicity of the cumulative sensitivity index curve, and having a continuous first derivative. This shapepreserving interpolation method has been used in several fields of soil sciences, such as the generation of soil particle size distribution curve (Shang, 2013) and estimation of soil hydraulic properties (Iden and Durner, 2007; Peters and Durner, 2008). The monotone piecewise cubic interpolation function for each time interval t (ti ≤ t ≤ ti+1 , i = 0, 1, · · · , n − 1), Zci (t), is Zci (t) = Zi+1 di+1
3hs2 − 2s3 h3 − 3hs2 + 2s3 + Z + i h3 h3 s2 (s − h) s(s − h)2 + di 2 h h2
(5)
where h = Δti ; s = t − ti ; and di and di+1 denote the slopes of the interpolant at knots ti and ti+1 , respectively. The slope at a knot can be estimated from the lengths and the first divided differences of two adjacent
intervals (Fritsch and Carlson, 1980; Moler, 2004). Piecewise linear interpolation and monotone piecewise cubic interpolation were accomplished using the interp1 function of Matlab (Gilat and Subramaniam, 2011): Zd = interp1(t, Z, td , ‘method’)
(6)
where Zd is the interpolated cumulative sensitivity index corresponding to td ; td is an array representing a desired division of the crop growth season; t is the (n + 1) – dimensional array of time calculated from Eq. 3; Z is the cumulative sensitivity index calculated from Eq. 2; and ‘method’ refers to the interpolation method. If ‘method’ is specified to ‘linear’ or ‘cubic’, it means that the piecewise linear method or the monotone piecewise cubic method was used in interpolation. After interpolation, the sensitivity index in a specified time interval [u, v] in the growth season can be calculated from: λ[u,v] = Zd (v) − Zd (u)
(7)
where λ[u,v] is the sensitivity index in the time interval [u, v]; and Zd (v) and Zd (u) are the interpolated cumulative sensitivity indexes at time v and u, respectively. The monotonicity of the interpolated curve can guarantee positive values of sensitivity index at arbitrary time intervals. Since the interpolation curve of cumulative sensitivity index passes through all original data, the observed and interpolated sensitivity indexes in each growth stage are exactly the same. CASE STUDIES The proposed method was used to estimate sensitivity indexes in irrigation intervals for main food crops in central Shanxi Province of China. In this region, corn and wheat are two main food crops, and water sensitivity indexes are available for growth stages (Table I) from irrigation experiments (Wang and Sun, 2003). Corn in the study region usually grows from early May to mid-September lasting for about 140 d, and the growth season can be divided into 4 stages with the duration from 20 to 51 d (Table I). Corn growth is most sensitive to water stress in the shooting stage. Wheat in the study region usually grows from late September to late June lasting for about 280 d. The growth season can be divided into 6 stages with the duration from 19 to 110 d (Table I). The main characteristic of wheat growth is that there exists a long frost period of about 110 d with very low sensitivity to soil water stress. Wheat growth is more sensitive to water stress in the shooting, greening, and heading stages.
DOWNSCALING CROP WATER SENSITIVITY INDEX
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TABLE I Sensitivity indexes at different growth stages for corn and wheat in central Shanxi Province of China a) Growth stage
Seedling Frost Greening Shooting Heading Grain-filling Total a) Cited
Corn
Wheat
Starting date
Duration
Sensitivity indexa)
Starting date
Duration
Sensitivity indexa)
May 1
d 51
0.0903
20 26 43 140
0.4873 0.2802 0.1548 1.0126
Sep. 22 Nov. 20 Mar. 10 Apr. 20 May 13 Jun. 1
d 59 110 41 23 19 28 280
0.0469 0.0154 0.2800 0.3332 0.2470 0.1067 1.0292
Jun. 21 Jul. 11 Aug. 6
from Wang and Sun (2003).
Considering the practical decision making for irrigation scheduling, an irrigation interval of a week or about 10 d is usually preferred, which is much shorter than the duration of a growth stage. Therefore, the sensitivity index should be downscaled from the growth stage to irrigation intervals of about 10 d for irrigation practice. Thus, a whole month in the growth season can be divided into three irrigation intervals, with the first and second intervals of 10 d and the third 8 to 11 d depending on total days in the month. The duration of the first and last irrigation intervals of the growth season may differ from 10 d depending on the sowing and harvesting dates. Downscaled sensitivity indexes of corn and wheat in central Shanxi Province of China From sensitivity indexes of different growth stages in Table I, the cumulative sensitivity index was calculated and interpolated with piecewise linear and monotone piecewise cubic methods (Fig. 1) and was then used to estimate sensitivity indexes in irrigation intervals of about 10 d (Fig. 2). Although these two interpolated curves seemed similar, the estimated values of the sensitivity indexes in shorter intervals were apparently different, especially when the difference of sensitivity indexes at two adjacent growth stages was great. The interpolated cumulative sensitivity index curves using the piecewise linear method are piecewise straight lines, which are continuous but not smooth at the boundary points of the growth stages (Fig. 1). As a result, the sensitivity index of a growth stage is evened over the stage through piecewise linear interpolation (Fig. 2), and the variation of crop sensitivity to water stress within a stage is not considered. On the contrary, the interpolated cumulative sensitivity index curves using the monotone piecewise cubic method are smooth in the whole growth season (Fig. 1). For corn, the downscaled sensitivity index (Fig. 2) in-
Fig. 1 Experiment and interpolated cumulative sensitivity index curves for corn (a) and wheat (b) in central Shanxi Province of China.
creased slowly in the seedling stage, then increased rapidly in the early shooting stage and reached the peak value in the late shooting stage, and finally decreased until crop maturation. For wheat, the downscaled sensitivity index was lower in the seedling stage and frost period. The lowest value was observed in midJanuary when the air temperature also recorded the lowest annual value. The sensitivity index increased rapidly after greening, reached a peak value in the late shooting stage, and then decreased to lower values in the grain-filling stage. For both corn and wheat, peak values of sensitivity index and minimum values in the initial and final stages were significantly different from
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Fig. 2 Downscaled sensitivity index in irrigation intervals of about 10 d for corn (a) and wheat (b) in central Shanxi Province of China.
S. H. SHANG
(F-M and F-S), respectively. The evapotranspiration processes of each stage with water stress were simulated with a daily water balance model (Shang and Mao, 2006) using data at the study site in 2003, and the initial soil moisture was regulated to obtain the desired water stress level. Using the Jensen model (Eq. 1), the relative crop yield was calculated from the experiment and downscaled sensitivity indexes (Fig. 3). Compared with the relative crop yield calculated from the experiment sensitivity indexes, relative errors of crop yield from the downscaled sensitivity indexes were usually small, being only 0.85% and 1.57% in average for the proposed and linear downscaling methods, respectively, which showed the rationality of downscaling water sensitivity index from growth stages to shorter intervals using an appropriate method of interpolation for the cumulative sensitivity index curve. Both the linear downscaling method proposed by Tsakiris (1982) and the downscaling method proposed in this study were applicable if the average relative error was concerned. However, the relative error of the linear downscaling method may exceed 5% in some water stress scenarios, which is far greater than that of the present method.
their evened values through piecewise linear interpolation. The variation of crop sensitivity to soil water stress within each growth stage was reflected in the results of monotone piecewise cubic interpolation, which more closely showed the impact of water stress on crop growth than the evened sensitivity index of piecewise linear interpolation. The downscaled sensitivity index in irrigation intervals was more appropriate to be used for irrigation scheduling. Assessment of downscaling methods To assess the effectiveness of the proposed downscaling method, several scenarios of water stress for wheat were used to calculate relative yield from the Jensen model with experiment and downscaled sensitivity indexes. Since wheat growth is not sensitive to water stress in the seedling and frost stages (Table I), only water stress in a single growth stage after greening was considered. Two water stress levels were designed: moderate and severe levels, which represented about 80% and 60% of the relative evapotranspiration in the concerning growth stage, respectively. Consequently, eight scenarios of water stress were used, including moderate and severe water stress levels in the greening stage (G-M and G-S), shooting stage (S-M and S-S), heading stage (H-M and H-S), and grain-filling stage
Fig. 3 Relative wheat yield for eight water stress scenarios including moderate and severe water stress levels in the greening stage (G-M and G-S), shooting stage (S-M and S-S), heading stage (H-M and H-S), and grain-filling stage (F-M and F-S) (a) and their relative error (b).
DOWNSCALING CROP WATER SENSITIVITY INDEX
The present downscaling method was more appropriate to downscaling water sensitivity index from growth stages to shorter intervals. CONCLUSIONS A downscaling method was proposed to estimate crop water sensitivity index in shorter (irrigation) intervals from that in growth stages, which was based on the monotone piecewise cubic interpolation of the cumulative sensitivity index curve. The proposed downscaling method was used to estimate sensitivity indexes in irrigation intervals of about 10 d for corn and wheat in central Shanxi Province of China. The downscaled sensitivity index could reflect the variation both at the growth stages and within each growth stage. Scenario analysis of water stress at a single growth stage of wheat showed the rationality of downscaling water sensitivity index from growth stages to shorter intervals using an appropriate method of interpolation for the cumulative sensitivity index curve, and the present downscaling method was superior to the traditional linear downscaling method. The downscaling method proposed could be easily incorporated into irrigation optimization models. REFERENCES Doorenbos, J. and Kassam, A. H. 1979. Yield Response to Water. FAO Irrigation and Drainage Paper 33. FAO, Rome. Ferreira, D. B. and Rao, V. B. 2011. Recent climate variability and its impacts on soybean yields in Southern Brazil. Theor. Appl. Climatol. 105: 83–97. Fritsch, F. N. and Carlson, R. E. 1980. Monotone piecewise cubic interpolation. SIAM J. Numer. Anal. 17: 238–246. Ganji, A. and Shekarriz fard, M. 2010. A modified constrained state formulation of stochastic soil moisture for crop water allocation. Water Resour. Manag. 24: 547–561. Georgiou, P. E. and Papamichail, D. M. 2008. Optimization model of an irrigation reservoir for water allocation and crop planning under various weather conditions. Irrig. Sci. 26: 487–504. Gilat, A. and Subramaniam, V. 2011. Numerical Methods: An Introduction with Applications using MATLAB. 2nd Ed. John Wiley & Sons, Inc., Hoboken. Han, S. J., Liu, Q. C., Wang, S. L. and Hu, Y. Q. 2010. Improvement and verification of cumulative function of crop water
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sensitive index. Trans. Chin. Soc. Agr. Eng. (in Chinese). 26(6): 83–88. Howell, T. A. and Hiler, E. A. 1975. Optimization of water use efficiency under high frequency irrigation: I. Evapotranspiration and yield relationship. Trans. Am. Soc. Agr. Eng. 18: 873–878. Iden, S. C. and Durner, W. 2007. Free-form estimation of the unsaturated soil hydraulic properties by inverse modeling using global optimization. Water Resour. Res. 43: W07451. Igbadun, H. E., Tarimo, A. K. P. R., Salim, B. A. and Mahoo, H. F. 2007. Evaluation of selected crop water production functions for an irrigated maize crop. Agr. Water Manage. 94: 1–10. Jensen, M. E. 1968. Water consumption by agricultural plants. In Kozlowski, T. T. (ed.) Water Deficit and Plant Growth. Volume II: Plant Water Consumption and Response. Academic Press, New York. pp. 1–22. Kaboosi, K. and Kaveh, F. 2012. Sensitivity analysis of FAO 33 crop water production function. Irrig. Sci. 30: 89–100. Kipkorir, E. C. and Raes, D. 2002. Transformation of yield response factor into Jensen’s sensitivity index. Irrig. Drain. Syst. 16: 47–52. Lhomme, J. P., Mougou, R. and Mansour, M. 2009. Potential impact of climate change on durum wheat cropping in Tunisia. Clim. Chang. 96: 549–564. Moler, C. 2004. Numerical Computing with MATLAB. The MathWorks, Inc., Natick. Peters, A. and Durner, W. 2008. Simplified evaporation method for determining soil hydraulic properties. J. Hydrol. 356: 147–162. Rao, N. H., Srma, P. B. S. and Chander, S. 1988. A simple dated water-production function for use in irrigated agriculture. Agr. Water Manage. 13: 25–32. Shang, S. H. 2013. Log-cubic method for generation of soil particle size distribution curve. Sci. World J. 2013: Article ID 579460. Shang, S. H. and Mao, X. M. 2006. Application of a simulation based optimization model for winter wheat irrigation scheduling in North China. Agr. Water Manage. 85: 314– 322. Stewart, J. I. and Hagan, R. M. 1973. Functions to predict the effects of crop water deficits. J. Irrig. Drain. Div.-ASCE. 99: 421–439. Tsakiris, G. P. 1982. A method for applying crop sensitivity factors in irrigation scheduling. Agr. Water Manage. 5: 335– 343. Wang, Y. R., Kang, S. Z., Li, F. S., Zhang, L. and Zhang, J. H. 2007. Saline water irrigation scheduling through a cropwater-salinity production function and a soil-water-salinity dynamic model. Pedosphere. 17: 303–317. Wang, Y. R. and Sun, X. P. 2003. Agricultural Water-Saving Theory and High Water Use Efficiency Pattern for Crops in Shanxi (in Chinese). China Science and Technology Press, Beijing.
Downscaling Crop Water Sensitivity Index Using Monotone Piecewise Cubic Interpolation∗1 SHANG Song-Hao∗2 State Key Laboratory of Hydroscience and Engineering, Tsinghua University, Beijing 100084 (China) (Received April 18, 2013; revised July 18, 2013)
ABSTRACT Crop-water production functions quantitatively describe the relationship between crop yield and field evapotranspiration. The crop water sensitivity indexes of crop-water production functions, a key factor for optimizing irrigation scheduling in case of water scarcity, are usually obtained from field experiments or other sources for crop growth stages, while their values in shorter intervals are preferred for practical irrigation scheduling. We proposed a method to downscale the sensitivity index from growth stages to shorter intervals by monotone piecewise cubic interpolation of the cumulative sensitivity index curve. This method was used to estimate sensitivity indexes in irrigation intervals of about 10 d for corn and wheat in central Shanxi Province of China. Results showed that the downscaled sensitivity index could reflect the impact of water stress on crop growth both at different growth stages and within each stage. Scenario analysis of water stress at a single growth stage of wheat showed the rationality of downscaling water sensitivity index from growth stages to shorter intervals through interpolation of cumulative sensitivity index, and this proposed downscaling method was superior to the traditional linear downscaling method. Key Words:
corn, crop-water production function, growth stage, water stress, wheat
Citation: Shang, S. H. 2013. Downscaling crop water sensitivity index using monotone piecewise cubic interpolation. Pedosphere. 23(5): 662–667.
INTRODUCTION Crop-water production functions quantitatively describe the relationship between crop yield and field evapotranspiration, which is useful for calculating crop yield under water stress conditions (Jensen, 1968). Meanwhile, crop yield or economic profit dependent on the yield is usually the objective function for the optimization model of deficit irrigation that determines the optimal irrigation date with limited water available for irrigation (Georgiou and Papamichail, 2008; Ganji and Shekarriz fard, 2010). Therefore, sensitivity index for the crop-water production functions is important for the optimization of irrigation scheduling in areas suffering from water scarcity. Besides, crop-water production functions can also be used in assessing the impact of climate change on crop yield (Lhomme et al., 2009; Ferreira and Rao, 2011). Crop-water production functions are usually expressed as empirical formulae relating crop yield to evapotranspiration in the whole growth season (Stewart and Hagan, 1973), at a specified growth stage ∗1 Supported
(Doorenbos and Kassam, 1979), or at different growth stages (Jensen, 1968; Howell and Hiler, 1975). The last type usually uses multiplicative (Jensen, 1968; Rao et al., 1988) or additive (Howell and Hiler, 1975) forms to express the combined effect of water stress at several growth stages. Sensitivity analysis of several cropwater production functions indicated that the multiplicative form is preferable to the additive form (Kaboosi and Kaveh, 2012). Among these crop-water production functions, the Jensen (1968) model of multiplicative form is often used in the optimization of irrigation scheduling with limited water supply (e.g., Shang and Mao, 2006; Wang et al., 2007; Georgiou and Papamichail, 2008; Ganji and Shekarriz fard, 2010). Sensitivity indexes of the Jensen (1968) model or other parameters of crop-water production functions can be determined from irrigation experiments in local environment (e.g., Wang and Sun, 2003; Igbadun et al., 2007), which is usually time-consuming and costly. Doorenbos and Kassam (1979) analyzed available field experimental data and proposed yield response factors for water stress at a particular growth stage for main
by the National Natural Science Foundation of China (No. 51279077) and the National Key Technology R&D Program of China (No. 2013BAB05B03). ∗2 Corresponding author. E-mail: [email protected].
DOWNSCALING CROP WATER SENSITIVITY INDEX
crops. Kipkorir and Raes (2002) proposed a function to transform the readily available yield response factors of Doorenbos and Kassam (1979) to the sensitivity indexes of the Jensen (1968) model. However, both yield response factors and sensitivity indexes are usually available in the time scale of growth stages. In irrigation scheduling, their values in shorter irrigation intervals are preferred rather than their values during a particular growth stage (Kipkorir and Raes, 2002; Georgiou and Papamichail, 2008). Therefore, there is a need to downscale the sensitivity index from growth stages to shorter irrigation intervals. To solve this problem, Tsakiris (1982) proposed a procedure for estimating sensitivity indexes in given time intervals using the cumulative values of the sensitivity index (cumulative sensitivity index) of the Jensen (1968) model. In this procedure, the cumulative sensitivity index is plotted against growth time with piecewise straight lines, and then sensitivity index in a given time interval can be estimated from the difference between the cumulative sensitivity index at the end and at the beginning of the considered time interval. This procedure has been used in several irrigation optimization models (e.g., Georgiou and Papamichail, 2008; Ganji and Shekarriz fard, 2010). The procedure is simple, but it does not consider the variation of sensitivity index within a growth stage. As a result, sensitivity index is evened in each stage, and irrigation intervals of the same duration within a growth stage have the same sensitivity index (Tsakiris, 1982), which is not sound for describing crop sensitivity to soil water stress. Besides, the rationality of downscaling sensitivity index from cumulative sensitivity index has not been fully investigated. To reflect the variation of the sensitivity index both at different stages and within each stage, some empirical formulae have been used to depict the cumulative sensitivity index curves, such as the logistic curve (Wang and Sun, 2003; Shang and Mao, 2006) and the power function-based sigmoid curve (Han et al., 2010). The sensitivity index estimated from the sigmoid curves is smaller in the initial and final growth stages and larger in the middle growth stages, which accords with the general variation pattern of crop sensitivity in response to water stress. However, the sigmoid curves are not flexible enough to depict the complex variation of cumulative sensitivity index in the whole growth season, especially for winter wheat with a long frost period and very low water sensitivity index in this period. The main objective of this study was to propose a sound method to downscale sensitivity index from
663
growth stages to shorter irrigation intervals, based on monotone piecewise cubic interpolation of cumulative sensitivity index. The proposed method was used to estimate sensitivity indexes in time intervals of about 10 d for corn and wheat in central Shanxi Province of China, and scenarios of water stress at a single growth stage of wheat were analyzed to demonstrate the rationality of the proposed downscaling method. DOWNSCALING METHOD FOR CROP SENSITIVITY INDEX The Jensen (1968) model for crop-water production function is as follows: ETi λi Y = Ym ETm,i i=1 n
y=
(1)
where y, Y , and Ym are the relative, actual, and maximum crop yields, respectively; ETi and ETm,i are the actual and maximum evapotranspiration during the growth stage i, respectively; n is the number of crop growth stages; and λi is the crop water sensitivity index at the stage i. Crop water sensitivity indexes of the above model are usually available for crop growth stages, while the cumulative sensitivity index curve (Tsakiris, 1982) provides the possibility to estimate the sensitivity index in an arbitrary time interval from available sensitivity indexes for growth stages. Since sensitivity indexes are greater than 0, the cumulative sensitivity index is a monotonic increasing function of growth time. If this function is known, the sensitivity index in an arbitrary time interval can be calculated from the difference between the final and initial values of the cumulative sensitivity index in the time interval. Moreover, the first derivative of the cumulative sensitivity index is the value of sensitivity index per unit time. Therefore, the main task to estimate sensitivity index in preferred irrigation intervals is to find an appropriate monotonic function to depict the cumulative sensitivity index curve from limited points in the curve. For a crop under consideration, the whole growth season is divided into n stages, and the duration and the sensitivity index of the ith stage are Δti and λi (i = 1, 2, · · · , n), respectively. The cumulative sensitivity index at time ti (i = 0, 1, · · · , n), Zi , can be calculated from: Z0 = 0, Zi =
i j=1
and
λj = Zi−1 + λi ,
i = 1, 2, · · · , n (2)
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t0 = 0,
S. H. SHANG
ti =
i
Δtj = ti−1 + Δti ,
j=1
i = 1, 2, · · · , n
(3)
Now, we have n + 1 points, (ti , Zi ), i = 0, 1, · · · , n, in the cumulative sensitivity index curve. Values of the cumulative sensitivity index at any other time may be estimated with appropriate interpolation methods. The procedure proposed by Tsakiris (1982) approximated the cumulative sensitivity index with the following piecewise linear interpolation function: Zl (t) = Zi
t − ti+1 t − ti + Zi+1 , ti − ti+1 ti+1 − ti
ti ≤ t ≤ ti+1 ,
i = 0, 1, · · · , n − 1
(4)
where Zl (t) is the piecewise linear interpolation function. The piecewise linear interpolation is simple and can preserve the monotonicity of the cumulative sensitivity index curve, but the first derivative of this interpolation function is piecewise constant. Consequently, the sensitivity index is evened over each growth stage, and the variation of crop sensitivity in response to water stress within the stage is not considered. To overcome the evenness of the sensitivity index in each growth stage by piecewise linear interpolation, higher order polynomial or piecewise polynomial interpolation is an alternative. However, commonly used higher order interpolation can not preserve the monotonicity of the cumulative sensitivity index curve. Here we used a monotone piecewise cubic interpolation (Fritsch and Carlson, 1980), which is characterized by passing through all n + 1 points, preserving the monotonicity of the cumulative sensitivity index curve, and having a continuous first derivative. This shapepreserving interpolation method has been used in several fields of soil sciences, such as the generation of soil particle size distribution curve (Shang, 2013) and estimation of soil hydraulic properties (Iden and Durner, 2007; Peters and Durner, 2008). The monotone piecewise cubic interpolation function for each time interval t (ti ≤ t ≤ ti+1 , i = 0, 1, · · · , n − 1), Zci (t), is Zci (t) = Zi+1 di+1
3hs2 − 2s3 h3 − 3hs2 + 2s3 + Z + i h3 h3 s2 (s − h) s(s − h)2 + di 2 h h2
(5)
where h = Δti ; s = t − ti ; and di and di+1 denote the slopes of the interpolant at knots ti and ti+1 , respectively. The slope at a knot can be estimated from the lengths and the first divided differences of two adjacent
intervals (Fritsch and Carlson, 1980; Moler, 2004). Piecewise linear interpolation and monotone piecewise cubic interpolation were accomplished using the interp1 function of Matlab (Gilat and Subramaniam, 2011): Zd = interp1(t, Z, td , ‘method’)
(6)
where Zd is the interpolated cumulative sensitivity index corresponding to td ; td is an array representing a desired division of the crop growth season; t is the (n + 1) – dimensional array of time calculated from Eq. 3; Z is the cumulative sensitivity index calculated from Eq. 2; and ‘method’ refers to the interpolation method. If ‘method’ is specified to ‘linear’ or ‘cubic’, it means that the piecewise linear method or the monotone piecewise cubic method was used in interpolation. After interpolation, the sensitivity index in a specified time interval [u, v] in the growth season can be calculated from: λ[u,v] = Zd (v) − Zd (u)
(7)
where λ[u,v] is the sensitivity index in the time interval [u, v]; and Zd (v) and Zd (u) are the interpolated cumulative sensitivity indexes at time v and u, respectively. The monotonicity of the interpolated curve can guarantee positive values of sensitivity index at arbitrary time intervals. Since the interpolation curve of cumulative sensitivity index passes through all original data, the observed and interpolated sensitivity indexes in each growth stage are exactly the same. CASE STUDIES The proposed method was used to estimate sensitivity indexes in irrigation intervals for main food crops in central Shanxi Province of China. In this region, corn and wheat are two main food crops, and water sensitivity indexes are available for growth stages (Table I) from irrigation experiments (Wang and Sun, 2003). Corn in the study region usually grows from early May to mid-September lasting for about 140 d, and the growth season can be divided into 4 stages with the duration from 20 to 51 d (Table I). Corn growth is most sensitive to water stress in the shooting stage. Wheat in the study region usually grows from late September to late June lasting for about 280 d. The growth season can be divided into 6 stages with the duration from 19 to 110 d (Table I). The main characteristic of wheat growth is that there exists a long frost period of about 110 d with very low sensitivity to soil water stress. Wheat growth is more sensitive to water stress in the shooting, greening, and heading stages.
DOWNSCALING CROP WATER SENSITIVITY INDEX
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TABLE I Sensitivity indexes at different growth stages for corn and wheat in central Shanxi Province of China a) Growth stage
Seedling Frost Greening Shooting Heading Grain-filling Total a) Cited
Corn
Wheat
Starting date
Duration
Sensitivity indexa)
Starting date
Duration
Sensitivity indexa)
May 1
d 51
0.0903
20 26 43 140
0.4873 0.2802 0.1548 1.0126
Sep. 22 Nov. 20 Mar. 10 Apr. 20 May 13 Jun. 1
d 59 110 41 23 19 28 280
0.0469 0.0154 0.2800 0.3332 0.2470 0.1067 1.0292
Jun. 21 Jul. 11 Aug. 6
from Wang and Sun (2003).
Considering the practical decision making for irrigation scheduling, an irrigation interval of a week or about 10 d is usually preferred, which is much shorter than the duration of a growth stage. Therefore, the sensitivity index should be downscaled from the growth stage to irrigation intervals of about 10 d for irrigation practice. Thus, a whole month in the growth season can be divided into three irrigation intervals, with the first and second intervals of 10 d and the third 8 to 11 d depending on total days in the month. The duration of the first and last irrigation intervals of the growth season may differ from 10 d depending on the sowing and harvesting dates. Downscaled sensitivity indexes of corn and wheat in central Shanxi Province of China From sensitivity indexes of different growth stages in Table I, the cumulative sensitivity index was calculated and interpolated with piecewise linear and monotone piecewise cubic methods (Fig. 1) and was then used to estimate sensitivity indexes in irrigation intervals of about 10 d (Fig. 2). Although these two interpolated curves seemed similar, the estimated values of the sensitivity indexes in shorter intervals were apparently different, especially when the difference of sensitivity indexes at two adjacent growth stages was great. The interpolated cumulative sensitivity index curves using the piecewise linear method are piecewise straight lines, which are continuous but not smooth at the boundary points of the growth stages (Fig. 1). As a result, the sensitivity index of a growth stage is evened over the stage through piecewise linear interpolation (Fig. 2), and the variation of crop sensitivity to water stress within a stage is not considered. On the contrary, the interpolated cumulative sensitivity index curves using the monotone piecewise cubic method are smooth in the whole growth season (Fig. 1). For corn, the downscaled sensitivity index (Fig. 2) in-
Fig. 1 Experiment and interpolated cumulative sensitivity index curves for corn (a) and wheat (b) in central Shanxi Province of China.
creased slowly in the seedling stage, then increased rapidly in the early shooting stage and reached the peak value in the late shooting stage, and finally decreased until crop maturation. For wheat, the downscaled sensitivity index was lower in the seedling stage and frost period. The lowest value was observed in midJanuary when the air temperature also recorded the lowest annual value. The sensitivity index increased rapidly after greening, reached a peak value in the late shooting stage, and then decreased to lower values in the grain-filling stage. For both corn and wheat, peak values of sensitivity index and minimum values in the initial and final stages were significantly different from
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Fig. 2 Downscaled sensitivity index in irrigation intervals of about 10 d for corn (a) and wheat (b) in central Shanxi Province of China.
S. H. SHANG
(F-M and F-S), respectively. The evapotranspiration processes of each stage with water stress were simulated with a daily water balance model (Shang and Mao, 2006) using data at the study site in 2003, and the initial soil moisture was regulated to obtain the desired water stress level. Using the Jensen model (Eq. 1), the relative crop yield was calculated from the experiment and downscaled sensitivity indexes (Fig. 3). Compared with the relative crop yield calculated from the experiment sensitivity indexes, relative errors of crop yield from the downscaled sensitivity indexes were usually small, being only 0.85% and 1.57% in average for the proposed and linear downscaling methods, respectively, which showed the rationality of downscaling water sensitivity index from growth stages to shorter intervals using an appropriate method of interpolation for the cumulative sensitivity index curve. Both the linear downscaling method proposed by Tsakiris (1982) and the downscaling method proposed in this study were applicable if the average relative error was concerned. However, the relative error of the linear downscaling method may exceed 5% in some water stress scenarios, which is far greater than that of the present method.
their evened values through piecewise linear interpolation. The variation of crop sensitivity to soil water stress within each growth stage was reflected in the results of monotone piecewise cubic interpolation, which more closely showed the impact of water stress on crop growth than the evened sensitivity index of piecewise linear interpolation. The downscaled sensitivity index in irrigation intervals was more appropriate to be used for irrigation scheduling. Assessment of downscaling methods To assess the effectiveness of the proposed downscaling method, several scenarios of water stress for wheat were used to calculate relative yield from the Jensen model with experiment and downscaled sensitivity indexes. Since wheat growth is not sensitive to water stress in the seedling and frost stages (Table I), only water stress in a single growth stage after greening was considered. Two water stress levels were designed: moderate and severe levels, which represented about 80% and 60% of the relative evapotranspiration in the concerning growth stage, respectively. Consequently, eight scenarios of water stress were used, including moderate and severe water stress levels in the greening stage (G-M and G-S), shooting stage (S-M and S-S), heading stage (H-M and H-S), and grain-filling stage
Fig. 3 Relative wheat yield for eight water stress scenarios including moderate and severe water stress levels in the greening stage (G-M and G-S), shooting stage (S-M and S-S), heading stage (H-M and H-S), and grain-filling stage (F-M and F-S) (a) and their relative error (b).
DOWNSCALING CROP WATER SENSITIVITY INDEX
The present downscaling method was more appropriate to downscaling water sensitivity index from growth stages to shorter intervals. CONCLUSIONS A downscaling method was proposed to estimate crop water sensitivity index in shorter (irrigation) intervals from that in growth stages, which was based on the monotone piecewise cubic interpolation of the cumulative sensitivity index curve. The proposed downscaling method was used to estimate sensitivity indexes in irrigation intervals of about 10 d for corn and wheat in central Shanxi Province of China. The downscaled sensitivity index could reflect the variation both at the growth stages and within each growth stage. Scenario analysis of water stress at a single growth stage of wheat showed the rationality of downscaling water sensitivity index from growth stages to shorter intervals using an appropriate method of interpolation for the cumulative sensitivity index curve, and the present downscaling method was superior to the traditional linear downscaling method. The downscaling method proposed could be easily incorporated into irrigation optimization models. REFERENCES Doorenbos, J. and Kassam, A. H. 1979. Yield Response to Water. FAO Irrigation and Drainage Paper 33. FAO, Rome. Ferreira, D. B. and Rao, V. B. 2011. Recent climate variability and its impacts on soybean yields in Southern Brazil. Theor. Appl. Climatol. 105: 83–97. Fritsch, F. N. and Carlson, R. E. 1980. Monotone piecewise cubic interpolation. SIAM J. Numer. Anal. 17: 238–246. Ganji, A. and Shekarriz fard, M. 2010. A modified constrained state formulation of stochastic soil moisture for crop water allocation. Water Resour. Manag. 24: 547–561. Georgiou, P. E. and Papamichail, D. M. 2008. Optimization model of an irrigation reservoir for water allocation and crop planning under various weather conditions. Irrig. Sci. 26: 487–504. Gilat, A. and Subramaniam, V. 2011. Numerical Methods: An Introduction with Applications using MATLAB. 2nd Ed. John Wiley & Sons, Inc., Hoboken. Han, S. J., Liu, Q. C., Wang, S. L. and Hu, Y. Q. 2010. Improvement and verification of cumulative function of crop water
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