Estimation of root water uptake and soil hydraulic parameters from root zone soil moisture and deep percolation

Agricultural Water Management 222 (2019) 38–47

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Estimation of root water uptake and soil hydraulic parameters from root zone soil moisture and deep percolation

T

⁎

Ickkshaanshu Sonkara, , Hari Prasad Kotnoora, Sumit Senb a b

Department of Civil Engineering, Indian Institute of Technology Roorkee, Roorkee, 247667, India Department of Hydrology, Indian Institute of Technology Roorkee, Roorkee, 247667, India

A R T I C LE I N FO

A B S T R A C T

Keywords: Root water uptake Soil moisture depletion Deep percolation Inverse approach

For efficient irrigation management practices, an accurate prediction of water uptake in the root zone and soil information are foremost important. The present study deals with the identification and estimation of root water uptake (RWU) and soil hydraulic parameters using inverse modeling. These parameters were estimated by minimizing the difference between observed and model simulated soil moisture and deep percolation during the crop growth period. The linked simulation optimization model is tested for three different objective functions using hypothetically generated observed data. Results indicate that the optimizer with objective function defined by soil moisture, failed to provide unique estimate of RWU and soil hydraulic parameters. Further, it has been observed that with the objective function defined by deep percolation, soil hydraulic parameters were uniquely estimated but RWU parameter was not estimated accurately. However, with the objective function, that includes both soil moisture and deep percolation, these parameters were uniquely estimated. A Lysimeter experiments were conducted with four crops i.e. berseem (Trifolium alexandrinum), wheat (Triticum aestivum), maize (Zea mays) and pearl millet (Pennisetum glaucum). Daily monitoring of soil moisture and deep percolation along with soil and crop parameter measurements were done for model validation. Inversely estimated soil hydraulic parameters were found to be in close agreement with laboratory obtained values. The results indicate that specifically for soils with high hydraulic conductivity, the information about deep percolation along with soil moisture is necessary for inverse estimation of root and soil parameters simultaneously. The moisture depletion pattern and deep percolation corresponding to optimized parameters for these crops were found to be in close agreement with observed values.

1. Introduction The hydrological cycle comprises of a number of key elements of which root water uptake (RWU) has a significant role in controlling water fluxes in soil profiles as it links the soil subsurface with the atmosphere in irrigated fields. The study of RWU is important because the roots withdraw most of the soil water from the root zone in response to transpiration (Chahine, 1992; Feddes et al., 2001). Zheng-feng et al. (2015) illustrated the various existing models that predict the root water extraction. The functioning of these models is broadly classified into two approaches i.e. microscopic approach and macroscopic approach (Feddes and Raats, 2004; Molz, 1981; Ojha et al., 2009). The microscopic approach describes the water movement through individual roots considering water flux in both the root system and soil in vicinity of roots, while in the macroscopic approach, the root system is considered as a single sink term representing the sum of water uptake

⁎

by individual roots. From the viewpoint of simplicity, the macroscopic based model are mostly favored in moisture flow models in the root zone (Couvreur et al., 2012). The root water extraction depends upon several factors, such as root density distribution, root hydraulic properties, soil water availability and soil salinity. Yu et al. (2007) highlighted that the high root density of maize plants is more vital than the maximum rooting depth for water uptake under water stress condition. Feddes et al. (1978) proposed a model that involves the prescribed function of the soil moisture pressure head that reduces potential transpiration to actual root water uptake under water stress condition. With the knowledge of root and soil hydraulic properties, the distribution of water extraction by the roots can be predicted by using these models (Feddes et al., 1976; Kang et al., 2001; Li et al., 1999; Molz et al., 1968; Ojha and Rai, 1996; Prasad, 1988). Numerous models based on macroscopic approach are used to determine the RWU taking care of water stress (Belmans et al., 1983;

Corresponding author. E-mail addresses: [email protected] (I. Sonkar), [email protected] (H.P. Kotnoor), [email protected] (S. Sen).

https://doi.org/10.1016/j.agwat.2019.05.037 Received 17 October 2018; Received in revised form 18 May 2019; Accepted 27 May 2019 Available online 03 June 2019 0378-3774/ © 2019 Elsevier B.V. All rights reserved.

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parameters required for model development.

Feddes, 1978; Jordan and Ritchie, 1971), soil salinity (Jordan and Ritchie, 1971; Li et al., 2015; Skaggs et al., 2006) and compensating effect (Couvreur et al., 2012; Javaux et al., 2013). From several studies, it has been found that the roots can compensate the water stress by withdrawing more water from non-stressed zone (Leib et al., 2006; Šimůnek and Hopmans, 2009; Vogel et al., 2013, 2013; Yadav et al., 2009). One of the simplest approaches was defined by Jarvis (1989). Jarvis (1989) introduces stress index, ω, which is the ratio of noncompensated to potential root water uptake. This simple empirical approach is widely used in modeling moisture flow in root zone (Šimůnek and Hopmans, 2009). However, field or laboratory measurement of crop root parameters like root density distribution is extremely challenging. Furthermore due to soil heterogeneity, the directly measured soil hydraulic properties on soil samples do not represent exactly the field conditions (Vereecken et al., 2010). Therefore, the root water extraction pattern predicted from forward modeling corresponding to directly measured soil hydraulic and root properties is not much precise. One of the ways to accurately estimate these parameters is by applying inverse estimation, which involves estimation of these parameters based on observed data. The soil water dynamics in the root zone is interrelated to RWU, soil water flux, and top and bottom boundary conditions. Thus, the information of root water uptake and soil hydraulic properties for the known initial, top and boundary conditions can be acquired by modeling dynamic process of RWU. The traditional inverse method is “trial and error” procedure, which involve manual comparison of computed and observed state variable (e.g. soil moisture content, deep percolation, soil water potential, etc.). The disadvantages of this method are time consuming, inaccuracy when several parameters are involved and non-uniqueness (Mbonimpa et al., 2015). An alternative to “trial and error” approach is to link numerical model with an optimization algorithm. Šimůnek et al. (1998) uses Marquardt-Levenberg optimization method to estimate soil hydraulic parameters from laboratory evaporation experiment. Vrugt et al. (2001) presented the 2-D RWU model based on Raats (1974) model. The developed model was linked to soil moisture flow model to optimize RWU parameters by minimizing the defined objective function related to soil moisture content. Sonkar et al. (2018a) estimated the non-linear root water uptake parameter by linking root water uptake model with the genetic algorithm (GA) based optimizer. The parameter was estimated by minimizing the difference between simulated and observed percentage soil moisture depletion. The inverse approach is also used to identify both RWU and soil hydraulic parameters simultaneously. Wöhling et al. (2013) use the AMALGAM search algorithm to estimate simultaneously soil hydraulic and plant parameters using soil moisture data. Similarly using bottom boundary flux data from lysimeter experiment Schelle et al. (2013) uniquely identified soil hydraulic and RWU parameters. However, from the results of Musters and Bouten (2000) and Musters and Bouten et al. (1999) it has been seen that the RWU parameters does not produce significant effect on soil moisture dynamics which might lead to identifiability problems for those parameters. Also, it has been observed by Hupet et al. (2003) that for soil with high hydraulic conductivity, the root water uptake parameters cannot be identified uniquely with the soil moisture status in the root zone. Similarly, Sonkar et al. (2018b) observed that with only soil moisture data, there are multiple solutions while estimating RWU and soil hydraulic parameters simultaneously. The first objective of the present study is to investigate the feasibility of simultaneous estimation of RWU and soil hydraulic parameters using root zone soil moisture data. For this, the developed RWU model is coupled with genetic algorithm (GA) based optimizer. The coupled model is tested by inverse estimation of these parameters using hypothetically generated observed soil moisture depletion and deep percolation rate. The second objective is to estimate these parameters using both soil moisture and deep percolation rate for different crops. Full crop season lysimeter experiments were conducted to obtain crop

2. Material and methods 2.1. Forward modeling 2.1.1. Governing equation The soil moisture and the deep percolation in the root zone is analysed by solving Richards equation (1931) with root water uptake as a sink term. The governing equation is given as

∂θ (ψ) ∂ψ ⎞ ∂K (ψ) ∂ ⎛ K (ψ) = − S (z , t ) + ∂t ∂z ⎝ ∂z ⎠ ∂z

(1)

where ψ = soil pressure head (L); θ = volumetric moisture content (L3L−3); K = unsaturated hydraulic conductivity (LT−1) ; t = time (T) ; z = vertical coordinate taken positive upward (L) and S(z,t) = sink term representing water uptake by roots expressed as volume of water per unit volume of soil per unit time (T−1). 2.1.2. Constitutive relationships To solve the non-linear Eq. (1), explicit expressions between the dependent variable (ψ ) and the nonlinear terms (K and θ) are required. In the present study, relationships proposed by van Genuchten (1980) are adopted, which are as follows: θ - ψ Relationship mv

1 ⎧ ⎤ ⎪⎡ Θ= ⎢ 1 + |α v ψ|nv ⎥ ⎣ ⎦ ⎨ ⎪ 1 for (ψ > 0) ⎩

for (ψ ≤ 0) (2)

−1

where αv (L ) and nv (dimensionless) are unsaturated soil parameters with mv = 1 − (1/nv) for nv > 1; and Θ = effective saturation (dimensionless) defined as:

Θ=

θ − θr θs − θr

(3) 3 −3

where θs = saturated moisture content (L L moisture content of the soil (L3L−3). K - θ Relationship:-

(

K (ψ) = K sat Θ1/2 [1 − 1 − Θ1/ mv

)m ]2 for ψ v

); and θr = residual

< 0

(4)

K (ψ) = K sat for ψ ≥ 0

(5) −1

where Ksat = saturated hydraulic conductivity (LT

).

2.1.3. Initial and boundary conditions Initially, the soil is assumed to have observed soil moisture content at the first day of planting the crop. Hence (6)

θini (z ) = θmeasured (z ) 0 ≤ z ≤ L, t= 0 3 −3

where θ measured (z) = measured soil moisture content (L L ), and L = length of the root zone (L). The soil surface is exposed to evaporation (Neumann boundary condition) and irrigation/precipitation (Dirichlet type boundary condition) i.e.

ψ (z ) = ψi/r z = L, during irrigation/rainfall

(7a)

∂ψ + 1⎞ = Es z = L, in absence of irrigation/rainfall − K (ψ) ⎛ ⎝ ∂z ⎠

(7b)

where ψ i/r = pressure head corresponding to saturated moisture content (ψ = 0), prevalent during irrigation or rainfall, and Es = Soil evaporation (LT−1). Since, the water table is deep in the study area and the study is focused in the root zone, the lower boundary condition was assumed to be gravity drainage which is Neumann or flux type boundary condition 39

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given by

∂ψ = 0 z = 0, t≥ 0 ∂z

⎧ 0 ψ ≤ ψw ⎪ (ψ − ψw )/(ψamc − ψw ) ψw < ψ < ψamc ⎪ f (ψ) = 1 ψamc ≤ ψ ≤ ψfc ⎨ ⎪ (ψ − ψa )/(ψfc − ψa ) ψfc < ψ < ψa ⎪0 ψ ≥ ψ a ⎩

(8)

The potential transpiration Tp (LT−1) can be directly computed by removing soil surface flux part (equal to Es) from crop evapotranspiration ETc [Eq. (9)], where ETc is obtained by multiplying reference evapotranspiration (ETo) by an appropriate crop coefficient. The ETo is calculated using the Penman-Montieth equation (Allen et al., 1998). This method uses climatic parameters such as maximum and minimum temperature, relative humidity, sunshine hour, and wind speed. The climate data were obtained from a weather station installed at National Institute of Hydrology (NIH), Roorkee.

Tp = ETc − Es

2.2. Numerical scheme A computer code in ‘FORTRON 77′ programming language has been developed for the implementation of RWU model. The governing partial differential Eq. (1) along with constitutive relationships (Eqs. 2–5) are solved by a mass conservative fully implicit finite difference scheme proposed by Celia et al. (1990). The length of root zone is divided into uniform segments. Finite difference approximations of Eq. (1) are written for each node. The resulting system of non-linear equations are linearized using Picards scheme, which is solved using Thomas algorithm. The iterative process is continued until the convergence is achieved i.e. the difference between pressure heads computed between two successive iterations falls within the specified tolerance limit. The solution provides pressure heads at each nodal point in the root zone at successive intervals of time. These pressure heads are used to compute the soil moisture depletion and deep percolation at successive time intervals.

(9)

where Es is calculated using experimentally monitored leaf area index (LAI) according to relation given by Belmans et al. (1983) as:

Es = ETc exp−0.6LAI

(10)

2.1.4. Root water uptake rate In the present study, the root water uptake calculation is done by using Ojha and Rai (1996) (herein after called O–R model). The potential root water uptake at depth z [Smax(z)] under the non-stressed condition is defined as follows:

2.3. Soil moisture depletion calculation

β

Smax (z ) =

Tp (1 + β ) ⎡ z 1 − ⎛ ⎞ ⎤ for 0 ≤ z ≤ zr ⎢ ⎥ zr z r ⎠⎦ ⎝ ⎣ ⎜

The soil moisture depletion [D(x)] at any discrete time interval is defined as the difference between the soil moisture content at the start and the end of time interval, given as

⎟

(11)

where β = root water uptake model parameters (dimensionless) and zr = root depth (m). Under soil moisture limiting condition, the actual extraction of soil water by the roots is lower than potential extraction and hence the water uptake by the roots depends on pressure head (Feddes et al., 1978). The actual root water uptake rate representing sink term in Eq. (1) is then obtained by modifying Smax with a reduction factor of soil moisture pressure heads. The net root water uptake rate at depth z [S (z)] is calculated from

S (z ) = f (ψ) Smax (z )

(13)

tnode − 1

⎤ ⎡ D (x ) = Δz ⎢ ∑ (θiini − θires ) ⎥ ⎦ ⎣ i = bnode + 1 Δz ini res ini res + − θtnode [(θbnode − θbnode ) + (θtnode )] 2

(14)

where θiini and θires are soil moisture contents at node i at the start and end of any discrete time interval respectively, bnode and tnode denote the bottom and top node of the layer x, and Δz = nodal distance. The total soil moisture depletion in the root zone is obtained as summation of depletions from each layer. The percentage soil moisture depletion MD* (x) in layer x is calculated as

(12)

where f (ψ ) = prescribed function of soil moisture pressure head. The pressure head lying between available moisture content (ψamc ) and wilting point (ψw ) or between field capacity (ψfc ) and anaerobiosis point (ψa ) defines water stress condition. In this condition the, f (ψ ) will be less than unity and interpolated linearly (Fig. 1). The water uptake by the roots is maximum (i.e. f (ψ ) = 1) for ψamc < ψ < ψfc . Below ψw and above ψa the plant will not be able to extract soil water from the roots (i.e. f (ψ ) = 0). The linear interpolation for f (ψ ) is formulated as:

MD* (x ) =

D (x ) nlayer

× 100

∑k = 1 D (k )

(15)

where nlayer = total number of layers in a root zone. 2.4. Inverse modeling Three objective functions are formulated (objective functions (16)–(18)) to estimate simultaneously the non-linear RWU parameter β and the soil hydraulic parameters Ksat, αv, and nv by minimizing the objective function ϕ . Objective function (16) is defined as sum of squares of normalized differences between the model predicted and field observed percentage soil moisture depletions for minimization, while objective function (17) is defined as sum of squares of normalized differences between model predicted and field observed deep percolation for minimization problem. Objective function (18) is defined as summation of normalized deviations between model predicted and field observed percentage soil moisture depletions and deep percolation. n

ϕ= Fig. 1. Definition sketch of f (ψ ) (Feddes et al., 1978).

i=1

40

2

MD* [ ] − MD' [ ] ⎞ ⎟ MD* [ ] ⎝ ⎠

∑ ⎜⎛

(16)

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I. Sonkar, et al. m

ϕ=

i=1 n

ϕ=

2

DP * [ ] − DP ' [ ] ⎞ ⎟ DP * [ ] ⎝ ⎠

∑ ⎜⎛

(17) 2

MD* [ ] − MD' [ ] ⎞ ⎟ + MD* [ ] ⎝ ⎠

∑ ⎜⎛ i=1

m

2

DP * [ ] − DP ' [ ] ⎞ ⎟ DP * [ ] ⎝ ⎠

∑ ⎛⎜ i=1

(18)

*

where MD [] and MD'[] are observed and simulated percentage soil moisture depletion respectively, DP*[] and DP'[] are observed and simulated deep percolation respectively, n is total number of observed percentage soil moisture depletion, and m is total number of observed deep percolation data. The normalization brings the magnitude of both the terms in objective function (18) to same order. Since the main objective of the present study is parameter identification, equal weightage is assigned to each function in objective function (18). The parameters are optimized with each of the defined objective functions (16)–(18) by minimizing ϕ using GA optimization technique. Unlike Levenberg-Marquardt method, which often gets stuck at local minima, GA is a derivative-free global search technique that assures global optimal solution (Majed and Houssem, 2009). However, it requires more computational operations than other non-linear optimization techniques. But, nowadays with the advancement of fast computing processors, the computational time is not of much concern. Since GA works on problem solution through a maximization, the minimization problem is converted into a maximization problem by defining the fitness function [f (x)] that is used in genetic operations (Deb, 2000). The fitness function is given by:

f (x) =

1 1+ϕ

Fig. 2. Variation of root depth during the crop season for wheat, berseem, maize and pearl millet.

exposed root is measured. For calculating the potential transpiration, the crop evapotranspiration is partitioned using leaf area index (LAI). The monitoring of the LAI was done using AccuPAR Ceptometer model LP-80 (Decagon Devices Inc., Pullman, WA, USA). The instrument based on the principle of radiation measurement thus accurately measures the LAI in real time. Figs. 2 and 3 shows the variation of root depth and LAI for four crops during crop growth period respectively. The root depth measurement was taken weekly while LAI was measured weekly during initial stage. However, from the development stage, the LAI observation was taken at an interval 3–4 days since LAI is observed to increase rapidly during development stage except for berseem crop. This is because, being a fodder crop, berseem required periodic harvesting during its growth period. The root depth increases till development stages and then afterward observed to be constant. The daily soil moisture status in the root zone was measured at different locations in the experimental site at 10 cm, 20 cm, 30 cm, 40 cm, 60 cm, and 100 cm depth below the soil surface. The in-situ measurement of soil water content was done using soil moisture probe (Profile Probe-PR2/6; Delta T Devices, Cambridge). The probe sensors measure the permittivity of the surrounding material using electromagnetic waves, which is used to measure soil moisture. The soil moisture at the mentioned depth was monitored by inserting the profile probe in the access tube installed in the field plots and lysimeters. To determine the soil properties, field representative samples were collected at three spots of the field plots at five depths from 0 to 140 cm and tested in the laboratory. Table 2 provides average measured soil characteristics at different depths. From Table 2 the soil texture in the root zone can be described by sandy loam type. The field capacity (FC) and permanent wilting point (PWP) refer to moisture content corresponding to matrix suction value of 33 Kpa and 1500 Kpa. These parameters were determined from the soil moisture characteristic curve obtained from pressure plate test. The observed FC and PWP are converted into ψfc and ψw respectively for determining f (ψ ). The soil retention parameters were obtained from Pressure plate test (Soil moisture corporation, Santa Barbara, California, USA). The experimentally observed soil moisture content and the corresponding equilibrium pressure are fitted with RETC model (van Genuchten, 1980) to

(19)

2.5. Experiments The field observations of soil moisture and deep percolation experiments for determination of soil and crop parameters were conducted at the irrigation laboratory of Civil Engineering Department, IIT Roorkee, India during November 2016 to October 2017. The climatic condition of the experimental site is described by humid subtropical type. The site is located in the geometric grid of 77° 53′ E and 29°52′ N at an altitude of 274 m above mean sea level. Two season lysimeter experiments were conducted on four crops (Table 1): two in Rabi season (wheat and berseem) followed by two in Kharif season (maize and pearl millet). The experimental site consists of two lysimeters and field plots. The crops were grown in the field plots as well as in lysimeters. The area of lysimeters is 1 m2 having a depth of 1.5 m repacked soil similar to that of the experimental field. An open chamber is built between both the lysimeter for collecting and measuring deep percolated water. This arrangement allows gravity drained water to drain towards outlet, which is finally collected in bucket kept in an open chamber. The quantity of water drained was measured using measuring cylinder. For complete growing season, the deep percolation was measured and recorded on daily basis. The root depth of the crops was measured from a field plot by a method similar to trench profile method in which a trench is excavated near to the crop root and the length of deepest Table 1 Details of crop duration and growth stages. Crop

Wheat Berseem Maize Pearl millet

Date of sowing

24.11.2016 25.11.2016 26.06.2017 26.06.2017

Date of final harvesting

13.04.2017 19.04.2017 04.10.2017 29.09.2017

Total duration (days)

141 146 101 96

Growth stagesa (days) I

II

III

IV

22 53 22 15

35 32 28 23

61 29 26 38

23 31 25 20

a Stages I, II, III and IV indicate initial, development, mid-season and maturity for wheat, maize, pearl millet and stages I, II, III, and IV refer to first, second, third, and fourth cuttings for berseem.

41

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Table 3 Soil hydraulic parameters based on RETC model. Depth (cm)

θr (%)

θs (%)

αv (m−1)

nv

R2

0–15 15–30 30–60 60–90 90–140 Average SD

4.9 4.6 4.2 3.4 4.6 4.3 0.6

35.7 39.9 38.1 39.0 39.1 38.4 1.6

1.6 0.6 1.3 2.2 1.8 1.5 0.6

1.493 1.741 1.545 1.366 1.473 1.523 0.138

0.9751 0.9827 0.9732 0.9750 0.9855 – –

Note: SD = standard deviation; R2 = coefficient of determination.

tested average soil parameters were assigned at each nodal point in the flow domain of the model. With the given soil, crop parameters and boundary conditions, the soil moisture content and the deep percolation was simulated at each time step of 15 min for 15 days. Finally, the layer wise percentage soil moisture depletion and daily deep percolation were calculated, which are treated as observed data for inversely estimating the RWU and soil hydraulic parameters. For generation of hypothetical data, the soil hydraulic parameters considered as average values; i.e., Ksat = 3.89 cm h−1, αv = 1.50 m−1, nv = 1.52, the RWU parameter β is considered as ‘2’, to reflect the nonlinearity in the RWU mechanism.

Fig. 3. Variation of leaf area index during the crop season for wheat, berseem, maize and pearl millet.

obtain soil retention parameters and are shown in Table 3. The saturated hydraulic conductivity Ksat was measured in situ by using Guelph Permeameter (Eijkelkamp Agrisearch Equipment) (Table 3). 2.6. Estimation of RWU and soil hydraulic parameters To estimate the RWU and soil hydraulic parameters using soil moisture data, the optimization technique based on GA has been implemented. The RWU model was coupled with GA based optimizer (Deb, 2000) for estimation of β, Ksat, αv, and nv. The identification/ estimation of the RWU and soil hydraulic parameters were done in two steps. In the first step, these parameters were simultaneously estimated using generated hypothetical generated soil moisture and deep percolation data with the objective functions (16)–(18). For this purpose, the coupled RWU-GA model was run for three times, each time with different objective function. The optimized parameters resulting from each run were checked with the true values of the parameters used to generate hypothetical data. In the second step, these parameters were inversely estimated for the four crops using observed soil moisture and deep percolation data with the best objective function among the three.

3. Results and discussion To check the feasibility of simultaneous estimation of RWU and soil hydraulic parameters using soil moisture data, the parameters were estimated using hypothetically generated data treating as observed data. Parameters estimation using objective functions (16)–(18) is discussed in the following sections. 3.1. Case 1: parameters estimation using soil moisture data The parameters were estimated by minimizing the normalized deviations between observed and computed percentage soil moisture depletion (Objective function (16)). The GA parameters value used in this problem were: chromosome length: 45, population size: 50, number of generations: 250, crossover probability: 0.6, and mutation probability: 0.08. To test the nonuniqueness in parameters identification, the parameters were optimized with three different seed values. Table 4 shows the true and estimated values of the parameters corresponding to different seed numbers. It can be seen from Table 4 that the parameters do not converge to their true values when objective function (16) is used for estimation. With three different seed values, the optimization results in three different sets of parameters indicating the nonuniqueness of the inverse procedure. The nonuniqueness in parameter estimation is explained as follows. The soil moisture movement in the root zone depends on both soil hydraulic parameters and root water uptake, and these two processes occur simultaneously during crop growth. The soil moisture depletion in the root zone is due to the combined effect of soil moisture movement resulting from unsaturated soil dynamics and root water uptake. Hence, using soil moisture depletion data alone (objective function (16)) is not sufficient to estimate both RWU and soil

2.7. Generation of hypothetical data For the generation of hypothetical percentage moisture depletion and deep percolation rate, a reference run of 15 days (for one irrigation interval) was simulated corresponding to actual growing condition of wheat crop. Root zone length of 150 cm is discretized into equally spaced nodal distance of 1 cm. The whole root zone was divided into 6 layers, each of 25 cm, for calculating the layerwise depleted soil moisture content from the simulated soil moisture. The field was irrigated on the first day, therefore initially, the soil moisture content is set to saturation followed by no irrigation/rainfall event until 15th day. The top boundary condition was assigned with soil evaporation flux and the lower boundary is considered as free drainage. The potential transpiration and the soil evaporation were calculated by partitioning crop evapotranspiration based on recorded LAI of the wheat crop. The interval was selected from 96 to 111 days after sowing (DAS). The root length of 1 m was taken as observed during this period. The laboratory Table 2 Average soil physical characteristics of the field plots. Depth (cm)

Sand (%)

Silt (%)

Clay (%)

Soil Type (USDA)

Bulk density (g/cm3)

Particle density (g/cm3)

FC (%)

PWP (%)

θs (%)

Ksat (cm day−1)

0–15 15–30 30–60 60–90 90–140

72.23 68.46 69.12 70.75 69.50

23.40 28.21 27.70 26.76 26.12

4.37 3.33 3.18 2.49 4.38

Sandy Sandy Sandy Sandy Sandy

1.56 1.55 1.53 1.54 1.59

2.57 2.50 2.49 2.45 2.56

19.25 20.05 19.26 19.85 20.20

6.20 7.05 6.49 6.39 6.89

38.52 39.20 39.23 39.00 39.12

91.55 92.05 92.17 94.97 93.89

Loam Loam Loam Loam Loam

Note: FC = field capacity; PWP = permanent wilting point. 42

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Table 4 Estimation of the parameters using hypothetically generated soil moisture and deep percolation data. Parameter

True value

Seed value = 0.30 β 2.0 αv (m−1) 1.50 −1 Ksat (cm h ) 3.89 nv 1.52 Seed value = 0.65 β 2.0 1.50 αv (m−1) −1 Ksat (cm h ) 3.89 nv 1.52 Seed value = 0.80 β 2.0 αv (m−1) 1.50 Ksat (cm h−1) 3.89 1.52 nv

Table 5 Parameter estimates for berseem, wheat, maize and pearl millet crop. Crop

β

αv (m−1)

Ksat (cm h−1)

nv

ϕ

Berseem Wheat Maize Pearl millet

1.55 1.72 1.30 1.83

1.488 1.504 1.510 1.502

3.950 3.800 3.754 3.806

1.511 1.536 1.529 1.527

0.131 0.122 0.163 0.144

Parameter estimation Objective function (16)

Objective function (17)

Objective function (18)

1.831 2.497 4.064 1.617

1.557 1.459 3.967 1.521

2.014 1.506 3.966 1.522

1.722 1.861 3.799 1.624

1.591 1.452 3.866 1.510

2.005 1.496 3.879 1.516

1.798 1.809 3.835 1.531

1.570 1.447 3.988 1.519

2.011 1.513 3.881 1.520

soil moisture depletion parts drives the optimization to provide unique estimate of RWU parameter β,

3.4. Estimation of RWU and soil hydraulic parameters from a lysimeters experiment From the preceeding discussion, it has been found that the observed data including soil moisture and deep percolation rate is best suited for unique estimation of a RWU and soil hydraulic parameters. In the present study, lysimeter experiments were conducted on four crops to determine soil hydraulic parameters (Ksat, αv, and nv) using field observed soil moisture status and deep percolation rate. Table 5 presents the optimized value of the parameters for all four crops. Results show that the optimized values of the soil parameters are found to be same for all the optimization runs for each crop. This has been expected as the crops were grown in the same field. The inversely estimated Ksat, αv, and nv values are in close agreement with those obtained from laboratory tests. Since, the plant roots poses different root architecture, root length, and density, the water extraction pattern differs from crop to crop. The optimal value of parameter β for berseem is 1.55, while for wheat, maize, and pearl millet are 1.72, 1.30 and 1.83 respectively. The simulated potential and actual RWU with the optimized parameters for each crop are shown in Fig. 4. Under non-stressed condition, i.e. when the soil moisture content is in between field capacity and available moisture content (Table 2), the actual RWU rate is equal to potential rate. From Fig. 4 it can be seen that, during the starting days after irrigation or precipitation event, the actual RWU is found to be same as potential rate but for end days, the actual RWU is found to be less than potential rate. From the temporally simulated soil moisture content, the layer wise percentage moisture depletions were calculated. These depletions were compared with the field observed percentage moisture depletion. Figs. 5–8 show the comparison between the observed and simulated soil moisture depletion for berseem, wheat, maize and pearl millet respectively. It has been observed from the moisture depletion pattern that most of the water uptake is from the top layers of the root zone. For a better analysis of depletion pattern, 0.25 m of thickness for each layer is considered. For the upper layers, the simulated soil moisture depletion is in close agreement with the observed depletion. However, for the lower layers, the differences between observed and computed percentage soil moisture depletion is more as compared to top layers. For berseem and wheat crop (Figs. 5 and 6), the predicted percentage moisture depletion corresponding to optimized parameters overestimates the measured moisture depletion in the lower layers. Similarly, for maize and pearl millet crop (Figs. 7 and 8), the simulated moisture depletions in the lower layers were not matched accurately with the observed depletion. This may be due to the slight heterogeneity in the lower root zone. The moisture observation taken at different depths were compared with corresponding simulated soil moisture for the four crops. Along with soil moisture profile, the RWU model also simulated daily deep percolation in a given crop growth period for all four crops. Deep percolation is the amount of gravity drainage water below crop root zone contributed to groundwater. The RWU model simulates the daily-accumulated deep percolation by calculating flux at bottom boundary of the considered soil domain. For observed data, the volume of deep percolated water is directly measured from lysimeter experiment and recorded daily. Measured and

hydraulic parameters uniquely. 3.2. Case 2: parameters estimation using deep percolation data In this case, the parameters were inversely estimated by minimizing the sum of squares of the normalized differences between observed and simulated deep percolation data (Objective function (17)). The GA parameters value adopted are same as in the case 1. The results of parameters identification using deep percolation data are summarized in Table 4. From Table 4 it is interesting to note that, with objective function (15) the soil hydraulic parameters (Ksat, αv, and nv) converged almost near to their true values with different seeds. However, the objective function failed to estimate the RWU parameter β uniquely. It is emphasized here that, deep percolation is the process which occurs well below the root zone and primarily depends on the unsaturated soil moisture dynamics and as such RWU dynamics does not affect much the deep percolation, Hence, inclusion of deep percolation data in the objective function facilitates identification of soil hydraulic parameters uniquely. However, it fails to estimate the RWU parameter β uniquely. 3.3. Case 3: parameters estimation using soil moisture and deep percolation data In this case, the parameters are optimized by minimizing the objective function defined by normalized differences of sum of squares of the observed and simulated percentage soil moisture depletion as well as deep percolation rate (Objective function (18)). Table 4 presents the parameters estimates obtained using objective function (18) for three different seed values. It is evident from Table 4 that using both soil moisture depletion and deep percolation data in the objective function results in unique estimation of both soil hydraulic parameters (Ksat, αv, and nv) and RWU parameter β with all these parameters converging very near to their true value with three different seeds. The ability of objective function (18) to uniquely estimate both RWU and soil hydraulic parameters is explained as follows. As already mentioned; the soil moisture depletion in the root zone is due to combined effect of unsaturated soil dynamics and RWU. However, the deep percolation which occurs well below the root zone, primarily depends on unsaturated soil dynamics influenced by the soil hydraulic parameters. Objective function (18) comprises two parts; (i) soil moisture depletion and (ii) deep percolation. The deep percolation part drives the optimization to estimate the soil hydraulic parameters (Ksat, αv, and nv) uniquely. With these unique estimates of soil hydraulic parameters, the 43

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Fig. 4. Comparisons between potential daily transpiration (Tpot) and root water uptake (RWU = actual transpiration, Tact) for (a) berseem, (b) wheat, (c) maize and (d) pearl millet.

was tested for inverse estimation of these parameters using hypothetically generated observed data sets. The hypothetical soil moisture and deep percolation data were generated based on the actual crop grown field condition of wheat. These data sets were treated as observed data and compared with simulated values in order to genetically optimize the nonlinear RWU parameter β of O–R model and soil hydraulic parameters Ksat, αv, and nv. The parameters were estimated using three defined objective functions. From the results obtained with the three objective functions, it was found that the optimizer failed to converge RWU and soil hydraulic parameters to their true values when using soil moisture data. With objective function, which includes deep percolation, only soil hydraulic

predicted rates of deep percolation for berseem, wheat, maize and pearl millet are presented in Figs. 9–12 respectively. The simulated deep percolation with the optimized parameters for all the four experiments showed good agreement with the field observed deep percolation. 4. Conclusions In this study, a RWU model is presented that simulates the deep percolation and soil moisture in the root zone for a known root and soil parameters. The developed model was linked with GA based optimizer to inversely estimate the RWU and soil hydraulic parameters using observed soil moisture status and deep percolation. The coupled model 44

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Fig. 5. Comparison of observed and simulated percentage soil moisture depletion at various intervals for berseem crop.

Fig. 6. Comparison of observed and simulated percentage soil moisture depletion at various discrete intervals for wheat crop.

Fig. 7. Comparison of observed and simulated percentage soil moisture depletion at various discrete intervals for maize crop.

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Fig. 8. Comparison of observed and simulated percentage soil moisture depletion at various discrete intervals for pearl millet crop.

Fig. 12. Comparison of observed and simulated daily deep percolation for pearl millet crop.

Fig. 9. Comparison of observed and simulated daily deep percolation for berseem crop.

moisture and deep percolation, the RWU and soil hydraulic parameters were uniquely estimated for different seeds as the optimized soil hydraulic parameters with deep percolation assisted in driving optimization of RWU parameter with soil moisture information. Hence, for simultaneous estimation of RWU and soil hydraulic parameters inversely, both soil moisture and deep percolation data is necessary. The present study also involves the estimation of β, Ksat, αv, and nv for a different cropping experiments using both soil moisture and deep percolation. With the optimized parameters, the soil moisture depletion and deep percolation were simulated and compared with observed data. The results show that the predicted percentage soil moisture depletion and deep percolation with the optimized RWU and soil hydraulic parameters using both soil moisture and deep percolation were in good agreement with observed data. In the present case, nonuniqueness in parameter estimation was observed while considering crop growth condition in the sandy loam soil. It has been in the recent research (Hupet et al., 2003) that for soils with high hydraulic conductivity, the RWU parameter identification problem is ill-posed, whereas for fine soils, the RWU optimization works well. Thus, inverse estimation of RWU parameters along with soil parameters also depends on the type of soil. Hence, prior to application of inverse method for real cases, feasibility assessment of inverse estimation of these parameters should be studied. We also recommend investigating the uncertainty in parameters estimation due to presence of noise in the observed soil moisture and deep percolation data.

Fig. 10. Comparison of observed and simulated daily deep percolation for wheat crop.

Fig. 11. Comparison of observed and simulated daily deep percolation for maize crop.

parameters converge near to their true values with different seed. Hence using deep percolation, inverse estimation of soil hydraulic parameters is assured but RWU parameter cannot be estimated uniquely as deep percolation is mainly associated with unsaturated soil moisture dynamics well below the root zone. However, with both soil

Acknowledgments The authors wish to acknowledge the Nature Environment Research Council (NERC), United Kingdom, and the Ministry of Earth Sciences (MOES), India, for supporting this research work through the research 46

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project, “Sustaining Himalayan Water Resources in a Changing Climate (SusHi-Wat).” The authors also thank the National Institute of Hydrology (NIH), Roorkee, Uttarakhand, India for providing climate data necessary for carrying out the present research.

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