Review and Analysis

H. Vereecken*
M. Weynants
M. Javaux
Y. Pachepsky
M. G. Schaap
M.Th. van Genuchten

Pedotransfer functions (PTFs) basedon the van Genuchten–Mualem (VGM)model have shown great value in thewidespread applica on of water flow
and solute transport models at thefield and larger scales. We reviewed thestate of the art with respect to using the
VGM model to describe hydraulic proper es and to develop PTFs.
H. Vereecken and M. Javaux, Agrosphere
Ins tute, Ins tute of Chemistry and Dynamics
of the Geosphere, Forschungszentrum Jülich,Jülich, Germany; M. Weynants and M. Javaux,
Earth and Life Ins tute— Environmental Sciences, Université Catholique de Louvain,
Louvain-la-Neuve, Belgium; Y. Pachepsky,
USDA-ARS, Environmental Microbial andFood Safety Lab., Beltsville, MD; M.G. Schaap,
Dep. of Soil, Water and Environmental Science, Univ. of Arizona, Tucson, AZ; and M.Th.
van Genuchten, Dep. of Mechanical Engineering, Federal Univ. of Rio de Janeiro, RJ,
Brazil. *Corresponding author (h.vereecken@
fz-juelich.de).
Vadose Zone J. 9:795–820
doi:10.2136/vzj2010.0045
Received 7 Mar. 2010.
Published online 27 Oct. 2010.
© Soil Science Society of America
5585 Guilford Rd. Madison, WI 53711 USA.

All rights reserved. No part of this periodical may
be reproduced or transmi ed in any form or by anymeans, electronic or mechanical, including photocopying, recording, or any informa on storage andretrieval system, without permission in wri ng fromthe publisher.

2010, Vol. 9

Using Pedotransfer FuncƟons
to EsƟmate the van Genuchten–
Mualem Soil Hydraulic ProperƟes:A Review
We reviewed the use of the van Genuchten–Mualem (VGM) model to parameterize soil
hydraulic proper es and for developing pedotransfer func ons (PTFs). Analysis of literature data showed that the moisture reten on characteris c (MRC) parameteriza on by
se ng shape parameters m = 1 − 1/n produced the largest devia ons between fi ed andmeasured water contents for pressure head values between 330 (log10 pressure head [pF]2.5) and 2500 cm (pF 3.4). The Schaap–van Genuchten model performed best in describing
the unsaturated hydraulic conduc vity, K. The classical VGM model using fixed parameters produced increasingly higher root mean squared residual, RMSR, values when the soil
became drier. The most accurate PTFs for es ma ng the MRC were obtained when using
textural proper es, bulk density, soil organic ma er, and soil moisture content. The RMSR
values for these PTFs approached those of the direct fit, thus sugges ng a need to improve
both PTFs and the MRC parameteriza on. Inclusion of the soil water content in the PTFs
for K only marginally improved their predic on compared with the PTFs that used only
textural proper es and bulk density. Including soil organic ma er to predict K had more
eﬀect on the predic on than including soil moisture. To advance the development of PTFs,
we advocate the establishment of databases of soil hydraulic proper es that (i) are derived
from standardized and harmonized measurement procedures, (ii) contain new predictors
such as soil structural proper es, and (iii) allow the development of me-dependent PTFs.
Successful use of structural proper es in PTFs will require parameteriza ons that accountfor the eﬀect of structural proper es on the soil hydraulic func ons.
Abbrevia ons: Bd, bulk density; HCC, hydraulic conduc vity curve; ME, mean error; MRC, moisture retenon characteris c; OM, organic ma er; PTF, pedotransfer func on; RMSD, root mean squared devia on;
RMSR, root mean squared residual; RMSRA, adjusted root mean squared residual; SSC, sand, silt, and clay;
VGM, van Genuchten–Mualem; WC, water content.

Knowledge of the unsaturated soil hydraulic properƟes is mandatoryfor describing and predicting water flow and solute transport in the vadose zone. In spiteof major advances in the development of measurement techniques, direct determinationis still expensive, time consuming, and, especially for larger scale applications, impractical. Therefore, PTFs have been developed that relate the soil hydraulic functions to moreeasily measurable soil properties. While various types of PTFs exist in terms of input andoutput (e.g., Pachepsky and Rawls, 2004), most PTFs relate parameters in the VGM modelto specific soil properties. The development of PTFs has greatly facilitated the widespread
application of water flow and solute transport models at the field and larger scales. Recent
developments in digital soil mapping and large-scale modeling again emphasize a needfor reliable PTFs.

6 Pedotransfer FuncƟons

Types of Pedotransfer FuncƟons

The past several decades have seen the development of different types of PTFs (e.g.,van Genuchten et al., 1992, 1999). Recent reviews of the state of the art in this field of
research have been given by Wösten et al. (2001), Pachepsky and Rawls (2004), and Shein
and Arkhangel’skaya (2006). Basically, two main types of PTFs can be distinguished:point and parametric PTFs. Point PTFs estimate values of soil moisture at fi xed pressure head values (e.g., de Jong and Loebel, 1982; Rawls and Brakensiek, 1982; Puckett
et al., 1985), whereas parametric PTFs estimate parameters of functions that describe
the observed data across a range of pressure heads (e.g., McCuen et al., 1981; Cosby etal., 1984; Wösten and van Genuchten, 1988; Vereecken et al., 1989). A few researchers

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have combined the latter approach with a direct fit of the parametric function, such as the VGM model, to the estimated soil
moisture values (e.g., Baumer, 1992; Tomasella and Hodnett,
1998; van den Berg et al., 1997; Tomasella et al., 2000). Most
efforts during the past several years have focused on the development of parametric PTFs because they provide a mathematical
function for the MRC or hydraulic conductivity curve (HCC)that can be used directly in mathematical models. Also, multimodeling approaches have been developed recently that combine
predictions with the different PTFs to either derive a single setof hydraulic parameters or aggregate the output of model runsthat were obtained for each of the individual PTFs (Guber et al.,2006, 2009). Guber et al. (2009) used 19 published PTFs and the
HYDRUS-1D code to model a field soil moisture regime. Theyevaluated different methods for combining the simulation resultsof the 19 different individual models to obtain the best accuracy
in terms of soil moisture content at different depths.

Pedotransfer FuncƟon Predictors
Extensive research in the past has focused on improving estimatesof the hydraulic properties using PTFs. One key area of researchhas dealt with the identification of additional soil informationthat could improve the accuracy of the PTFs, besides such classical PTF predictors as sand, silt, and clay contents, bulk density,
and C content. Additional information may involve more detailed
textural information (e.g., Vereecken et al., 1989, 1990; Børgesen
et al., 2008), terrain attributes and vegetation (e.g., Romano andPalladino, 2002; Sharma et al., 2006; Leij et al., 2004), soil structural information (e.g., Rawls and Pachepsky, 2002; Pachepsky andRawls, 2004; Pachepsky et al., 2006), the water content at selectedpressure heads (e.g., Al-Majou et al., 2008a; Børgesen and Schaap,
2005; Børgesen et al., 2008), morphological properties (e.g., Linet al., 1999), or taxonomic information (e.g., Lilly et al., 2008;
Al-Majou et al., 2008b).

Scaling Pedotransfer FuncƟons
Recent attempts to improve PTF predictions has involved thescaling of PTFs using coarse- or local-scale data such as the watercontent. Gribb et al. (2009) evaluated the quality of PTFs andlaboratory-scale measurement techniques (e.g., the multistep outflow method) to predict soil moisture dynamics at the field scale.
They found that inversely estimated hydraulic properties derived
from in situ measurements of the water content and the pressure
head provided better descriptions of the soil water dynamics than
PTFs and the laboratory-scale measurements. Model predictions substantially improved, however, by scaling the PTF andlaboratory-estimated soil moisture retention characteristics using
minimum and maximum values of the field soil water content.
Pringle and Lark (2007) and Pringle et al. (2007) stressed theneed for site-specific PTF development, while Jana et al. (2007)
used PTFs derived from very coarse-scale data (watershed or subwatershed data) to predict the soil water retention characteristics
at the plot and field scales.

Methods for PedotransferFuncƟon Development
We only briefly summarize here the various methods that may be
used to derive PTFs. An extensive review was given by Pachepsky
and Rawls (2004). Two main categories can be distinguished: statistical regression techniques (linear and nonlinear models) anddata mining and exploration techniques (artificial neural networks
and group methods of data handling). In general, methods basedon artificial neural networks have led to PTFs that performed bestin terms of basic indicators such as the RMSR. This is largely due
to the fact that this approach does not require an a priori functional form with which to relate PTF input to PTF output.
Recently, new methods such as nearest neighbor methods (Nemes
et al., 2006a,b; Jagtap et al., 2004), support vector machines(Lamorski et al., 2008; Twarakavi et al., 2009), and multiscale
Bayesian neural networks (Jana et al., 2007) have been introduced. Nemes et al. (2006a,b) applied a nearest neighbor method
to estimate the water retention at 330 and 15,000 cm using ahierarchical set of predictor variables. Th is method is based on
pattern recognition and similarity rather than fitting equations todata. Lamorski et al. (2008) introduced support vector machines(SVMs), an alternative data-driven method developed in the fieldof machine learning, to improve the efficiency of artificial neuralnetworks. This approach was successfully used to predict soil watercontents at various pressure head levels using simple soil properties.
Twarakavi et al. (2009) used a SVM-based approach to develop
PTFs for the VGM parameterization model using the databasefrom Schaap and Leij (1998a) and Schaap et al. (2001). Results of
their analysis have been included here.

Unexplained Variability
Despite much progress in identifying appropriate PTF predictorsthat explain the observed variability in the estimated soil hydraulic properties, substantial unresolved or unexplained variability
still exists at the level of the soil sample and the database used toderive the PTFs (Schaap and Leij, 1998a). This unexplained variability still plays an important role when functional aspects of soils
are being studied and analyzed using mathematical models (e.g.,
Christiaens and Feyen, 2001). These functional aspects include,e.g., vulnerability to leaching of contaminants (Wösten et al.,2001), workability (Wösten et al., 2001), moisture supply capacity (Vereecken et al., 1992), groundwater recharge (Vereecken etal., 1992), and aeration (Wösten et al., 2001). Wösten et al. (1999)
showed that in the case of workability and aeration, about 60% of
the total variability is caused by errors related to the estimationof soil hydraulic properties using PTFs. In terms of the prediction of the breakthrough of an inert tracer, this value was about
25%. Vereecken et al. (1992) showed that 90% of the variation in
the predicted moisture supply was attributed to estimation errors
in hydraulic properties using the PTFs developed by Vereecken
et al. (1989, 1990). Recently, Chirico et al. (2010) evaluated the

www.VadoseZoneJournal.org | 796

effect of PTF prediction uncertainty on the components of thesoil water balance at the hillslope scale. They found the simulated
evaporation to be much more affected by the PTF model error
than by errors due to uncertainty in the input data. Less information is available about the role of PTF uncertainty in the predictionof the leaching susceptibility of soils to chemicals and hazardous
substances. It is generally accepted that, in these cases, preferential
flow phenomena and soil structural properties may play an important role; however, frequently used PTFs do not account for theseeffects in the estimation of hydraulic properties.
Several reasons have been advanced to explain the presence of
this unresolved variance (Wösten et al., 2001; Minasny and
McBratney, 2002; Pachepsky and Rawls, 2004). First, predictors used in the PTFs are not always determined directly on thesoil sample used for measuring the hydraulic properties. Th is
becomes more important as the spatial and temporal variabilityof additional soils information increases and the information
content is not directly related anymore to the sample on which
the hydraulic properties were determined. Second, both Wösten
et al. (2001) and Pachepsky and Rawls (2004) pointed out thatthe database size should be large enough and contain all the necessary parameters that impact predictions of the soil hydraulicproperties. Most available databases contain only such classicalinformation as textural properties, organic C, and bulk density
while lacking relevant information related to structural properties, clay mineralogy, support scale, or other data. A third issue
is the appropriateness and accuracy of the functional relationships used to describe the measured soil moisture retention
and hydraulic conductivity data. Despite considerable researchon this issue, we feel that too much attention has focused on
PTFs that use empirical unimodal equations to describe the soil
moisture retention characteristic. Because of the relatively successful performance of the traditional VGM parameterization,
most studies of the hydraulic conductivity have focused on thismodel, while alternative options have not been explored enough,especially for structured media. A fi nal issue is the accuracy,
consistency, and quality of the measurement methods. Accurate
measurement of the MRC and HCC remain very difficult at highand low pressure heads, thus causing a lack of reliable data inthese ranges. Wösten et al. (2001) pointed out that accurate measurement of the hydraulic properties is probably the single mostimportant factor that could improve the accuracy and reliabilityof PTFs. Noticeably lacking thus far, however, are standardized
methods that could be used to ascertain the quality of measured
data and which could be considered as benchmark techniques.
To materially advance the development of PTFs, we believe that
progress needs to be made in the four areas of research identified
above. In this review, we have addressed issues associated with thestatus and further development of PTFs with regard to the VGM
parameterization of the hydraulic properties. In terms of the VGM
parameterization, we have focused especially on issues related tothe dry and wet moisture range of soils. In the following, we define
the dry range as the range of pressure head values larger than pF 3.0
and the wet range as the range of values smaller than the inverseof the α parameter in Eq. [1] (i.e., water content values close tosaturation). Our choice to focus on the traditional VGM model
is justified by the fact that this parameterization has evolved to ade facto standard, including its frequent use as a benchmark forcomparison with other parameterizations.

The van Genuchten–Mualem
Model ParameterizaƟon
The VGM model (van Genuchten, 1980) was developed by combining an empirically based power law equation describing therelationship between pressure head, h, and moisture content, θ,with the Mualem (1976) predictive pore-size distribution modelfor the unsaturated hydraulic conductivity. The van Genuchten
model for θ(h) is written asS=θ − θrn −m= ⎡⎢1 + (α h ) ⎤⎥⎦θs − θr ⎣[1]

where S is effective saturation (dimensionless), h is the pressure
head (cm), considered here to be positive under unsaturated conditions, θ r is the residual moisture content (dimensionless), θ s is thesaturated moisture content (dimensionless), α is a parameter corresponding approximately to the inverse of the air-entry value (cm−1),
and n and m are shape parameters. In its most general formulation,
the Mualem model can be written as (Mualem and Dagan, 1978)⎛ θ⎞2⎜⎜ ∫ 1/ h1+b d θ ⎟⎟⎟⎟K ( θ ) = K s S l ⎜⎜ θ0⎜⎜ sat⎟⎟1+b⎜⎝ ∫0 1/ h d θ ⎠⎟⎟[2]

where Ks is the saturated hydraulic conductivity, l is the pore sizeinteraction term, and b is the tortuosity. Based on a data set of 45
disturbed and undisturbed samples, Mualem (1976) derived an
optimal value of 0.5 for l when using measured Ks values as thematching point and setting the value of b = 1. To obtain a relatively
simple closed-form expression for the hydraulic conductivity, vanGenuchten assumed that m = 1 − 1/n. He obtained, in that case,the closed form equation⎫
1−1/ n ⎤ 2 ⎪⎪⎧ ⎡K ( S ) = K s ⎪⎨ S l ⎢1 − 1 − S n /(n−1)
⎥ ⎪⎬
⎪⎪ ⎢⎣
⎥⎦ ⎪⎪
⎩
⎭

(

)

[3]

EvaluaƟon of Pedotransfer FuncƟons
and the Direct Fits
Donatelli et al. (2004) reviewed various indicators that havebeen used to evaluate the performance of PTFs for predicting
soil water contents and unsaturated hydraulic conductivities. The
most frequently used indicator for PTF performance using the
VGM model is the RMSR (see Tables 1 and 2). Th is indicator has

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Table 1. Overview of van Genuchten–Mualem parameterizations used to describe the moisture retention characteristic and to develop pedotransfer
functions (PTFs), including the mean error (ME), the root mean squared residual for the direct fit (RMSR), and the root mean squared residualadjusted for the direct fit (RMSRA).

Study

Method,parameters†

No. ofsamples

Databasen

Pressurehead levels

Method, sample
state, sample size‡

ME

Soil
properties
used forRMSR RMSRA PTFs§−0.008

0.04

0.053

T, Bd,OM
T, Bd,
OM,
HOR,
WC

cm
320

7

10, 33, 100,
PPM; UD; 30–70 cm3330, 1000,
3300, 15,000

M2, Børgesen and NN; m = 1 − 1/n Børgesen and
Schaap (2005)
Schaap (2005)

1618

5

10, 30, 100,
HW (10–100, UD);–¶
1000, 15,000
PPM (400–1000,UD);
PPM(15,000,
D); 100 cm3

0.012

0.026#

M3, Cornelis etal. (2005)

RA; 5 parameters Cornelis et al.
(2005)

48

9

Five between
Sand box (10–100, UD); –10 and 100,
PPM (200/300, UD);and 200, 300,
PPM (1000/15000,
1000, 15,000
D); 100/20 cm3

0.007

0.011

M4, Cornelis etal. (2005)

RA; m = 1 − 1/n Cornelis et al.
(2005)

48

9

Five between
Sand box (10–100, UD); –10 and 100,
PPM (200/300, UD);and 200, 300,
PPM (1000/15000,
1000, 15,000
D); 100/20 cm3

0.0101 0.014

M5, Nemes etal. (2003)

NN; m = 1 − 1/n Nemes et al.
(2001)

235

10–
0, 2.5, 10, 30,
100, 200, 500,
2500, 15,000,
1.5 × 106

–

0.013

0.018

T, Bd,OM,
WC

M6, Schaap etal. (1998)

NN; m = 1 − 1/n Rawls (Schaap
1209
and Leij, 1998a)

6–13 –

–

–

–

0.02

T, Bd,WC

M7, Schaap andLeij (1998a)

NN; m = 1 − 1/n UNSODA(Schaap et al.,
1998)

554

6–52 multiple ranges
0–1.5 × 10 4

12 different methods(pressure based, HW,
tensiometer-based,
and sand box)––

0.0404# T, Bd

M8, Schaap andLeij (1998a)

NN; m = 1 − 1/n Rawls (Schaap
1209
and Leij, 1998a)

6–12 multiple ranges
0–1.5 × 10 4

mixed

–

–

0.0158# T, Bd

M9, Schaap andLeij (1998a)

NN; m = 1 − 1/n Ahuja (Schaap
371
and Leij, 1998a)

6–15 multiple ranges
0–1.5 × 10 4

mixed

–

–

0.0113# T, Bd

M10, Schaap andBouten (1996)

NN; m = 1 − 1/n Schaap andBouten (1996)

204

9

3.5, 7.5, 13,
20, 30, 40,
60, 90, 150

hanging water column––

0.011#T, Bd

M11, Schaap andBouten (1996)

NN; m = 1 −1/n, θ r = 0204

93.5, 7.5, 13,
20, 30, 40,
60, 90, 150

hanging water column––

0.016#T, Bd

M12, Schaap andLeij (2000)

NN; m = 1 − 1/n UNSODA

235

6–52 multiple ranges
0–1.5 × 10 4

12 different methods(pressure based, HW,
tensiometer based,
and sand box)

–

–

0.012

T, Bd

M13, Schaap etal. (2001)

NN; m = 1 − 1/n ROSETTA

2134

6–52 multiple ranges
0–1.5 × 10 4

–

?0

0.012

0.0214T, Bd WC

M14, Twarakavi
et al. (2009)

SVM; m =1 − 1/n

ROSETTA2134

6–52 multiple ranges
0–1.5 × 10 4–?0,phdiff.††0.012

0.0214T, Bd,
WC

M15, Weynants
et al. (2009)

RA; m = 1

Vereecken etal. (1989)

166

9

0, 3, 10, 31,
98, 196, 619,
2461, 15,542

Sand box (0–98); PPM(196–619); PPM(2461.5–15,542);
100 cm3–

0.0097 0.013(0.091)T, Bd,
OM

M16, Weynants
et al. (2009)

RA; m = 1 − 1/n Vereecken etal. (1989)

166

9

0, 3, 10, 31,
98, 196, 619,
2461, 15,542

Sand box (0–98); PPM(196–619); PPM(2461.5–15,542);
100 cm3–

0.0134 0.018(0.099)T, Bd,
OM

M17, Scheinost
et al. (1997)

RA; m = 1397

84, 30, 60, 300, PPM (4–600, UD), 365
600, 3000,
cm3; PPM (600–
6000, 15,000
15,000, UD), 10 cm3––

M1, Al-Majou etal. (2008a)

RA; m = 1 − 1/n, Al-Majou etθ r = fi xedal. (2008a)

Schaap andBouten (1996)

Scheinost etal. (1997)

www.VadoseZoneJournal.org | 798

RMSD: T, Bd,
0.019
OM

Table 1. Continued.

Study

Method,parameters†

No. ofsamples

Database

Pressurehead levels

n

ME

Soil
properties
used forRMSR RMSRA PTFs§

suction plate (Hartge
and Horn, 1971)

–

–

–

T, Bd,
WC

0.008#T, Bd

Method, sample
state, sample size‡cm
M18, Stumpp
et al. (2009)

NN; m = 1 − 1/n Stumpp et al.
(2009)

1257

M19, Ye et al.
(2007)

NN; m = 1 − 1/n Hanford (Ye etal., 2007)70

15–25 multiple ranges
0–1.5 × 10 4

multistep outflow (eachstep to equilibrium),pressure plate––

M20, Schaap andLeij (1998b)

NN; m = 1 − 1/n UNSODA (Leijet al., 1996)554

6–52 multiple ranges
0–1.5 × 10 4

12 different methods(pressure based, HW,
tensiometer based,
and sand box)

–

0.013

M21, Wösten
et al. (1999)

RA; m = 1 − 1/n Hypress

5521

14

multiple methods––

0, 3, 10, 31.6,
100, 316,
15,800

0–1.6 × 10 4T, Bd,
WC–

T, Bd,OM,
HOR† RA = regression analysis, NN = neural network analysis, SVM = support vector machines; n and m are shape parameters, θ r = residual moisture content.
‡ UD = undisturbed, D = disturbed, HW = hanging water, PPM = pressure plate extraction, numbers in parentheses refer to pressure head levels (cm).
§ T = texture, OM = organic matter, Bd = bulk density, HOR = horizon, WC = water content.
¶ No information provided for the study.
# Made available by the researchers.
†† phdiff = ME available as a function of the pressure head.

Table 2. Overview of different van Genuchten–Mualem parameterizations used to describe the hydraulic conductivity curve (HCC) and to the predict HCC
using pedotransfer functions (PTFs), including the mean error (ME) and the root mean squared residual for the direct fit of the VGM model (RMSR).
RMSR

Methods

Soil properties
used for PTFs§

1.4

mixed

–¶

?0

0.41

mixed

T, Bd, WC

245

–

1.08

mixed

T, Bd, WC

UNSODA

245

–

0.77

mixed

texture, Bd, WC

NN, VGM, Ks andl = 0.5 fi xed

UNSODA

235 (at least 5
K points)

–

1.31#

mixed

–

K6, Schaap and Leij (2000)

NN, VGM, Ks
flexible, l = −1

UNSODA

235 (at least 5
K points)

–

0.75#

mixed

T, Bd

K7, Schaap and Leij (2000)

NN, VGM, Ks
and l flexible

UNSODA

235 (at least 5
K points)

–

0.41#

mixed

T, Bd

K8, Børgesen et al. (2008)

NN, VGM, Ks
and l flexible

152 (4 geometric –mean values)

0.81#

constant-head and drip method T, Bd, OM, WC(up to 80 cm) + PTF of
HYPRES for 100, 400, 600 cm,
20- by 20-cm cylindrical cores

K9, Weynants et al. (2009)

RA, VGM, Ks
and l flexible

Vereecken etal. (1989)

166

–

0.73#

crust method and hot air method T, Bd, OM

K10, Schaap and van
Genuchten (2006)

modified VGM

UNSODA

235 (at least 5
K points)

phdiff.†† 0.26#

Study

Parameterization†

Database

No. of samples‡

K1, Schaap et al. (2001)

NN, VGM, Ks
and l fi xed

ROSETTA

235 (4117 data
points)K2, Schaap et al. (2001)

NN, VGM, Ks
and l fitted

ROSETTA

235 (4117 data
points)K3, Schaap and Leij (1998b)

NN, VGM, Ks
fitted, l = 0.5

UNSODAK4, Schaap and Leij (1998b)

NN, VGM, Ks
fitted, l = −1

K5, Schaap and Leij (2000)

ME

mixed

–

† NN = neural network analysis, VGM = van Genuchten–Mualem model, RA = regression analysis, Ks = saturated hydraulic conductivity, and l = tortuosity parameter.
‡ K = conductivity.
§ T = texture, Bd = bulk density, OM = organic matter, WC = water content.
¶ No information provided.
# RMSR adjusted using Eq. [5].
†† phdiff. = ME available for specific pressure head ranges.

www.VadoseZoneJournal.org | 799

also been used frequently to evaluate the performance of the VGM
model in terms of describing the measured soil hydraulic properties. The RMSR is defined asN∑ i =1( Ei − M i )RMSR =

2[4]
N
where Ei is the ith estimated value, Mi is the ith measured value,
and N is the number of data pairs consisting of Ei and Mi. Whenthe number of parameters is not small compared with the numberof available data points, the RMSR can be adjusted according to⎛
N ⎞⎟⎟RMSRA = RMSR ⎜⎜⎜[5]⎟⎝⎜ N − p ⎠⎟where p is the number of parameters.
The adjusted root mean squared residual (RMSRA) is especially
useful to compare direct fits of the VGM model to MRC data
because the number of measurements per curve for some data sets
is often quite low. The adjustment transforms the RSMR into an
unbiased estimator of the goodness-of-fit that can be compareddirectly with the pure error or variability in the data (Whitmore,1991). Differences between the RSMRA and RSMR become substantial only when p is comparable to N.
A few researchers have used the root mean squared deviation(RMSD) as defined by Tietje and Tapkenhinrichs (1993):
RMSD =

2b
1
Ei − M i ) d h
(
∫
b−a a

[6]

We have include RMSD values in the tables whenever available; however, there is no simple conversion to RMSR A to
perform comparisons.
Bias is often evaluated in the literature by calculating the meanerror (ME), which is defined as
NEi − M iN
i =1

ME = ∑[7]

Negative ME values indicate an average underestimation of theobserved moisture content or hydraulic conductivity values, while
positive values indicate overestimation of the target variables. As
errors may cancel out, this indicator is of limited use to evaluate
bias in the MRC and HCC. The ME has also been used in the past
to show the bias of fits or PTF predictions for stratified data such as
textural class or pressure head class (Schaap et al., 2001; Stumpp etal., 2009; Twarakavi et al., 2009). With the ME it is also possible
to bias-correct the RMSR (i.e., RMSR with the bias removed) (e.g.,
Al Majou et al., 2008a). In this way, the PTF error can be split into
a bias and a level of “uncertainty.”
Other indicators such as the Akaike information criterion(Cornelis et al., 2005), mean absolute error (Weynants et al., 2009),

and coefficient of determination (Cornelis et al., 2005) have also
been used (as shown in Table 1) to evaluate the performance of
direct fits of Eq. [1] to measured data. This has not been done in asystematic enough manner, however, to make it possible to compare results across databases.

6 van Genuchten–Mualem

ParameterizaƟon of theHydraulic ProperƟes

Here we review available information on the performance of Eq. [1]and [3] in describing measured hydraulic properties in databases ofsoil hydraulic properties and basic soil properties that were established to derive PTFs The performance was evaluated mainly using
the RMSRA and RMSR values, for reasons explained above. We
also included ME values when stratified data were available (e.g.,different pressure heads).

Performance of the Classical van Genuchten–
Mualem ParameterizaƟon
The VGM model defined by Eq. [1] and [3] has been widely usedin the past to describe measured hydraulic functions and is now
the most common parameterization of the hydraulic properties
in mathematical models for flow and transport in porous media.
Equation [1] has been successfully used to parameterize measured
soil moisture retention data for the development of PTFs for the soil
hydraulic properties. As discussed above, its attractiveness resides
in part in the fact that for specific assumptions with respect to its
parameters, a closed-form solution as given by Eq. [3] is obtained.
In most comparative studies, the van Genuchten model with fiveparameters has been found to provide the largest flexibility indescribing observed moisture retention data (Wösten and van
Genuchten, 1988; Vereecken et al., 1989; Cornelis et al., 2005).
Although Eq. [1] with five independent parameters (including m)
provides the most flexibility in terms of describing measured data(van Genuchten and Nielsen, 1985), it is not the most efficient modelfor deriving PTFs (e.g., Cornelis et al., 2005). Reducing the numberof parameters from five to four, for example by assuming m = 1 − 1/nor m = 1, generally leads to improved PTF predictions at the expenseof less flexibility in describing the observed data (Vereecken et al.,
1990; Cornelis et al., 2005; Lennartz et al., 2008). The difference
between the two most often used four-parameter parameterizations
is illustrated in Fig. 1. Figure 1a shows that setting m = 1 creates asymmetrical MRC shape, while S is always equal to 0.5 when h =α−1 (i.e., αh = 1). On the other hand, assuming m = 1− 1/n leads toasymmetrical MRC plots, as shown in Fig. 1b.

Comparison of van Genuchten–Mualem Model
Performance across Databases
Tables 1 and 2 provide an overview of the different databases thatwere used to either evaluate the quality of Eq. [1] to describe MRC
data or to derive PTFs. Each study or database is referred to below

www.VadoseZoneJournal.org | 800

1, 3, and 4, were converted to centimeters by multiplying
by a factor of 10. We used this factor (rather than 9.788)
for reasons of convenience.
The results from different databases used to fit Eq. [1] tomeasured soil moisture retention data are given in Table
1. In each case, the calculated or reported RMSR, ME,
and RMSRA values were averaged across N samples. The
RMSRA values ranged between 0.008 and 0.053, withan average value of 0.019 and a standard deviation of
0.011. The lowest RMSRA was obtained for Eq. [1] withfive parameters (i.e., independent m and n values), whichreflects the considerable flexibility of the five-parameterversion in describing observed retention data. Despite thisFig. 1. Effect of different van Genuchten–Mualem (VGM) parameterizations on theflexibility, this parameterization has not been used in the
shape of the moisture retention curve, where S is effective saturation, h is the pressure
literature to develop PTFs. Several reasons are probable.
head, and α, n, and m are VGM shape parameters: (a) effect of the value of n assumingFirst, no clearly defined relationship between m and easilym = 1: n = 1.1 (C), n = 1.3 (K), n = 1.5 (G), n = 1.7 (I), and n = 1.9 (8); (b) differences between two four-parameter versions: n = 1.3, m = 1 (C); n = 1.3, m = 1 − 1/nmeasured properties could be derived (e.g., Cornelis et al.,(@); n = 1.7, m = 1 (G); and n = 1.7, m = 1 − 1/n (X).
2001). Second, substantial correlations often exist among
the VGM parameters, often leading to sets of nearly equiby using M or K followed by a number, depending on whether the
final combinations of fitted parameters, which in turnstudy referred to the moisture retention characteristic (M) or theincreases ambiguity in the relationships between the VGM paramhydraulic conductivity (K). A distinction was made between dataeters and soil properties. Third, in most cases only a marginal loss
sets used for developing the PTFs and those used for PTF validain the RMSR occurred when describing the MRC with four rather
tion. Throughout this review we have used pressure head having athan five parameters (see also van Genuchten and Nielsen, 1985).
unit length (cm) to be consistent with Eq. [1]. Matric potentials in
Finally, no closed-form equation for K(h) exists when five indepenkilopascals, as used in some of the studies that are listed in Tables
dent parameters are used in the VGM model. This hampers the
Table 3. Accuracy of pedotransfer functions (PTFs) for the moisture retention characteristic expressed as the root mean squared residual (RMSR)
using different combinations of predictor variables, including textural class (TC), sand, silt, and clay content (SSC), bulk density (Bd), water content(WC), and organic matter content (OM); best refers to the PTF using WC values providing the lowest RMSR. Numbers in parentheses after the RMSR
indicate pressure head values of the water contents that were used in the PTF.
RMSR
Study

TC

SSC

SSCBd
–

SSCBdOM

SSCBdWC (best)

0.04

–

SSCBdOMWC (best)

M1, Al-Majou et al. (2008a)

0.045

–

M2, Børgesen and Schaap (2005)–

0.048†

M3, Cornelis et al. (2005)

–

–

–

–

–

–

M4, Cornelis et al. (2005)

–

–

–

–

–

–

0.058§

0.042§

0.041§

0.026§ (330, 15000)?0.02 (100, 330, 15,000)

0.02–0.026 (100, 330, 15,000)

M5, Nemes et al. (2003)

0.042†‡–
0.038 (100, 1000, 15000)†‡

M6, Schaap et al. (1998)

0.107

0.104

0.087

–

0.058 (100, 330)–

M7, Schaap and Leij (1998a)––

0.096 (0.093)¶

–

–

–

M8, Schaap and Leij (1998a)––

M9, Schaap et al. (1998a)
M12, Schaap and Leij (2000)––

0.086 (0.093)¶

–

–

–

0.061 (0.093)¶

–

–

–

–

–

–

–

M13, Schaap et al. (2001)

0.078

0.076

0.068

–

0.044 (330, 15800)–

M14, Twarakavi et al. (2009)

–

0.062

0.053

–

0.038 (330, 15800)–

M15, Weynants et al. (2009)

–

–

–

0.036

–

–

0.109

0.096

–

0.068 (100, 330)–

M20, Schaap and Leij (1998b)† Seven textural classes were used.
‡ Information on soil horizon was included.
§ Provided by the researchers. Values are calculated as averages across pressure head levels.
¶ Overall average value obtained for the three databases (M7, M8, and M9).

www.VadoseZoneJournal.org | 801straightforward calculation of unsaturated hydraulic conductivity values.

Table 4. Accuracy of pedotransfer functions (PTFs) expressed as the root mean squared residual (RMSR)
for the hydraulic conductivity curve for different sets of predictors: textural class (TC), sand, silt, and clay
contents (SSC), bulk density (Bd), water content (WC), and organic matter content (OM); best refers to
the lowest RMSR found for this combination. Numbers in parentheses after the RMSR indicate pressure
head values of the water contents that were used in the PTF.

The databases with fewer than 1000
samples typically provided RMSRA
RMSR
values <0.02. Two reasons can beTC
SSC SSCBd SSCBdOM SSCBdWC (best) SSCBdOMWC (best)
invoked to explain this finding. First, Study
–
–
–
–
–
–
additional samples may create addi- K1, Schaap et al. (2001)
K2,
Schaapetal.
(2001)
1.06
1.02
1.05
–
0.9
(330,
15000)–
tional variability because the space of
1.76 1.7–
1.62 (100, 330)–
the soil properties is more extensively K3, Schaap and Leij (1998b) –sampled. Second, smaller databases K4, Schaap and Leij (1998b) –1.36 1.29–
1.18 (100, 330)–
have typically used the same measure- K5, Schaap and Leij (2000) –
–
–
–
–
–
ment methodology for all MRCs, thus K6, Schaap and Leij (2000) –
–
1.16†
–
–
–
eliminating the effect of measurement K7, Schaap and Leij (2000) –
–
1.18†
–
–
–
method and possibly also minimizing K8, Børgesen et al. (2008) –
–
0.71
0.598
–
0.614 (100)operator errors. The effect of the mea- K9, Weynants et al. (2009) –
–
–
1.07†
–
–
surement method needs to be further† RMSR adjusted using Eq. [5].
substantiated. For example, Børgesen
and Schaap (2005) and Al Majou et al.
(2008a; M1 and M2 in Tables 1 and 3)
used a standard set of methods but obtained larger RMSRA values
Samples subsequently are moved into low-pressure chambers forthan the RMSRAs found by Schaap and Leij (1998a), who analyzed
measurements up to approximately 500 cm (pF 2.7). Finally, higha database consisting of hydraulic properties obtained with 12 difpressure chambers are commonly used to determine moisture
ferent measurement techniques.
content values up to the wilting point and above using disturbed
soil samples. Sometimes small undisturbed samples taken from
In addition to differences in the measurement method, the datalarger samples are used at the lower pressure heads. In employbases in Table 1 show considerable variations in the number ofing the different pressure head steps, it is not clear how long thedata points used to determine the MRC, the values of pressure
equilibration procedure has to be, and generally no or very little
head used to determine the soil moisture contents, and the size of
information about this is available in the literature or the databases.
the sample used at the different pressure head values. Treatment of
In case the experiments are terminated too early at high pressure
the samples and operation at medium and high pressure head levels
head levels equilibrium conditions do not prevail and moisture
was also found to deviate considerably among researchers. Some
contents are prone to be systematically biased. A lack of equilibresearchers used undisturbed soil samples up to 15,800 cm (pF 4.2),
rium may also be an issue, however, at lower pressures in pressure
whereas others used small disturbed soil samples at this pressurechambers when relatively large soil cores are used. Unfortunately,head (e.g., Vereecken et al., 1989). It is not clear how much theseinformation about the duration that was used to obtain equilibfactors affect a quality assessment of the VGM model in describrium conditions at each imposed pressure head was usually not
ing moisture retention data and how much variability could begiven in the databases.
explained by these various measurement and sampling factors.
Values of the ME for direct fits have, in a few cases, been reported
In addition, the measurement method itself often differed among
also, either for the entire database (Schaap et al., 2001) or as a funcresearchers. Typically, a combination of three or more techniquestion of the pressure head (e.g., Twarakavi et al., 2009). Mean error
was used to obtain the complete soil moisture retention characresults obtained at different pressure heads are discussed below.
teristic in the laboratory (Dane and Topp, 2002). Table 1 shows
We included in this analysis data from Weynants et al. (2009) tothe different methods that have been used to measure the moisprovide a broader base for comparison.
ture retention data. For composite databases such as UNSODA,
a mixture of different methods was used for the various soil
The various databases shown in Table 1 differ substantially insamples. Other databases (e.g., Vereecken et al., 1989) used theterms of their textural composition, as shown in Fig. 2. The datasame set of measurement techniques for all samples. Combining
bases of Nemes et al. (2003) and Stumpp et al. (2009) contained
different measurement techniques may well introduce substantial
many more fi ne-textured soils and fewer coarse-textured soils
inconsistencies in the data. For pressure head values up to 100 cm,
than the other databases. Extreme examples are the databases
soil cores are typically equilibrated on a sandbox apparatus or aof Schaap and Bouten (1996) and Ye et al. (2007), which conpressure cell or plate at selected pressure heads. Alternatively, or
tained only sandy materials. The Rawls database (Schaap andadditionally, some have used the hanging water column technique.
Leij, 1998a) also contained a large number of coarse-textured

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soils and practically no silty soils. The differences in textural composition of the various
databases partly explain the observed variation
in RMSRA values given in Table 1. Hydraulicproperties of certain textural classes are sometimes more difficult to estimate by direct fittingof Eq. [1] to the available retention data or fromderived PTFs, which suggests that the percentage share of these textural classes in the overall
database may affect the overall RMSRA. For
example, silts often show the largest RMSRA
values when fitting Eq. [1] to retention data.
Schaap and Leij (2000) found an RMSRA of
0.0141 for silts compared with 0.0096 for clays.
Soils with a sandy and loamy texture produced
values of 0.0122 and 0.0119, respectively.

Fig. 2. Variation in textural composition of samples in the different databases used to
fit the van Genuchten–Mualem (VGM) parameterization and to develop pedotransfer
functions (PTFs). M7 refers to UNSODA, and M9 refers to the Ahuja database (Schaapand Leij, 1998a). M8 (Schaap and Leij, 1998a) and Schaap et al. (1998) both refer to the
Rawls database.

Pedotransfer function estimates of the hydraulicparameters also tend to differ among textural classes. Schaap et al.
(1998) showed that water content values could be estimated wellfor sandy loam and clay loam soils, with an RMSRA of 0.05. The
RMSRA values for other textural classes ranged between 0.052
and 0.08, with the largest values obtained for sands and clays.
Clays and sands made up only 12.6% of the samples in their database. As such, the proportion of sandy loam and clay loam soils
was much larger (34.2%), thus influencing the overall RMSRA
more than other textural classes. Results obtained with different
databases may therefore be strongly affected by the proportion of
the various textural classes. While similar effects can be expectedwith respect to the distribution of organic C content and the bulk
density of soils in the various databases, information about thisis mostly missing. Literature data mainly provide information on
the range and mean values of C content and bulk density but noton the frequency distributions of these properties in the databases
and variations in the RMSRA for specific ranges.
Table 2 summarizes RMSR values obtained by directly fitting the
VGM model to hydraulic conductivity data. For the HCC, there istypically only a small difference between the RMSR and
RMSRA values due to the large number of data points
compared with the number of parameters. In the caseof the HCC, a direct comparison between the RMSRA
and RMSR is therefore allowable.
In general, less information is available on the performance of the VGM for describing hydraulic conductivity
compared with the moisture retention data. Basically,four databases have been used to evaluate direct fits of
the VGM model to measured data: ROSETTA (Schaap
et al., 2001), UNSODA as a subset of ROSETTA(Schaap and Leij, 1998a), and the databases of Børgesen
et al. (2008) and Vereecken et al. (1989).

Table 2 shows that the best results for the hydraulic conductivity data were obtained when both K s and l were fitted tothe observed hydraulic conductivity data. In many cases, thisimproved flexibility has led to physically unrealistic values of l,as pointed out by Kosugi (1999), Schaap and Leij (2000), andSchaap and van Genuchten (2006). In an analysis of a subsetof the UNSODA database using a similar approach to thatof Mualem (1976), Schaap and Leij (2000) found an optimalvalue of −1 for l. Comparisons of the performance of the VGM
model using UNSODA have been partly hampered by the factthat different subsets with various sample sizes and proportionsin textural composition have been used for the evaluations.
Information on how these subsets differ in the composition ofsoils and soil properties is not available in the literature. Figure
3 shows that mostly sandy soils have been used in the past toevaluate the VGM parameterizations. Schaap and Leij (2000)
provided data about variations in the RMSRA across texturalclasses. They found that clays and silts produced the largest
RMSRA values (0.481 and 0.403, respectively) when the VGM
was fitted directly to the data, compared with 0.395 and 0.398
for sands and loams, respectively. The data set of Børgesen et al.

Fig. 3. Distribution of the main textural fractions within the different databases forthe hydraulic conductivity curve.

www.VadoseZoneJournal.org | 803

(2008) also contained mostly sandy material but no quantification of variations in the RMSRA across textural classes.
Similarly as for the retention data, most databases comprised mixed
methods for measuring the hydraulic conductivity. The databases of
Børgesen et al. (2008) and Vereecken et al. (1989) are the only ones
that have hydraulic properties that were measured on every sample
with the same measurement techniques within the entire database.
An evaluation of the possible effect of measurement technique on
the RMSRAs is hence not possible due to a lack of data.

Analysis of the van Genuchten–Mualem
Parameteriza on at Diﬀerent Pressure Heads

Earlier work by Vereecken et al. (1989) using 182 measured
soil moisture retention data sets of soils with different textures
showed that the restriction m = 1 − 1/n led to a somewhat worse
description than the five-parameter model or the model with m= 1 as originally proposed by Brutsaert (1966). Vereecken et al.
(1989) therefore suggested not to use the restriction proposed by
van Genuchten but to fi x m at 1.0 and leave n as a free parameter.
As a consequence, n values <1 were frequently obtained for fi netextured soils, which implies that the differential soil moisturecapacity goes to infi nity, as discussed by Nielsen and Luckner(1992). The main reason for these low n values were caused by the
sharp drop in the soil moisture content in the very wet end of the
moisture retention characteristic, probably caused by structuralproperties producing a heterogeneous porous system. The presence of a heterogeneous porous system may not have been detected,
however, due to a lack of sufficient retention data pairs close tosaturation and the effect of sample height. Indeed, as shown byPeters and Durner (2006), when measuring retention curves, thewater content should be explicitly considered as an integral of theequilibrium water content distribution with depth.

The Soil Moisture Reten on Characteris c. Several studies
have pointed out weaknesses in the VGM parameterization for
describing the wet and dry parts of the soil moisture retention
characteristic (e.g., Khlosi et al., 2008). To verify these findings,
we analyzed literature data that provided information about the
RMSRA obtained by fitting the VGM model to observed data andplotting the RMSRA as a function of the pressure head. Only afew studies provided this information for the MRC, while only
Figure 5 shows the ME for different parameterizations of Eq. [1]two published studies could be found where this information
as obtained for two models derived from one database. Weynants
was available for the hydraulic conductivity. For the purpose of
et al. (2009) fitted Eq. [1] with m = 1 and m = 1 − 1/n to the datathis review, we derived the variation in the RMSRA with presbase of Vereecken et al. (1989). Both parameterizations tend tosure head using the data of Weynants et al. (2009). Figure 4 showsunderestimate the observed soil moisture content close to saturaRMSRA values for the MRC at different pressure heads obtainedtion. This may be explained by the inability of Eq. [1] to account forfor four data sets: UNSODA (Nemes et al., 2003), Tomasella etmacroporous flow domains. In the very dry range, Eq. [1] tendedal. (2003), the Vereecken database (Weynants et al., 2009); andto overestimate the observed moisture content. In the medium
Børgesen and Schaap (2005). For Eq. [1] with m = 1 − 1/n andpressure head range, the parameterization with m = 1 provided
m = 1, the RMSRA tended to increase with increasing pressure
the lowest ME values.
head. The largest RMSRA values were found for pressure heads
between 500 (pF 2.7) and 2500 cm (pF 3.4), with values ranging
An accurate description of the MRC in the wet range remains abetween 0.015 and 0.037. The largest RMSRAs were obtained forchallenge and has not yet been resolved. Th is is due not only tothe data from Børgesen and Schaap (2005). Relatively low RMSRA
problems with the parameterization itself but also to difficulties invalues for Eq. [1] were found at pressure heads in the very wet range,
precisely determining the soil water content at low pressure heads
with the lowest values obtained by Schaap et al. (2001) for m =near saturation. Incomplete saturation may occur because of wet1 − 1/n, and for m = 1 obtained by Weynants et al. (2009) using
tability problems and the presence of entrapped or dissolved air,the data of Vereecken et al. (1989). In addition,
the lowest variation in RMSRA among thedifferent databases was found at pressure headvalues between 2.5 (pF 0.4) and 30 cm (pF 1.5).
The VGM parameterization (Eq. [1]) applied to
the UNSODA data of Nemes et al. (2003) gavethe largest deviation for the moisture content
at saturation. The researchers attributed thisdeviation to the presence of macroporosity.
Durner (1994) previously pointed out that forheterogeneous pore size domains, the classicalVGM parameterization may fail. The limitationof the restriction m = 1 − 1/n in fitting the wet
Fig. 4. Root mean squared residual adjusted (RMSRA) values for the direct fit of Eq. [1] topart was also shown in the data of Vereecken et
moisture retention characteristic data as a function of the pressure head.
al. (1989) and Cornelis et al. (2005).

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both of which will affect the measured saturated water content, θ s . In addition, exact
measurement of changes in the water content
at low pressure heads is very difficult, particularly when the pressure head is less than
the sample height. Moreover, inverse methods such as the multi-step outflow method
are not model free and typically invoke a
relationship between the pressure head andmoisture content that ignores the presenceof macroporous regions.
We note that air entrapment is generally
far more of an issue at the plot or field scale,thus causing laboratory measurements of thesaturated water content, or even of the laboratory-measured satiated water content that
accounts for air entrapment (e.g., Luckner etal., 1989; Hillel, 2004), not to be representative of field conditions. This also suggests thatextrapolation of laboratory-measured θ s andKs values, as well as PTFs derived from suchmeasurements, to transient field moisture conditions may be problematic.

Fig. 5. Mean error (ME) of the water content at different pressure head values for two different
parameterizations of the van Genuchten–Mualem (VGM) model.

Weaknesses in describing the dry end of
the MRC has been mainly attributed to thefact that water retention in that part is noFig. 6. Root mean squared residual (RMSR) values of the unsaturated hydraulic conductivity (K)longer determined by capillary forces but by
for the direct fit of the van Genuchten–Mualem (VGM) model to hydraulic conductivity curve
adsorptive forces (Li and Wardlaw, 1986;
data as a function of the pressure head (pF). “Gardner, Vereecken” refers to the Gardner equation(Eq. [8]) as it was used by Vereecken et al. (1990) and Weynants et al. (2009).
Lenormand, 1990; Nimmo, 1991). In addition, there are only a few data sets containing
measurements for pressure head data beyond
15,800 cm (pF 4.2). Khlosi et al. (2008) pointed out difficulties inof Schaap and van Genuchten (2006) to account for the effects
measuring moisture contents at these high values. Several studies
of macroporosity near saturation. The RMSR values of Schaap ethave shown that the residual moisture content in Eq. [1] is diffial. (2001) were taken from their Fig. 4. We included in Fig. 6 also
cult to determine using inverse methods. As such, Nimmo (1991),
the results for the Gardner equation of the hydraulic conductivas well as many others (e.g., van Genuchten and Nielsen, 1985)
ity because this equation was found to perform equally well orsuggested considering θ r, which strongly influences the MRC
better than the VGM parameterization when evaluated across thedescription in the dry range, as a VGM fitting parameter.
entire database (Vereecken et al., 1990; Schaap and Leij 1998b).
The Gardner HCC is given by
The above considerations suggest that the invoked experimental
Ks
techniques, procedures, and protocols for measuring moisture
K (h ) =[8]n
1 + ( Bh ) g
retention data probably contribute to the RMSRA when fitting
Eq. [1] to data and that they may introduce inaccuracies whenestimating parameters.
where B and ng are fitting parameters.
The Hydraulic Conduc vity Curve. Figure 6 shows variations
in RMSR values for the hydraulic conductivity as a function of
the pressure head for five cases. The particular cases involve theclassical VGM parameterization with fi xed Ks and l parameters,
the VGM with adjustable Ks and l values (Schaap et al., 2001;
Weynants et al., 2009), and the modified VGM parameterization

Except for the classical VGM model with fixed Ks and l parameters,
all parameterizations in Fig. 6, including the Gardner HCC, generated lower RMSR values with increasing pressure heads up to avalue of 2500 cm (pF 3.4). The best results across the entire rangeof pressure head values were obtained with the model proposed by
Schaap and van Genuchten (2006) that included provisions for

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flow through macropores (see also below).
All models, except for the classical VGM
model, produced the highest RMSR in the
wet and very dry ranges. The lowest RMSR
values were obtained in the pressure headrange between 100 (pF 2) and 2500 cm(pF 3.4). This differs from analyses of the
MRC, which showed the largest residuals
in this pressure head range (Fig. 4). Noticethat the Gardner equation performed
quite well in the very wet range, providing RMSR values that were very similar
to those of the improved model of Schaapand van Genuchten (2006). This may be
explained by the presence of the B parameter in Eq. [8], which may be interpreted ina way similar to the α parameter in Eq. [1].

Fig. 7. Mean error (ME) values of the unsaturated hydraulic conductivity (K) at different pressure
heads (pF) for the direct fit of the van Genuchten–Mualem (VGM) model using a flexible saturated
hydraulic conductivity Ks and tortuosity parameter l to observed hydraulic conductivity data (Weynants et al., 2009).

Figure 7 shows the variation in ME across different pressure headlevels for the database of Vereecken et al. (1989) as analyzed by
Weynants et al. (2009) using the VGM model with variable Ks
and l parameters. At relatively low pressure head values, the VGM
model clearly underestimated the observed hydraulic conductivity
as indicated by the negative ME values. The ME for pressure headvalues between 30 (pF 1.5) and 200 cm (pF 2.3) changed from anegative to a positive value. For larger pressure head values (except
at pF 4.2), the ME remained positive. This was confirmed by the
analysis of Schaap and van Genuchten (2006) of the UNSODA
database, which revealed a substantial underestimation between 0
and 30 cm (pF 1.5), a slight overestimation between 30 and 1000 cm,
and again an underestimation beyond 1000 cm (pF 3) (Fig. 7). They
also showed that the traditional VGM model substantially overestimated the hydraulic conductivity between 0 and 1000 cm andseverely underestimated the hydraulic conductivity beyond 1000 cm.

Extensions of the Classical van Genuchten–
Mualem Model
The following is an overview of several approaches that were developed
to overcome the limitations of the VGM model in describing and predicting the hydraulic properties across the entire range of moisture
contents, with specific attention to the wet and dry ends of the curves.

Heterogeneous Pore Size Systems
and Pore Geometry
The use of the VGM model assumes that the pore size distributionof the soil can be described by a single unimodal pore size distribution function. Durner (1994) emphasized that undisturbed
soils frequently have pore systems that cannot be described witha unimodal pore size distribution function. Aggregation and biological soil-forming processes may lead to secondary pore systems
involving larger pores. To describe the presence of a heterogeneous
pore size system in a satisfactory manner, Durner (1994) proposeda multimodal retention function by superposition of subcurves
described by Eq. [1]. Using example soils, he showed that conductivity estimates based on the classical VGM model can differ byorders of magnitude compared with the van Genuchten modelwith a multimodal pore size distribution. Recently, Priesack andDurner (2006) extended this approach of heterogeneous pore systems to the mathematical description of the VGM HCC.
In addition to a unimodal pore size distribution function, theVGM parameterization assumes that the pore system can be
described as a bundle of capillaries with flow processes governedby the principles of capillary flow. Moreover, capillaries are either
fi lled with water or with air, without accounting for fi lm flowthat may dominate in relatively dry soils. Tuller et al. (1999) proposed a new concept of pore space geometry that was composedof angular pore cross-sections connected to slit-shaped spaces withan internal surface area to led to a better description of liquid distribution, water flow, and solute transport in partially saturated
porous media. Their approach provides a better means of combining adsorption and capillary processes and is more consistent withthe observed pore shapes obtained from soil core slices.
Tuller and Or (2001) compared their approach with the VGM
model to describe experimental data of the hydraulic conductivity. They considered the MRC to consist of different ranges basedon liquid-filling stages, each described using a simple algebraic
expression relating the relative liquid saturation to the pressure
head. They fitted the parameters of these algebraic expressions to
the measured soil moisture retention data, which were then used as
input parameters in the derived sample-scale hydraulic conductivity
equations. Very good agreement was found between the predictions
and measured data. For each of the analyzed soil samples, additional
measured or predicted soil properties were available, which makes
it possible to derive PTFs between the MRC parameters and clay
content, specific surface area, and porosity or other data. Four model

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parameters were adjusted to the data: a dimensionless slit lengthparameter β, a γ distribution parameter ω, the air entry value μ b, anda distribution overlap parameter λ that relates a dimensionless slitspacing parameter α s to the largest pore and the mean pore length.
Analysis of their data (Tuller and Or, 2001, p. 1268, Tables 2 and 3)
showed high positive correlation between the β and λ parameters
and the clay content, while a negative correlation was found between
μ b and the surface area. These results are encouraging and suggestthat further work is needed to relate these parameters to MRC dataobtained on natural soil samples.

Improving the Descrip on in the Wet Range
Two problems may arise close to saturation and question the use of
the VGM model: (i) structural pores in the soil are activated andmay lead to noncapillary flow, for which the VGM model is not
appropriate; and (ii) noncontinuity of the air phase may lead toerroneous estimates of the hydraulic conductivity.
Macroporous Flow and the Air-Entry Value. Different solutions have been proposed for the first problem. Dual-permeability
models have been developed over the years in attempts to predictflow and transport in macroporous soils and fractured rock (Tsangand Pruess, 1987; Gerke and van Genuchten, 1993; Larsbo et al.,2005; Jarvis, 2008). These models assume that structured media
consist of two overlying but intertwined continua at the macroscopic scale: a macropore or facture pore system and a permeable
matrix pore system. The Richards equation with VGM parameterization is typically used for the matrix domain, while different
options exist for simulating water flow in the macropore domain.
Gerke and van Genuchten (1993) also used VGM parameterizations and the Richards equation for the high-permeability domain,along with a coupling term to allow water exchange between themacropore and matrix domains. Jarvis (2008) based his model on
the generalized Kozeny–Carman equation, along with a pore-scale
tortuosity factor initially suggested by Fatt and Dykstra (1951) andthe soil water retention function model of Ross et al. (1991). Usinga data set of 12 Swedish soils, Jarvis (2008) was able to relate thepore size distribution index in the Ross et al. (1991) model to clay
content and macroporosity. Unfortunately, broad applicability ofthis approach using PTFs is questionable because of the current
lack of reliable hydraulic data in the wet range (see also below).
Using the saturated hydraulic conductivity as a matching factor forpredicting the unsaturated hydraulic conductivity function with
Eq. [3] or some other equations for K(h) is problematic and of little
value in macroporous soils (e.g., Ehlers, 1977; van Genuchten andNielsen, 1985; Jarvis et al., 2002). Based on an analysis of field
data, Mohanty et al. (1997) suggested that the VGM parameterization should only be used for capillary-dominated flows at pressure
heads >3 cm (pF 0.5). When gravity flow dominates (pressure
heads <3 cm), other more appropriate models for retention andthe hydraulic conductivity should be used. Th is again supports

findings that the VGM model does not perform well for macroporous soils or soils with heterogeneous pore size distributions nearsaturation (e.g., Durner, 1994). Mohanty et al. (1997) used visualinspection of the hydraulic functions to identify a breakpoint in
the pressure head that separates the flow domain into capillarydriven and a non-capillary-driven regimes. The VGM hydraulicparameters were used to estimate the hydraulic properties of thecapillary flow domain, whereas an exponential equation with threeparameters was used to represent the non-capillary-flow domain.
These parameters included the breakpoint soil water pressure head,its corresponding hydraulic conductivity, and a fitting parameter
representing the effective macroporosity or other structural features contributing to the non-capillary-dominated flow domain.
Finally, Schaap and van Genuchten (2006) introduced a small
but constant pressure head in the VGM model, as suggested by
Vogel and Cislerova (1988), to account for the effects of macroporosity on K(h) near saturation, thereby removing several of theshortcomings in the original VGM model. Using a detailed neuralnetwork analysis of part of the UNSODA database, they found themacropore contribution to the hydraulic conductivity to be mostsignificant between 0 and 4 cm (pF 0.6), although the effects ofmacroporosity were visible until pressure heads of about 40 cm(pF 1.6). The modified conductivity model significantly reduced
the RMSRA for all textures (range 0.242–0.275), while also producing only very small systematic errors between 0 and 15,800 cm(pF 4.2) (Fig. 6). Schaap and van Genuchten (2006) used a valueof 4 cm (pF 0.6) for the minimum capillary height, ha , in theirmodified model. Børgesen et al. (2006) found a similar value forha in their analysis of a different database. They similarly focusedon the HCC near saturation by implementing a scaling approachto account for soil structural effects on K. The value of 4 cm (pF
0.6) is similar to the value of 6 cm (pF 0.8) used by Weynants et al.
(2009) to exclude the effects of macroporosity in their analysis of
the Vereecken database, and the values of 6 (pF 0.8) to 10 cm (pF 1)
used by Jarvis (2007) as limits of the macropore region (equivalent
to macropores having equivalent cylindrical diameters of 0.3–0.5
mm, as suggested by the Laplace–Young equation).
In addition, introduction of an air-entry value provides better
numerical stability to numerical solutions of the Richards equation, in particular when applied to the hydraulic properties ofvery fi ne-textured soils, when n is less than about 1.2. The discontinuity in the differential soil water capacity at saturation leads
to strong nonlinearities in the hydraulic conductivity function,
K(h), near saturation (Vogel and Cislerova, 1988; Luckner et al.,
1989; Fuentes et al., 1992). Therefore, Vogel and Cislerova (1988)
and Vogel et al. (2000) introduced a very small non-zero minimum capillary height, ha , in Eq. [1]. Very similar to the approach
by Vogel et al. (2000), Ippisch et al. (2006) introduced an airentry value in Eq. [1] for cases when n < 2 or αha > 1. These
modifications do not cause a major change in the retention function as such but lead to improved predictions of the hydraulicwww.VadoseZoneJournal.org | 807

conductivity function while also improving the performance of
inverse methods for measuring the hydraulic properties of fi netextured soils. Weynants et al. (2009) compared the model of
Ippisch et al. (2006) with the MVG model, the Mualem–Kosugi
model (Kosugi, 1996), and the Gardner (1958) model using thedatabase of Vereecken et al. (1989). For the MRC, they foundthat Eq. [1] with m = 1 and the MVG parameterization as used by
Weynants et al. (2009) provided the lowest RMSRA values. For
the hydraulic conductivity data, only values for pressure heads>6 cm (pF 0.8) were used. In this case, the lowest RMSRA wasfound for the model of Ippisch et al. (2006). They found, however,that introduction of the air-entry value improved the predictionof only the HCC and not the MRC.
Desatura on and Pore Size Distribu on. Discontinuities in the
non-wetting (air) phase in the near-saturated part of the MRC maylead to erroneous estimates of the hydraulic conductivity, irrespective of whether one or many pore size distribution models are used
(Corey, 1992; Jana et al., 2007). This is due to the fact that, at highwater contents (above approximately 85% saturation), air is not free
to enter all parts of the medium. As explained by Corey (1992),
regions containing large pores may be entirely isolated from the airphase by surrounding water-filled pores. One may therefore question
whether data close to saturation should be used to predict hydraulic conductivities based on capillary phenomena only, and whether
or not two-phase flow models should be used. The uncertainty indetermining the very wet part of the MRC motivated Corey (1992)
to extrapolate retention values down to the saturated value. Earlier
work by Henderson (1949) and White et al. (1972) demonstrated
that retention of soil water in the very wet range is controlled by theboundary area to volume ratio of the sample but not by the pore size
distribution. At water contents small enough to provide a finite airpermeability, the curves are no longer affected by the boundary area
to volume ratio. The presence of boundary effects seems to have been
confirmed by lattice Boltzmann simulations of pore-scale air andwater redistribution in a glass bead system on wetting and drying(Schaap et al., 2007). Better correlations between water retention
and hydraulic conductivity curves were obtained by excluding watercontent data >85% of the saturated value.

Improving the Descrip on in the Dry Range
Several researchers have reported difficulty in describing the dry
end of the MRC using Eq. [1]. The residual soil moisture content, being the moisture content when the pressure head goesto infi nity, has been found to be an ill-defi ned parameter, withvalues often becoming negative during optimization unless special precautions are taken (e.g., Khlosi et al., 2008). To prevent
negative values, the lower boundary of the residual moisture content is usually fi xed to zero, which frequently leads to inaccurate
descriptions of the observed data and unrealistic paths of thecurve (Pachepsky et al., 1996; Khlosi et al., 2006; Haverkamp
et al., 2005). Børgesen and Schaap (2005) showed that RMSR

values obtained with Eq. [1] were generally larger than the
RMSRs obtained using the PTF point estimation method. They
attributed this to imperfect fits of the VGM parameterization toretention data at 15,000 cm (?pF 4.2).
Several studies have addressed the dry region issue by proposing
improved MRC descriptions across the whole range between saturation and oven dryness. Improvements in the dry end have beenobtained by incorporating an additional expression that describes
the adsorption of water (Campbell and Shiozawa, 1992; Fayer andSimmons, 1995; Cornelis et al., 2005).
In a more recent study, Khlosi et al. (2008) compared the performance of eight different MRC models (including those that accountfor adsorption of water) in describing retention data between saturation and oven dryness. They found that a combination of theadsorption equation of Campbell and Shiozawa (1992) with themodel of Kosugi (1999) provided the best description for 137 MRCs
taken from the UNSODA database. Rossi and Nimmo (1994) also
proposed two models for the dry end, both based on the observationthat the retention curve in the dry range is approximately proportional to the logarithm of the suction. They indicated that theirmodels can be easily added to the VGM parameterization.

Flexible Interpreta ons of the Classical van
Genuchten–Mualem Parameteriza on
Since its publication in 1980, the VGM model has been the subjectof many studies analyzing its capability to describe the unsaturated
hydraulic conductivity of natural soils. Mualem (1976) initially tested
his predictive hydraulic conductivity model using a limited databaseinvolving 45 MRC and HCC data sets mostly obtained from coarsetextured, uniformly packed soil materials. By fitting the model withl as a variable parameter, he found an average value of 0.5. The modelof Mualem (1976) was tested on different soils and found to performbetter than other theoretical models such as the models proposed by
Burdine (1953) and Childs and Collis-George (1950) (Mualem, 1976;
van Genuchten and Nielsen, 1985; Vereecken, 1995).
Using the VGM model with l = 0.5 to predict the measured unsaturated hydraulic conductivity, however, frequently leads to over- or
underpredictions (Schaap and Leij, 2000). In addition, considering l as a fitting parameter brings much more flexibility, as shownin Fig. 8. An increase in l causes a steeper decrease in the relative
hydraulic conductivity as a function of relative saturation, S. The
decrease in the hydraulic conductivity near full saturation is controlled by the value of n to a much larger extent than by the valueof l (e.g., Wösten and van Genuchten, 1988).
Therefore, many studies have revisited the value of l in Eq. [2]and [3]. For example, Vereecken (1995) found values of l strongly
deviating from 0.5, including many negative values, using a setof 185 measured hydraulic conductivity data sets determined on

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conductivity can be described. The Gardner equation was also foundto better describe the hydraulic conductivity data of 245 K(h) data
sets from the UNSODA database (Schaap and Leij, 1998a). The
RMSR of the direct fit was equal to 0.22, compared with 1.08 for
the VGM model with l = 0.5, and 0.77 for the VGM model with l =−1. The PTFs using textural properties and soil moisture data were
found to better predict the unsaturated hydraulic conductivity than
the traditional VGM parameterization. Decoupling of the MRC
and HCC functions was also suggested by Fuentes et al. (1992).

Fig. 8. Effect of the tortuosity parameter l on the relative hydraulicconductivity (K/Ks) computed with the van Genuchten–Mualem(VGM) model at different effective saturation (S) values. Results
are plotted for shape parameter n values of 1.3 (solid lines) and 1.7(dashed lines).

undisturbed soil columns. Negative values for l in the Mualem
model were also found by Schuh and Cline (1990), Kaveh and van
Genuchten (1992), Yates et al. (1992), Kosugi (1999), and Schaapand Leij, (2000), among others. A negative value of l implies thatthe connectivity of water-fi lled pores increases with decreasing soil
moisture content, which intuitively does not seem realistic.
Improved descriptions of measured hydraulic conductivity data are
possible by treating both l and Ks as adjustable parameters in the
VGM parameterization. Schaap and Leij (2000) showed that optimization of both parameters improved the RMSR by a factor of 3.2.
The parameters l and Ks hence should be treated as empirical shapefactors without much physical meaning. This ultimately implies thatthe model of Mualem (1976), as well as related statistical pore-size
distribution models published previously or later, provides a simplistic conceptualization of the hydraulic conductivity of a porous
medium (Schaap and Leij, 2000; Tuller and Or, 2001).
Treating Ks, similarly to the saturated or satiated water content,
θ s, as an empirical parameter with only limited physical meaning
had already been advocated earlier by van Genuchten and Nielsen(1985) and Luckner et al. (1989). In a more recent study, Wagner
et al. (2001) compared different PTFs to predict the unsaturated
hydraulic conductivity. They found that the best estimates of K(h)were obtained when Ks was predicted from soil properties rather
than using the measured saturated values.
Another option to improve descriptions of the soil water retention
and unsaturated hydraulic conductivity functions is by decoupling
the retention and conductivity predictions. This decoupling would
provide more flexibility in describing both properties. For example,
and as mentioned above, Vereecken et al. (1990) showed that the
Gardner equation for K(h) provides better prediction of the hydraulic conductivity than the VGM model. This is partly due to the
Gardner (1958) equation providing more flexibility in fitting thewet range. By letting B in Eq. [8] become >1, sharp drops in hydraulicPachepsky et al. (1999) noted that while much effort has focused on
the parameter l in Mualem’s predictive model for the unsaturated
hydraulic conductivity, almost no research has been performed on
the tortuosity parameter b in Eq. [2]. Pachepsky et al. (2000) related
the unsaturated hydraulic conductivity to soil water retention using
a capillary bundle model with a microscopic connectivity correction,γ, expressed as a power law function of the pore radii:
θ

− 2−γ

K = K s ∫ h (τ)
0

dτ

[9]

where τ is the integration variable. The term Sl was excluded to
avoid using two different terms in an equation to simulate the same
phenomenon of tortuosity. The model was applied to data from147 soil horizons, providing an accurate approximation so that the
RMSR in the estimated log10(K) was 0.21. Two parameters of themodel appeared to be closely correlated so that only one connectivity parameter was needed. Reducing the number of parametersfrom two to one and refitting the model to the data decreased theaccuracy of the estimates, as the RMSR increased from 0.21 to
0.31; however, the RMSR remained about 1.5 to 2 times lower thanthe RMSR of the best saturated hydraulic conductivity estimatesobtained with the UNSODA database.

6 Accuracy and ValidaƟon of

Pedotransfer FuncƟons for thevan Genuchten–Mualem Model
Accuracy

Several reviews about the accuracy and reliability of PTFs for the
VGM model have been provided (e.g., Wösten et al., 2001; Schaap,
2004; Donatelli et al., 2004). In the literature, the term accuracy isrelated to the comparison between predicted and measured values ofmoisture content or hydraulic conductivity that were used to develop
the PTF. Reliability is related to the evaluation of PTFs on measured
values that are different from those that were used to develop the PTFs
(Wösten et al., 2001). Reliability studies are typically validation studies
such as those performed by Tietje and Tapkenhinrichs (1993), Wösten
et al. (2001), and Wagner et al. (2001).
As for the evaluation of the direct fit of the VGM model to hydraulicdata, evaluations of the performance of PTFs have been performed

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mainly in terms of the RMSR. Other indicators are the ME (Nemes
et al., 2003), the coefficient of determination (e.g., Stumpp et al.,2009; Cornelis et al., 2001), the RMSD (Tietje and Tapkenhinrichs,
1993), the mean absolute error (Weynants et al., 2009), the mean
relative error (Weynants et al., 2009), and the model efficiency(Weynants et al., 2009). In our analysis, we have focused on the
RMSR because this indicator has been reported widely in the literature. In addition, we have included data on the ME as a function of
the pressure head for both the MRC and the HCC.
Pedotransfer functions providing point estimates of the watercontent or hydraulic conductivity tend to be more accurate thanestimates based on parametric PTFs (e.g., Pachepsky et al., 1996).
For example, Tomasella et al. (2003) compared two different
approaches for obtaining analytical expressions of the whole MRC.
In the first approach, moisture content values at specific pressure
heads were estimated by PTFs, after which the VGM model was
fitted to these data. In a second approach, the VGM model was
directly fitted to the data, after which the fitted parameters wererelated to soil properties. They found that the RMSR of the direct
estimation technique (the second approach) was 0.098, whereas
the RMSR of the first approach was 0.036. The largest differencein RMSR between the two approaches was found for pressure headvalues of 0, 60 (pF 1.8), and 15,800 cm (pF 4.2). The RMSR valuesof the development data set for the two approaches were 0.04 and0.182, 0.035 and 0.111, and 0.025 and 0.182, respectively. Of thevalidation data set, the RMSRs for the parametric approach wereagain larger, but the pressure heads at which this occurred were in
the midrange, i.e., 100 cm (pF 2), 330 cm (pF 2.5), and 1000 cm(pF 3). Tomasella et al. (2003) suggested that moisture contents at
different pressure heads may be controlled by different soil properties and thus that using PTFs for a specific pressure head allows abetter combination of these properties. The large errors obtainedwith the parametric methods for specific pressure heads may wellbe due to the inappropriate parameterization in these regions.
Table 3 shows that the largest improvements in predictions of the
MRC have been obtained when water content information was
included in the PTF. The RMSR values for that case are still considerably larger than the values obtained for the direct fit as given
in Table 1. The potential of combining organic matter (OM) andthe soil water content to improve a PTF has not yet been fully
explored. Only a few researchers (e.g., Nemes et al., 2003; Børgesen
et al., 2008) have evaluated the combined effect of using OM andwater content data in addition to textural data and bulk density. In
the study of Børgesen et al. (2008), the RMSR obtained with thisinformation was not better, or was even slightly worse, than PTFs
based on either sand, silt, and clay (SSC), bulk density (Bd) and OM(SSCBdOM) or SSC, Bd, and water content (WC) (SSCBdWC). It
should be noted that also information on the type of soil horizon wasused as a predictor but its impact on reducing the RMSRs appeared
to be relatively small based on the data presented in Børgesen et al.
(2008). A similar result with respect to OM was obtained by Nemes

et al. (2003). They obtained the lowest RMSR value of all the studies
presented in Table 3 using three moisture content values in addition
to SSC and Bd. The combined use of OM and soil water content, in
addition to texture and bulk density, showed a somewhat higher
RMSR than the case with only SSCBdWC (see Nemes et al., 2003,
p. 1097, Fig. 2a). The data show that OM as a predictor variable
relative to the introduction of WC data seems to be less important;however, not all moisture content values are equally effective in
reducing the RMSR. Børgesen et al. (2008) also showed that, ingeneral, the lowest RMSR values (highest accuracy) were obtained
for the data set of Nemes et al. (2001) compared with the values
obtained for a European data set derived from HYPRES, an intercontinental data set, or a combination of these data sets using PTFs
with different input levels.
Study M6 (Schaap et al., 1998) provided information on the distribution of the RMSRA across the USDA textural classes. For the
PTFs using the highest level of input data (SSC, Bd, and WC at 330cm), the lowest RMSRA was obtained for the sandy loam and clay
loam classes with a value of 0.050. The overall proportion of the two
textural classes in the data set was 34%, thus influencing strongly
the overall RMSRA value. The largest RMSRA was found for sandy
soils (0.080). This can be explained by the large variation in particle
sizes (fine to coarse) within the sand fraction. This will lead to alarger air-entry value for the sandy soils within the coarser fraction.
Table 4 shows that even the best PTF models for the HCC still
showed RMSR or RMSRA values that were twice as large as the
RMSR or RMSRA values obtained for the best direct fit (Table
2). Using moisture content values and a more detailed particle size
distribution (e.g., Børgesen et al. [2008], not shown in the table)
did not necessarily improve the estimates. This is different from thecase of the PTFs used to estimate soil moisture contents, as presented in Table 3. Here the introduction of moisture content andorganic matter substantially reduced the RMSR values. The RMSR
or RMSRA values of PTFs using textural information, bulk density, and organic matter (K8 and K9 in Table 4) were of the sameorder as the PTF using only textural information and Bd. Although
Weynants et al. (2009) only considered hydraulic conductivity datain the matrix flow domain, their obtained RMSR value was considerably larger than the value obtained in Study K8. On the other
hand, K8 used only a limited number of measured data points forthe hydraulic conductivity, eventually leading to lower RMSR values.
Studies K1 and K5 could not be evaluated because no PTF was
required due to the fact that Ks was fixed to the measured value andl was fi xed at 0.5. Only Study K6 provided information on the distribution of the RMSRA across textural classes. No distinct pattern
could be identified. For the highest level of input that included the
VGM parameters, the lowest RMSRA (0.67) was obtained for theclass of silty soils. In the case of Study K8, the use of seven texturalclasses and water contents measured at 100 (pF 2), 500 (pF 2.7), and
15,000 cm (?pF 4.2) did not improve the results compared with the
PTF using the water content at 100 cm.

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In the following, we analyzed the accuracy of PTFs to estimate specific ranges of the MRC and HCC with due attention to the wet anddry ranges using RMSR or RMSRA values as the basic indicator.
Wösten et al. (2001) provided RMSR values for moisture contents at
100 (pF 2) to 330 cm (pF 2.5) and 15,000 cm (?pF 4.2) of 19 different PTFs based on point estimation. The mean value of the RMSR
across all PTFs, MRMSR, was approximately 0.05. Evaluating only
the RMSR for the directly estimated soil water contents at 330 and
15,000 cm resulted in a MRMSR of 0.049 (SD = 0.02, n = 7) and
0.45 (SD = 0.03, n = 11). Calculating the MRSR for the complete
moisture retention characteristic gave a value of 0.059 (SD = 0.03, n= 10). A comparison of the RMSR for the point estimations with the
MRMSR for the complete MRC obtained from the Ahuja (0.061),
Rawls (0.086), and UNSODA (0.096) databases, all cited in Schaapand Leij (1998a), shows that the latter are still larger. This may be due
to inappropriate parameterization or having a combination of different MRC measurement methodologies. Inspection of the figures in
Schaap and Leij (1998b) suggests that the largest uncertainties still
occur in the wet and dry range extremes.
Rajkai and Várallyay (1992) used regression equations to estimate
soil water contents at different pressure heads for 270 soil samples.
They found that the largest uncertainty, expressed as the coefficientof determination (R2), occurred at intermediate pressure heads (30[pF 1.5] to 1000 cm [pF 3]), with the lowest value of R 2 around100 cm (pF 2). They did not make a distinction between texturalproperties of the soils. Donatelli et al. (2004) developed a set ofbasic indices to evaluate 14 PTFs to estimate moisture content
values at 330 and 15,000 cm. The indices were mainly used to rank
the different PTFs in terms of their ability to estimate moisture
content values. No evaluation of the VGM approach related to thecomplete MRC and HCC was performed.

intervals coincided with the number of data available. Texturalclasses with more data (sandy loam and loam) resulted in narrower
confidence intervals than textural classes with fewer data (e.g., clay,
silty loam). In addition, they suggested that not only the numberof samples is important but also the variability in soil hydraulicproperties within a textural class. It should be noted, however, thatthe largest uncertainty occurred for the fi ner textured soils. As
mentioned above, the standard VGM parameterization does not
perform well for those soils because of pore heterogeneity and possible limitations in the VGM model for small n values. It would
therefore be worthwhile to examine different parameterizations
that may perform better in describing the wet range of the MRC.
In addition to a statistical analysis of the residuals between themeasured and the fitted values, the VGM parameterization may
provide insight into the soil properties influencing the behavior of
the MRC in the wet range. This requires, of course, that relevant
structural information is available.
Analyzing Schaap and Leij (1998a, Fig. 4) showed that, in general, either the dry or wet range of the MRC showed the largest
uncertainty. The range between 30 (pF 1.5) and 1000 cm (pF 3)
showed, in most cases, the lowest uncertainty, indicating that
relevant information was available and that the VGM parameterization performed well. This contradicts the findings of Rajkai and
Várallyay (1992) and Nemes et al. (2003), among others.
Figure 9 shows the variation in RMSR values as a function of pressure head for PTFs using texture and bulk density as predictor
variables. As expected, the highest RMSR values were found for
PTFs that used only soil texture (Fig. 9). For most of the PTFs, weobserved an initial increase in the RMSR with pressure head, followed by a decrease in the RMSR for pressure head values >2500
cm (pF 3.4). This behavior was also observed for the RMSRA in
the case of the direct VGM fit to measured moisture retention
data. The PTFs of Stumpp et al. (2009) and Nemes et al. (2003)
showed a slight decrease in the RMSR with increasing pressure
head in the very wet range. The higher values near saturation

Schaap and Leij (1998a) analyzed the accuracy of PTFs at estimating the MRC based on the VGM parameterization using threedifferent data sets. As predictors for the PTFs they required theavailability of sand, silt, clay, and bulk density, and they required
at least six water retention points for
the MRC. They found that calibrated
PTFs predicted the soil moisture content with RMSRs between 0.06 and0.10 cm3 cm−3. Cross-validating PTFs
among the data sets showed that the
RMSR increased to values between
0.08 and 0.13 cm 3 cm−3 . The extent
of the confidence intervals for water
retention data was found to vary among
the textural classes. The intervals were
found to be relatively narrow for the
loamy sand and loam samples (0.04
cm3 cm−3) but increased to 0.10 cm3
Fig. 9. Variation in the root mean squared residual (RMSR) value as a function of pressure head forcm−3 for the silty loam and clay sampedotransfer functions using texture and bulk density as predictors: SSC = sand, silt and clay; Bd =bulk density; TC = textural class; TC(7) = textural distribution using seven classes.
ples. They also showed that narrower

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may be due to the inability of Eq. [1]to capture the effects of macroporosity.
The values in the dry range were on thesame order as in the wet range. The only
exception was for the data measured by
Stumpp et al. (2009). Th is may partly
be explained by the different nature of
their soils compared with most databases. The database of Stumpp et al.
(2009) contained a large contribution offine-textured soils, especially clay soils,
as well as a large number of soils withvery low bulk densities of about 0.2 g
cm−3 and very high organic matter contents (up to 50%) (see also Fig. 2).

Fig. 10. Variation in the root mean squared residual (RMSR) value as a function of pressure head for pedotransfer functions using texture, bulk density, and organic matter in various combinations: SSC = sand,silt and clay; Bd = bulk density; OM = organic matter; TC(7) = textural distribution using seven classes.

Using bulk density improved predictions at the lower pressure head values(Sharma et al., 2006; Schaap et al.,2001). The RMSR values at 15,800 cm(pF 4.2) were very similar for all PTFs
except Børgesen and Schaap (2005). In
the case of Stumpp et al. (2009), we didfind very large RMSR values comparedwith the other databases. Evaluationof the ME showed that water contents
were underestimated at all pressure
head levels. The high RMSR values for
the PTFs using sand, silt, and clay were
found mainly for soils rich in organic
Fig. 11. Variation in the root mean squared residual (RMSR) value as a function of pressure head formatter. Cornelis et al. (2001) analyzed
pedotransfer functions using texture, bulk density, organic matter, and soil water content in different
combinations: SSC, sand, silt and clay; Bd, bulk density; OM, organic matter; TC(7), textural distribuvarious PTFs for the VGM model (seetion using seven classes; WC, water content at pressure head values of 30 cm (3), 320 cm (32), 330 cm
Table 2) and found that most of them(33), 15,000 cm (1580), and 15,800 cm (1580).
also underestimated the MRC. The
mean difference varied between −0.058
and 0.028, with the largest mean differin this range. Th is pressure head value falls within the range ofence value being for the PTFs derived by Scheinost et al. (1997)pressure head values where the largest RMSRs were observed.
and the smallest one for Rawls and Brakensiek (1985).
Therefore, the moisture content corresponding to this pressure
head value was very effective in reducing the RMSRA. The best
Figure 10 shows the effect of including organic matter on the
PTFs produced RMSR values very close to those of the direct fit,
RMSR across pressure heads. Including organic matter led toshowing the same order of magnitude.
improved predictions, with the lowest RMSR values occurring in
the wet range and the very dry range for the models of Vereecken
Sharma et al. (2006) provided information on the variation
et al. (1989) and Weynants et al. (2009) using the same database.
in the RMSRA across textural classes for PTFs using texture
All models showed an increase in the RMSR between 10 (pF 1)
and bulk density with different sets of remotely sensed paramand 500 cm (pF 2.7). The RMSRs tended to decrease or remain
eters. The PTFs were derived from the database establishedstable at the larger pressure heads, depending on the model.
by Mohanty et al. (1997). The remotely sensed data used in
the PTFs were slope, aspect, normalized difference vegetaFigure 11 shows the effect on the RMSR of including the soil watertion index, aspect, f low accumulation, and digital elevation.
content at certain pressure heads. In general, including the water
Figure 12 shows the RMSR for the PTF using the largest set
content produced the lowest RMSRs in the range between 100 (pF
of remotely sensed predictors in addition to texture and bulk2) and 3000 cm (pF 3.5). Apparently the water content at 330 cm
density. A substantial improvement between 23 and 27% was(pF 2.5) pressure head was very effective in reducing the RMSR
obtained for the RMSR in the medium pressure head range

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(330–8000 cm) compared with when only
textural information and bulk density
were used. The improvement decreased
with increasing pressure head, indicating that remotely sensed information isless capable of improving the predictions
for dry soils. For the dry and wet ranges,
practically no improvement was obtained(compare with Fig. 9).
Figure 13 shows variations in the RMSR/
RMSRA as a function of the pressure headfor PTFs predicting VGM hydraulic conductivity using textural information, bulk
density, and organic matter. The modelof Weynants et al. (2009) excluded values
below a pressure head of 6 cm. The PTFs
using textural information, bulk density,and organic matter or a combination ofthese predictors showed the lowest RMSR
values for pressure heads between 10 (pF 1)
and 500 cm (pF 2.7) (Fig. 13). A comparisonof the results of Weynants et al. (2009) and
Schaap et al. (2001) shows that the effect ofincluding organic matter is the largest in thedry range.
Figure 14 shows the effect of including watercontent at different pressure head values as
an additional variable to texture and bulk
density. We included the data of Schaap etal. (2001) for the PTF using texture andbulk density as a reference. Including thewater content at 330 cm (pF 2.5) did not
improve the hydraulic conductivity prediction. Both lines are on top of each other (Fig.
14). Introducing the water content at 15,800
cm (pF 4.2) mainly improved the RMSR
values in the dry range. For this reason, it
seems that organic matter is a better predictor of the hydraulic conductivity in the dry
range than the water content at the wilting
point (see also Fig. 12). Still, care should betaken in extrapolating these results because
we compared the outcomes of two different databases. Further research is therefore
needed to confirm these findings.
In addition to the RMSR, we also evaluatedthe dependency of the ME on the pressure
head for the PTFs of Weynants et al. (2009)
for predicting the MRC and HCC. Figure
15 shows the ME across different pressure

Fig. 12. Variation of the root mean squared residual (RMSR) value as a function of pressure headfor pedotransfer functions using textural data (sand, silt, and clay content [SSC]), bulk density(Bd), and a maximum set of remotely sensed data (RS) consisting of slope, aspect, normalized difference vegetation index, flow accumulation, and digital elevation.

Fig. 13. Variation in the root mean squared residual or root mean squared residual adjusted (RMSR
or RMSRA) of the unsaturated hydraulic conductivity (K) as a function of pressure head (pF)
for pedotransfer functions using textural information (textural class [TC] or sand, silt, and clay
[SSC]), bulk density (Bd), and organic matter (OM) as predictors. Weynants et al. (2009) used
the RMSRA and Schaap et al. (2001) the RMSR.

Fig. 14. Variation in the root mean squared residual (RMSR) of the unsaturated hydraulic conductivity (K) as a function of pressure head (pF) for pedotransfer functions using textural information,bulk density and water content: SSC = sand, silt and clay; Bd = bulk density; WC2.5 and WC2.5,
4.2 = water content at pressure head levels of only pF 2.5 (330 cm) and pF 2.5 and 4.2 (15,800cm), respectively.

www.VadoseZoneJournal.org | 813

head values for selected PTFs. The results
indicate that the ME values were lowest
for PTFs using SSCBdOM or SSCBd
and water content data. Including OM
or water content information led to ME
values that evened out across the zero
axis. Pedotransfer functions based on
texture alone or texture and bulk density tended to underestimate the watercontents. Figure 16 shows the ME as afunction of pressure head for PTFs predicting the HCC. The largest deviations
were found in the wet and the very dry
range, reflecting the difficulty of PTFs tocorrectly estimate hydraulic conductivity near saturation and at pressure headvalues around 15,800 cm (pF 4.2).

Fig. 15. Mean error (ME) values for water content as a function of pressure head for pedotransfer functions using various combinations of predictors: SSC = sand, silt, and clay; Bd = bulk density; Om =organic matter; WC33 = water content at a pressure head of 330 cm; WC33, 1500 = water content atpressure heads of 330 and 15,000 cm.

ValidaƟon of PedotransferFuncƟons Using the van
Genuchten–Mualem
ParameterizaƟon
We reviewed the data available for evaluating the accuracy and performance ofa PTF to predict the VGM moistureretention and hydraulic conductivity
functions using independent data sets.

Valida on of Pedotransfer
Func ons for the Moisture
Reten on Characteris c

Fig. 16. Plot of the mean error (ME) of the unsaturated hydraulic conductivity (K) as a function ofpressure head (pF) for two pedotransfer functions using either sand, silt, and clay content, bulk density, and organic matter (SSCBdOm) or the van Genuchten–Mualem (VGM) model with unsaturated
hydraulic conductivity (Ks) and a tortuosity parameter (l).

Table 5 provides an overview of thevarious PTF validation studies that wereconducted for the VGM soil water retention function using independent data sets. Overall, the RMSR for
PTF validation ranged between 0.037 and 0.143. Again, the lowest
values were obtained for those PTFs that used soil moisture content, such as in the case for the Børgesen and Schaap (2005) and
Børgesen et al. (2008) data sets. These values were at the same level
as the RMSR obtained for the development data set using the PTFs
that included soil moisture. The other PTFs using either (i) texture
and bulk density or (ii) texture, bulk density, and organic matterproduced similar RMSR values as those obtained for the data set
used for PTF development. This indicates that PTFs show a highreliability when applied to independent data sets. It appears, however,that there is a difference between PTF validation based on splitting
the data set and validation based on completely independent data.
This can be seen by comparing the RMSR values of Børgesen and
Schaap (2005) and Børgesen et al. (2008) with the data obtainedby Weynants et al. (2009) using the database of Cornelis et al.
(2001). The values for the PTF validation derived from Vereecken
et al. (1989), HYPRES, and ROSETTA ranged between 0.091 and

0.1402, whereas the RMSR values of Børgesen and Schaap (2005)
and Børgesen et al. (2008) ranged between 0.037 and 0.064. Thissuggests that validation of PTFs is best performed on completely
independent data sets rather than on data sets that are split into a
subset for development and a subset for validation.
In attempts to validate different PTFs, Cornelis et al. (2001) foundthat clay soils showed the largest RMSD values for all PTFs basedon the VGM parameterization, except for the PTF of Scheinost etal. (1997). The latter had the largest deviations for the loamy sandand sand textural classes. The better performance of Scheinost etal. (1997) for clay soils may be due to the larger number of clay soils
compared with the sandy soils. Scheinost et al. (1997) developed
their database and PTFs for specific regional conditions involving
soils having high organic matter contents (up to 8%), substantial
gravel content, a large range in bulk density (0.8–1.85 g cm−3), anda textural composition with clay, silt, and sand contents of up to 60,
76, and 97%, respectively. They did not include the saturated water

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Table 5. Literature findings dealing with pedotransfer function (PTF) validation using the van Genuchten–Mualem (VGM) parameterization for the
moisture retention characteristic, including the root means squared residual (RMSR) or root mean squared deviation (RMSD).
Study

PTF validated, parameterization, type†

Database

No. ofsamples

RMSR

Weynants et al. (2009)

Weynants et al. (2009), m = 1 − 1/n, RA

Cornelis et al. (2001)

48

0.099

Weynants et al. (2009)

Vereecken et al. (1989), m = 1, RA

Cornelis et al. (2001)

48

0.091

Weynants et al. (2009)

HYPRES, m = 1 − 1/n, RA

Cornelis et al. (2001)

48

0.117

Weynants et al. (2009)

ROSETTA, m = 1 − 1/n, NN

Cornelis et al. (2001)

48

0.140

Weynants et al. (2009)

Wösten et al. (1999), m = 1 − 1/n, RA

Vereecken et al. (1989)

166

0.046

Weynants et al. (2009)

Schaap et al. (1999), m = 1 − 1/n, NN

Vereecken et al. (1989)

166

0.059

Tietje and Tapkenhinrichs (1993)–

0.05 (RMSD)

Tietje and Tapkenhinrichs (1993) Vereecken et al. (1989), m = 1, RA
Cornelis et al. (2001)

Vereecken et al. (1989), RA

Cornelis et al. (2001)

298

0.0312(RMSD)

Cornelis et al. (2001)

Wösten et al. (1994), m = 1 − 1/n, RA

Cornelis et al. (2001)

298

0.033 (RMSD)

Cornelis et al. (2001)

Scheinost et al. (1997), m = 1, RA

Cornelis et al. (2001)

298

0.0373 (RMSD)

Cornelis et al. (2001)

Schaap et al. (1999), m = 1 − 1/n, NN

Cornelis et al. (2001)

298

0.0553 (RMSD)

Cornelis et al. (2001)

Wösten et al. (1999), m = 1 − 1/n, RA-cont

Cornelis et al. (2001)

298

0.0388 (RMSD)

Cornelis et al. (2001)

Wösten et al. (1999), m = 1 − 1/n, RA-class

Cornelis et al. (2001)

298

0.0389 (RMSD)

Schaap et al. (1998)

Schaap et al. (1998), m = 1, RA

Vereecken et al. (1989)

1209

0.098‡

Schaap and Leij (1998a)

Schaap and Leij (1998a), m = 1 − 1/n, NN

Rawls (Schaap and Leij (1998a)

1209

0.087‡ (0.127/0.119)§

Schaap and Leij (1998a)

Schaap and Leij (1998a), m = 1 − 1/n, NN

Ahuja (Schaap and Leij (1998a)

371

0.062‡ (0.092/0.079)§

Schaap and Leij (1998a)

Schaap and Leij (1998a), m = 1 − 1/n, NN

UNSODA (Schaap and Leij, 1998a)

554

0.087‡ (0.105/106)§

Børgesen and Schaap (2005)

Børgesen and Schaap (2005), m = 1 − 1/n, NN

Børgesen and Schaap (2005)

1608

0.038–0.055¶

Børgesen et al. (2008)

Børgesen et al. (2008), m = 1 − 1/n, NN

Børgesen and Schaap (2005)

1608

0.037–0.064¶

Scheinost et al. (1997)

m = 1, RA

Scheinost et al. (1997)

396
0.017/0.035 (RMSD)#(3 replicates)† RA = regression analysis, RA-cont = regression analysis–continuous PTF, RA-class, regression analysis–class PTF, NN = neural network analysis, m and n = shapeparameters.
‡ RMSR adjusted using Eq. [5].
§ Cross-validation between data sets.
¶ Data range refers to different predictor variables (three [Børgesen and Schaap (2005)] or seven textural classes [Børgesen et al., 2008]).
# Data range refers to different validation databases.

content when fitting the VGM equation. The samples that wereused were quite large (365 cm3), which raises concerns about thelong time periods needed to reach equilibrium, especially for theclay soils. They did not provide information about the equilibration time used at the different pressure heads.
For the Vereecken PTFs using the VGM model, Tietje andTapkenhinrichs (1993) found that, on average, the moistureretention data were underestimated. Deviations from the measured moisture content data were largest for soils having a low
bulk density. Samples with high organic matter content were still
well predicted, although no information on the range of organic
C was given (the database had a range of 0–22%). Cornelis et al.
(2001) found that the PTFs listed in Table 2 generally underestimated the measured soil moisture content, except the PTFs ofScheinost et al. (1997). Th is may have been due to different measurement techniques. Wösten and van Genuchten (1988) used theevaporation method of Boels et al. (1978) and Bouma (1983) todetermine the MRC for pressure head values <800 cm (pF 2.9).
For the higher pressure heads, standard pressure chambers wereused. Vereecken et al. (1989) used the methods given in Table 1
to determine the MRC. For Schaap et al. (1998) and Wösten etal. (1999), a great variety of methods and sample sizes was usedin these large multi-institute databases. The possible effects ofdifferent measurement techniques on the determination of the
MRC were already discussed above. Differences in sample size, theduration of equilibration, and water removal procedures (pressure
increase vs. imposed evaporative flux) may be key factors.
Clearly, the lack of information on the type of methods used and thedifference in accuracy of the methods themselves makes it difficult toevaluate the performance of the PTFs and the VGM parameterization. More information is needed on the impact of the measurement
method on the MRC and HCC. Only a few studies have addressed
the issue of measurement technique comparison (Stolte et al., 1994;
Mallants et al., 1997, 1998; Gribb et al., 2004). In addition, different
methods have been used to determine the predictor variables, which
may also have an effect. This is especially the case for the texturalcomposition of soils (e.g., Weynants et al., 2009) but also for thedetermination of the bulk density. Benchmark studies have shown

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large differences in determining the sizeof the clay fraction. A thorough analysisof the impact of these different methods
on the textural predictors used in PTFs
has, to our knowledge, not yet been performed. Studies that compare different
measurement methods for the hydraulicproperties in terms of their suitability todevelop PTFs are lacking. Studies of thistype have focused mainly on comparisonsof field measurement techniques withlaboratory-scale methods.

Table 6. Validation of pedotransfer functions (PTFs) expressed as the root mean squared residual (RMSR)
for the van Genuchten–Mualem model to predict hydraulic conductivity curve using independent data.
No. of data
pointsRMSR

Study

PTF validated, type†

Database

Schaap and Leij (1998b)

Schaap and Leij (1998b), l = −1

UNSODA

1.65–1.77‡

Schaap and Leij (1998b)

Schaap and Leij (1998b), l = 0.5

UNSODA

1.24–1.37‡

Weynants et al. (2009)

HYPRES, Wösten et al. (1999)

Vereecken et al. (1989)

136

2.54§

Weynants et al. (2009)

ROSETTA, Schaap et al. (2001)

Vereecken et al. (1989)

136

3.01§

† l is the tortuosity parameter.
‡ Ranges evaluated the effect of different predictor sets.
§ RMSR adjusted using Eq. [5].

Valida on of Pedotransfer Func ons for theUnsaturated Hydraulic Conduc vity Func on
Only a few studies are available that have attempted to validate a
PTF for the VGM unsaturated hydraulic conductivity function
using independent data sets (Table 6). Independent refers to datathat have not been used in the development process. As such, independent data sets may include the effects of measurement method,
differences in soil type, soil composition, and sampling size, anddifferences in soil physical and chemical properties (texture, structure, C content, bulk density, and clay mineralogy). This is, of course,
also valid and applicable to data sets for the MRC. Weynants etal. (2009) found that the PTFs from HYPRES performed better
than the PTFs derived from ROSETTA. They suggested that thiswas due to the fact that HYPRES represents mainly European soils
and as such is more affi liated with the database of Vereecken et al.
(1989) than ROSETTA, which represents a mixture of predominantly European and American soils. Probably the distribution ofsoil types and soil properties and the underlying correlations in thedata are substantially different among the three databases. Figure 1
already indicated that substantial differences among databases exist
with respect to their textural composition.

PTFs. Here we give a few examples of possible strategies for further
improving PTFs in the future.

Obtaining More Reliable
Hydraulic Measurements
To be predicted accurately, measurements of the hydraulic parameters must be repeatable, scale consistent, and associated with thelowest possible experimental error. If these three criteria are notmet, measurements result in incompatible data, which complicates PTF development and decreases PTF predictive power. For
example, Ks has been known to be intrinsically scale dependent(Timlin and Pachepsky, 1998; Pachepsky and Rawls, 2004). This
was also reported for water retention (Pachepsky et al., 2001).
The type of experiment and experimental setup may also affectthe assessment of hydraulic properties, particularly with inverse
modeling (Javaux and Vanclooster, 2006). Some experimental
procedures may also result in non-unique identifiable soil properties. Standardization of soil hydraulic characterization methods
is sorely needed. Developing databases with harmonized support
scales and experimental protocols could significantly improve thequality of PTF predictions.

Developing More Comprehensive Databases
Procedures for the development and validation of PTFs may welldiffer also depending on the invoked regression and optimization
approach. For example, Schaap and Leij (1998a) used the bootstrap
method to select data sets for development and validation from thesame database. Because many of the samples in the database were
not completely independent (e.g., most databases consist of clusters of samples taken from the same soil or research area), however,the data sets for development and validation were not completely
independent and some bias may exist.

6 Prospects in PedotransferFuncƟon Development

In this review, we have highlighted some of the approaches thathave been used to develop PTFs for the VGM hydraulic property
parameterization and indicated possible areas that need improved

Current soil PTF databases do not homogeneously cover all soil
types. Larger and more complete databases, with a uniform distribution of samples among soil classes, should increase the predictive
capability of a PTF obtained with these databases, in particularfor fine-textured soils.

New Data ExploraƟon Techniques
In addition to establishing comprehensive databases, there is stilla need to further evaluate and analyze the existing databases bydeveloping and applying new data exploration methods witha specific emphasis on sparse or unevenly distributed data sets.
Techniques such as nearest neighbor methods (e.g., Nemes et al.,2006a,b) and support vector machine approaches (e.g., Twarakavi
et al., 2009) based on machine learning methods seem to be very
promising in this respect.

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Defining Comprehensive Metadata
The term metadata refers to information about the data. To
improve database quality, it is important that users know precisely
the experimental conditions under which the data have been collected. Databases should include information on sample scale andexperimental protocols, but also on the experimental error associated with the measurements, the number of replicates, and theuncertainty associated with the data. For example, the hydraulicproperties of the upper soil horizon may depend on soil management. Information on the period of the year, the state of the soil,
the history of tillage practices, and the type of crop or vegetation
should therefore be indicated to be able to further develop time- or
structure-dependent PTFs.

Adding Relevant Predictors
Like the measured hydraulic properties, predictors need also to be
unique for a given spatial location and optimally should be obtained
directly on the sample used for the determination of the hydraulicproperties. In addition, they need to be easily identified (in termsof time and experimental efforts) and bring relevant information.
Predictors that are expensive to obtain and are not unique or areuser dependent should be avoided. Various studies have shown thepotential of predictors such as soil structure, clay mineralogy, andtaxonomic information, for the prediction of soil hydraulic properties(e.g., Pachepsky and Rawls, 2004). No unified approach exists thus far
on how to best include structural properties in PTFs. Soil structure
predictors in particular can suffer from a lack of uniform protocol
or definition, or may depend on the experience of the observer. Thisis typically the case for soil aggregation. Recent developments in thefield of noninvasive methods may be valuable and provide proxies for
improved PTFs such as x-ray tomography, spectral induced polarization, and nuclear magnetic resonance. Further research is neededon how to extract relevant information from these new techniques
and to fully explore their potential in improving PTF predictions.
Electromagnetic signals may contain relevant information on soil
pore distribution and hydraulic properties. Photographs can also give
relevant information after image analysis or as metadata. Pictures ofsimple dye tracer experiments could be used similarly. Current computer data storage capacities allow the addition of such informationin databases with almost no limitations.

Toward Time-Dependent
Pedotransfer FuncƟons
Pedotransfer functions have traditionally been developed for theupper soil layers, which are prone to rapid changes due to humanactivities. With the inclusion of more information on the history of
the soil management practices at a site, the development of temporalpedotransfer functions (TPTFs) could explicitly take into accountsoil tillage practices to predict soil hydraulic properties at given times.
To include a temporal component in PTFs would require a priori
an understanding of the temporal variation of the MRC and HCC.

6 Conclusions
Analysis of the RMSRA values obtained by directly fitting the
VGM parameterization to measured moisture retention data
produced an average value of 0.017 for the analyzed databases.
The accuracies of PTFs for the moisture retention characteristic
obtained from databases in the literature are now approaching
those obtained with direct VGM model fits to moisture retention
data but the RMSRA values are still two to five times larger thanthe RMSRA obtained with direct fits. The lowest values wereobtained for PTFs using textural properties, bulk density, organic
matter, and water content. The accuracy of the best PTF for the
HCC was approximately twice as large as the accuracy obtainedwith the direct VGM fit to hydraulic conductivity data. The mostaccurate PTFs are based on flexible Ks and l parameter while allowing negative values for l.
A pressure-head-dependent analysis of the RMSRA showed thatthe largest deviations occurred at intermediate pressure heads
for the MRC in the case of a direct fit, as well as for PTFs using
combinations of soil texture, bulk density, and organic matter as
predictors. Including soil moisture measurements as a predictor
in PTFs was shown to improve the prediction of soil moisture,
although the improvements were not equally effective for all pressure heads. Including soil measurements at intermediate pressure
head values (between approximately pF 1.5–3.0) was successful in
reducing uncertainty in that specific range. In the case of the HCC,
the largest deviations were found in the wet and dry ranges forboth the direct fits and for all PTFs. This suggests that particularattention should be paid to improving both the parameterizationof the HCC as well as the accuracy of PTFs in these ranges.
Our analysis also indicated that soil organic matter is an important predictor for hydraulic parameters and soil moisture contents,
especially in the wet and dry ranges. Organic matter may help
more in reducing RMSRs in the dry soil moisture range (pressure head values around the wilting point) than remotely sensed
attributes, compared with cases where only soil texture and bulk
density were used. Soil organic matter reduced substantially the
RMSRA of PTFs predicting the hydraulic conductivity comparedwith cases where only soil texture and bulk density were used. Thisis especially the case for the dry range (around the wilting point).
Including soil moisture content in PTFs predicting the HCC was
found to be less effective in reducing the RMSR values than theeffect of soil moisture in PTFs for the MRC.
The RMSRs of PTFs for the MRC in validation studies were foundto be similar to the RMSRs obtained from the data sets for the development of PTFs using only soil texture and bulk density. For the
HCC, the RMSRs obtained in validation analyses were larger thanthe values obtained from the data sets used for PTF development.
These differences may be as large as a factor of seven. Only a few PTF
validation studies for the HCC exist in the literature. Databases that

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have been used for the development of PTFs for the HCC at present
contain mostly coarse-textured soils. There is a need to include moresoils with finer textures and different mineralogies.
The contribution of measurement errors to the RMSR obtained bydirectly fitting the VGM parameterization to measured data has
not yet been quantified. More attention should be paid to this issue
because this may well lead to much improved PTF predictions.
Databases used to identify the quality of the VGM model indescribing the soil hydraulic properties and to develop PTFs differ
considerably in terms of the proportion of soil textural classes, theavailability soil properties, and information on measurement methods used to determine the hydraulic properties. This may lead todifficulties in comparing results derived from different databases.
Parameters that describe the structural properties of a sample onwhich the MRC and HCC were measured are lacking completely.
A need therefore exists to extend the soil properties in the databases with soil structural parameters determined on the sample
used for all the other measurements. This may require the establishment of a new, comprehensive, and perhaps universal databasefor developing improved PTFs. We feel that, at present, further
improvement of PTFs is mainly limited by a lack of new information such as soil structure. It seems that with currently available
soil data, a limit to improving the PTFs based on the VGM model
may have been reached. The development of such a new databaseshould be based on the use of a consistent set of measurement techniques and protocols, as well as the inclusion of an enlarged set ofsoil properties, including structural properties, determined at thescale of the measurement volume for the hydraulic properties. The
derivation of the hydraulic data should be model free. In addition,
MRC and HHC measurements should be performed on the same
sample, preferably simultaneously.
To evaluate the performance of the direct fit of the VGM model
as well as of the PTFs, a consistent set of performance indicators
should be defined. Analysis of the available data has shown thatparameters with respect to bias are only rarely included in the evaluation. A pressure-head-dependent analysis of bias indicators or
any other relevant information may especially provide new insights
for the improvement of PTFs.

Acknowledgments

Attila Nemes (Agricultural Research Station, Beltsville, MD), Wim Cornelis (University of Ghent, Belgium), Christine Stumpp (University of California, Riverside),
Navin Kumar Twarakavi (University of Alabama at Auburn) and Christen Børgesen (University of Aarhus, Denmark) are gratefully thanked for providing theoriginal data. We thank the Transregional Collaborative Research Centre, TR-32,“Patterns in Soil–Plant–Atmosphere Systems: Monitoring, Modelling and Data Assimilation” for supporting this research.

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