free-form estimation of soil hydraulic properties using wind’s … · 2008. 11. 21. · free-form...

Free-Form estimation of soil hydraulic properties usingWind’s method

S. C. IDEN & W. DURNER

Institut fur Geookologie, Technische Universitat Carolo–Wilhelmina zu Braunschweig, Langer Kamp 19c, 38106 Braunschweig, Germany

Summary

Transient evaporation experiments offer the potential to determine simultaneously the soil hydraulic

properties necessary to simulate water flow in unsaturated soils. We present a new algorithm for deter-

mining the retention and conductivity curve from evaporation experiments which uses Wind’s method

with a free-form soil water retention function. Our algorithm estimates nodal values of volumetric water

content and derives a smooth and monotone retention curve by cubic Hermite interpolation. A multilevel

routine increases the number of nodes and their adequate number is identified by a performance criterion

which balances goodness of fit, the cross correlation between the estimated water contents and the number

of degrees of freedom. We calculate point values of unsaturated hydraulic conductivity by the instanta-

neous profile method and discard unreliable conductivity estimates by a statistical filter criterion. Results

for three synthetic data sets including an uncertainty analysis of the estimated retention curves show that

the algorithm is suitable to identify, both correctly and precisely, the soil hydraulic properties. An appli-

cation to a real data set confirms these results. In order to enable the free-form functions to be used in

numerical flow simulations, we extrapolate the retention function to the dry range and compute a coupled

conductivity function based on the Mualem model. Major advantages of the proposed method are the

enormous flexibility provided by the free-form functions, the low level of parameter cross-correlation in

comparison with classic parametric functions, and the possibility of assessing the uncertainty of the

retention curve individually in different ranges of pressure head.

Introduction

The current standard for simulating transientwater flow in soil is

the Richards equation (van Dam et al., 2004). Solving the

Richards equation for a given domain and problem-dependent

initial and boundary conditions relies on the accurate determi-

nation of the soil water retention curve and the unsaturated

hydraulic conductivity function. Standard methods to deter-

mine these soil hydraulic properties (SHP) in the laboratory

and in the field have been reviewed extensively in the literature

(Dane & Topp, 2002). The two most important experiments

allowing the simultaneous determination of both properties

under transient flow conditions in the laboratory are the multi-

step outflow and the evaporation method. The latter was

introduced by the early work of Wind (1968) who presented

the experimental procedure and the first algorithm to deter-

mine the SHP from measurements of pressure head in the col-

umn and the loss of weight caused by evaporation across the

upper boundary of a soil column. The state-of-the-art of the

method is described in Halbertsma & Veerman (1994) and

Arya (2002). While the experimental design has remained

more or less the same over the last four decades, new mathe-

matical procedures and computer algorithms to calculate the

SHP have been developed which improve accuracy, stability or

simplicity (Tamari et al., 1993; Wendroth et al., 1993; Schin-

dler & Muller, 2006). Moreover, potential errors of the

method have been assessed (Mohrath et al., 1997). In parallel,

the inverse modelling approach (Hopmans et al., 2002) first

applied to outflow experiments has yielded promising results

when applied to the evaporation method (Simunek et al.,

1998a; Romano & Santini, 1999). Although it has the advan-

tage of a process model based evaluation of water flow (in

most cases by use of the Richards equation) without relying on

simplifying assumptions such as those of linearity in the spa-

tial distribution of pressure head or water content, the method

remains computationally expensive, and suffers from numeri-

cal difficulties for relatively coarse materials.

The objective of this article is to present a new algorithm for

estimating SHP byWind’s evaporationmethod in which the soil

water retention curve is not restricted to an a priori definedCorrespondence: S. C. Iden. E-mail: [email protected]

Received 27 September 2007; revised version accepted 8 May 2008

1228# 2008 The Authors

Journal compilation # 2008 British Society of Soil Science

European Journal of Soil Science, December 2008, 59, 1228–1240 doi: 10.1111/j.1365-2389.2008.01068.x

shape. While the original algorithm of Wind used polynomial

functions to describe the soil water retention curve, later ver-

sions used parametric functions of the SHP, e.g. the van Gen-

uchten equation (van Genuchten, 1980) or more flexible

composite functions accounting for effects of soil structure on

the SHP, e.g. the bimodal van Genuchten model of Durner

(1994) to describe the water content – pressure head relationship

(Halbertsma&Veerman, 1994). Irrespective of the experimental

design and evaluation method selected to identify the SHP, it

remains a challenge to identify the most appropriate among the

available parameterizations (Russo, 1988; Vrugt et al., 2003),

and testing a multitude of (if not all) available closed-form

expressions significantly increases the work load of studies

dealing with unsaturated flow problems. As an example, the

study of Leij et al. (1997) investigated the performance of 14

retention and 11 unsaturated hydraulic conductivity functions

to describe adequately measured hydraulic properties of soils

leading to a comprehensive dataset, which is impressive in its

extent. Currently, more and more closed-form expressions are

presented in the literature claiming to be in some sense more

generally applicable than existing models. Although the objec-

tive is to get more appropriate parametric descriptions of the

SHP, potentially to be used in pedotransfer functions, the

occurrence of ever more model approaches exacerbates the

work load problem outlined above. One alternative to develop-

ing more adequate parametric functions is to apply spline inter-

polation or spline regression methods. Among the most

sophisticated ideas of this type appears to be the free-form

approach of Bitterlich et al. (2004), introduced in the framework

of inverse modelling of column outflow experiments. Because it

does not assume a predefined shape and provides enormous

flexibility of the SHP, the approach effectively minimizes

parameterization errors. Moreover, as shown by Iden & Durner

(2007), it may help as well to overcome identifiability problems

because the estimated parameters show remarkably small cross-

correlation in comparison with classic parametric functions.

We have adapted the improved free-form estimation algo-

rithm of Iden & Durner (2007) to Wind’s evaporation experi-

ment and show the applicability of the algorithm by analyzing

three synthetic data sets obtained from numerically simulating

evaporation experiments and one real evaporation experiment

conducted on a sandy loam soil. The issue of selecting an

efficient parameterization is addressed within a multi-level

framework using information about goodness of fit, parameter

cross-correlation and the number of estimated parameters.

Wind algorithm using free-form functions

Free-form approach

We adopt that part of the improved free-form algorithm pre-

sented by Iden & Durner (2007) that estimates the soil water

retention curve. It partitions the pF interval coveredby anunsat-

urated flow experiment into sub-intervals of unequal length (pF

denotes the base-10 logarithm of the water suction given in cm).

The sub-intervals are separated by r þ 1 nodes located on the

pF axes. After assigning values of the volumetric water con-

tent y [m3 m–3] to the nodes, the soil water retention curve is

calculated by applying cubic Hermite interpolation. The pF

interval partitioned by the r þ 1 nodes is bounded at its lower

end by a pF value of zero. The upper interval limit is set to

pFmax, which is defined as the pF value corresponding to the

lowest observed pressure head during the experiment. Since

the evaporation method yields only limited information about

the shape of the SHP close to saturation, we fix the saturated

volumetric water content ys to an independently determined

value and assign it to the pF value of zero. The imposed air

entry value of �1 cm is a reasonable choice for unsaturated

flow problems (Vogel et al., 2001). At the remaining r nodes,

the volumetric water contents are allowed to vary freely, lim-

ited only by the following inequality constraints which are

introduced in order to ensure the monotonicity of the reten-

tion curve (Bitterlich et al., 2004)

ys � p1

pj � pjþ1 1 � j � r � 1ð1Þ

where the volumetric water contents are denoted by p to

emphasize their nature as estimated model parameters.

Determination of the soil water retention curve using

Wind’s method

Figure 1 gives an overview of the variables occurring in a typical

evaporation experiment evaluated with Wind’s method. A soil

column of length L [m] is allowed to evaporate only from the

Figure 1 Schematic overview over the experimental setup and the

major variables occurring in Wind’s evaluation of the evaporation

method using four tensiometers (T1-T4). See text for explanation of

variables.

Free-Form estimation using Wind’s method 1229

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Journal compilation # 2008 British Society of Soil Science, European Journal of Soil Science, 59, 1228–1240

top and is instrumented with a number of nt tensiometers pro-

viding measurements of pressure head hi,j [m] from inside the

soil sample at different times, where i indicates the position of

the measurement (increasing from bottom to top) and j indicates

the observation time. Simultaneous determination of the mass

of the column yields an average or effective volumetric water

content of the entire sample at time j, denoted as yobsj. The num-

ber of times at which measurements are taken is denoted by nd.

Weusenonlinear regressionanalysis to estimate the parameters

of the free-form soil water retention curve y(h). Given the

observed pressure head readings hi,j, volumetric water contents

per compartment yi;j are calculated from the free-form soil

water retention function y(h). The column-averaged volumetric

water content at time j is calculated as the weighted mean of the

water contents of the single compartments with length Li [m]

Æyæj ¼1

L+nt

i¼1

Liyi; j: ð2Þ

This procedure relies on the assumption that the observed

pressure head hij is representative of the corresponding com-

partment i with length Li (with the tensiometer being located

at its centre) and that the soil column is homogeneous, i.e. that

the relationship y(h) is identical for all compartments.

The parameter vector defining the soil water retention curve

p ¼ (p1, . . ., pr)T, where T denotes matrix transpose, is esti-

mated by minimizing an objective function formulated accord-

ing to the ordinary least squares approach

OðpÞ ¼ +nd

j¼1

ðÆyæj�yobsj Þ2: ð3Þ

We obtain the minimum of O(p) by applying an adapted

version of the Shuffled-Complex-Evolution algorithm (SCE-

UA) of Duan et al. (1992) that accounts for the inequality con-

straints given by Equation (1). The modifications are described

in Iden & Durner (2007).

Calculation of unsaturated hydraulic conductivities

After estimating the water retention curve by the free-form

approach, unsaturated hydraulic conductivities are calculated

by the instantaneous profile method for all numbers of nodes

r. Based on the mass balance equation of soil water formu-

lated for each compartment and the fact that water flow across

the lower boundary of the column is zero, volumetric water

flow [m s–1] from compartment i to i þ 1 at time j þ 1/2 can be

calculated as

qi; jþ1=2 ¼ � 1

Dtjþ1=2+i

k¼1

Lkðyk; jþ1 � yk;jÞ

1 � i � nt ^ 1 � j � nd � 1: ð4Þ

where yk;j denotes the model-predicted water content of com-

partment k at time j and Dtjþ1/2 ¼ tjþ1 – tj [s] is the corre-

sponding time difference. In order to infer the unsaturated

hydraulic conductivity function K(h) [m s–1], one assumes that

the flow process can be mathematically described by the

Darcy-Buckingham law

qi; jþ1=2 ¼ � Kð�hiþ1=2; jþ1=2Þ�

h�iþ1; jþ1=2 � h�

i; jþ1=2

Dziþ1=2þ 1

�ð5Þ

where Dziþ1/2 [m] is the vertical distance between the two tensi-

ometers i and i þ 1, hi, jþ1/2* denotes the geometric mean of

pressure head at position i and times j and j þ 1, and the

matching pressure head is calculated as the geometric mean

over space and time (Simunek et al., 1998a)

�hiþ1=2; jþ1=2 ¼ � ðhi; j � hi; jþ1 � hiþ1; j � hiþ1; jþ1Þ0:25: ð6Þ

The unsaturated hydraulic conductivity function is obtained

at discrete values of pressure head calculated from Equation

(6) by combining Equations (4) and (5) and solving for

Kð�hiþ1=2; jþ1=2Þ.One problem of evaporation experiments is that they provide

insufficient information about the conductivity function in the

moisture range close to saturation (Tamari et al., 1993; Wen-

droth et al., 1993). Any evaluation using Wind’s method suf-

fers from the same drawback caused by the fact that gradients

in hydraulic potential are relatively small in comparison with

their measurement error close to saturation. Similar to the

study of Tamari et al. (1993), we filter the calculated conduc-

tivity values by analyzing the measurement error of the corre-

sponding hydraulic gradient. This error is calculated from the

error of the pressure head readings and the uncertainty of the

vertical position of the tensiometers by applying Gauss’s law

for the propagation of random error. We assume all errors to

be uncorrelated and discard those calculated conductivity val-

ues for which the calculated gradient is smaller than 3 times its

error standard deviation.

Multi-level procedure, performance criteria and uncertainty

analysis

In practice, the number of nodes used for cubic Hermite inter-

polation exerts a strong influence on the outcome of parameter

estimation and thus on the determined soil water retention

curve. More nodes imply an increased flexibility in the retention

curve and therefore an improved match to the observations. In

contrast to this, increasing the number of nodes results in

a higher cross-correlation of the estimated parameters. We

increased the number of nodes by using the multilevel routine

of Iden & Durner (2007) which inserts a single node per step of

the cascade. After the insertion, the nodes are distributed evenly

in the unit plane spanned by a normalized pF axes and a normal-

ized water content axes.

1230 S. C. Iden & W. Durner

# 2008 The Authors


For comparing the goodness of fit for different values of r, we

use the minimum value of the objective function Omin as

defined by Equation (3) or its pendant, the root mean squared

error (RMSE) of the water contents. The interactions between

the estimated model parameters are quantified by means of the

parameter collinearity index g of Brun et al. (2001) which is

calculated by computing the nd � r sensitivity matrix S ¼ sk;lcontaining the partial derivative of the kth model-predicted

column-averaged water content with respect to the lth model

parameter

sk;l ¼@Æyæk@pl

; ð7Þ

at the optimumparameter vector by a first-order approximation

using central differences with a relative parameter perturbation

of � 1%. Whenever necessary, the perturbation is decreased to

satisfy the nonequality constraints imposed by Equation (1)

(Iden & Durner, 2007). A new matrix U is calculated by nor-

malizing the columns of the sensitivity matrix by their individ-

ual L2-norms to compensate for different magnitudes of the

sensitivities. The collinearity index g is the reciprocal of the

square root of the smallest eigenvalue of the matrix UTU (Brun

et al., 2001).

We define a heuristic test variable P* similar to the one used

by Iden & Durner (2007) to identify the optimal number of

nodes r

P� ¼ Omin

nds2yþ g

5þ r

6ð8Þ

where sy is the standard deviation of the water content error

which is a combined result of the accuracy of the scale ss andthe error of the pressure head measurements sh that propagate

into an error of the predicted water content for each compart-

ment. Assuming that both error components are independent,

sy can be calculated by applying Gauss’ law for the propaga-

tion of random error

sy ¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffis2s þ

�@y@h

shffiffiffiffint

p�2

s: ð9Þ

Because of the nonlinearity of the soil water capacity function

C(h) ¼ @y/@h, sy is a function of the pressure head. Similarly,

the error sh usually depends on the magnitude of the measured

pressure head (Romano & Santini, 1999). For the sake of sim-

plicity, we chose reasonable average values of the a priori

unknown function C(h) and the varying error sh for calculat-

ing an average sy to be used in Equation (8).

The test variable P* defined by Equation (8) seeks to find

a balance between the goodness of fit, the cross correlation of

the estimated parameters (identifiability) and their number

(principle of parsimony). In case the residual sum of squares

Omin normalized by the number of data nd is approximately

equal to the error variance s2y, the first term will be unity. In

parallel, the second term yields a value of unity whenever the

lower limit of the critical range of g reported by Brun et al.

(2001) is reached, and the third term yields a value of unity

whenever the number of degrees of freedom reaches six. This

accounts for the fact that retention curves estimated by the

free-form approach based on more than 6 degrees of freedom

tend to exhibit rugged shapes (Iden & Durner, 2007). Despite

some similarity to other statistical model selection criteria, P*

is a purely heuristic measure to identify the best parameteriza-

tion when applying the free-form approach. The optimum

number of nodes is indicated by a minimum value of P*.

The uncertainty of the identified soil water retention curve is

calculated by linear error propagation. The parameter covari-

ance matrix is computed from the sensitivity matrix S

Cp ¼Omin

nd � rðSTSÞ�1 ð10Þ

and confidence intervals for the estimated model parameters are

calculated from the best fit values and the diagonal elements of

Cp by use of Student’s t-distribution. Since the estimated

model parameters are the volumetric water contents at nodal

values of pF, the confidence interval of the entire function y(h)for a given level of significance can be inferred easily by cubic

Hermite interpolation. If necessary, we adjust the confidence

intervals at the nodes to the constraints given by Equation (1),

to guarantee monotone uncertainty bounds of y(h) (Iden &

Durner, 2007). Note that the calculated uncertainty intervals

have a local meaning because the model parameters exert

a local influence on the retention curve. Consequently, it is pos-

sible to assess how precisely y(h) can be identified in different

ranges of pressure head (Iden & Durner, 2007). This is in con-

trast to the uncertainty estimates of the soil hydraulic properties

obtained from fitting parametric models (Vrugt et al., 2003;

Durner et al., 2008) in which the model parameters exert

a global influence on the shape of the hydraulic properties.

Extrapolation of the retention curve

To enable water flow simulations in pressure head ranges not

covered by the evaporation experiment, we extrapolate the esti-

mated soil water retention curve towards dryness by use of the

Brooks&Corey functionwith the improved extension to the dry

range of Fayer & Simmons (1995), given as

yðhÞ ¼ yawðhÞ þ ½ys � yawðhÞ��

h

hc

��l

if h< hc

ys if h � hc

8<:

ð11Þ

where hc [m] is an air entry pressure head, l [-] and ya [-] are

shape parameters, and the function w(h) defined as

wðhÞ ¼ 1 � log10ð�h=cmÞ6:9

ð12Þ


# 2008 The Authors


describes adsorption of water as function of pressure head. We

fit the three model parameters hc, l, and ya to water contents

from the free-form retention curve in the pressure head range

corresponding to the pF-interval 0.9 pFmax to pFmax by non-

linear least squares and use the obtained parameter estimates

to predict y(h) in the pF range pFmax to 6.9 by Equation (11).

Fitting coupled conductivity functions

The numerical simulation of unsaturated water movement

requires a representation of both the retention and hydraulic

conductivity curve in either functional or tabular form. In gen-

eral, it is possible to apply the free-form approach to the pre-

dicted point values of unsaturated hydraulic conductivity aswell

and derive an optimal function K(h). Because there is insuffi-

cient information about K(h) close to saturation and in the dry

range, we couple the extrapolated free-form retention curve

with the pore connectivity model of Mualem (1976) which can

be written as

KðSÞ ¼ KsSt

RS0

h�1 ds

R10

h�1 ds

26664

377752

¼ KsSt xðSÞ2 ð13Þ

where s and S [-] both denote degree of saturation, and t [-] is

a shape parameter. As the Fayer & Simmons (1995) model

predicts a water content of zero at pF 6.9, we calculate the

degrees of saturation as S ¼ y/ys. We solved the integral occur-

ring in Equation (13) numerically and determined the two

unknowns t and Ks by linearizing Equation (13) to

log10 KðSÞ � 2 log10 xðSÞ ¼ log10 Ks þ t � log S ð14Þ

and applying linear regression analysis.

Application

Identification using synthetic data

Three synthetic data sets reflecting different SHPwere generated

by simulating evaporation experiments under laboratory condi-

tions using the Hydrus-1D software (Simunek et al., 1998b). We

assumed a soil column which was 6 cm high and instrumented

with four tensiometers at 0.75, 2.25, 3.75 and 5.25 cm depth

from the top of the column. An atmospheric upper boundary

condition was specified with a constant potential evaporation

rate of 2 mm day–1 and a critical pressure head at the surface

of -10000 cm. In Hydrus-1D this implies that the boundary

condition is switched from a flux condition to a Dirichlet con-

dition as soon as the pressure head reaches this critical value

(Simunek et al., 1998b). The initial condition was specified in

the pressure head by assuming a hydrostatic distribution with

a pressure head of �1 cm at the bottom of the column. The

three soils analyzed were a loamy sand (S1), a clay loam (S2)

and a structured soil (S3). The corresponding soil hydraulic

properties were parameterized by use of the Mualem-van

Genuchten (van Genuchten, 1980) and the bimodal Mualem-

van Genuchten (Durner, 1994) models . The parameters are

summarized in Table 1. We defined 100 recording times for

each scenario, equally distributed on affiffit

p-axis, and calculated

column-averaged soil water contents as the differences between

the initial amount of water in the columns and the cumulative

evaporation across the upper boundary. The resulting volumet-

ric water contents were corrupted with independently, normally

distributed noise with a standard deviation of 6�10–5 reflecting

a precision of the scale used for weighing the sample of 0.01 g

(the assumed diameter of the column was 6 cm). The simulated

pressure heads were corrupted with noise according to the the

error model of Romano & Santini (1999). The standard devi-

ation of the error was calculated as

shi¼ 0:1 cm þ 0:004 � jhij: ð15Þ

The virtual experiments were simulated until the upper tensi-

ometer reached apressure headof�700 cm.After generating the

data, the Free-Form Wind algorithm was run with the number

of nodes r ranging from 1 to 12. During the optimizations, the

saturated water contents were fixed to their known true values.

The number of complexes during the SCE-UA optimizations

was set to 2 r and the number of points per complex was set to

2 r þ 1. The optimization was stopped when either the maxi-

mum relative bandwidth of all model parameters or the rela-

tive bandwidth of the objective function values from all

complexes was lower than 0.001. For the uncertainty analysis

of y(h), the uncertainty of ys was inferred from the precision of

the scale noted above. During the identification, we specified

Table 1 Parameters of the Mualem-van Genuchten (1980) and the bimodal Mualem-van Genuchten (Durner, 1994) models used for generation of

synthetic data sets

Scenario ys yr

a1

n1

w1 a2

n2 Ks t/ cm–1 / cm–1 / cm day–1

S1 (loamy sand) 0.39 0.046 0.039 2.10 1.0 � � 199 0.5

S2 (clay loam) 0.41 0.095 0.019 1.31 1.0 � � 6.24 0.5

S3 (structured, bimodal) 0.45 0.050 0.040 1.80 0.9 0.4 8.0 240 0.5


# 2008 The Authors


the measurement error of the pressure head readings accord-

ing to Equation (15) and assumed the installation depth of the

tensiometers to be free of error and a water content error syof 5�10–4 to be used in Equation (8). The resulting hydraulic

conductivity estimates were filtered by the criterion presented

above.

Identification using laboratory data

A laboratory experiment was conducted on a loamy sand

column (LS) sampled from the plough horizon at a field of

the Federal Agricultural Research Centre in Braunschweig,

Germany.The soil columnwas 6 cmhigh and8.5 cm indiameter,

the sampling depth was 10 cm and the soil consisted of 64%

sand, 30% silt, and 6% clay. Four tensiometers were installed

at the same depths as described above. The evaporation rate was

increased by blowing air over the top of the column using a fan.

This resulted in a constant evaporation rate of approximately

8.3 mm day–1. The duration of the experiment was slightly less

than 22 hours. Water contents on 53 occasions were deter-

mined by weighing the column. The convergence criteria of the

SCE-UA optimization were set as stated above. During the

parameter estimation procedure, the saturated water content

was fixed to the gravimetrically determined value of 0.354 and

its error was specified in accordance with the precision of the

scale. In addition to applying the Freeform-Wind algorithm,

we estimated retention function parameters of the unimodal

and bimodal van Genuchten models (van Genuchten, 1980;

Durner, 1994) and the model of Ross & Smettem (1993).

Table 2 gives an overview over the corresponding model equa-

tions, estimated parameters and parameter transformations

during the optimizations. Again, ys was fixed during the opti-

mization and the SCE-UA algorithm was used for nonlinear

fitting. We assumed an error standard deviation for the pres-

sure heads as given by Equation (15) and an error in the verti-

cal position of the tensiometers of 0.1 cm for filtering the

calculated unsaturated hydraulic conductivity data. The per-

formance of the optimal free-form parameterization was com-

pared with that of the three parametric functions by checking

the corrected Akaike Information Criterion AICc (Burnham &

Anderson, 2002), the RMSE of the water contents, the num-

ber of runs in the residuals nrun and the collinearity index g.The AICc was calculated as (Burnham & Anderson, 2002)

AICc ¼ nd ln

�Omin

nd

�þ 2np þ 2npðnp þ 1Þ

nd � np � 1ð16Þ

where np is the number of estimated model parameters. The

last term in Equation (16) corrects for small sample sizes and

vanishes if nd np. Burnham & Anderson (2002) recom-

mended that the difference of the AICc should be computed

from its minimum value among all tested models

DAICc ¼ AICc � AICc;min ð17Þ

and state that models with an DAICc greater than 10 fail to

explain some substantial variation in the data and might be

omitted from further consideration. We fitted unsaturated

hydraulic conductivity functions based on Equation (13) in the

case of the three parametric models of the SHP used (Table 2).

The integral x(S) was computed either by the closed-form ex-

pressions of van Genuchten (1980) and Priesack & Durner (2006),

or by numerical integration. Table 3 summarizes the corre-

sponding mathematical expressions. The two unknowns log10 Ks

and t were estimated by linear regression using Equation (14).

Results and discussion

Synthetic data sets

The results for the three synthetic data sets are shown in

Figure 2. The three left-hand plots show the generated data

for the loamy sand (S1), the clay loam (S2) and the structured

soil (S3) (from top to bottom). For the clay loam, unsaturated

hydraulic conductivity is sufficiently high to sustain a constant

evaporation rate across the soil surface throughout the virtual

experiment. Accordingly, the column-averaged water content

decreases linearly with time. This compares with the loamy sand

(S1) and the structured soil (S3) which both exhibit a two-stage

behaviour: firstly a constant evaporation rate followed by a

decreased rate caused by the switch in the upper boundary con-

dition when the critical pressure head at the surface is reached.

The three plots in the centre column show the soil water reten-

tion curves used for simulating the experiments and the identi-

fied retention curves using the Free-Form Wind algorithm

together with the 95% confidence interval of the estimated

Table 2 Parametric models, their mathematical expressions for the relative saturation, and estimated model parameters for fitting the real data set

MODEL Expression for SðhÞ ¼ yðhÞ � yrys � yr

Estimated parameters

van Genuchten (1980) 1) (1 þ (–a h)n)–m yr, log10 a, log10 (n – 1)

Ross & Smettem (1993) 1) 2) w1ð1�a1 hÞea1h þ w2ð1þð�a2 hÞnÞ�m yr, w2, log10 a1,log10 a2, log10 (n – 1)

bimodal (Durner, 1994) 1) 2) +2

i¼1

wið1 þ ð�ai hÞni Þ�mi yr, w2, log10 a1, log10 a2,log10 (n1 – 1), log10 (n2 – 1)

Additional constraints1) mi ¼ 1 – 1/ni2) wi þ w2 ¼ 1


# 2008 The Authors


y(h). In fact, the uncertainty interval is invisible, because the

uncertainty is very low. This reflects the high information con-

tent of the experimental design, the quality of the data and the

effectiveness of Wind’s evaluation method. We show the esti-

mated functions yielding minimum values of the heuristic test

variable P*. In all cases, the match between the true and esti-

mated retention curves is excellent. Only in the case of the

structured soil (S3) do slight deviations become apparent at

conditions which are close to saturation. This is caused by the

fact that the columns were not fully saturated at the beginning

of the evaporation experiment but initially covered the pressure

head interval between -1 and -7 cm. The mismatch is therefore

not a shortcoming of the Free-Form Wind algorithm.

The three plots on the right-hand side of Figure 2 show the

true hydraulic conductivity functions and the conductivity val-

ues obtained from the instantaneous profile method. The agree-

ment between the true functions and the estimates is excellent for

the data points which fall into a pressure head range with suffi-

cient information to determine K(h). Obviously, the chosen fil-

ter criterion is appropriate because the discarded points show

both significant scattering and systematic deviations from the

true functions. The point clouds do not scatter symmetrically

around the true functions because the calculated conductivity

is proportional to the reciprocal of the gradient (Tamari et al.,

1993). In addition, since positive gradients implying down-

ward flow have to be filtered out because they yield negative

conductivity estimates, the statistical distribution of gradients

is cut off.

Figure 3 illustrates how to identify an appropriate number of

nodes r. It shows (from left to right) the RMSE of the water

contents used for parameter estimation, the collinearity index

g and the heuristic test variable P* for the three synthetic data

sets (first three rows) and the real data set (LS) (bottom row).

Increasing r leads in almost all cases to a decrease in the

RMSE because more parameters (estimated water contents)

provide more flexibility in the retention curve. However, the

monotone decrease with increasing r is not guaranteed,

because the positions of the nodes used for interpolation are

shifted on the pF axes when new nodes are added in the multi-

level routine (Iden & Durner, 2007, see the supplementary

information for examples). In contrast to the decrease in the

RMSE, increasing r yields stronger cross correlations between

the model parameters, as indicated by higher values of g. Forthe three synthetic data sets S1, S2 and S3, maximum values of

g are 1.9, 4.8 and 2.9, respectively, which are still low in com-

parison with the critical range of 5-20 reported by Brun et al.

(2001). For values of r smaller than 10, values of g are remark-

ably small, indicating that the parameter estimation problem is

well posed. The fact that data set S2 shows the highest values

of g is probably caused by the smaller water content range

covered by the experiment, as compared with S1 and S3 (see

Figure 2). As pointed out above, identifying an optimal num-

ber of nodes provides a compromise between goodness of fit,

parameter interaction and the number of degrees of freedom.

From Figures 2 and 3 it can be inferred that the heuristic test

variable P* yields minimum values for the three test cases (9, 7

and 7 for S1, S2 and S3, respectively) while simultaneously

yielding soil water retention curves that perfectly match the

true y(h)-relationships in the pressure head range covered by

the experiments.

Real evaporation experiment

The measured values of column averaged water content and

pressure head in four depths for the real column are depicted

in Figure 4. The linear decrease of the column-averaged water

content indicates that the evaporation rate remained constant

throughout the experiment. The results of fitting using the Free-

Form Wind algorithm are summarized in the lower most row

of plots in Figure 3. An adequate number of nodes is eight

because this gives a minimum value of P*. The values of the

collinearity index g are very small up to r ¼ 10, indicating

a well posed parameter estimation problem. However, g showsa distinct jump afterwards, reaching values of more than 50

for r ¼ 11 and r ¼ 12. This means that parameter correlation

impedes a unique determination of single-parameter values in

these cases.

The performance of the Free-FormWind algorithmwith r ¼ 8

is compared with the three parametric models in Table 4. In

comparison with the parametric fits, the Free-Form Wind

algorithm results in a markedly lower minimum value of the

RMSE and a higher number of runs in the residuals. The

bimodal van Genuchten (Durner, 1994) and Ross & Smettem

(1993) models perform in a similar way, but the respective

RMSE is ca four times greater than that of the free-form

model. Although the number of estimated parameters is

greater for the latter, the parameter collinearity index g takes

a value of only 1.45. This is significantly lower than the critical

value of g and the values for the three parametric models.

According to the corrected Akaike criterion AICc, the free-

form model is the optimal model structure among the tested

alternatives and the three parametric models should be

Table 3 Mathematical expressions for theMualem integral x as definedby Equation (13) used for fitting a coupled conductivity function for

the real data set

MODEL Expression for x

van Genuchten (1980)

1� 1� S

n

n�1!1�1=n

Ross & Smettem

(1993)

numerical integration

bimodal

(Durner, 1994) 1) 2)

+2

i¼1

wi aif1�ð�aihÞni�1½1 þ ð�aihÞni ��mig

+2

i¼1

wi ai

Free-Form numerical integration

Additional constraints1) mi ¼ 1 – 1/ni2) wi þ w2 ¼ 1


# 2008 The Authors


(1980) type or a combination of a bimodal van Genuchten

(Durner, 1994) curve with a Ross & Smettem (1993) macro-

porosity function in order to yield a fit of the same quality as

the free-form model. However, as the amount of data in the

pressure head range lower than 1.5 is so small, the parameter

estimation problem would suffer from even greater identifi-

ability problems than those already occurring for the fits using

the bimodal van Genuchten and Ross & Smettem models. The

upper left-hand plot in Figure 5 also shows the 95% confi-

dence interval of the retention curve estimated with the free-

form approach. As in the case of the synthetic data, the uncer-

tainties are remarkably low.

The right-hand column in Figure 5 shows plots of the pre-

dicted unsaturated hydraulic conductivity values obtained from

the instantaneous profile method for the free-form model with

r ¼ 8 and the three parametric models. The filter criterion

means that all conductivity points for pF values smaller than

about 1.98 were discarded. Conversely, 92 point values of K(h)

were accepted and used for fitting conductivity functions. The

different symbols of the accepted points denote different posi-

tions of the calculated gradients. It becomes evident that con-

ductivities calculated from gradients between the upper

compartments lie approximately on the same line, whereas

those stemming from the two lower most compartments have

smaller values. This clustering is caused either by heterogene-

ity, e.g. due to consolidation of the soil, by errors in the instal-

lation depths of the tensiometers or calibration errors of the

pressure transducers (Mohrath et al., 1997).

The discrepancies of the predicted hydraulic conductivities

between the four models are small. This is in agreement with

the small discrepancies in the identified retention curves in the

pressure head range supporting the determination of K(h). The

larger differences of the estimated retention functions close to

saturation do not lead to discernible differences in the calcu-

lated conductivities because the corresponding data are filtered

out. However, when coupling the estimated retention curves

with the Mualem (1976) model, the small discrepancies close

to saturation become amplified in the conductivity prediction.

This is clearly shown by the four fitted curves in Figure 5. As

can be seen in Table 6, the estimated saturated hydraulic con-

ductivities differ by more than half an order of magnitude. In

agreement with the results for the retention function, the RMSE

log 1

0 R

MS

Elo

g 10

RM

SE

log 1

0 R

MS

Elo

g 10

RM

SE

3 6 9 12−4

−3

−2

−1

0

−4

−3

−2

−1

0

−4

−3

−2

−1

0

−4

−3

−2

−1

0

γ3 6 9 12

0

2

4

6

log 1

0 P

*lo

g 10

P*

log 1

0 P

*lo

g 10

P*

S1

3 6 9 120

1

2

3

3 6 9 12

γ

3 6 9 120

2

4

6

S2

3 6 9 120

1

2

3

3 6 9 12

γ

3 6 9 120

2

4

6

S3

3 6 9 120

1

2

3

r3 6 9 12

γ

r3 6 9 12

0

2

4

6

r

IPM

3 6 9 120

1

2

3Figure 3 An overview of the results obtained

from applying the Free-Form Wind algorithm

to the three synthetic data sets and to the real

data set (from top to bottom). Data shown

from left to right are the RMSE values, the val-

ues of the collinearity index g and the values of

the test variable P* for different numbers of

nodes r.


# 2008 The Authors


values of the base-10 logarithms of hydraulic conductivity

are almost identical for the bimodal (Durner, 1994), Ross &

Smettem (1993) and free-form models, whereas the unimodal van

Genuchten (1980) function results in a higher value (Table 6).

This result is supported by Figure 5 which shows that the

Mualem-van Genuchten model clearly does not fit data.

Summingup, the free-form results showonly small differences

to the bimodal vanGenuchten (Durner, 1994) andRoss&Smet-

tem (1993) models for the soil tested, if the estimated properties

are compared in the pF-range greater than 1.5. However, dis-

crepancies in the SHP become evident close to saturation, in

particular when a capillary bundle model is applied in order to

predict the unsaturated hydraulic conductivity curve. To what

degree the discrepancies of the SHP obtained from the different

models propagate into differences in flow predictions is highly

dependent on the respective initial and boundary conditions and

is therefore not the subject of the present paper. For the real data

set, the free-formmodel is the optimalmodel structure according

to the AICc-statistics and should be preferred over the para-

metric models tested. The resulting optimal soil hydraulic

functions are depicted in Figure 6, including their extrap-

olations to the dry range. They can be used easily in software

tools for solving the Richards equation that can handle SHP

in tabular form.

Conclusions

We have developed a new algorithm for estimating SHP using

Wind’s evaporationmethod that uses a free-formretention func-

tion. It was applied to synthetic data and laboratory measure-

ments and identified soil hydraulic properties of varying shapes

both correctly and with small uncertainty. Since the uncertainty

intervals of the free-form functions have a local meaning, they

allow us to assess how precisely y(h) can be identified in differ-

ent ranges of pressure head covered by the experiment. The

free-form approach effectively avoids parameterization errors

in the retention function, and thus eliminates the need to test

different parametric models from which the most adequate has

to be selected. We have shown that the free-form model pa-

rameters have extremely small cross-correlations, both abso-

lutely and relative to those of three parametric models tested

on the real data set. In practice, strong parameter correlation

poses problems for optimization algorithms and leads to non-

uniqueness of estimated parameters. This is an important

problem, because unique parameter estimates are required if

pedotransfer functions are applied in order to relate soil

hydraulic parameters to more easily measurable soil properties

or land surface characteristics (Vrugt et al., 2003). In order to

Table 4 Comparative statistics for different parametric models of the soil water retention curve and the Free-FormWind algorithm with 8 degrees of

freedom for the real data set

MODEL np RMSE /10�4 nrun g AICc DAICc

van Genuchten (1980) 3 24.5 4 17.7 �630.9 216.2

Ross & Smettem (1993) 5 10.5 6 116.5 �716.0 131.1

bimodal (Durner, 1994) 6 11.5 8 50.3 �703.8 143.3

FF r ¼ 8 8 2.82 9 1.45 �847.1 0.0

Table 5 Estimated model parameters for the real data set. The saturated water content was fixed to ys ¼ 0.354 during the optimizations

MODEL yr

a1n1 w2

a2n2/ cm–1 / cm–1

van Genuchten (1980) 0.166 0.012 1.535 � � �Ross & Smettem (1993) 0.178 0.019 � 0.498 0.002 3.074

bimodal (Durner, 1994) 0.194 0.002 5.753 0.799 0.014 1.961

h /m

Time /days0 0.2 0.4 0.6 0.8 1.0

−0

−2

−4

−6

−8

θ

T1

T3

θ

T2

T4

0.20

0.25

0.30

0.35

Figure 4 The measured column averaged water contents and pressure

heads in four depths for the real data set.


# 2008 The Authors


use the estimated properties in numerical flow simulations,

and to circumvent the problem of lacking information about

the conductivity function close to saturation, we have extrapo-

lated the free-form retention function with the Fayer & Sim-

mons (1995) extension of the Brooks & Corey model and have

coupled the free-form retention curve with the capillary bundle

model of Mualem (1976). Since the validity of the prediction

model close to saturation must be carefully tested, additional

measurements of hydraulic conductivity near saturation should

be considered to obtain a reliable conductivity function. This

need is not specific to the free-form estimation technique, but

relates equally to parametric functions. Accordingly, testing

the validity of the hydraulic conductivity function in the dry

range requires attention because diffusion of water vapour and

film and corner flow are not accounted for in the applied capil-

lary bundle model.

−1 −10 −100 −1000

0.23

0.27

0.31

0.35

θ

observed

fitted

FF 95%

−1 −10 −100 −1000

0.23

0.27

0.31

0.35

θ

observed

fitted

−1 −10 −100 −1000

0.23

0.27

0.31

0.35

θ

observed

fitted

−1 −10 −100 −1000

0.23

0.27

0.31

0.35

θ

h /cm

observed

fitted

−1 −10 −100 −1000

−2

−1

0

1

2

log 1

0 (K

/cm

day

−1 )

log 1

0 (K

/cm

day

−1 )

log 1

0 (K

/cm

day

−1 )

log 1

0 (K

/cm

day

−1 )

FF r = 8

Mualem disc.

bottom centre top

Mualem disc.

bottom centre top

Mualem disc.

bottom centre top

Mualem disc.

bottom centre top

−1 −10 −100 −1000

−2

−1

0

1

2

vanGen.

−1 −10 −100 −1000

−2

−1

0

1

2

Ross &Smettem

−1 −10 −100 −1000

−2

−1

0

1

2

van Gen.bimodal

h /cm

Figure 5 The identified soil hydraulic proper-

ties for the real data set. The left-hand plots

show the estimated retention functions (solid

black lines) and the column-averaged volumet-

ric water contents plotted against the arithmetic

average of the pressure head (gray squares).

The right-hand plots show the point values of

K(h) (coloured symbols indicate different posi-

tions of accepted values, grey circles denote dis-

carded values) and the estimated conductivity

functions based on the Mualem model (dashed

lines).


# 2008 The Authors


Acknowledgements

We thank Torsten Eckelt for providing the data for the evapo-

ration experiment and Andre Peters for useful discussions dur-

ing the development stage of the software.

Supporting Information

Numerical values of selected statistical measures of misfit and

parameter interaction for all data sets, and estimated soil water

retention curves for different numbers of degrees of freedom

are provided in the supplementary information available from

the web.

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# 2008 The Authors


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