Soil & Tillage Research 79 (2004) 145–152
www.elsevier.com/locate/still

Estimating the water retention curve from soil properties:
comparison of linear, nonlinear and concomitant variable methods
Kálmán Rajkaia,*, Sándor Kabosa, M.Th. van GenuchtenbaResearch Institute for Soil Science and Agricultural Chemistry, Hungarian Academy of Sciences, P.O. 35, 1025 Budapest, HungarybGeorge E. Brown Jr. Salinity Laboratory, USDA, ARS 450W, Big Springs Road, Riverside, CA 92507, USA

Abstract
The unsaturated soil hydraulic functions involving the soil–water retention curve (SWRC) and the hydraulic conductivity
provide useful integrated indices of soil quality. Existing and newly devised methods were used to formulate pedotransfer
functions (PTFs) that predict the SWRC from readily available soil data. The PTFs were calibrated using a large soils databasefrom Hungary. The database contains measured soil–water retention data, the dry bulk density, sand, silt and clay percentages,
and the organic matter content of 305 soil layers from some 80 soil profiles. A three-parameter van Genuchten type function was
fitted to the measured retention data to obtain SWRC parameters for each soil sample in the database. Using a quasi-random
procedure, the database was divided into ‘‘evaluation’’ (EVAL) and ‘‘test’’ (TEST) parts containing 225 and 80 soil samples,
respectively. Linear PTFs for the SWRC parameters were calculated for the EVAL database. The PTFs used for this purpose
particle-size percentages, dry bulk density, organic matter content, and the sand/silt ratio, as well as simple transforms (such as
logarithms and products) of these independent variables. Of the various independent variables, the eight most significant wereused to calculate the different PTFs. A nonlinear (NL) predictive method was obtained by substituting the linear PTFs directly
into the SWRC equation, and subsequently adjusting the PTF parameters to all retention data of the EVAL database. The
estimation error (SSQ) and efficiency (EE) were used to compare the effectiveness of the linear and nonlinearly adjusted PTFs.
We found that EE of the EVAL and the TEST databases increased by 4 and 7%, respectively, using the second nonlinear
optimization approach. To further increase EE, one measured retention data point was used as an additional (concomitant)variable in the PTFs. Using the 20 kPa water retention data point in the linear PTFs improved the EE by about 25% for the TEST
data set. Nonlinear adjustment of the concomitant variable PTF using the 20 kPa retention data point as concomitant variable
produced the best PTF. This PTF produced EE values of 93 and 88% for the EVAL and TEST soil data sets, respectively.
# 2004 Elsevier B.V. All rights reserved.
Keywords: Soil–water retention function; Pedotransfer functions; Unsaturated flow; Soil quality

1. Introduction* Corresponding author.
E-mail address: krajkai@rissac.hu (K. Rajkai).

The unsaturated soil hydraulic functions are important parameters in many soil, hydrological, ecologicaland agricultural studies, are critical input parameters0167-1987/$ – see front matter # 2004 Elsevier B.V. All rights reserved.
doi:10.1016/j.still.2004.07.003

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K. Rajkai et al. / Soil & Tillage Research 79 (2004) 145–152

in models for variably-saturated flow and contaminanttransport, and often serve as integrated indices for soil
quality (e.g., National Research Council, 1993; Lin,
2003). Unfortunately, direct measurement of thesefunctions is impractical for most applications inresearch and management, especially for relatively
large-scale problems. For this reason, many applications rely on pedotransfer functions (PTFs) toindirectly estimate the hydraulic properties from moreeasily measured or more readily available information(e.g., soil texture and bulk density). Reviews of studiesof this type are given by Tietje and Tapkenhinrichs,
1993, Bruand et al. (1997) and van Genuchten et al.
(1999).
In this paper, we study how successfully the soil
water retention curve (SWRC) can be predicted with
PTFs that were derived using conventional andmodified statistical approaches. A soil hydraulicdatabase from Hungary will be used to demonstrate
and test the proposed method. The database for ourstudy was divided into ‘‘evaluation’’ (EVAL) and‘‘test’’ (TEST) parts. A three-parameter van Genuchten type function (van Genuchten, 1984) will be used
for the SWRC. We first use relatively standard
approaches to derive PTFs for estimating the retention
parameters. In attempts to improve the SWRC
predictions, and following previous work by Scheinost
et al. (1997), the linear PTFs are substituted directly
into the SWRC, after which the parameters are further
adjusted nonlinearly using all retention data of the
EVAL database. Since the linear PTFs of the three
SWRC parameters each contained nine unknows, thissecond nonlinear approach will involve 27 adjustableparameters. To further improve the accuracy of thePTFs, additionally one measured retention data point
will be used as a concomitant variable in the PTF
models. The retention data point that most significantly improves the SWRC predictions will be used asconcomitant variable for the final PTFs.

curve at tensions of 0.1, 0.25, 1, 3.2, 10, 20, 50, 250
and 1500 kPa. Retention values between 1 and 50 kPa
were obtained with the hanging water column method
using sand- and kaolin-plate boxes (Várallyay, 1973),
while the higher suction values (250 and 1500 kPa)
were obtained using pressure chambers (Buzás, 1993).
Particle-size fractions were measured with theconventional pipette method using 0.5 N Na2P2O5
to facilitate dispersion. Particle-size limits were set at0.002, 0.005, 0.01, 0.02, 0.05 and 0.25 mm. The
coarsest particle fraction between 0.25 and 2 mm wasdetermined by wet sieving. The dry bulk density of thesoils was measured on core samples, while the soilorganic matter content was obtained using conventional oxidation methods (Buzás, 1989).
The majority of soil samples in the Hungarian
database (225 of 305) are representative of the mainsoil types of the Hungarian lowland (Rajkai et al.,1981), while 75 samples are from hill slope areas.
About 75% of the soils are medium, 15% fine, and
10% coarse textured. The medium-textured soils
formed mostly on loess parent materials. The finertextured soils are mostly from the Trans–Tisza area
where loess materials settled into a wet alluvial basin,
and from hill slope areas where clay formation was animportant part of the soil genesis process. The coarsetextured soils were collected from sandy areas of the
Hungarian Lowland. Table 1 summarizes the main soil
characteristics of the EVAL and TEST subsets of the
Hungarian soils database.

2. The Hungarian soils database

Table 1
Selected properties of the evaluation (EVAL) and test (TEST) arts ofthe Hungarian soils database

The Hungarian database contains data from 305
soil samples. Water retention data were measured oncore samples collected in 100 cm3 cylinders having aheight of 5 cm and an outer diameter of 5 cm.
Retention data were obtained along the main drying

3. Soil–water retention model
The following three-parameter van Genuchten
function was used to the describe water retention
data (van Genuchten, 1984):
u¼

Case

us

(1)

½1 þ ðahÞn ð11=nÞ

Sand Silt Clay OM (%) r (mg/m3) Pressure head (kPa)1

EVAL 25.4 47.1 27.5 1.4
TEST 26.8 47.8 26.8 1.6

1.44
1.43

20

1500 N46.6 35.6 17.0 225
47.0 36.9 17.9 80

K. Rajkai et al. / Soil & Tillage Research 79 (2004) 145–152

where u is the soil moisture content (m3/m3), h is thesoil water tension (cm), and us, a, and n are model
parameters. Eq. (1) was fitted to the measured data ofeach soil sample using a nonlinear least-squares optimisation approach (Marquardt, 1963) that minimized
the sum of squared deviations (SSQ) between
observed and fitted water contents.

4. Pedotransfer functions
The fitted SWRC parameters were correlated withsoil properties in the Hungarian database, as well as
several derived soil properties. For the original
properties we used the bulk density, r, the organic
matter content,OM, the sand fraction, S, the silt
fraction, Si, and the clay fraction, C. As derived soil
properties we selected the logarithm of the clay
fraction, ln (C), the sand/silt ratio, S/Si, the squares ofthe original properties (e.g., r2 and C2), and several

147

simple products (e.g., (rC)2 and r2C2) as shown in
Table 2 listing the different pedotransfer functions.
Linear regression techniques were subsequently used
to correlate the SWRC parameters with the mostsignificant soil properties as predictors (Rajkai andVárallyay, 1992). As dependent variables for the linear
PTF regressions we used us and the log transforms of aand n.
To make the linear regression approach more
flexible, all eight independent explanatory variables
were entered and maintained in the PTF; this approach
is further referred to as the LR8 predictive model.
Next, the goodness of prediction of the LR8 model
was improved by adding one measured water retention
data point to the PTFs. Rawls and Brakensiek (1989)
among others suggested that the wilting point should
be used for this purpose. We also investigated theeffect of using other measured retention data, andfound that using retention data close to the SWRC
inflection point (at about 20 kPa for most of our soils)

Table 2
Linear and nonlinearly adjusted pedotransfer functions obtained for the evaluation (EVAL) data baseLinear and nonlinearly adjusted PTFs

Predictive2

us = 118.76  60.02  r  0.25  OM  0.0007  C  1.99  ln (C) + 9.78 
r2  0.04  rS + 0.116  S/Si + 0.00078  r2C2
us = 123.76  65.37  r  0.28  OM  0.000048  C2  1.99  ln (C) + 12.46 
r2  0.054  rS + 0.14  S/Si + 0.00049  r2C2
us = 134.88  0.127  FC  74.4  r  0.19  OM  0.00027  C2  2.83  ln (C) +14.37  r2  0.053  rS  0.0074  S/Si  0.00087  r2C2
us = 107.95  51.5  FC  0.00008  r  0.086  OM  0.042  C2 + 0.0003  ln (C) 
0.38  r2  2.07  rS + 7.9  S/Si + 0.067  r2C2
ln (a) = 27.18 + 0.15  rSi + 0.18  C  13.32  ln (Si)  0.024  r2C + 0.314 
Si  0.0027  C2  0.75  S/Si  0.0013  r2Si2
ln (a) = 16.97 + 0.12  rSi + 0.22  C  9.34  ln (Si)  0.039  r2C + 0.21 
Si  0.0029  C2  0.435  S/Si  0.00093  r2Si2
ln (a) = 31.74  0.26  FC  0.37  rSi + 0.81  S/Si + 0.157  C + 0.01  r2C +13.52  ln (Si)  0.01  Si  0.0016  C2 + 0.0014  r2Si2
ln (a) = 13.89 + 4.67  FC  0.0085  rSi + 0.81  S/Si + 0.213  C  0.033  r2C 
0.002  ln (Si)  0.12  Si + 0.00048  C2 + 0.294  r2Si2
ln (n) = 0.287 + 0.47  r  0.008  OM  0.00007  C2 + 0.06  ln (C)  0.00046 
rS  0.01  r2C  0.0068  S/Si + 0.00015  r2C2
ln (n) = 0.069 + 0.32  r  0.007  OM  0.000009  C2 + 0.00147 ln (C)  0.00011  rS  0.0064  r2C + 0.0015  S/Si + 0.000081  r2C2
ln (n) = 0.67 + 0.0039  FC + 0.59  r  0.01  OM  0.0001  C2  0.0005 
rS + 0.11  ln (C)  0.012  r2C + 0.013  S/Si + 0.00018  r2C2
ln (n) = 0.36 + 0.000045  FC + 0.0116  r  0.002  OM + 0.16  C2  0.0737 
rS + 0.0000105  ln (C)  0.00046  r2C  0.0055  S/Si  0.0035  r2C2

R2

method

N

0.84

LR8

225

NLR8

2025

LR8FC

225

NLR8FC

2025

LR8

225

NLR8

2025

LR8FC

225

NLR8FC

2025

LR8

225

NLR8

2025

LR8FC

225

NLR8FC

2025

0.85

0.23

0.64

0.61

0.64

r = bulk density (mg/m3); OM = organic matter (%); S = sand fraction (>50 mm Si = silt fraction (502 mm); C = clay fraction (<2 mm); ln (Si) =logarithm of silt fraction; ln (C) = logarithm of clay fraction; S/Si = sand–silt ratio; FC = field capacity retention data.

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K. Rajkai et al. / Soil & Tillage Research 79 (2004) 145–152

produced the best predictions (results not further
shown here). The predictive model using an independently measured retention point at this 20 kPa tension(approximately field capacity, FC) will be referred toas LR8FC.
To further improve the LR8, and LR8FC methods,
we carried out an additional nonlinear adjustment
procedure as outlined by Scheinost et al. (1997) andMinasny et al. (1999). For this purpose we generalized
the estimated linear PTFs as listed in Table 2 (i.e.,
those for LR8 and LR8FC) by allowing the fitted
coefficients to become unknowns in a new set ofregressions. For example, the LR8 PTF for us was
generalized to (Table 2):
us ¼ a1 þ a2 r þ a3 OM þ a4 C2 þ a5 lnðCÞ
2

2 2

þ a6 r þ a7 rS þ a8 ðS=Si Þ þ a9 r C(3)Linear regression equations of this type for each of
the three unknown retention parameters were substituted directly into Eq. (1) to yield the following
general equation
a1 þ a2 r þ a3 OM þ a4 C2 þ a5 lnðCÞ
þ a6 r2 þ a7 rS þ a8 ðS=SiÞ þ a9 r2 C 2
u¼
f1 þ ½hexpðb1 þ b2 rSi þ b3 C þ b4 lnðSi Þ þ
b5 r2 C þ b6 Si þ b7 C 2 þ b8 ðS=SiÞ þ b9 r2 S2i Þl gd
(4a)
where

Fig. 1. Fitted (VG3) and predicted 3-parameter van Genuchten
functions and measured water retention values at various pressure
heads of a soil sample of the TEST data set.

not the individual PTF calibration parameters). SSQ
values decreased substantially when Eq. (4) was usedrather separate linear regressions. We will use theprefix NL to refer to this improved nonlinear global
optimization approach. The NL-adjusted PTFs for the
LR8 and LR8FC methods are given in Table 2. The
effects of NL global optimization of the different PTFs
are clearly demonstrated in Fig. 1. The mean prediction errors (ME) for the SWRCs of the EVAL and
TEST soil data sets are given in Table 3.

l ¼ exp½c1 þ c2 r þ c3 OM þ c4 C 2 þ c5 lnðCÞ
þ c6 rS þ c7 r2 C þ c8 ðS=Si Þ þ c9 r2 C 2 (4b)and d = 1  1/l. The exponential functions in Eq. (4a)and Eq. (4b) arise because ln (a) rather than a andln (n) rather n were used in the regressions. Application of Eq. (4) to the EVAL retention data sets leads to2025 nonlinear equations (225 data sets each havingnine retention points) involving 3  9 = 27 or 3  (9 +1) = 30 unknowns (the latter when a concomitantvariable is added). The resulting multivariate nonlinear optimization (inverse) problem may in general
present severe uniqueness and convergence problems.
However, our objective was not to find a uniqueinverse solution, but rather to improve (lower) the
sum-of-squares error, SSQ, from the case when thethree unknown retention parameters were regressed
separately (i.e., only the PTF end result is important,

5. Goodness of the model predictions
Following Rajkai et al. (1996), we consider aSWRC prediction ‘good’ if the mean estimation error(ME) for all measured retention data is less than 2.5%.
This means that the mean absolute difference between
the estimated and measured retention data values isless than 2.5%. The residuals are expressed in terms ofvolumetric water content percentages. The estimation
efficiency (EE) of the different PTF models is definedas the percentage of soils for which the predictions are‘good’.
The accuracy of a prediction can also be expressedusing the mean prediction error PEh at a particulartension h for the databases as a whole:pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ
PEh ¼ ðSSQ=NÞ
(4)

K. Rajkai et al. / Soil & Tillage Research 79 (2004) 145–152

149

Table 3Model selection criteria and statistical data of the predictive water retention models
Statistical

Predictive models

Factors

Fitted model
EVAL

TEST

EVAL

TEST

EVAL

TEST

EVAL

TEST

EVAL

TEST

Residual SSQ
Model DF
N
ME
AIC
SBC

2814
675
2025
0.39
2016
5805

1062
240
720
0.40
760
1859

21893
27
2025
1.10
4875
5026

11453
27
720
1.33
2046
2170

14601
30
2025
0.90
3839
4004

6190
30
720
0.98
1609
1746

19540
27
1810
1.04
4644
4796

10186
27
720
1.25
1962
2085

9284
30
1810
0.71
3019
3184

4348
30
720
0.82
1355
1492

LR8

LR8FC

NLR8

NLR8FC

SSQ: the sum of square error; model DF: the total number of degrees of freedom of the optimization (number of models or equations  number ofmodel parameters); N: the total number of water retention data (number of soil samples  number of retention data per sample); AIC: the Akaike
information criteria and SBC: the Schwartz and Bayes information criteria.

where SSQ is the sum of square error and N thenumber of data sets in the database. In addition we
calculated the mean prediction error, ME, for a retention curve as follows
ME ¼

(8)

and

X PEh

(5)

n

where n is now the number of retention data points.
The form of the regression model isyi ¼ f ðxi ; bÞ þ ei ;(Kass and Raftery, 1995)
"
#
N
1X
2
ðyi ŷi Þ þ 2P
AIC ¼ Nln
N i¼1

i ¼ 1; :::; N(6)where yi and xi are known measured values of the
independent and dependent variables, respectively, b
is the unknown parameter vector, and ei is an independent zero-mean Gaussian error term. The log-likelihood function for this model (without the negligible
constants, ei) isN
1X
2ln LðbÞ ¼ N lnðyi  ŷi Þ2
N i¼1

!
(7)

where L(b) is the Gaussian likelihood function andŷi ¼ f (xi, ML-estimator of b).
The usual goodness-of-fit measure, R2, and the Ftest of significance have a drawback in that they are
relatively insensitive to the number of parameters (i.e.,the dimension of the parameter vector b) involved in
the regression equation. While numerous criteria havebeen proposed to remedy this drawback, we use in thisstudy Akaike’s information criterion (AIC) and the
Schwarz and Bayesian criterion (SBC) as follows

"

#
N
1X
2
ðyi  ŷi Þ þ PlnðNÞ
SBC ¼ Nln
N i¼1

(9)

where P is the number of parameters used. According
to Akaike (1973) the AIC is based on a predictive
point of view, which makes the predictive distribution
conditional on a model and its estimated parameters
using the maximum likelihood method (Linhart andZucchini, 1986). The SBC model-selection criterion
above is that of Schwarz and Bayes (Schwarz, 1976)
which places more emphasis on the number of parameters in a model. Schwarz (1976) introduced thiscriterion to select the model with the highest posterior
probability using the asymptotic normality of theestimator of b with the maximum likelihood estimator
as its mean, and using Fisher-information as inverse of
the variance. When models are compared in terms of
their AIC and SBC, the better model is that model
which has a smaller value of the invoked criterion.
Values for AIC and SBC were calculated using the
SPSS statistical software package (SPSS, 1998).

6. Results and discussion
Fig. 1 shows that the nonlinear (NL) global
optimization approach did not improve the predicted

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K. Rajkai et al. / Soil & Tillage Research 79 (2004) 145–152

Fig. 2. Estimation efficiency (EE) of the SWRC predictive methods
for the TEST data set.

SWRC close to the measured retention value at h =1500 kPa in case of the NLR8FC PTF, but that thepredictions are close to the other retention data. Fig. 1
also shows the performance of the individually fitted
SWRCs.
Fig. 2 shows that the estimation efficiency, EE, isbetween 53 and 78% for the TEST soils for the LR8
and the LR8FC PTFs. Results indicate that the NLR8
and NLR8FC PTFs for the TEST soils have a 7–10%
higher EE than their linear versions (LR8 and
LR8FC).
The mean SWRC prediction error, ME, is plotted inFig. 3 for the different predictive models. Notice that
the LR8FC model performed better for the TEST soil

Fig. 3. Mean errors of the SWRC predictive methods and the fitted
van Genuchten function (VG3).

database, and that NL optimization did improve theresults somewhat further for the TEST data set as well.
Using field capacity as a concomitant variable
significantly improved the results; furthermore, NL
adjustment did lead to substantial reductions in the
Mean Error, ME.
The two model selection criteria (AIC and SBC) for
the different predictive methods are listed in Table 3.
As expected, the individually fitted retention data sets
produced the smallest AIC values. However, while the
AIC values suggest that all PTFs generate results thatare worse than the individually fitted models, the SBC
criterion qualifies almost all PTFs as being more
effective than the individually fitted SWRCs in termsof Akaike’s statistical stability criterion. The biggest
difference between the AIC and SBC criteria is their
sensitivity to the degrees of freedom (DF) of the PTF.
When the SBCs of the predicted SWRCs arecompared to each other, a rank from weak to goodcan be established. This shows that the nonlinearly(NL) adjusted models performed better than theirlinear versions.
The NL adjusted PTFs in Table 3, using FC asconcomitant variable, for the EVAL and TEST data
sets suggest that the predictive PTFs are fairly general.
The best predictive model (NLR8FC) produces very
reasonable results, giving ME values for the TEST
soils that are only about two times higher than thosefor the individually fitted SWRCs (VG3) in Fig. 3.
The prediction efficiency (EE) represents the ratioof ‘‘good" predictions to the total number of data setsin the database. ME, together with EE, demonstrates
the power of a particular PTF (Fig. 2). The relatively
high ME and low EE values associated with the LR8model indicate limited flexibility and accuracy of thismodel. The use of one measured retention data point(FC) in the PTF significantly decreased ME andincreased EE. Using the field capacity in the LR8model increased the EE by about 20% as compared to
the LR8 (Fig. 2). Nonlinear adjustment of the LR8FC
model improved the EE up to 95% for the EVAL soils,
and 88% for the TEST database. This means that
nonlinear adjustment leads to almost identical
performance of the predictive model for the EVAL
and TEST data sets. We conclude from this that thewater retention data of the Hungarian soils are best
predicted using the LR8FC PTF. However, the ME of
the best PTF is still two times larger than the ME of theK. Rajkai et al. / Soil & Tillage Research 79 (2004) 145–152

fitted SWRC model. It seems that this is probably thebest result that can be achieved using the traditional
statistical methods we have used. Recently several
neural-network based methods have been used toderive pedotransfer functions (Pachepskij et al., 1996;
Schaap et al., 1998). It would be interesting to make adetailed comparison of the prediction accuracy of
those neural networks with the PTFs derived in thisstudy.

7. Summary and conclusions
A three-parameter van Genuchten type model wasused to describe the water retention curves of
Hungarian soils. Fitted SWRC parameters wereregressed linearly with a large number of soil physicalproperties in the database. To increase the flexibility of
the linear PTFs we used not only the statistically mostsignificant original soil properties, but also several
derived properties (e.g., the logarithms, squares, andproducts of the original soil properties). Altogether weused eight measured and transformed soil properties,
as well as one measured retention point to construct
PTF’s for the three retention parameters.
The linear PTFs were substituted into the SWRC
and further adjusted nonlinearly using all measured
retention data of the EVAL soil database. The
estimation efficiency (EE), being the number of good
estimations as a percentage of all data sets in the entire
database, improved with this nonlinear adjustment
from 4 to 7%. The EE further improved when one
retention data point was added as a concomitantvariable in the PTF analysis. Using the 20 kPa (nearfield capacity) retention point for this purpose in the
nonlinear analysis increased the EE up to 88% of the
independent TEST data set, and produced a mean
estimation error of less than 1% for the TEST
database. The applied techniques significantly
improved the PTF prediction accuracies, and should
be applicable to a much wider array of soils than only
the Hungarian database. The PTFs derived in thispaper provide improved relationships for estimating
the soil water retention curve from soil texture andrelated properties, and as such may prove useful instudies dealing with assessments of physical soil
quality, as well as for application to many other largerscale soil, hydrological and agricultural problems.

151

Acknowledgements
This study was partially supported by the Hungarian Scientific Research Fund (OTKA) Nr. T038412,
and the SAHRA Science Technology Center as part of
NSF Grant EAR-9876800.

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