Published online 23 January 2008
Published in Vadose Zone J 7:31-38 (2008)
DOI: 10.2136/vzj2007.0025
© 2008 Soil Science Society of America
677 S. Segoe Rd., Madison, WI 53711 USA
ORIGINAL RESEARCH
Parameter Estimation of a Root Water Uptake Model under Salinity Stress
Haruyuki Fujimakia,*,
Yoshitake Andoa,
Yibin Cuib and
Mitsuhiro Inouec
a Univ. of Tsukuba, 1-1-1 Tennodai, Tsukuba, Ibaraki 305-8572, Japan
b School of the Environment, Nanjing Univ., Nanjing 210093, China
c Arid Land Research Center, Tottori Univ., 1390 Hamasaka, Tottori 680-8550, Japan
* Corresponding author (fujimaki{at}sakura.cc.tsukuba.ac.jp).
All rights reserved. No part of this periodical may be reproduced or transmitted in any form or by any means, electronic or mechanical, including photocopying, recording, or any information storage and retrieval system, without permission in writing from the publisher.
Received 30 January 2007.
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ABSTRACT
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Accurate prediction of root water uptake under salinity stress contributes to efficient water management in arid and semiarid lands. We present a cost-effective and reliable method to determine parameter values in a widely used macroscopic root water uptake model. We conducted column experiments using soybean [Glycine max (L.) Merr.] in a greenhouse. Six columns with one plant each were used: three were under salinity stress, the others provided potential transpiration. Three time-domain reflectometry probes were inserted into each of the three columns to observe water content and electrical conductivity. In the daytime, the soil surface was covered to prevent evaporation. Weight of the columns was manually measured to obtain daily transpiration. After the stress period, root density distributions were obtained by dismantling the columns. Two parameter values were inversely determined by minimizing the sum of square difference between observed and calculated daily transpiration rates. Water uptake at each depth and time was calculated by substituting linearly interpolated osmotic potential into a stress response function. Optimized daily transpiration agreed well with the observations. In Addition, deviation in the three optimized response functions was small at low-to-moderate stress, indicating the reliability of the method.
Abbreviations: EC, electrical conductivity • RWU, root water uptake • TDR, time domain reflectometry
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INTRODUCTION
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Irrigation-induced salinity is a major limiting factor for crop production in arid and semiarid lands. Despite water scarcity, irrigation rate must exceed evapotranspiration rates to leach out excess salts (Ayers and Westcot, 1985; Abrol et al., 1988). To minimize water use while avoiding yield reduction, it is essential to accurately predict root water uptake (RWU) under salinity stress.
Macroscopic RWU submodels using stress response functions that describe the dependence of reduction coefficients on matric and/or osmotic potential at each depth are widely employed in user-friendly simulation models of water and solute movement in soils such as HYDRUS (
im
nek et al., 2006) and SWAP (Van Dam et al., 1997). In the submodels, the rate of water uptake, S (s–1), being a sink term in the continuity equation, is given by multiplying the reduction coefficient, 
, by the potential rate of water uptake, Sp (s–1) (Feddes et al., 1978; Feddes and Raats, 2004):
 | [1] |
In macroscopic RWU models, several variants differing in expressing combined water and salinity stress have been proposed (Homaee et al., 2002; Feddes and Raats, 2004). For example, HYDRUS (imnek et al., 2006) provides two options, termed additive (Eq. [2]) and multiplicative approaches (Eq. [3]), for expressing reduction in water uptake under the coexistence of drought (water) and salinity stress:
 | [2] |
 | [3] |
where w and s are reductions caused by drought and salinity stress, respectively, h and ho are pressure head and osmotic head respectively, and h50, ho50, p, p1, and p2 are fitting parameters (van Genuchten, 1987). Note that h50 and ho50 are heads when water uptake is 50% of its potential rate and therefore represent simple indices of tolerance of crops. It is still under question which approach can describe water uptake more accurately. Potential uptake rate is proportional to both the normalized root density or root activity, β (cm–1), and potential transpiration rate, Tp (cm d–1):
 | [4] |
The normalized root density, β, is defined as the density of active roots that is normalized such that integrating over the root zone yields unity:
 | [5] |
where 
r is the density of active roots, often represented by root length density (cm–2), and z is the depth (cm). Since transpiration rate, Tcal, is the integral of S over the root zone, Tcal is calculated from
 | [6] |
Despite widespread acceptance, a reliable method for determining parameter values in the response function has not been established. With regard to salt tolerance, the widely cited tables describing the relationship between relative yield and electrical conductivity (EC) of saturated extract, averaged over the root zone as well as the entire growing season, are quite approximate and carry significant uncertainty as pointed out by Maas (1990). An alternative salt tolerance index (i.e., parameter values) applicable to modern water–soil management, and methods to routinely determine such an index are required. Furthermore, parameter values in root water uptake models can be universal indices of the tolerance of plants to stress; therefore, the establishment of the model and the methodology of parameter determination are also important for breeding.
Homaee et al. (2002) obtained the reduction coefficients from a time domain reflectometry (TDR)–equipped column experiment with alfalfa (Medicago sativa L.) grown in a greenhouse. They determined the parameter values by fitting the relationship between the mean potential over the root zone and relative transpiration (i.e., ratio of actual to potential transpiration). They also assumed that relative transpiration equals the reduction coefficient in the macroscopic RWU model ( = T/Tp). If the distribution of root activity were uniform, and the reduction function were linear, this approach would be mathematically correct. In general, however, root activity is not uniform and reduction functions are nonlinear. It is noteworthy that the reduction function of Feddes et al. (1978) is composed of three linear segments but is nonlinear as a whole. Additionally, they used transpiration of nonstressed, well-grown plants as the "potential" values, although the potential transpiration rate depends not only on meteorological conditions but also on growth level such as leaf area index. Long-term stress causes large differences in growth compared with nonstressed plants. In such cases, it is difficult to obtain potential transpiration. If transpiration rate from nonstressed plants having larger leaf area were defined as Tp, plants with smaller leaf area as a result of previous stress could never attain Tp even when they are relieved from stresses. This situation violates the model described above. Therefore, the reliability of their method remains questionable.
Dudley and Shani (2003) obtained optimized parameter values by trial and error and evaluated RWU models including potential flow model by Nimah and Hanks (1973), another widely used RWU formulation. Dudley and Shani (2003) used the difference between predicted and measured yields, assuming that yield was proportional to cumulative transpiration. Skaggs et al. (2006) also assumed that relative yield equals the reduction coefficient to derive parameters in the reduction function from literature listing the dependence of relative yield on mean EC of soil for a variety of crops. This assumption introduces additional uncertainty since transpiration may differ day to day, even if the cumulative transpiration at harvest is the same.
Several studies have determined the parameter values by minimizing the residual between predicted and observed water content (e.g., Rasiah et al., 1992; Vrugt et al., 2001; Hupet et al., 2002; Skaggs et al., 2006). They assumed that root activity distribution follows prescribed simple functions and optimized parameter values of the functions. Zuo and Zhang (2002) presented an inverse technique to estimate irregular root density distribution and reported that estimated distributions matched with directly sampled ones (Zuo et al., 2004). They also used water content data in the objective function. As assessed by Hupet et al. (2002) and Zuo et al. (2004), such inverse methods are sensitive to errors in hydraulic properties of the soil. Zuo et al. (2004) reported that the uncertainty in hydraulic properties gave acceptable error in the estimated distribution of root water uptake. However, errors in hydraulic properties can often be much larger than they assumed. Moreover, instead of using directly measured potential transpiration, they used a simple meteorological formula for estimating potential transpiration, which could be another source of error.
The objective of this study was, therefore, to present a cost-effective and reliable method to determine the response function. Our experimental design can be performed by a single operator and does not require expensive, large weighing lysimeters, thus allowing the routine evaluation of the tolerance of local crops.
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Materials and Methods
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Column Experiment in a Greenhouse
Six columns with 29 cm i.d. and 39 cm in height were placed in a greenhouse located in the Agricultural and Forestry Research Center, University of Tsukuba, Japan. To monitor water content and EC in the soil, three TDR probes (CS605, Campbell Sci, Logan UT) were inserted horizontally into each of three columns (A, B, C) such that the depths of the middle rods were located at 5, 15, and 30 cm, respectively, as illustrated in Fig. 1
. Thermocouples were inserted at the same depths as TDR probes to observe temperature. Three other columns were used in the experiment but were not equipped with TDRs or thermocouples. Since the columns were placed aboveground, the columns were wrapped with glass wool of 5 cm thickness to minimize temperature fluctuation and horizontal nonuniformity in temperature. Three water-filled ceramic pipes with 2.1 cm o.d. and 15 cm in length were placed at the bottom of each column to simulate natural drainage condition by applying pressure equivalent to field capacity (about –40 cm for this experiment). Air-dried Tottori sand (sand 99.7%, silt 0.3%) was packed to a bulk density of 1.55 g cm–3. To minimize nonuniformity during packing, the mass of every 5 cm of packed soil was measured before adding the next soil increment. Three seeds of soybean [Glycine max (L.) Merr., Bon-odori, Kaneco Shubyou Inc, Maebashi, Japan] were sown on 29 May 2003. Two weeks later, the seedlings were thinned to one plant per column. The stress period started after healthy plants had grown by irrigating sufficient amount of tap water (EC = 0.6 dS m–1) with 2000 fold-diluted liquid fertilizer (Hyponex Japan, Osaka, Japan). Resulting concentration of N, P, and K were 50, 22, and 42 mg L–1, respectively.
Irrigation with NaCl solution started at sunset on 31 July 2003. During the salinity stress period, the irrigation interval was 2 d, and irrigation amounts were determined for each column to replenish the loss by transpiration (not including loss by drainage) since the previous irrigation. In such a scheme, salt is not leached out and gradually accumulates in the root zone while avoiding drought stress. By gradually increasing the salinity, we can observe the response of transpiration across a wide range of salinity. The evapotranspiration was measured by manually weighing the columns and discharge amount through the ceramic pipes. The weights were measured at sunrise and sunset. During the daytime, the soil surface was covered with a white-colored, 1-cm-thick styrene foam to prevent evaporation from the soil, thus isolating transpiration. During the nighttime, the soil surface was uncovered to ensure the respiration of roots. The NaCl concentration of irrigated water was initially about 2000 mg L–1. Since no apparent reduction in transpiration was observed until 3 August, the concentration was increased to about 3000 mg L–1 from the irrigation on 4 August. The other three columns and plants were used to provide potential transpiration by continuing irrigation with sufficient amount of tap water. As noted above, TDR and thermocouples were not used for the columns. Hourly solar radiation was also observed. The experiment was terminated when the relative transpiration, the ratio of actual to potential transpiration, became less than half.
Calibration of the TDR Probe
To accurately measure volumetric water content, 
, from the dielectric permittivity measurement, we conducted a soil-specific calibration. Sand was packed in the same column as above with the same bulk density to the thickness of 14 cm from the bottom. After saturation with 3000 mg L–1 or 5000 mg L–1 NaCl solution, the solution was slowly drained through the ceramic pipes by applying suction. Assuming that the average in the column calculated from the observed discharge amount is equal to around the rods, the relationship between (the root of) permittivity, 
0.5, and was obtained as shown in Fig. 2
. The 0.5 at air-dry was measured before saturation. Since changing the concentration to 5000 mg L–1 did not obviously affect the relationship in the low-to-middle water content range, data from 5000 mg L–1 solution were fitted to the following equation:
 | [7] |
where a and b are the fitting parameters, which were 0.126 and 0.242, respectively, for the Tottori sand. Note that the linear relationship between (apparent) 0.5 and did not hold in the higher water content range. Because observed 0.5 was less than 3.5 in most of the salinity stress period, we fitted Eq. [7] in < 0.2. Since the effect of temperature on the permittivity of Tottori sand is negligible for low-to-middle water content values, we did not consider the temperature effect.
At the same time, we could obtain the dependence of bulk EC, 
b, on which is required to calculate the EC of the soil solution, w, from b and measured with TDR. Assuming that b is proportional to w at a given , we fitted the dependence of the relative EC, b/w, on using a power function:
 | [8] |
where a and b are the fitting parameters, which were presented in Fig. 3
. Note that the curve-fitting was performed for < 0.2 for the same reason as noted above.
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FIG. 3. Dependence of bulk electrical conductivity on water content for Tottori sand. Electrical conductivity of the soil solution was 5.4 dS m–1.
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Estimation of Osmotic Potential Profile
Observed bulk EC, b, using TDR was converted to that of soil solution, w, using Eq. [8]. Because w also depends on temperature, w was normalized to that at a reference temperature using the following equation (U.S. Salinity Laboratory Staff, 1954)
 | [9] |
where Ts is the soil temperature (K), w25 is w at Ts = 298.2 K.
We estimated the profile of w25 by linearly interpolating and extrapolating the values at the depth of TDR measurement. The depth increments of the calculation was 0.5 cm. The values of w25 were converted to salt concentration, c (mg cm–3), assuming that NaCl is the only solute:
 | [10] |
where ac and bc are the fitting parameters. Measuring w25 at various concentrations and curve fitting gave values of ac and bc as 0.465 and 1.08, respectively. The osmotic potential, ho (cm), of the solution can be estimated from (Campbell, 1985)
 | [11] |
where 
is a unit-conversion factor (10.2 cm kg J–1), 
is the number of ions per molecule, C is the molar concentration of the solute (mol kg–1), 
is the osmotic coefficient, which was assumed to be unity, and R is the universal gas constant (8.31 J mol–1 K–1).
Root Distribution
After the end of the experiment, all sand was taken at every 5-cm layer. The roots were cut at the boundary of the layers using a pair of scissors and a sharp knife. The roots were obtained by sieving the samples through an 0.8-mm screen after air-drying. The segments of the roots were scanned at 600 dpi to obtain PNG-format images. The total length of the root segments for each layer was automatically calculated using a program (SimpleDigitizer, www.sakura.cc.tsukuba.ac.jp/
fujimaki/download/SimpleDigitizer) that implements the intersection method (Newman, 1966).
Estimation of the Response Function
Parameter values in the response function, ho50 and p2, were inversely estimated with Levenberg–Marquardt's maximum neighborhood method (Marquardt, 1963). The objective function to be minimized in the algorithm was
 | [12] |
where b is the vector of the optimized parameters, ho50 and p2, N is the duration of the stress period (day), and 
cal,i and , are calculated and measured daily transpiration (mm) of ith day, respectively. Daily transpiration were calculated by integrating hourly calculated transpiration rates, Tcal (mm h–1), during the daytime:
 | [13] |
where 
t is the time increment (1 h in this study) of the calculation.
Transpiration rate, Tcal, was calculated by integrating uptake rate, S, across the root zone. S at each depth was given by multiplying the reduction coefficient, , by the potential rate of water uptake, Sp (Eq. [1]). Sp at each depth was obtained by multiplying the normalized root density, β, obtained as described above, by the potential transpiration rate, Tp, obtained as described below. In the calculation of , we have incorporated drought stress using Eq. [3] in addition to the inverse analysis, assuming that w was unity. Since drought stress imposed in this experiment was small, simultaneous optimization of h50 and p1 would be erroneous. We thus assumed that h50 = ho50 and p1 = p2.
Volumetric water content, , at each depth was estimated by linearly interpolating and extrapolating at the depth of the TDR measurement. The matric head at each depth was calculated using the retention function of the Tottori sand (Fig. 4
). A bimodal function (Zurmühl and Durner, 1998) was used to describe retention curve of the Tottori sand:
 | [14] |
where w, 1, 2, n1, and n2 are the fitting parameters. We obtained the following parameter values: sat = 0.429; w = 0.736; 1 = 0.0469; 2 = 0.141; n1 = 4.87 and n2 = 1.33 for main drying curve. We used only the main drying curve, neglecting hysteresis since the drying and wetting curves converge in the low matric head (i.e., dry) range, which is critical to water uptake. Since the above equation cannot be solved algebraically for h, h was obtained by the bisection method.
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Results and Discussion
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Observed air temperature and humidity in the greenhouse during the stress period are shown in Fig. 5
. Figure 6
shows observed water content using TDR probes for the three columns during the stress period. Water content at z = 5 cm responded quickly to irrigation and dropped to near air-dry during a few hours before the irrigation. As shown in Fig. 4, the matric head of Tottori sand is higher than –1,000 cm even when is 0.025. Therefore, except for very close to the soil surface, drought stress would have been small throughout the duration. We thus may assume that w was unity for most of the depth and time. However, since plants may have been imposed to mild drought stress during a few hours before irrigation, we included drought stress in the inverse analysis as described below.
Time evolutions in w25 at each probe depth are presented in Fig. 7
. The value of w25 at z = 5 cm largely fluctuated in response to salt accumulation due to water uptake while leaving salts and leaching of the salts by irrigation. On the other hand, w25 at z = 15 cm and 30 cm almost steadily increased with time. Most of the salt remained in the root zone since drainage had almost ceased when salt reached near the bottom of the column.
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FIG. 7. Time evolution of electrical conductivity of soil solution. Arrows indicate when irrigations were performed.
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Table 1
lists the total length for each plant in columns A, B, C. Dividing the total length by the volume of the layers gave the distribution of root length density. Figure 8
shows the normalized root length density profile, β(z), for columns A, B, C.
Estimation of Potential Transpiration Rate
Figure 9
shows the time evolution of relative transpiration which is the ratio of actual to potential daily transpiration, ri.
 | [15] |
where pi is the potential daily transpiration of ith day (mm). If the meteorological conditions are kept constant and the leaf area does not significantly change, just before the stress period can be used as p. Otherwise, if this kind of experiment is performed in a large field, equations used to estimate potential evapotranspiration may be applied. Either approach, however, can be expensive. We thus estimated p from of control (nonstressed) columns (plants) having a similar leaf area. Leaf area indexes are also listed in Table 1. Because p inevitably differs from plant to plant due to nonuniformity in micrometeorological condition as well as genetic differences, we introduced a growth correction factor, 
, for each plant.
 | [16] |
where ctrl is the average transpiration of the three control columns (D, E, F):
 | [17] |
The is thus the average value of the ratio of i to ctrl during the 8 d just before the stress period. During the stress period, p was estimated by multiplying by ctrl assuming that the growth correction factor is invariant over the stress period:
 | [18] |
Since the stress period was relatively short and the leaf area did not significantly change, this assumption may be justified. During the stress period, no increase in leaf area was measured, but rather small decreases (less than 10%) were obtained due to the falling of old leaves. The values of for the plants are also listed in Table 1. As shown in Fig. 9, ri decreased with time in response to the accumulation of salt in the root zone. Observed ctrl is also shown in Fig. 9 (right axis). Units of the transpiration were converted from the mass to the depth of water by dividing the mass by the area of the soil surface of the column and the density of water.
Potential transpiration rate, Tp, substituted into Eq. [6] to estimate Tcal, was estimated assuming that transpiration rate is proportional to solar radiation, Ra (kW m–2):
 | [19] |
To verify the assumption, we measured hourly transpiration rates of the two columns for 2 d. Offsets of the regression lines between solar radiation and hourly transpiration rates were nearly zero with correlation coefficients of 0.93 and 0.88. These results support the validity of the assumption. This assumption may, however, be less valid in the field where the effect of wind is larger.
Response Function
The inversely estimated response function for salinity stress is shown in Fig. 10
. Optimized parameter values are listed in Table 2
. Thin curves indicate results obtained neglecting drought stress (i.e., assuming that w is unity). Large difference were not seen among curves considering and neglecting the effect of drought stress, indicating that the drought stress was very limited. Since slight difference did exist, however, we will discuss below results from inverse analyses considering drought stress. There were some difference in the parameter values, but the curves nearly overlap at high s, which would be the realistic range under saline irrigation for crop production. The coefficient of variation (standard deviation divided by the average, CV) for ho at s = 0.85 was 0.05, which is less than 25% the CV (0.22) for ho at s = 0.5 (i.e., ho50). Pursuing precision at lower s (i.e., severe stress) would be less important when applied to irrigation management. We may therefore conclude that the reliability of the method is satisfactory. Still, given the variation in the estimated parameter values, replicated experiments as performed in this study are recommended.
The calculated daily transpiration using the optimized parameter values agreed well with the measured values as shown in Fig. 11
. The root mean square errors (RMSE) are given in Table 2. Although the macroscopic RWU model is less physically based compared with models that consider the potential and resistance of water flow in plants such as a model by Nimah and Hanks (1973), the model was found to be able to predict daily transpiration with sufficient accuracy, if the parameter values are set properly, the distribution of root and matric and osmotic potentials in the root zone are accurately predicted, and Tp is estimated accurately.
Figure 12
shows examples of the ho profiles at two moments, 12:00 p.m. on 2 August and 11 August. Obtained ho was substituted into Eq. [3], giving an osmotic reduction coefficient, s, at a set of parameters, ho50 and p2, at each iteration level in the inverse parameter estimation. Figure 12 shows that because of the low ho in the upper layer during the early stage (12:00 p.m. on 2 August), sβ was estimated to have been significantly lower than β in the upper layer. In the later stage (12:00 p.m. on 11 August) when the salt had been transported into the deeper layer, sβ was lower than β throughout the root zone.
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FIG. 12. Examples of the profiles of osmotic head, normalized root density, and sβ (column C) at 12:00 p.m. on August 2 and August 11.
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Conclusions
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We have presented a method to determine parameter values in a widely used macroscopic root water uptake model. We conducted a column experiment using soybean in a greenhouse. Two parameter values, ho50 and p2, were inversely determined by minimizing the sum of square difference between observed and calculated daily transpiration rates. Water uptake at each depth and time was calculated by substituting linearly interpolated osmotic potential into a stress response function. Estimated parameter values are therefore not affected by uncertainty in soil hydraulic properties. Optimized daily transpiration agreed well with the observations. In addition, deviation in the three optimized response functions was small, and calculated transpiration agreed well with the measured values at low-to-middle stress level, indicating the reliability of the method.
We have presented a method using relatively short columns instead of field experiments using expensive, large weighing lysimeters. Further studies are needed to determine whether restricting rooting depth of deeper rooting crops to about 40 cm provides different estimated parameter values than in field conditions where roots can freely penetrate into deeper layers. In this study, we could not evaluate the applicability of this method to different sets of experimental conditions. As for the growth stage or plant size, this method would be applicable to younger or more mature plants, simply by selecting the appropriate sizes of TDR probes and columns.
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ACKNOWLEDGMENTS
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The programs used in this study are freely distributed under the general public license: SimpleDigitizer, used for image analysis of the root length, available at http://www.sakura.cc.tsukuba.ac.jp/fujimaki/download/SimpleDigitizer; and PERF, used for parameter estimation of the response function, available at http://www.sakura.cc.tsukuba.ac.jp/fujimaki/download/perf.
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TOP
ABSTRACT
INTRODUCTION
