Published in Vadose Zone Journal 3:271-277 (2004)
© 2004 Soil Science Society of America
677 S. Segoe Rd., Madison, WI 53711 USA
ORIGINAL RESEARCH PAPERS
A Generalized Function of Wheat's Root Length Density Distributions
Qiang Zuoa,
Feng Jiea,
Renduo Zhang*,b and
Lei Menga
a Dep. of Soil and Water Sciences and Key Laboratory of Plant–Soil Interactions, MOE, College of Resources and Environment, China Agricultural University, Beijing 100094, P.R. China
b State Key Lab. of Water Resources and Hydropower Engineering Science, Wuhan University, Wuhan 430072, China, and Dep. of Renewable Resources, University of Wyoming, Laramie, WY 82071-3354, USA
* Corresponding author (renduo{at}uwyo.edu).
Received 13 February 2003.
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ABSTRACT
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The root length density (RLD) is an important parameter to model water and nutrient movement in the vadose zone and to study soil–root–shoot–atmosphere interactions. However, it is difficult and time-consuming to measure and determine RLD distributions accurately. Especially RLD distributions change with different soil environment, plant species, growing seasons, and climatic conditions. In this study, measured data sets of wheat RLD distributions were collected from the literature and transformed into normalized root length density (NRLD) distributions. A total of 610 values of wheat NRLD distributions were pooled together. These data showed a general trend, independent of soil environment, wheat species, growing seasons, and climates. A generalized function was established to characterize the NRLD distributions versus normalized root depths. To verify the generalized function, we measured RLD distributions of winter wheat (Triticum aestivum L.) using laboratory and field experiments for different soils, growing stages of wheat, atmospheric conditions, and water supplies. Using the generalized function, we predicted winter wheat RLD and compared the predicted results with the experimental data and with results using other NRLD functions. The comparison showed that the generalized function predicted RLD distributions more accurately than the other functions. Although simulated results of soil water dynamics in soil–wheat systems were similar for the different NRLD functions, the generalized function should be advantageous for applications that require accurate information of root development and distribution.
Abbreviations: RLD, root length density • NRLD, normalized root length density
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INTRODUCTION
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ROOT DEVELOPMENT AND DISTRIBUTION in soils are important information for root–water and nutrient-uptake studies in soil–plant systems (Asseng et al., 1997). However, it is difficult and costly to measure root distributions accurately, because root distributions change with time as well as with different soil environment, plant species, growing seasons, climatic conditions, and other factors. In current root–water and nutrient-uptake models, RLD distributions are more commonly incorporated than root weight density distributions (Prasad, 1988; van Noordwijk and van de Geijn, 1996; Jamieson et al., 1998; Wu et al., 1999; Musters and Bouten, 2000; Vrugt et al., 2001). Root length density distributions are often utilized to analyze soil–root–shoot–atmosphere interactions (Smit et al., 1994; Asseng et al., 1997; Zubaidi et al., 1999; Liedgens et al., 2000; Chassot et al., 2001). Enormous efforts have been made to obtain spatial and temporal distributions of RLD using experimental measurements and simulations.
Experimental measurement approaches of RLD distributions include the root sampling method (Kumar et al., 1993) and rhizotron (or minirhizotron) method (Ephrath et al., 1999). The root sampling method is direct and reliable, however, time-consuming and destructive. The rhizotron method can be used to monitor root development under almost undisturbed conditions by comparing a series of root photographs taken during successive time periods. Nevertheless, reliability of the rhizotron technique has yet to be fully assessed. Many factors, such as insertion angles of observation tubes for photographs and the calibration curve between root count and RLD can affect the accuracy of the RLD distribution. Therefore, the accurate and effective measurement of transient RLD distributions is still a challenging task.
Simulation approaches of RLD distributions include root architecture models (Diggle, 1988; Grabarnik et al., 1998; Thaler and Pagès, 1998; Bidel et al., 2000), plant growth models (e.g., AFRCWHEAT2, CERES–Wheat, Jamieson et al., 1998; Jamieson and Ewert, 1999), and shoot and root models (Thornley, 1995, 1998). Almost all the root growing simulation models, which are usually comprehensive and complicated, contain a set of production rules and parameters based on various assumptions, such as potential root water uptake, root restriction factor, assimilation and/or photoassimilation partitioning and C allocation, and root biomass/root length ratio (Thaler and Pagès, 1998; Thornley, 1998; Jamieson and Ewert, 1999; Bidel et al., 2000). However, it is difficult to define or evaluate these assumptions and parameters. Major issues remain concerning the mechanisms and integration of uptake activities within a soil–root system and in modeling root development when interacting with a complex soil environment (van Noordwijk and van de Geijn, 1996).
Fortunately, a large amount of data has been published for RLD distributions of different crops, especially for wheat. It is essential to collect the available data of RLD distributions and to establish general rules for root growth of wheat. Wu et al. (1999) introduced the concept of NRLD distribution (Lnrd) and analyzed Lnrd of wheat, maize (Zea mays L.), cotton (Gossypium hirsutum L.), and bean (Phaseolus vulgaris L.) based on data of RLD in the literature. Their results showed that NRLD distributions for each crop at different growth stages are quite similar and that it is feasible to use a single Lnrd function for each crop. Nonetheless, the results need additional examinations because of the limited data used.
To further explore and apply the concept of NRLD distribution, our objective of this study was to establish a generalized function, based on available data of wheat RLD distributions in the literature. The generalized function of wheat RLD distributions was used to predict spatial and temporal RLD distributions, which were compared with experimental data measured under different conditions. The generalized function was also applied in water flow simulations in a soil–plant system. The simulated soil water distributions were compared with field experimental data.
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MATERIALS AND METHODS
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Normalized Root Length Density Distribution
The following model is often employed to simulate one-dimensional vertical soil water flow under evaporative condition and in the presence of root-water uptake (van Genuchten, 1987):
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Here h is the soil matric potential (cm), C(h) the soil water capacity (cm–1), K(h) the soil hydraulic conductivity (cm d–1), h0(z) the initial soil matric potential in the profile (cm), E(t) soil surface evaporation rate (cm d–1), hL(t) the matric potential at the lower boundary (cm), L simulated depth (cm), z vertical coordinate originating from the soil surface and positive downwards (cm), and S(z,t) is the root-water uptake rate (d–1). The root-water uptake rate can be defined by (van Genuchten, 1987; Wu et al., 1999; Musters and Bouten, 2000):
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 | [6] |
where zr(= z/Lr) is the normalized rooting depth ranging from 0 to 1, where Lr denotes the maximum rooting depth, 
(h) is a dimensionless reduction function simulating water stress, Smax(zr, t) the maximal specific water extraction rate under optimal soil water conditions (d–1), Tp the potential transpiration rate (cm d–1), h1 and h2 are threshold values of matric potential (cm), and ß is a fitting parameter. Assuming that the specific water extraction rate is proportional to RLD for optimal soil water conditions (Feddes et al., 1978; Prasad, 1988), Wu et al. (1999) defined the normalized root length density distribution Lnrd as:
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where Ld(zr) is the RLD at zr (cm cm–3). Since Lnrd(zr) is normalized by rooting depth, it is independent of growth stages and can be used to characterize RLD distributions with a single function.
Literature Data
We searched for all possible data of wheat RLD distributions in the literature using the following databases: CABI (Center for Agriculture and Biology International), AGRICOLA, AGRIS, Water Resources Abs, CJFD-CNKI (Chinese Journal Full-text Database—China National Knowledge Infrastructure), WangFang Data (Chinainfo), and CALIS (China Academic Library and Information System), using the retrieval keywords "root length density + wheat" and the retrieval period from January 1995 to April 2002. From this, we selected papers that documented complete (along the whole root depth) and measured data of RLD distributions. The procedure resulted in 89 data sets from 10 papers (Asseng et al., 1997; Lotfollahi et al., 1997; Anderson et al., 1998; Feng and Liu, 1998; Feng et al., 1998; Ephrath et al., 1999; Li et al., 1999; Wu et al., 1999; Zubaidi et al., 1999; Fan et al., 2000). The data sets resulted in a total of 610 data points measured for different climate types in the world (Australia, China, Germany, Israel, etc.), wheat species [Triticum aestivum L. (cv. Factor, Molineux, etc.), Triticum turgidum L. conv. durum, etc.], growth stages and cropping systems (both in the field and in the laboratory), soils (sand, fine dune sand, loamy sand, loam, sandy loam clay, clay), water, and nutrient supplies (N, P, K, and other microelements), among others. All experiments were conducted under surface irrigation.
The data were transformed into the NRLD distributions (Eq. [7]) using the following procedure. First, the measurement depth z was normalized as zr = z/Lr according to the rooting depth of each case. Then the integration 
10Ld(zr)dzr was calculated numerically with the trapezoidal formula. The numerical integration step was set to be 1/Lr and values of RLD between two successive measured points were interpolated linearly to guarantee the accuracy of the numerical integration.
Experiments
Experiments in soil columns and in the field were conducted to measure distributions of winter wheat (Triticum aestivum L. cv. Nongda 189) RLD and soil water content. The column experiment was performed in a greenhouse using polyvinyl chloride (PVC) columns that were 45 cm high and 10 cm in diameter. Fifty-four columns were used for the experiment with three treatments (CW0, CW1, and CW2) of different water stresses (Table 1). For each treatment, 15 columns were setup to measure wheat–root distributions at different soil layers and time during the growing period. Each column was split vertically into two halves. At the beginning of the experiment, the split columns were taped together and all columns were sealed with PVC back covers at the bottom. The columns were packed with a fine sandy soil (bulk density of 1.64 g cm–3). Three soil columns for each treatment were setup to monitor soil matric potential changes and the amount of supplied water by installing five tensiometers at depths of 5, 10, 15, 25, and 35 cm.
Winter wheat was planted in the columns using a seed density of four plants per column similar to that in the field (4.5–6.0 million plants per hm2). The top of each column was filled with 3 cm of quartz sand to reduce surface evaporation. Sufficient nutrients were supplied to all the treatments. The experiment lasted for 42 d (from 27 Apr. to 8 Jun. 2001). Root samples were taken every 6 d and for five times during the experimental period. At each sampling time, three duplicate columns for each treatment were opened to extract soil cores. The soil cores were cut into 4-cm soil layers, each of which was washed to collect roots.
The field experiment in wheat was conducted between the stage of turning green (15 Mar. 2002) and the harvest stage (10 Jun. 2002) with three treatments (FW1, FS1, and FS2, Table 1) and two duplicate plots. The plots were irrigated 3 or 4 times, depending on the treatments. To measure root density in the field, soil cores were sampled using a 15-cm tall auger with a 10-cm i.d. The soil cores were washed to collect roots. Roots were sampled four times during the experimental period on 16 Apr., 8 May, 18 May, and 2 Jun. 2002.
The roots collected from each soil layer in the column and field experiments were scanned with a SNAPSCAN 1236 scanner (AGFA, Germany) and analyzed using the WinRHIZO Pro software package (Regent Instruments Inc., Canada), from which RLD distributions were determined. The measured values of RLD distributions were transformed to NRLD using Eq. [7].
In the field experiment, the soil profiles consisted of two layers. The texture of the upper 70-cm layer was a sandy loam, whereas the lower 70- to 200-cm layer was a fine sand. Neutron probes were installed in all plots to measure soil water content at depths of 10, 20, 30, 40, 50, 70, 90, 120, 150, and 180 cm. Soil water retention curves were measured with the pressure plate method in the laboratory using soil samples taken from the field with six duplicates for each soil (Klute, 1986). Values of the saturated hydraulic conductivity for the two soil types were measured in the field using the steady infiltration method. Soil water retention and unsaturated hydraulic conductivity functions were described using the van Genuchten (1980) expression. The hydraulic parameters are summarized in Table 2.
Soil surface evaporation E(t) was measured daily, using microlysimeters (Boast and Robertson, 1982; Evett et al., 1995). Three microlysimeters were installed in each plot and weighed at the same time every morning. The potential evapotranspiration rate ETp (cm d–1) and potential evaporation rate Ep (cm d–1) were evaluated using Penman-Monteith (Monteith, 1965) and the modified Penman (Ritchie, 1972) equations, respectively. The required meteorological data were collected from a nearby meteorological station. The potential transpiration rate Tp was calculated from Tp = ETp – Ep.
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RESULTS AND DISCUSIONS
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All 610 data points of wheat NRLD distributions from the literature were pooled in Fig. 1
. As shown, the NRLD distributions [Lnrd(zr)] versus the normalized root depth converged to a general trend, indicating that a generalized function for wheat root distributions is feasible. The values of Lnrd(zr) are maximal close to the soil surface, decrease gradually downwards, and reach zero when zr = 1 (corresponding to the rooting depth Lr). We propose the following generalized function (GF) for the NRLD distribution:
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where a, b, c, and d are fitted coefficients. Using a nonlinear optimization procedure to the pooled data, we obtained values for a = 4.522, b = 5.228, c = 9.644, and d = 2.426. The fitted curve is shown in Fig. 1, with a coefficient of determination (r2) of 0.72.
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Fig. 1. Data of normalized root length density of wheat and curves of the generalized function (GF), linear function (LF), and Wu et al.'s function (1999) (WF). The triangles denote the 39 points used by Wu et al. (1999).
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For comparison, we also fitted the NRLD data to a linear function (r2 = 0.57):
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which is also shown in Fig. 1. The linear function matches the NRLD data poorly in the upper (0 < zr < 0.2) and lower (0.8 < zr < 1.0) regions of the normalized depth. Especially as zr > 0.84, values of the NRLD calculated from the linear function become negative, which is physically meaningless.
Based on 39 data of wheat RLD distributions in the literature, Wu et al. (1999) introduced the following third-order polynomial equation to describe the NRLD distribution:
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The curve of Eq. [10] is also shown in Fig. 1. Although Eq. [10] fitted the 39 data points very well (with r2 = 0.96), the coefficient of determination (r2) between the estimated results of Eq. [10] and the 610 data that included the 39 points, was only 0.63. The function does not fit the NRLD data well within the upper region of the normalized rooting depth.
Generated from much large population of samples, the statistical result in this study (Eq. [8]) should be more representative than Eq. [10]. The mean of 95% uncertainty intervals of the generalized function along the normalized depth was within ±0.29.
Root growth is influenced by many factors, such as soil type, soil water content, nutrient level, plant species, and atmospheric condition, among others. Rooting depths and RLD distributions change with time and space continuously. However, the generalized function for wheat NRLD distributions is only dependent on the normalized root depth and independent of other factors. Therefore, the generalized function can be used to describe wheat root distribution and root growth under various conditions, thereby providing useful information for root–water and nutrient–uptake simulations and soil–root–shoot–atmosphere interaction studies.
Measured rooting depths (Lr) for treatment CW1 and FW1 at different time are listed in Table 3. The NRLD distributions Lnrd(zr) of winter wheat in the column and field experiments are compared with Eq. [8], [9], and [10] in Fig. 2
. The generalized function (Eq. [8]) was in good agreement with the measured data for the column and field experiments with determination coefficients r2 = 0.90 and 0.89, respectively, whereas Eq. [9] and [10] did not match the measured Lnrd(zr) data well.
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Fig. 2. Comparison between experimental data of normalized root length density distributions of wheat and estimated results using the generalized function (GF), linear function (LF), and Wu et al.'s function (1999) (WF).
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Root mean squared errors (RMSE1) between the measured Lnrd(zr) and calculated values using Eq. [8], [9], and [10] for different treatments are compared in Table 4. The averaged RMSE1 values were 0.565, 0.970, and 0.900 for Eq. [8], [9], and [10], respectively. The experiments were conducted under various conditions, for example, different water supplies and water sources, different growth stages of winter wheat, and atmospheric conditions. Therefore, the results further demonstrated that it is useful and reasonable to utilize the generalized function to describe the NRLD distribution.
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Table 4. Root mean squared errors (RMSE1) between measured and calculated Lnrd(zr) values using the generalized function (GF), the linear function (LF), and the Wu et al.'s function (1999) (WF). Root mean squared errors (RMSE2) between measured and estimated root length density values using the GF, LF, and WF, and root mean squared errors (RMSE3) between measured and simulated soil water content values using the GF, LF, and WF.
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In practice, if the rooting depth Lr and RLD at one depth (e.g., near the soil surface) are known at several growing stages, the RLD distribution at other depths can be estimated easily using Eq. [7]. We demonstrated the estimation using the measured data for treatments of CW1 in the column experiment and FW1 in the field experiment. With the rooting depths Lr in Table 3, each observation depth z was normalized to zr, that is, zr = z/Lr. For treatment CW1 with measured RLD at z = 2 cm, we had zr = 2/Lr and calculated the Lnrd(zr) value using Eq. [8]. The integration 10Ld(zr)dzr, which is a constant along the normalized depth, was evaluated using Eq. [7], that is, 10Ld(zr)dzr= Ld(zr)/Lnrd (zr = 2/Lr). According to Eq. [7], the root length densities at different depths were estimated by
 | [11] |
in which Lnrd(zr) for different depths was calculated using Eq. [8]. The estimated RLD distributions were compared with the measured values in Fig. 3a
. Similarly, for treatment FW1, using Eq. [7], Eq. [8], and the measured RLD at z = 7.5 cm, we estimated RLD distributions and compared with the measured values in Fig. 3b. Following the same procedure, we estimated the RLD distributions of winter wheat for treatments CW1 and FW1 at different time using Eq. [9] and [10] and compared the estimated and measured values in Fig. 3. In both examples, the estimated RLD distributions from the generalized function (Eq. [8]) compared reasonably well with the measured data, having the smallest root mean squared errors (RMSE2) among the three functions (Table 4). The comparable agreement indicated that the generalized function is capable to estimate wheat RLD distributions effectively with only two parameters: the rooting depth Lr and the measured RLD at one depth (usually near the soil surface for convenience). The required information is minimal and can be determined relatively easily.
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Fig. 3. Comparison between experimental data of root length density distributions of wheat and estimated results using the generalized function (GF), linear function (LF), and Wu et al.'s function (1999) (WF) for treatments of (a) CW1 in the column experiment and (b) FW1 in the field experiment.
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The development of the generalized function for NRLD distributions is useful when simulating soil water movement in a soil–wheat system, without observed RLD distributions. The required information is the rooting depths at different growing seasons, which can be acquired relatively easily. The following example applied the generalized function to simulate soil water flow for the scenario of the field soil–wheat experiment (treatment FW1). The simulated depth L was 180 cm (L > Lr), and the soil matric potential at the lower boundary was interpolated linearly in time using measured values. The threshold values h1 and h2 in Eq. [6] were set to –1500 and –64 cm, respectively (Musters and Bouten, 2000) and the index ß was optimized to a value of 0.35 using the measured RLD distribution (Zuo and Zhang, 2002). Based on the above parameters [including the hydraulic parameters, E(t), Tp, ß, h1, and h2], the rooting depths in Table 3, and the Lnrd(zr) function of Eq. [8], the soil water flow equations (Eq. [1]–[6]) were solved numerically using the implicit finite difference method with a numerical spatial step of 
z = 1 cm and a time step of tj+1 = 1.25tj. Simulation of soil water flow in treatment FW1 started on 3 Apr. 2002 (t = 0) after irrigation. The measured soil water content profiles on 9 Apr. and 27 Apr. 2002 were compared with the simulation results in Fig. 4
. Using the two Lnrd(zr) functions of Eq. [9] and [10], we repeated the simulations and present the simulated soil water content profiles in Fig. 4 as well. The root mean squared errors (RMSE3) between the measured soil water contents on 9 and 27 Apr. 2002 and the simulated values using the three Lnrd(zr) functions are listed in Table 4. In general, the simulated soil water content profiles with the different Lnrd(zr) functions were similar and matched the measured soil water distributions reasonably well (Fig. 4 and Table 4).
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Fig. 4. Comparison of measured soil water content distributions in FW1 soil–wheat system and simulated profiles using the generalized function (GF), linear function (LF), and Wu et al.'s function (1999) (WF) for duration from 3 to 27 Apr. 2002.
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It appears that the simulated soil water content profiles with root–water uptake were insensitive to the type of NRLD distributions. In other words, different root–water uptake models can be used to simulate nearly identical soil water content distributions in the soil–plant system. Nevertheless, accurate information of root distribution and development may be more critical in other applications. In a review of the value of using process-oriented models of water and nutrient uptake in improving integrated agriculture, van Noordwijk and van de Geijn (1996) summarized the studies in plant growth and demand for water and nutrients, models of root growth and uptake of water and nutrients, models to evaluate uptake capacity given a certain root development, and models with root growth interacting with the shoot. They concluded that models for nutrient and water uptake based on actual root development are necessary to successfully improve the agronomic efficiency of fertilizer and water use. They stated that the increasing concern for unavoidable water stress, nutrient insufficient and surplus situations, and environmental consequences of farm management practices in the present integrated agricultural systems requires a more comprehensive understanding of root performance, such as root–water and nutrient–uptake and soil–root–shoot–atmosphere interaction. For such applications, models require a more accurate description of root development and distribution. Undoubtedly, the RLD distribution should be a critical parameter in such models.
The RLD comprises the total length of all collected roots in a soil layer, including effective and ineffective roots. Only effective roots actively take up water from the soil. However, this distinction is not made in the RLD distribution. Thus, we should be careful in relating the RLD distribution to active root water uptake. The relationship between RLD distribution and root–water–uptake functions needs further research.
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CONCLUSIONS
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Data sets of RLD distribution of wheat were collected from the literature to develop a generalized function to characterize a NRLD distribution. The data sets were measured for a wide range of conditions, such as soil type, wheat species, climates, water and nutrient applications, and cropping systems. A total of 610 data points of RLD distributions was normalized. The resulting NRLD distributions [Lnrd(zr)] of wheat showed a consistent trend, when plotted versus the normalized rooting depth. The data were fitted to an exponential function with a coefficient of determination (r2) of 0.72. Because of the large amount of data used, the generalized exponential function should be more representative and practical than the polynomial equation of Wu et al. (1999). To examine the accuracy of the generalized function, NRLD distributions of winter wheat were measured from extensive experiments in soil columns and in the field, conducted using different levels of irrigation water and wheat growing stages. The generalized function matched the measured Lnrd(zr) distributions well with coefficients of determination (r2) of 0.89 and 0.90, respectively, for the field and column experiments. Compared with a linear model and the polynomial equation of Wu et al. (1999), the generalized function provided a much better estimation of RLD distribution. However, simulated results of soil water dynamics in soil–wheat systems were similar using any of the three functions.
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ACKNOWLEDGMENTS
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The authors thank Drs. B. Li and L. Ren for their useful discussions and suggestions and thank Drs. L. Ren and G. Huang to provide the soil hydraulic parameters of the field experiment. This study was supported partly by the National Key Basic Research Special Funds (NKBRSF), China (grant no. G1999011700), Hi-Tech Research and Development Program of China, and Huo Ying Dong Educational Fund Council.
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