calibrating the spatiotemporal root density …...research article 10.1002/2017wr020452 calibrating...

RESEARCH ARTICLE10.1002/2017WR020452

Calibrating the Spatiotemporal Root Density Distributionfor Macroscopic Water Uptake Models UsingTikhonov RegularizationN. Li1 and X. Y. Yue2

1School of Water Resources and Environment, China University of Geosciences, Beijing, China, 2School of MathematicalSciences, Soochow University, Suzhou, China

Abstract Macroscopic root water uptake models proportional to a root density distribution function(RDDF) are most commonly used to model water uptake by plants. As the water uptake is difficult and laborintensive to measure, these models are often calibrated by inverse modeling. Most previous inversionstudies assume RDDF to be constant with depth and time or dependent on only depth for simplification.However, under field conditions, this function varies with type of soil and root growth and thus changeswith both depth and time. This study proposes an inverse method to calibrate both spatially and temporallyvarying RDDF in unsaturated water flow modeling. To overcome the difficulty imposed by the ill-posedness,the calibration is formulated as an optimization problem in the framework of the Tikhonov regularizationtheory, adding additional constraint to the objective function. Then the formulated nonlinear optimizationproblem is numerically solved with an efficient algorithm on the basis of the finite element method. Theadvantage of our method is that the inverse problem is translated into a Tikhonov regularization functionalminimization problem and then solved based on the variational construction, which circumvents the com-putational complexity in calculating the sensitivity matrix involved in many derivative-based parameterestimation approaches (e.g., Levenberg-Marquardt optimization). Moreover, the proposed method featuresoptimization of RDDF without any prior form, which is applicable to a more general root water uptakemodel. Numerical examples are performed to illustrate the applicability and effectiveness of the proposedmethod. Finally, discussions on the stability and extension of this method are presented.

1. Introduction

Plant root water uptake is an important mechanism that dramatically affects water content distribution invegetated soils and exerts a considerable influence on the water flux to atmosphere and groundwater (Jav-aux et al., 2008; Vrugt et al., 2001). Various mathematical models that simulate the root water uptake havebeen proposed in the past decades, as can be seen in many comprehensive reviews (e.g., Cardon & Letey,1992; Feddes & Raats, 2004; Green et al., 2006; Mathur & Rao, 1999; Molz, 1981; Skaggs et al., 2006; Wang &Smith, 2004). These studies are unusually divided into two main categories: the macroscopic approachwhich concentrates on the integrated properties of the entire root system and the microscopic approachwhich relies on the transport mechanism through single roots. Because it is practically impossible to mea-sure the detailed geometry of the root system, most root water uptake functions have been developedusing a macroscopic rather than a microscopic approach (Molz, 1981; Vrugt et al., 2001).

In the macroscopic approach, the water extraction by plant roots is represented by a volumetric sink termand simply added to the Richards equation for variable saturated water flow:

@hðhÞ@t

5@

@z

�KðhÞ @h

@z21

� ��2Sðz; tÞ; (1)

where h is the water content [L3 L23], h is the pressure head [L], t is the time [T], z denotes the verticalcoordinate that is positively oriented downward, K(h) is the unsaturated hydraulic conductivity [L T21], andS(z, t) is the volume sink term describing the volumetric water uptake by plant roots from a unit volume ofsoil per unit time [L3 L23 T21]. One major difficulty posed by equation (1) stems from the unknown form ofthe sink term S(z, t). The most common formulations (as reviewed by Green et al., 2006; Molz, 1981) that

Key Points:� Root density distribution function

varying with time and depth iscalibrated from pressure headdynamics in the soil profile� Tikhonov regularization constraints

are introduced to deal with the ill-posedness of the inverse problem� The inverse problem is reformulated

as a functional minimization problemand efficiently solved on the basis ofthe finite element method

Correspondence to:N. Li,[email protected]

Citation:Li, N., & Yue, X. Y. (2018). Calibratingthe spatiotemporal root densitydistribution for macroscopic wateruptake models using Tikhonovregularization. Water ResourcesResearch, 54, 1781–1795. https://doi.org/10.1002/2017WR020452

Received 23 JAN 2017

Accepted 22 FEB 2018

Accepted article online 26 FEB 2018

Published online 10 MAR 2018

VC 2018. American Geophysical Union.

All Rights Reserved.

LI AND YUE CALIBRATION OF A ROOT WATER UPTAKE MODEL 1781

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describe this sink term are based on the ideas of the very influential work by Gardner (1964), whichdescribes the microscale physics of water flow from the soil to plant roots. In these formulations, the rootwater uptake is assumed to be directly proportional to a root density distribution function (RDDF), to thedifference of matric potential between the soil and the root, and to the hydraulic conductivity of the soil. Ingeneral, they take the form of

Sðz; tÞ5LKðhÞðh2hrÞ; (2)

where L represents the RDDF usually defined as length of roots per unit soil volume [L22] and hr is the waterpressure head in the root xylem and is assumed to be constant over the root zone [L]. There are many mac-roscopic models that employ expressions similar to equation (2) but with different symbols for the term L(e.g., Bresler et al., 1982; Feddes et al., 1974; Gardner, 1964; Herkelrath et al., 1977; Hillel et al., 1976; Nimah& Hanks, 1973; Rowse et al., 1978; Skaggs et al., 2006; Vogel et al., 2012; Whisler et al., 1968). Someapproaches are proposed to evaluate L using field and laboratory experiments (e.g., Coelho & Or, 1999;Feddes, 1971; Feddes et al., 1974; Nimah & Hanks, 1973; Zuo et al., 2006). However, the complexity of thesoil-root system makes the experiments difficult and labor intensive, and therefore the RDDF is often deter-mined by inverse modeling. Note that in most studies, the RDDF is often assumed to be constant withdepth and time (reviewed by Hopmans & Guttierez-Rav�e, 1988) or accommodated as certain function ofdepth, such as exponential (e.g., de Rosnay & Polcher, 1998; Feddes et al., 1974; Luo et al., 2003, 2008) orother (e.g., Li et al., 2001; Zuo et al., 2004) functions. However, it is clear that the RDDF L under natural fieldconditions will vary with the type of soil, which changes with depth, and plant root growth, which usuallychanges with depth and time (e.g., Cai et al., 2017; Feddes et al., 1976; Hopmans & Guttierez-Rav�e, 1988;Oswald et al., 2008; Raats, 2007; Vrugt et al., 2001; Wang & Smith, 2004).

This paper aims to calibrate the macroscopic root water extraction model from observations of soil waterpressure head (or soil water content). We consider identifying a both spatially and temporally varying RDDFL(z, t) in macroscopic model, that is, the value of RDDF will be identified in a number of gridcells and eachvalue will also vary over time. This calibration is associated with the inverse problem for the Richards equa-tion, which is a complex task because the inverse problem is usually ill-posed, that is, possible lack of identi-fiability and stability is more severe: even small measurements errors may lead to large deviations in thesolution (Bitterlich & Knabner, 2002; Jardani et al., 2007; Vocciante et al., 2016). Such problems in vadosezone research have been approached in various ways from Monte Carlo-based stochastic methods (e.g.,Duan et al., 1992; Durner, 1994; Hupet et al., 2002; Schelle et al., 2012; Vrugt et al., 2004) to derivative-baseddeterministic methods (e.g., Bitterlich & Knabner, 2002; Cockett et al., 2015; Finsterle & Kowalsky, 2011;Finsterle & Zhang, 2011; �Simunek & van Genuchten, 1996). For highly parameterized inverse problemswhose forward simulation is computationally expensive, the derivative-based deterministic method is oftenmore practical and computationally efficient compared to the stochastic method (Finsterle & Kowalsky,2011). For example, the Levenberg-Marquardt method with calculation of the sensitivity matrix is com-monly used for the inverse Richards equation. Nevertheless, the computation is challenging because thesensitivity matrix is generally a large and dense matrix and its computation requires a substantial amount oflinear algebra and storage (Cockett et al., 2015; Finsterle & Kowalsky, 2011), often making the methodimpractical when there are many parameters to estimate, as is the one presented here. Furthermore, ineffi-cient or inaccurate computation of the sensitivity matrix can lead to an extremely slow inversion.

To address these issues—heavy computation and ill-posedness, we propose an inverse method that effec-tively copes with the ill-posedness and the nonlinearity in the inverse problem with good computationalefficiency. First and foremost, we formulate our inverse problem in the framework of the Tikhonov regulari-zation theory (Tikhonov & Arsenin, 1977; Tikhonov et al., 1995), adding smoothing factors to the originalminimization problem and thus adding additional constraints to reduce the solution space. Then, startingwith a semidiscretization scheme for the forward problem, we translate the inverse problem into a func-tional minimization problem and frame it in a spatially continuous system, which circumvents the computa-tional complexity in calculating the sensitivity matrix. Finally, we apply a numerical algorithm based on thefinite element method to effectively achieve the numerical solution of the nonlinear equation formulatedby the inversion. Notice that the Levenberg-Marquardt method also involves the idea of Tikhonov regulari-zation. The presented method differs from it in the use of regularization, in the formulation of the objectivefunction, and in the numerical algorithm used for the optimization problems. We formulate a constraint

Water Resources Research 10.1002/2017WR020452


optimization problem by adding Tikhonov regularization terms to the original inverse problem instead ofsolving the nonlinear least squares (as Levenberg-Marquardt does) and computes the optimization problemby variational construct and the finite element method.

The study is proposed for optimization of a spatially and temporally variable RDDF. It is efficient and conve-nient to identify the unknown parameter over a high-resolution solution space (spatiotemporal scalesherein) because it avoids complex computations of the sensitivity matrix involved in many derivative-basedparameter estimation approaches. Moreover, this study features optimization of a general root water uptakemodel because the RDDF is not necessarily assumed to have any prior function form and will be identifiedin its spatiotemporally discrete form, with which one can use linear (or higher order) interpolation schemesto get the intermediate values or fit an analytical formula to L(z, t).

This paper proceeds as follows. In section 2, we present the forward problem. In section 3, we formulate theinverse problem to calibrate the RDDF based on soil water pressure head dynamics. Synthetic examples arepresented for both infiltration and evaporation experiments in order to illustrate the implementation anddemonstrate the performance of the proposed method in section 4. Then, further discussions on the stabil-ity and extension of this method are demonstrated in section 5. Finally, conclusions are given in section 6.

2. Forward Problem: Unsaturated Water Flow in the Soil-Plant System

Substituting equation (2) to (1) and considering a spatial and temporal-dependent RDDF, we can expressthe unidimensional Richards equation with root water uptake as

@hðhÞ@t

5@

@z

�KðhÞ @h

@z21

� ��2Lðz; tÞKðhÞðh2hrÞ; t � 0; z 2 X; (3)

where X5½0; zd� with zd being the depth of the unsaturated soil column.

To solve equation (3) one needs to define the soil hydraulic properties including the soil water retentioncurve hðhÞ and the unsaturated hydraulic conductivity function K(h). In this study, these properties areparameterized by one of the most frequently used models—van Genuchten model (van Genuchten, 1980)as follows:

hðhÞ5hr1ðhs2hrÞ½11ðajhjÞn�2m; (4)

KðhÞ5Ksf12ðajhjÞn21½11ðajhjÞn�2mg2

½11ðajhjÞn�m2

; (5)

where hs and hr are the saturated and the residual water contents, respectively, [L3 L23], a is the parameterrelating to the mean pore size [L21], n and m (dimensionless) are empirical parameters determining theshape of the soil water retention curve with m5121=n, and Ks is the saturated hydraulic conductivity[L T21].

Equation (3) is called the mixed form of Richards equation because both pressure head h and moisture con-tent h are remained. Taking advantage of the differentiability of the soil retention function, the Richardsequation can be expressed in terms of pressure head h, leading to a h-based form of Richards equation:

CðhÞ @h@t

5@

@z

�KðhÞ @h

@z21

� ��2Lðz; tÞKðhÞðh2hrÞ; (6)

where CðhÞ � dh=dh is the so-called soil water capacity [L21], representing the slope of the soil water reten-tion curve. Our inverse problem will be developed on the basis of this h-based form, which is valid in bothsaturated and unsaturated regions and is applicable to both homogeneous and heterogeneous soils.

Initial and boundary conditions are also required to solve this equation. Here we impose a time-dependentflux boundary (e.g., evaporation) on the surface and a pressure head boundary at the bottom of the soil pro-file, respectively. The initial and the boundary conditions are

hjt505h0; (7)



2KðhÞ @h@z

21

� ��z50

5EðtÞ; (8)

hjz5zd5hd: (9)

Other boundary conditions can also be used to formulate the inverse problem.

3. Inverse Problem

The spatial and temporal variability of soil water flow processes depends on the root water uptake in vege-tated soil profiles. As such, measurements of the dynamic water content or pressure head at various soildepths should contain information about the water uptake. The inverse problem framed herein is to iden-tify the RDDF, L(z, t), from dynamics of pressure heads throughout the profile, hobs, given known hydraulicproperties. That is, we consider the following inverse problem: find function L(z, t) that satisfies

hðz; t; LÞ5hobsðz; tÞ for all z; t: (10)

3.1. A Tikhonov Regularization ApproachIn this section, we use a Tikhonov regularization approach to formulate the inverse problem. We start with asemidiscretization scheme for the Richards equation (6). Let t0 < t1 < t2 < � � � < tN with tn5t01nDt andDt5ðtN2t0Þ=N, and denote the current time, the current pressure head, and the current RDDF by tn, hn, andLn

unknown, respectively, where hnð�Þ � hð�; tn; �Þ, and Lnunknownð�Þ � Lunknownð�; tnÞ. The discretized approxima-

tion to equation (6) using a fully implicit backward Euler scheme with explicit linearization of C and K leadsto a sequence of stationary problems:

Cðhn21Þ hn2hn21

Dt5@

@z

�Kðhn21Þ @hn

@z21

� ��2Ln

unknownðzÞKðhn21Þðhn2hrÞ (11)

with boundary conditions:

2Kðhn21Þ @hn

@z21

� ��z50

5En; (12)

hnjz5zd5hd; (13)

where En is the current value of the time-dependent evaporation rate, En5EðtnÞ.

Now we can view the inverse problem as finding LnunknownðzÞ that satisfies

hnðLnunknownÞ5hn

obs for all z; (14)

where hnobsð�Þ � hobsð�; tnÞ is the current observations.

Here we frame the inverse problem as an optimization problem, and further add Tikhonov regularizationterms to handle the nonuniqueness and instability, that is, minimizing a data misfit and regularizationterms. Given L0ð�Þ � Lð�; t0Þ, we construct a sequence of objective functionals Jn and of optimal solutions Ln,n51; 2; � � � ;N, with Ln � Lð�; tnÞ being the approximate solution for Ln

unknown. For each n, givenL0; L1; � � � ; Ln21 2 L5H1ðXÞ � fu 2 L2ðXÞ : @zu 2 L2ðXÞg, where H1ðXÞ denotes any differential or piecewisedifferential function on X, and L2ðXÞ denotes any square integrable function on X, we introduce the objec-tive functional:

JnðLÞ5 c1

2DtjjL2Ln21jj2L2

1c2

2jj@z Ljj2L2

112jjhnðLÞ2hn

obsjj2L2; (15)

where the typical measure, L2 norm is used:

k �kL2 5

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiðXð�Þ2dz

s: (16)

In this functional, the first two terms related to temporal and spatial constraints respectively are Tikhonovterms that control model regularization, where c1 and c2 are known as Tikhonov regularization parameters.



Based on the assumption that the RDDF is always smooth or piecewise smooth in space, the stronger norm,that is, norm of the first order of derivative, jjLjj1 � jjL0jjL2

is chosen as the penalty term. This form was origi-nally proposed by Phillips (1962) and Tikhonov (1963) (see also Franklin, 1974; Kirsch, 2011). The third termminimizes the data misfit between the simulated and the observed data. We aim to solve the following con-trol problem: given L0; L1; � � � ; Ln21 2 L, find Ln 2 L that satisfies

Ln5 arg minL

JnðLÞ: (17)

Here Ln is the optimal solution of problem (17), i.e., the functional JnðLÞ reaches its minimum at Ln. In orderto calculate the variational of Jn with respect to L, l5lð�Þ 2 L is introduced and denotes the increment ordecrement of Ln. For any l 2 L, we denote LE5Ln1El; E 2 R, and define

jðEÞ5JnðLEÞ: (18)

Note that jðEÞ reaches its minimum at E50, that is

j0ð0Þ5 ddE

JnðLEÞjE5050 (19)

from which we can derive the following necessary condition that Ln satisfiesðX

c1

DtðLn2Ln21Þl1c2@z Ln@z l

� dz1

ðX

hnðLnÞ2hnobs

�x hnðLnÞ; Ln; lð Þdz 5 0; 8 l 2 L ; (20)

where x satisfies

Cðhn21Þ xDt

5@

@z

�Kðhn21Þ @x

@z

�2Kðhn21Þ½Lnx1lhnðLnÞ2lhr � (21)


Kðhn21Þ @x@z

��z50

50 (22)

and

xjz5zd50: (23)

The proof of equations (20)–(23) is in Appendix A. The minimization problem now becomes a variational problem(equation (20)). In next section, this variational problem is solved numerically after a finite element discretization.

3.2. Numerical Solution for the Inverse ProblemIn this section, we present numerical procedures to derive the RDDF LnðzÞ from equation (20). As it is diffi-cult to find this function in continuous space, here we discretize equation (20) in a finite-dimensional space,that is, we attempt to find concrete formulae for a series of finite-dimensional linear problem whose solu-tion will approximately solve the original equation. To complete the discretization, we need to choose a setof basis of L: wi, i50; 1; . . . ;M, where M is the total number of discretization elements. Next, we replace theinfinite-dimensional parameter space L with a finite-dimensional subspace Ls � L, whereLs5spanfw0;w1; . . . ;wMg, and then find Ln

s ðzÞ 2 Ls such that Lns ðzÞ satisfies equation (20) for any l 2 L.

Here it is natural to choose the piecewise linear basis function (tent function). Given a regular discretizationin space: 05z0 < z1 < z2 < � � � < zM5zd with zi5z01iDz and Dz5zd=M, for each point zi (i51; 2; . . . ;M21),we choose wi whose value is 1 at zi and 0 at every zj, j 6¼ i, i.e.,

wiðzÞ5

z2zi21

zi2zi21; zi21 � z � zi

zi112zzi112zi

; zi � z � zi11

0; otherwise

:

8>>>>><>>>>>:

(24)

For i 5 0 and i 5 M,



w0ðzÞ5z12z

z12z0; z0 � z � z1

0; otherwise; wMðzÞ5

z2zM21

zM2zM21; zM21 � z � zM

0; otherwise

:

8><>:

8><>: (25)

Then, for each i, equation (20) can be approximated as

c1

DtðLn

i 2Ln21i Þ

ðXwiwi dz

1c2 Lni21

ðXrwi21rwi dz1Ln

i

ðXrwirwi dz1Ln

i11

ðXrwi11rwidz

� �

52

ðX

hn

�XM

j51

Lni wj

�2hn

obs

!x hn

�XM

j51

Lni wj

�;XM

j51

Lni wj;wi

!dz

: (26)

The proof of equation (26) is in Appendix B. Because equation (26) is nonlinear, an iterative procedure isrequired at each time step to solve this equation. We will illustrate the procedure in the following subsection.

Now we can conclude the inversion process as a workflow in which several components interact and inte-grate, as shown in Figure 1.

3.3. Computational AlgorithmNote that the optimal solution (26) is determined by a system of equations (11)–(13), (21)–(23). Let tn be thecurrent time, Dt > 0 be the step size of time, and t0 < t1 < t2 < � � � < tN with tn5t01nDt. Denote

Lnð�Þ5Lð�; tnÞ; hnð�; �Þ5hnð�; �; tnÞ; xnð�; �Þ5xnð�; �; tnÞ; hnobsð�Þ5hn

obsð�; tnÞ: (27)

Supposing L0 is given, we inductively solve for Ln, n51; 2; . . . ; N through an implicit discretization of equa-tion (26). Let Ln;k ; xn;k , and hn;k be the values at the kth iteration of the nth time step. The iteration proce-dure for the nth time step is elaborated as follows.

Figure 1. Schematic description of the inversion method, including inputs, formulation of the inversion, and thenumerical solution of the inverse problem.



1. Choose a tolerance g > 0 and an initial guess: Ln;05Ln21; hn;05hn21, and xn;05xn21, and set k 5 1.2. Solve for Ln;k according to an implicit discretization of equation (26):

c1

DtðLn;k

i 2Ln21i Þ

ðXwiwidz

1c2 Ln;ki21

ðXrwi21rwi dz1Ln;k

i

ðXrwirwi dz1Ln;k

i11

ðXrwi11rwidz

� �

52

ðXðhn;k212hn

obsÞxn;k21dz

: (28)

Application of equation (28) to each note i (i50; 1; . . . ;M) leads to M 1 1 linear equations for Ln;k :

AiLn;ki211Bi L

n;ki 1CiL

n;ki1152Fn;k21

i ; (29)

where

Ai52r1; (30)

Bi52r211; (31)

Ci52r1; (32)

Fni 5Ln212

ðXðhn;k212hn

obsÞxn;k21dz (33)

with r15c2DtDz2 ; r252c2

DtDz2 1c1.

3. Use the updated Ln;k to solve for hn;kð�; �Þ from equations (11)–(13).4. Use the updated Ln;k and hn;kð�; �Þ to solve for xn;kð�; �Þ from equations (21) to (23).5. Let Ln5Ln;k ; hn5hn;k and stop if k Ln;k2Ln;k21 k< g. Otherwise set k5k11 and go to step 2.

A flow chart is summarized in Figure 2 to depict the computationalalgorithm. Note that the solution depends on the values of the regu-larization parameters. In theory, the smaller c1 and c2, the more accu-rate the solution would be. However, decreasing the values of suchparameters may cause numerical oscillation. In our simulations, wechoose equal value for c1 and c2, the value varies for different exam-ples, and the value is always very small. Besides, the solution alsodepends on the initial guess L0ð�Þ. It is preferred that some prior infor-mation of the RDDF is available to estimate the required initial condi-tion. In case that no such information is available, one canalternatively choose c150 in the first time step to estimate the initialguess.

4. Numerical Examples and Tests

In this section, we illustrate the formulated inverse model and dem-onstrate its performance by means of artificial examples. Soil waterpressure heads generated via forward simulations are used as obser-vations to calibrate the root water uptake function. We denote thetrue values of RDDF by ‘‘Ltrue’’ and the calibrated values from theinverse model by ‘‘Lcali’’ in the following text. Moreover, water pres-sure head dynamics simulated with the obtained ‘‘Lcali’’ will bedenoted by ‘‘hsimu,’’ which are compared to the ‘‘observed’’ pressureheads ‘‘hobs.’’

As to measures for the error between true and the calibrated values,the Pearsons product-moment correlation coefficient (R2) and the

Figure 2. Flow chart of the numerical algorithm (iterative process) to solve theinverse problem.



root-mean-square error (RMSE) are used to determine how well the model simulates the observed data(Krause et al., 2005; Legates & McCabe, 1999):

R25

XN

i51

ðLi 2Li ÞðLi2Li Þ

XN

i51

ðLi 2Li Þ2

!1=2 XN

i51

ðLi2Li Þ2 !1=2

0BBBBB@

1CCCCCA

2

; (34)

RMSE5

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1N

XN

i51

ðLi 2LiÞ2vuut ; (35)

where N is the total number of elements, L and L are the calibrated and true values of RDDF, respectively,L and �L are the mean values of L and L, respectively. RMSE indicates the scatterness around 1:1 line whenplotting L against L. A larger RMSE means a more scattered relationship, whereas a smaller RMSE meansthat the model is more appropriate.

4.1. Application to Evaporation and Infiltration-Evaporation ExperimentsIn this section, we present numerical experiments for vertical unsaturated flow processes including evapora-tion and infiltration (here infiltration means downward movement of water into soil), which dominate waterexchanges in most agricultural fields. We consider flow problem in a soil column of 1.2 m, which is the maindomain the crop roots grow. The computation unit for the numerical simulations is divided into 20 gridcells, thus the discretization used is 0.06 m. The van Genuchten parameters for sandy loam, including satu-rated hydraulic conductivity (Ks) and shape parameter (a) which often show a high degree of spatial variabil-ity, are generated as random fields with exponential correlations to characterize the heterogeneity of thesoil profile. The geometric means of Ks and a are 0:15 m h21 and 3:367 m21, respectively. The other vanGenuchten parameters are assumed to be constants because of their moderate or weak variability as fol-lows: hs 5 0.3658, hr 5 0.0386, and n 5 2.

We first consider a soil water evaporation flow problem and assume that the evaporation experiment is per-formed after the soils reach the water content near field capacity, that is, the initial condition is assigned ash 5 21 m. The soil surface is imposed with a time-dependent flux boundary:

h521m; 0 � z � 1:2m; t50

KðhÞ @h@z

21

� �5EðtÞ; z50; t > 0

h521m; z51:2m; t > 0

;

8>>><>>>:

(36)

where EðtÞ520:0000001sin ðp=212pt=1440Þ. We use time discretization with 200 time steps and a totaltime of 600 h.

For root water uptake, we assume a time-dependent and space-dependent RDDF given by the equation:

Lðz; tÞ5be2bzðkt1nÞ (37)

with b 5 200, b50:11; k50:01, and n50:2. The values of this function are taken as the true values of theRDDF, denoted by Ltrue and shown in Figure 3a.

On the basis of these true values, we generate the ‘‘observed’’ data, a total of 10,000 measurements as func-tion of space and time, illustrated in Figure 3b. From these ‘‘observed’’ measurements, the formulatedinverse method is used to calibrate the RDDF. Figure 3c shows the calibrated values of L compared to thetrue values. We can see from this figure that the calibrated and the true values are in excellent agreement,indicated by R2 and RMSE listed in Table 1. Moreover, with the calibrated L in both cases, the simulatedwater potentials are equally in good agreement with the observed values, as can be seen in Figure 3d. Wealso apply the inversion to an infiltration example with h521 m as the top boundary condition within600 h. To save space, we only present the errors for this calibration in Table 1.



Given that the weather and soil conditions near the soil surface may change frequently during the plantgrowing period, we then consider a soil water flow problem with changing top boundary conditions, thatis, the top boundary condition switches from head-controlled (infiltration) to flux-controlled (evaporation):

h5210 m; 0 � z � 1:2 m; t50

h521 m; z50; 0 < t � T1

2KðhÞ @h@z

21

� �5EðtÞ; z50; T1 � t � T2

h5210 m; z51:2m; t > 0

;

8>>>>>><>>>>>>:

(38)

where T1 5 300 h and T2 5 600 h. This is a typical water infiltration intoan initially dry soil profile and then evaporation when the soil is wet.

Figure 4 shows the calibration results. The comparisons between thecalibrated and the true RDDF, and the comparisons between the sim-ulated and the observed water pressure heads are illustrated in Fig-ures 4a and 4b, respectively. Also, good results are indicated by thecorresponding errors (R2 and RMSE) listed in Table 1.

t (h)

z (m

)

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0 100 200 300 400 500 600

0

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)

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z (m

)Observed h (t=150 h)Simulated h (t=150 h)Observed h (t=300 h)Simulated h (t=300 h)Observed h (t=450 h)Simulated h (t=450 h)Observed h (t=600 h)Simulated h (t=600 h)

Figure 3. The calibration results for the evaporation experiment case. (a) The true values of root density distribution Ltrue, (b) the observed pressure heads, (c) thecalibrated values Lcali compared to the true values Ltrue (dashed lines denote Lcali while solid lines denote Ltrue), and (d) the simulated pressure heads compared tothe observed values.

Table 1Errors Between the Calibrated and the True Root Density Function for theEvaporation, Infiltration, and Infiltration-Evaporation Cases, Respectively

Case

Regularizationparameterðc15c2Þ

Errors

NIaR2 RMSE (m22)

Evaporation 0.01 0.98 2:3831023 6Infiltration 0.0001 0.99 1:9531023 4Infiltration-evaporation 0.001 0.97 2:7131023 5

aNI denotes the number of iterations for each time step (averaged).



In the simulations, the regularization parameters c1 and c2 are prescribed to be the same value denoted byc. As we can expect, the residual error (jjhðLÞ2hobsjj2L2

) and the penalty term (jj@Ljj2L2) are increasing and

decreasing functions of c, respectively. If c is chosen too small then the residual error is small while the pen-alty term can be large. Although the inverse problem is very close to the original ill-posed problem in such

situation, one should expect instabilities (arise when c � 0:00001 inthe infiltration example). If the regularization parameter is chosen toolarge then the residual error becomes large and the problem solvedhas only a little connection with the original inverse problem. Wenotice that both (c, jj@Ljj2L2

) and (c, jj@Ljj2L21jjhðLÞ2hobsjj2L2

) provide anL-shape curve on which its corner can be used as a suitable regulariza-tion parameter. Considering the balance between the two terms, inthe numerical examples, we choose the regularization parameterwhere the sum of the two terms reaches a minimum. In the presentedexamples, the regularization parameters c1 and c2 are both prescribedas 0.0001 in the infiltration period and 0.01 in the evaporation period,respectively. We observe from the simulation that this value cannot betoo small for the cases involving evaporation process. This could beexplained by the increasingly drier soil conditions that lead to greaternonlinearity and thus to higher degree of ill-posedness during theevaporation process.

In all the simulations we choose the convergence criteria for the itera-tion g50:00001. We notice that the number of iterations is larger(10) for the first dozen time steps and then smaller (<5) for the sub-sequent time steps. We also list the average numbers of iteration forthe evaporation, infiltration, and infiltration-evaporation cases, respec-tively, in Table 1.

Apart from those stochastic inverse methods, which generally includeMonte Carlo-based sampling algorithms and always examine theobjective function over an extended region in the parameter space,the present study proposes a deterministic inverse approach and usesthe regularization algorithm to ensure the uniqueness and stability ofthe inverse problem. Although the optimized solution is deterministicgiven certain numerical algorithms, a similar parameter space can be

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hobshsimu

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Figure 4. Calibration results when the infiltration-evaporation experiment case is conducted. (a) Comparisons between the true and the calibrated root density dis-tribution function (dashed lines denote calibrated L while solid lines denote the true values) and (b) comparisons between the simulated and the observed waterpressure heads.

Figure 5. Identification errors for different number of the discretization ofsolution spaces.

Table 2Errors Between the Calibrated and the True Root Density Function for the ThreeCases With Different Levels of Measurement Noise

Case R2 RMSE (m22)

Case I (r50:01) 0.88 1:7231022

Case II (r50:05) 0.85 2:1131022

Case III (r50:1) 0.78 2:5931022



given by adjusting some prescribed numerical parameters includingthe regularization parameters and convergence criteria for the itera-tion within prescribed accuracy of the inverse problem.

5. Discussions

5.1. Stability of the Inverse AlgorithmIn this section, two aspects are investigated to indicate the stability ofthe inverse algorithm.

First, a multilevel calibration is investigated when different number ofdiscretization elements in space for LnðzÞ (n51; 2; � � � ;N) is consid-ered, i.e., different freedom for the spatial distribution of the RDDF isconsidered. With the same initial and boundary conditions as in theinfiltration-evaporation case but with Ltrue52e20:11zð0:01t10:02Þ, weperform a series of identification for M53; 5; 7; 9; 11; 15; 20; 25; 30;35; 40; 45; 50; 60; 70; 80, where M is the number of discretization ele-ments in space. The identification errors for these experiments areshown in Figure 5. We observe that R2 increases and RMSE decreasesfor M � 15 while M increases. Such variation in performance for thecases wih M � 15 might be mainly caused by large numerical errorswhen larger discretization interval is applied. In addition, both R2 andRMSE become very stable for M � 20. This behavior is a sign that theill-posedness does not increase as the degrees of freedom increase.

Second, apart from the experiments above carried out with the exactobservations, we consider the measurements errors by adding aGaussian distributed noise to the ‘‘observed’’ data, that is,hobs5hobsð11rNð0; 1ÞÞ. Then, the perturbed data hobs is used for cali-bration. Also, for the above infiltration-evaporation experiment, threedifferent levels of noise are added: r 5 0.01, 0.05, and 0.10. The result-ing disturbed data serve as observations for the identification. Theerrors (R2 and NRMSE) between the calibrated and the true RDDF arelisted in Table 2, indicating that a growing Gaussian noise leads to anincreasing discrepancy between the calibrated and the true values.Nevertheless, a reasonable reconstruction is also obtained in the caseof a high noise level, indicated by the simulated water pressure headsat different time during the infiltration and evaporation periods asshown in Figure 6, which verifies the stability of our inversealgorithms.

Besides, we also test the stability of the inverse problem against timeresolution, that is, within the same total simulation time different timestep sizes (Dt51, 2, 3, 4, 5, 6, 7, 8, 9, 10 h, respectively) are assigned toconduct the inverse simulations. Comparisons of the performance(indicated by R2 and RMSE) of the inverse model between the tencases show that almost no significant differences are found under dif-ferent time resolutions, possibly indicating the stability of the inverseformulation against time solution.

5.2. Possibility of Simultaneous Optimization of L(z, t) and hrðtÞNotice that in this study, the root pressure head hr is assumed to be aknown constant. This seems like a limitation in practical applicationsbecause hr would be generally time variable and extremely time con-suming to measure in field. Actually, this parameter as a function oftime can also be incorporated into the inverse model besides the

Figure 6. Comparisons of the simulated and the observed water pressureheads when different levels of measurement errors are considered: (a) r50:01,(b) r50:05, and (c) r50:1.



RDDF, that is, two functions (hrðtÞ and L(z, t)) are simultaneously optimized from the observed pressureheads via inversions. The new inverse problem could be naturally formulated by adding additional constrainon hr to the objective functional as the constrain on L(z, t). Notice that in the original inverse model, becauseof the nonlinear relationship between L(z, t) and water pressure head h, Tikhonov regularization is necessaryand iterative algorithm is applied to solve the formulated inverse problem. However, this regularization isnot necessarily required when calibrating hr because of the linear relationship between hr and h, that is, hr

could be directly identified via inversion of the pressure head. Therefore the calibration of hr is simpler thanL(z, t) and one can easily add hrðtÞ as additional unknown parameter to be estimated. Actually, only onemore equation is needed at each time step when hr is simultaneously calibrated via the inversionprocedures.

6. Conclusions

This study proposes an inverse method to calibrate the RDDF, an important factor in the macroscopic rootwater uptake model in the unsaturated water flow modeling. In contrast to most existing literature, in whichthe RDDF is taken to be constant in space and over time or dependent on only soil depth, this study featuresoptimization of a spatially and temporally varying RDDF, whose value is identified in a number of gridcellsand each value also varies over time, from measurements of soil water potential dynamics. No prior form isassumed for the RDDF, thus our algorithm attempts to calibrate a more general root water extraction model.

As the inverse problem is typically ill-posed and nonlinear, it is formulated by the use of the Tikhonov regulari-zation and then effectively solved by application of an efficient numerical algorithm on the basis of the finiteelement method. The advantage of our method is that the inverse problem is framed as a functional minimiza-tion problem and solved based on the variational construction, which circumvents the computational com-plexity in calculating the sensitivity matrix involved in many derivative-based optimization approaches.

The formulated inverse model is tested in extensive numerical examples with artificially generated data setsfor both water infiltration and evaporation processes in heterogeneous soil profiles. The calibrated RDDFshows a good agreement with the true values, with high values of R2. Also, with the optimized RDDF, theobserved and simulated water potentials are in good agreement.

The inverse formulation is based on the assumption that the parameter to be estimated is a smooth func-tion or a piecewise smooth function with respect to depth. Otherwise, if the RDDF is piecewise constant,i.e., as constant within each soil layer but allowed to vary between these, one can choose other appropriateconstraints such as solution norm to construct the optimization problem. To extend the results of this study,we recommend applying a similar analysis to other macroscopic conceptual root water uptake models.Finally, extending the proposed inverse method to the optimization of soil properties or further integrationof both soil properties and root water uptake model parameters simultaneously needs more work in afuture study.

Appendix A

Substituting equation (15) to (19) yields

ddE

ðX

c1

2DtðLE2Ln21Þ21

c2

2ð@z LEÞ2

� dz1

ðX

12

hnðLEÞ2hnobs

�2dz

� ��E50

50 ; (A1)

i.e., ðX

c1


� dz1

ðX

hnðLnÞ2hnobs

� dhnðLEÞdE

��E50

dz50 : (A2)

We denote the derivative of the pressure head with respect to the parameter E (i.e., the variation dhn) byxðz; tÞ, i.e., set

xðz; tÞ5 dhnðLEÞdE

��E50

; (A3)

which can be written as



xðz; tÞ5 limE!0

hnðLn1ElÞ2hnðLnÞE

: (A4)

Noticing that both hnðLn1ElÞ and hnðLnÞ satisfy equations (11)–(13), characterizing the distribution of pres-sure head at the current time, we can derive that x satisfies the following equation (denoted by ‘‘derivativeequation FðxÞ50’’):

Cðhn21Þ xDt

5@

@z

�Kðhn21Þ @x

@z

�2Kðhn21Þ½Lnx1lhnðLnÞ2lhr � (A5)


Kðhn21Þ @x@z

��z50

50; (A6)

xjz5zd50: (A7)

Because x depends on the values of hnðLnÞ, Ln, and l, we denote it as xðhnðLnÞ; Ln; lÞ. Therefore, the calibra-tion problem is framed asð

X

c1


� dz1

ðX

hnðLnÞ2hnobs

�x hnðLnÞ; Ln; lð Þdz50; 8 l 2 L : (A8)

Appendix B

In the calibration problem 27, the current root density distribution LnðzÞ function approximated by Lns ðzÞ

2 Ls can be written as

Lns ðzÞ5

XM

i50

Lni wiðzÞ; (B1)

where Lni � LnðziÞ.

Then, 8 ls 2 Ls , let lsðzÞ5wiðzÞ; i50; 1; . . . ;M, for each i, equation (27) can be approximated as

ðX

c1

Dt

XM

j51

ðLnj 2Ln21

j Þwjwi1c2

XM

j51

Lnj rwjrwi

!dz

1

ðX

hn

�XM

j51

Lnj wj

�2hn

obs

!x hn

�XM

j51

Lnj wj

�;XM

j51

Lnj wj;wi

!dz50

: (B2)

Taking the advantages of this choice of basis that the inner products

<wi ;wj> 5

ðXwiwjdz (B3)

are zero whenever j 6¼ i, and the integrands of

/ðwi ;wjÞ5ð

Xrwirwj dz (B4)

are zero whenever ji2jj > 1, then, for each i, we can obtain equation (26) from equation (20).

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