Comparison of Models for Indirect Estimation of Water Retention
and Available Water in Surface Soils
Marcel G. Schaap,* Attila Nemes, and Martinus Th. van Genuchten

Reproduced from Vadose Zone Journal. Published by Soil Science Society of America. All copyrights reserved.

ABSTRACTin the past (for reviews see Rawls et al., 1991; Wösten
et al., 2001) generally have a strong degree of empiricism
in that they contain model coefficients that are calibrated on existing soil hydraulic databases. A PTF canbe a simple look-up table that gives hydraulic parameters according to textural class (e.g., Carsel and Parrish,1988), include linear or nonlinear regression equations(Rawls and Brakensiek, 1985; Minasny et al., 1999), or
make use of neural network analysis (Pachepsky et al.,
1996; Schaap and Bouten, 1996). Most PTFs are intended to be used when no, or at most limited, soil
hydraulic data are available. One of the crucial requirements of PTFs is therefore their ability to give reasonable estimates for a wide range of soils. Other important
issues include the required input data (which may limit
the applicability of a PTF when not available), model
complexity (which may determine the ease in which a
PTF can be implemented in larger computer models or
databases), and desired output data. The first objectiveof this study was to test 11 PTFs for estimating soil
water retention parameters. While these PTFs generally
have somewhat larger errors than PTFs that estimate
specific retention points (e.g., Schaap and Bouten, 1996;
Minasny et al., 1999), they enable the estimation of soil
water contents at any pressure head, thus facilitating
their use in numerical applications. The PTFs selectedin this study were evaluated using the soil survey database of the USDA-NRCS (Soil Survey Staff, 1995) that
contains pertinent data about 21 680 soil profiles acrossthe United States. The PTFs were evaluated (i) using measured water contents at several soil water pressure heads
and (ii) for their ability to estimate available water content, a parameter that is often required in large-scalehydrological studies.
Before testing the PTFs we first address one subtle
issue regarding parameter-based PTFs. Our concern involves the fact that parametric PTFs are often developed
using objective functions based on parameter values
but evaluated in terms of water contents (the observed
quantity). Parametric PTFs are generally constructed inthree steps. First, water retention equations are chosen
and fitted to observed water content–pressure head data,
thereby yielding fitted retention parameters. The objective function used for fitting is usually defined in termsof squared differences between measured and observed
water contents, here referred to as a “water content
criterion.” The second step in the development of para-

Quantitative knowledge of the unsaturated soil hydraulic properties is required in most studies involving water flow and solute transport in the vadose zone. Unfortunately, direct measurement of suchproperties is often difficult, expensive and time-consuming. Pedotransfer functions (PTFs) offer a means to estimate soil hydraulic properties
based on predictors like texture, bulk density, and other soil variables.
In this study, we focus on PTFs for water retention and show thatsystematic errors in five existing PTFs can be reduced by using watercontent–based objective functions, instead of parameter value–based
objective functions. The alternative analysis was accomplished byestablishing offset and slope coefficients for each estimated hydraulicparameter. Subsequently we evaluated these and six other PTFs for
estimating water retention parameters using the NRCS soils database.
A total of 47 435 records containing 113 970 observed water contents
were used to test the PTFs for mean errors and root mean squareerrors. No overall superior model was found. Models with many calibration parameters or more input variables were not necessarily better
than more simple models. All models underestimated water contents,
with values ranging from ⫺0.0086 to ⫺0.0279 cm3 cm⫺3. Average root
mean square errors ranged from 0.0687 cm3 cm⫺3 for a PTF thatprovided textural class average parameters to 0.0315 cm3 cm⫺3 for amodel that also used two water retention points as predictors. Available soil water content for vegetation was estimated with errors ranging from 0.058 to 0.080 cm3 cm⫺3, depending on the model and thedefinition of available water.

T

he hydraulic properties of the vadose zone exert
a strong control over the movement of water anddissolved solutes between the soil surface and the groundwater table. However, because of their strong nonlinearity, direct measurement of the hydraulic properties (water retention and unsaturated hydraulic conductivity) is
difficult, expensive, and subject to considerable experimental limitations. Our ability to obtain meaningful data
is further complicated by the fact that the hydraulicproperties often exhibit significant spatial variability.
These problems indicate the need for PTFs as inexpensive and rapid tools to estimate soil hydraulic properties
using indirect methods based on correlations between
the hydraulic quantities and other more easily measurable soil or sediment variables, such as the particle-size
distribution, bulk density, and/or organic matter content.
The wide variety of PTFs that have been developed
M.G. Schaap and M.Th. van Genuchten, George E. Brown, Jr. Salinity
Laboratory, USDA, ARS, 450 W. Big Springs Road, Riverside, CA
92507; A. Nemes, Research Institute for Soil Science and Agricultural
Chemistry of the Hungarian Academy of Sciences, Herman Ottó u. 15,
H-1525 Budapest, Hungary, currently at USDA-ARS Hydrology andRemote Sensing Laboratory, USDA, ARS, 10300 Baltimore Ave., Bldg.
007, BARC-West, Beltsville, MD 20705. Received 9 Apr. 2004. OriginalResearch Paper. *Corresponding author (mschaap@ussl.ars.usda.gov).

Abbreviations: AWC, available water content; COS1, Cosby PTF
based on univariate regression; COS2, Cosby PTF based on bivariate
regression; ME, mean error; PTF, pedotransfer function; RBC, Rawls
and Brakensiek PTF with Brooks–Corey parameters; RMSE, root
mean square residual; RVG, Rawls and Brakensiek PTF with van
Genuchten parameters; URMSE, unbiased root mean square residual;
VER, Vereecken PTF; WOE, Wösten PTF.

Published in Vadose Zone Journal 3:1455–1463 (2004).
© Soil Science Society of America
677 S. Segoe Rd., Madison, WI 53711 USA

1455

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VADOSE ZONE J., VOL. 3, NOVEMBER 2004

metric PTFs is the selection of plausible predictors andan appropriate mathematical structure (tables, linear or
nonlinear equations, or neural networks) that link thepredictors to the fitted retention parameters. In the third
step, the PTF coefficients are calibrated, generally using
an objective function that is defined in terms of sumof squares of differences between fitted and estimatedretention parameters (here called a “parameter criterion”). However, parametric PTFs are often evaluatedwith a water content criterion (e.g., mean errors or root
mean square errors) that is based on observed water
retention points. Due to the nonlinearity of the invokedhydraulic functions and the inherent imprecision of PTF
estimates, a PTF that provides optimum estimates ofparameters may not necessarily give optimum estimatesin terms of water contents. For completeness we notethat the term fitted parameter implies the result of an
optimization of a retention function to observed watercontents, while estimated parameter refers to a PTF estimate.
Scheinost et al. (1997) circumvented the water content
vs. parameter criterion problem by calibrating parameter-based PTFs using an objective function that was
defined in terms of a water content criterion. For thispurpose they substituted simple expressions that estimated retention parameters from texture directly into
the van Genuchten (1980) retention equation. The advantage of this approach is that the resulting equation
can be optimized directly to a database of observed water
retention characteristics. No fitted hydraulic parameters
are necessary in this approach, yet they are generated
automatically when the PTF is used. Minasny et al. (1999)
adopted this method and found its accuracy equivalent
to that of a neural network approach.
Unfortunately, because of the mathematical complexity of neural network optimization, it is difficult to employ the approach used by Scheinost et al. (1997) andMinasny et al. (1999) to the neural network models
developed by Schaap and Leij (1998) and implemented
in the computer program Rosetta (Schaap et al., 2001).
Minasny and McBratney (2002) solved this problem by
first calibrating a neural network PTF using the parameter criterion, and subsequently reoptimizing the calibrated neural network coefficients with a water content

criterion by means of Levenberg–Marquardt optimization. Using this method, called the Neuro-m method,they obtained a 13% better performance than neuralnetwork results without the Neuro-m reoptimization.
As a second objective in this study we follow a different approach and assume that retention parameters thatare optimal according to the parameter criterion differ
only modestly from those that would be optimal according to the water content criterion. We propose theuse of simple linear expressions to modify parametersof existing models in Rosetta such that they provide
better estimates of the water contents. Although applicable to any parameter-based PTF, we will apply theparameter translation procedure only to the models in
Rosetta, using the database that was used for its calibration. The proposed procedure is not applied to the other
PTFs since we do not have access to their respective
calibration databases.

PEDOTRANSFER FUNCTIONSEVALUATED IN THIS STUDYThis section briefly describes the most important characteristics of the PTFs that are evaluated in this study.
Some of these characteristics are represented in Table
1; we refer the reader to the references for more indepth descriptions of the PTFs. Most of the PTFs arealso able to estimate the saturated hydraulic conductivity, and some even the unsaturated conductivity. However, performance for these properties will not be tested
in this study since the NRCS database lacks the necessary conductivity data.

Rosetta (Models H1 through H5)
Rosetta (Schaap et al., 2001) may be used to estimate
water retention parameters in van Genuchten’s (1980)
equation, given by
␪(h) ⫽ ␪r ⫹

␪ s ⫺ ␪r
[1 ⫹ (␣h)n]m[1]where ␪(h) is the measured volumetric water content(cm3 cm⫺3) at pressure head h (cm, in this study taken
positive for unsaturated conditions), the parameters ␪rand ␪s are residual and saturated water contents, respec-

Table 1. Overview of the most important characteristics of the pedotransfer functions evaluated in this study.†
Required input
Model

Reference

H1
H2
H3
H4
H5
RBC
RVG
COS1
COS2
VER
WOE

Schaap et al. (2001)

Rawls and Brakensiek (1985)
Cosby et al. (1984)
Vereecken et al. (1989)
Wösten et al. (1999)

Sand
x
x
x
x
x
x
x
x
x
x

Silt

x
x
x
x

x
x

Clay
x
x
x
x
x
x
x
x
x
x
x

␳b

OC

Hor

x
x
x
x
x
x
x

␪330

x
x

x
x

x

␪15 000x

Nc

Output

NRCS

48
52
58
64
70
35
35
6
9
15
44

VG
VG
VG
VG
VG
BC
VG
BC
BC
VG
VG

%
100.00100.00100.00
99.97
99.97
99.99
99.99
100.00100.0097.85
97.85† ␳b, bulk density; OC, organic C or organic matter; Hor, horizon; ␪330 and ␪15 000, water contents at 330 and 15 000 cm pressure; Nc, number of model
coefficients; Output, type of retention function (VG: van Genuchten, BC: Brooks–Corey); NRCS, percentage of the 47 435 selected NRCS records forwhich a particular model could be run. See text for further explanation.

1457

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www.vadosezonejournal.org

tively (cm3 cm⫺3); ␣ (⬎0, in cm⫺1) is related in an approximate manner to the inverse of the air entry pressure,and n (⬎1) is a pore-size distribution parameter (vanGenuchten, 1980). The parameter m is equal to 1 ⫺ 1/n.
Rosetta uses a hierarchical approach that allows users
to estimate water retention parameters (␪r, ␪s, log␪,logn) using limited to more extended sets of predictors(Schaap et al., 1998, 2001). The first model (H1) is aclass PTF, consisting of a look-up table that provides
parameter averages for each USDA soil textural class(see also Table 1). The second model (H2) uses sand,silt, and clay percentages as input and, as opposed to
H1, provides hydraulic parameters that vary continuously with texture. The third model (H3) includes bulk
density as an additional predictor, while the fourthmodel (H4) also uses the water content at 330-cm pressure head. The last model (H5) includes the water content at 15 000 cm pressure in addition to the input variables of H4. The choice of pressure heads in models H4and H5 was determined by their availability in the
NRCS database (Soil Survey Staff, 1995). Model H1
provides the mean values of ␪r, ␪s, log␣, and logn foreach of the 12 USDA textural classes. This model thus
contains 48 model coefficients (12 classes ⫻ 4 parameters). Models H2 through H5 provide the same parameters and contain a larger number of coefficients without
a well-defined meaning (Schaap and Bouten, 1996). The
objective function used for the calibration minimized
the variance of all hydraulic parameters simultaneously
according to4NOp ⫽ 兺 兺 (␨i,j ⫺ ␨i,j⬘ )2[2]

i⫽1 j⫽1

where N is the number of samples and ␨ and ␨⬘ arefitted and estimated parameters (␪r, ␪s, log10␣, log10n,
indexed by i), respectively.

M

NOi ⫽ 兺 兺 [␪i,j ⫺ ␪⬘(hj,␪″r,i,␪″s,i,␣″i ,n″i )]2[4]i⫽1 i⫽1

where M is the number of water retention characteristic,
N the number of measured points for each retention
characteristic, and ␪ and ␪⬘ are measured and estimated(with modified parameters) water contents, respectively. The offset parameter ai for ␪r was set to zero toconstrain ␪r to positive residual water contents. Higherorder alternatives (e.g., polynomials) to Eq. [3] werealso considered but were more difficult to optimize andoften yielded nonunique coefficients. Although the outlined procedure leads to different retention parametervalues, we stress that all parameters retain their traditional meaning.

Rawls and Brakensiek (RBC and RVG)

Modified Rosetta Models(Models H1m through H5m)
We employ a parameter modification procedure basedon the hypothesis that the parameter estimates obtainedwith Models H1 through H5 can be modified to Models
H1m through H5m, such that a minimum error in termsof water contents can be obtained. We assume that asimple linear translation of the estimated parameters
is sufficient to reach the minimum variance between
estimated and observed water contents. For this purpose, each of the estimated retention parameters ␨i⬘ is
modified to ␨i″ according to␨i″ ⫽ ai ⫹ bi␨i⬘Fig. 1. Textural distribution of the 2134 samples used to calibrate
Models H1 through H5. The textural classes are given in Fig. 2.

[3]Each retention parameter and each PTF H1m throughH5m has its own parameters ai and bi. Using the originalcalibration database employed for models H1 throughH5 with 2134 samples and 20 574 individual retention
points (Schaap and Leij, 1998; Schaap et al., 2001; atextural distribution is given in Fig. 1), we simultaneously optimized parameters ai and bi for all four retention parameters for each PTF using the objective function

Rawls and Brakensiek (1985) presented a multivariate PTF for estimating parameters of the Brooks andCorey (1964) equation, given by

冦

␪s

冧

h ⱕ hb␭h␪(h) ⫽ ␪ ⫹ (␪ ⫺ ␪ ) b h ⬎ h
r
s
r
b
h

冢 冣

[5]

where hb is the air entry pressure head, and ␭ a poresize distribution parameter comparable to n in Eq. [1].
Input variables to the model, here identified as RBC, areporosity, and sand and clay percentages. The saturated
water content was not directly estimated with this PTF
but set equal to the porosity, φ. In this study the porosity
is derived from the bulk density (␳b) as φ ⫽ 1 ⫺ ␳b/2.65,
where 2.65 is the assumed density of the solid phase(g cm⫺3). In addition to the RBC model, Rawls andBrakensiek (1985) presented a conversion of Brooks–
Corey parameters to van Genuchten parameters, assuming the approximations ␣ ⫽ 1/hb and n ⫽ ␭ ⫹ 1. Thisversion of the RBC model will be referred to as RVG.
Although Rawls and Brakensiek (1985) indicated that

Reproduced from Vadose Zone Journal. Published by Soil Science Society of America. All copyrights reserved.

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VADOSE ZONE J., VOL. 3, NOVEMBER 2004

their models were valid for sand contents between 5
and 70% and clay contents between 5 and 60%, weapplied their models to all selected data. Reasons forthis were that the differences in errors were minor (notfurther shown here) and because otherwise only 71.5%of the selected NRCS data could be evaluated. Although
no explicit statement was found in Rawls and Brakensiek(1985) and related publications, we assume that theircalibration database contained several thousands ofsamples from agricultural soils in the USA.

Cosby (COS1 and COS2)
Cosby et al. (1984) presented two PTFs based on univariate or bivariate regression, identified as COS1 and
COS2, respectively. The models are very simple—they
contain only six or nine regression parameters in total—and estimate parameters of the Campbell (1974) water
retention equation. For the purposes of this study weinverted the Campbell equation to yield the Brooks–
Corey (1964) equation, but with ␪r in Eq. [5] set to 0.
Cosby et al. (1984) calibrated their PTFs using data
derived from Holtan et al. (1968) and Rawls et al. (1976),
totaling 1448 soil samples.

Vereecken (VER)
Vereecken et al. (1989) presented several regression
equations for estimating parameters in a modified van
Genuchten curve in which m ⫽ 1 and n ⬎ 0 (Eq. [1]).
The model we selected (referred to as VER) uses sandand clay percentages, bulk density, and organic C content as input. Another model by Vereecken et al. (1989)
relied on principal component analysis of the textural
distribution, but it could not be tested in this study
because the required data were not available. We notehere that estimates of the ␣ and n parameters in Vereecken et al. (1989) are based on natural logarithms,
not base 10 logarithms, as suggested in the publication.
The VER model was calibrated using data from duplicate samples derived from 182 soil horizons in northern Belgium.

Wösten (WOE)
Based on an analysis of the European soil hydraulicdatabase HYPRES, Wösten et al. (1999) presented a
PTF for estimating van Genuchten parameters. The
multivariate model with 44 parameters estimates ␪s, ␣,and n (as well as parameters in the Mualem–van Genuchten equation) using silt, clay, organic matter content,bulk density, and the Boolean variable “topsoil” vs.
“subsoil.” In this study ␪r was set to 0.01 (Wösten et al.,1999) and the organic matter content was estimated bymultiplying the organic C content with 1.72 (Nelson andSommers, 1982). Wösten et al. (1999) classified all A
and E horizons as topsoils. For the purposes of this study
we assumed topsoils for all samples taken at depths
shallower than 30 cm.

THE NRCS SOIL
CHARACTERIZATION DATABASEThe NRCS soil characterization database (Soil Survey
Staff, 1995) contains detailed data of 21 680 soil profiles

across the USA and served to test all PTFs selected forthis study. The data are available on a soil horizon basis,
totaling 136 620 records involving some 250 possible
attributes covering a broad range of soil physical, chemical, and mineralogical characteristics. No single record
contains a complete set of data and, unfortunately, only
a limited number of water contents are available. Water
retention at 15 000 cm is available for 90.1% of thesamples in the NRCS database, while the 330 cm watercontent is available for 37.1% of the cases. Water contents at 60, 100, 1000, and 2000 cm are only available
for a small fraction of the database. All retention data
were measured using pressure plates, with the 60-, 100-,and 330-cm water contents obtained on undisturbed
clods (w6cld, w10cld, w3cld, respectively, in Soil Survey
Staff 1995). The water contents at 1000, 2000, and 15 000
cm (w1bair, w2bair, w15ad, respectively, in Soil Survey
Staff, 1995) were measured on initially air-dry sieved
samples. We refer the reader to Soil Survey Staff (1996)
for more information. Water contents in the NRCS database are given on a gravimetric basis and were converted
to volumetric units using the bulk density measured at
330 cm pressure (db_13b, Soil Survey Staff 1995).
To test the selected PTFs we required that at least
soil texture, bulk density, and the 330- and 15 000-cm
water contents were available. As shown in Table 1, not
all data could be used for all PTFs. Most often this wasdue to missing organic C data (needed for the VER and
WOE models), low or extremely high bulk densities(densities ⬍0.5 and ⬎2.0 g cm⫺3 were excluded) or inconsistent data (e.g., sometimes the volumetric watercontent at 330 cm was smaller than that at 15 000 cm).
Occasionally unrealistically high volumetric water contents were found (e.g., ⬎1 cm3 cm⫺3), presumably due
to data entry errors in the NRCS database. To prevent
unreasonable results, we required that all volumetricwater contents should be smaller than 0.6 cm3 cm⫺3, areasonable assumption given that the smallest availablepressure head in the NRCS database was 60 cm. Together with other constraints on the data, this left 47 435
records and 113 970 water retention points for testingof the PTFs. We note that most retention points werelocated at 330- and 15 000-cm pressure heads; values at
60, 100, and 1000 cm were available for only 2.7, 15.4,
and 0.7% of the selected records, respectively. A textural distribution of the selected NRCS data appears in
Fig. 2. Note that these data are presented in the somewhat unusual unit of number of samples per square
percentage. Plotting all individual points as in Fig. 1
would have made the graph unreadable.

EVALUATION CRITERIA
Three criteria were used to quantify errors in the PTF
estimates of the water contents. The most commonlyused criterion in PTF-related work is probably the root
mean square residual (RMSE), defined as
RMSE ⫽冪N 兺 (␪⬘⫺ ␪)
1 N

i⫽1

2

[6]

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www.vadosezonejournal.org

Fig. 2. Textural distribution of 47 435 samples used to evaluate thepedotransfer functions selected for this study. Displayed is thenumber of samples at intervals of 1, 5, 10, 15, 20 samples per square
percentage. S, sand; lS, loamy sand; sL, sandy loam; scL, sandy
clay loam; sC, sandy clay; L, loam; siL, silty loam; Si, silt; sicL,
silty clay loam; siC, silty clay; cL, clay loam; C, clay.

where N is the size of the (sub)set of observations forwhich the RMSE is computed, and ␪i and ␪i⬘ are measured and estimated water contents, respectively. The
value of ␪i⬘ is computed by evaluating the appropriate
retention function at the observed pressure head withthe estimated retention parameters. The RMSE may beviewed as giving the accuracy of the model in terms ofstandard deviations.
When systematic errors exist, the RMSE values arebiased and do not reflect the true zero-mean variance.
Systematic errors are often an artifact of the calibration
database (Schaap and Leij, 1998) and could render acomparison of PTFs based on RMSE values difficult.
We therefore decompose the RMSE into a mean error
and an unbiased RMSE (Hastie et al., 2001).
Mean errors (ME) may be used to quantify systematic
errors between measurements and model estimations:
1 N
ME ⫽ 兺 (␪⬘⫺ ␪)
[7]
N i⫽1
ME values are negative when the PTF underestimateswater contents. Unbiased RMSE (URMSE) values wereused by Tietje and Hennings (1996) and have the meanerrors removed according to
URMSE ⫽冪

1 N[(␪⬘⫺ ME) ⫺ ␪]2
N i兺
⫽1

[8]

URMSE values should always be equal to or smaller
than the corresponding RMSE values; RMSE, ME, and
URMSE values in this study are all given in cubic centimeters per cubic centimeter.

RESULTS AND DISCUSSION
Modification of the Parameters Estimatedby Models H1 through H5Figure 3 shows the mean errors for nine pressure headclasses available in the calibration database (between

1459

boundaries at 0, 3, 10, 30, 100, 300, 1000, 3000, 10 000,
and 30 000 cm) for the original H1 through H5 models(Fig. 3a) and after parameter modification (Fig. 3b).
Also shown are the results of the direct fit of the van
Genuchten equation (Eq. [1]) to the data (FIT). The
direct fit shows no large systematic errors in this pressure range, indicating that water retention can be described adequately with Eq. [1]. However, the meanerrors of H1 through H4 before modification were considerable (between ⫺0.045 and 0.025 cm3 cm⫺3). Water
contents for pressure heads smaller than 3 cm were
underestimated, overestimations are present between 3
and 10 cm, and strong underestimations are found beyond 30 cm. The patterns exhibited by H1 through H3
are virtually identical, indicating that the algorithm used
for PTF development (textural class averages for H1
vs. a neural network approach for H2 and H3) is not
the cause of the systematic errors. Models H4 and H5have less severe systematic errors since they use one or
two measured retention points as input. The parameter
modification procedure removed most systematic errors
for pressure heads higher than 30 cm (Fig. 3b). However,the systematic errors were not removed near saturation;that is, the underestimations between 0 and 3 cm andthe overestimations between 3 and 10 cm remained. The
modified PTFs H1m through H5m estimate retention
near saturation with a pressure-dependent database average bias between ⫺0.02 and 0.02 cm3 cm⫺3.
Table 2 provides the coefficients found for the parameter modifications (Eq. [3]) as well as ME and RMSE
values for the original and modified models. By far thestrongest effect is seen for the parameter n in Eq. [1];the slope coefficient for this parameter ranges from0.655 for Model H2 to 0.877 for Model H5. These results
indicate that the modified log10n values were reduced
substantially with regard to the original values. The effect of this change is that, especially for higher pressure
heads, estimated water contents will be higher than before, which in turn, leads to smaller underestimations(Fig. 3b). Smaller changes in offset and slope (relative to
0 and 1, respectively) were found for the other retention
parameters. Most likely these changes account for correlations that are commonly present among water retention parameters—when n is modified, changes in theother parameters are necessary to maintain a good matchwith observed water retention data. Table 2 shows thatmean errors are one (H5 vs. H5m) to almost two orders(H1 vs. H1m) of magnitude smaller for the modified
parameters compared with the original estimates. Corresponding RMSE values were reduced modestly: approximately 0.005 (H5m) to 0.008 (H2m) cm3 cm⫺3 compared with the original models. Expressed in percentages
the reductions in RMSE ranged between 8% (H1m andH2m) and 13% (H4m). These reductions are very similar to those found by Minasny and McBratney (2002)
for the Neuro-m method for reoptimization of neuralnetwork–based PTFs. Because of the reduced errors(especially for ME), we will use Models H1m throughH5m in the remainder of this study rather than H1
through H5.

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VADOSE ZONE J., VOL. 3, NOVEMBER 2004

Fig. 3. Mean errors for the H1 through H5 models in Rosetta (a) before and (b) after parameter transformation. The mean errors were computed
for nine pressure head classes with boundaries at 0, 3, 10, 100, 300, 1000, 3000, 10 000, and 30 000 cm. The results were plotted at the midpointof each pressure class (e.g., 1.5, 6.5, 20, 65 cm). For reference, the direct fit to the original retention data is also shown (FIT).

Evaluation of PTFs on Data
from the NRCS Database
Table 3 shows ME and URMSE results for the 11
PTFs for six different pressure heads for the NRCS soil
characterization database. Also shown are the average
ME, URMSE, and RMSE errors, weighted according
to the number of observations for each pressure head.
Note that only few observations were available for the
60- and 1000-cm pressure heads, whereas more observations were available for 100 and 2000 cm (7562 and 10431
data points, respectively). Most or all water contents at
330 and 15 000 cm could be used for our analysis.
The weighted ME values show that all models underestimated water retention with values ranging from⫺0.0279 cm3 cm⫺3 for H1m to ⫺0.0043 cm3 cm⫺3 for
COS2. This underestimation may be a result of an overestimation of measured volumetric water contents, whichwere derived from gravimetric water contents and thebulk density at a pressure head of 330 cm (see the section
about the NRCS database). For individual water retention points the most negative ME (⫺0.0421 cm3 cm⫺3)
was found for the RVG model at 330 cm, while thelargest ME (0.0118 cm3 cm⫺3) was found for Model H5m
at 60 cm. Table 3 also substantiates setting ␪s to theporosity, φ, in the RBC and RVG models. Both models
show negative ME values at 60 cm. If we had accounted
for air entrapment (e.g., by assuming ␪s ⫽ ␤φ, with ␤⬍1), a more negative ME would have resulted. Table
3 also shows that setting ␪r to zero (COS1, COS2) orto a small value (WOE) does not lead to large ME
values at 15 000 cm. The COS1 and COS2 models evenslightly overestimated water contents at this pressure
head. The COS1 and VER models were also tested by
Kern (1995) using a smaller subset of the NRCS database for pressure heads at 100, 330, and 15 000 cm. Our
ME values for the COS1 model are somewhat smaller

Table 2. Translation coefficients for Models H1 through H5 for the van Genuchten (1980) water retention parameters for Rosetta’s
calibration database (Schaap and Leij 1998). Mean errors (ME) and biased root mean square residuals (RMSE) for original andmodified models are also shown. The offset parameter for ␪r was set to zero (see text).
Offset parameters (ai)

H1
H2
H3
H4
H5

Slope parameters (bi)Original

Modified␪s

log10␣

log10n␪r␪slog10␣

log10n

ME

RMSE

ME

RMSE

⫺0.013⫺0.0480.002
0.011⫺0.020

0.198
0.159
0.162
0.067
⫺0.040⫺0.003
0.004
0.003
0.006
⫺0.003

0.957
0.969
0.995
1.156
1.220

1.008
1.103
1.000
0.974
1.045

1.086
1.050
1.028
1.039
0.956

0.696
0.655
0.660
0.796
0.877

⫺0.022⫺0.021⫺0.022⫺0.019⫺0.013

0.078
0.076
0.068
0.047
0.044

0.00030.00060.00080.00100.0013

0.072
0.070
0.060
0.041
0.039

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Table 3. Mean error (ME) and unbiased root mean square errors (URMSE) for six retention points. Also listed are the number ofavailable water contents per pressure (Nobs) and the weighted ME, URMSE, and (biased) RMSE errors (in italics) for the characteristics.
Negative ME values indicate underestimated water contents.
Pressure (cm):
Nobs :

60
1288

100
7309

330
47162

1000
352

2000
10429

15000
47435

0.00530.0014⫺0.01440.00680.0077⫺0.0246⫺0.0251
0.00350.0102⫺0.0145⫺0.0077⫺0.0398⫺0.0352⫺0.0408⫺0.0303⫺0.0231⫺0.0298⫺0.0301⫺0.0060⫺0.0021⫺0.0127⫺0.0137⫺0.0227⫺0.0197⫺0.0209⫺0.0173⫺0.0133⫺0.0074⫺0.0074
0.00430.00690.0107⫺0.0025⫺0.0279⫺0.0240⫺0.0275⫺0.0141⫺0.0086⫺0.0233⫺0.0258⫺0.0084⫺0.0043⫺0.0065⫺0.0131

–
–
–
–
–
–
–
–
–
–
–
RMSE

0.05570.05030.05190.05590.0496
0.0376
0.03790.03440.03810.03320.0367

0.0611
0.0561
0.06170.05990.04690.04940.04950.04850.05220.05750.05220.04930.04520.04770.04520.02520.03980.03980.04330.04640.0491
0.04370.06250.05850.05740.03960.02920.0571
0.05620.05990.0616
0.0636
0.0551

0.06870.06350.0641
0.04270.03150.06340.06390.0616
0.0626
0.06570.0575Weighted error

Reproduced from Vadose Zone Journal. Published by Soil Science Society of America. All copyrights reserved.

ME
cm3 cm⫺3
H1m
H2m
H3m
H4m
H5m
RBC
RVG
COS1
COS2
VER
WOE

⫺0.0057⫺0.0018⫺0.01310.00760.0118⫺0.0036⫺0.0235
0.00230.0090⫺0.0071⫺0.0154⫺0.0198⫺0.0168⫺0.0158
0.00470.0116⫺0.0228⫺0.0339⫺0.0181⫺0.0118⫺0.0090⫺0.0153⫺0.0326⫺0.0278⫺0.0334⫺0.0110⫺0.0045⫺0.0385⫺0.0421⫺0.0207⫺0.0153⫺0.0220⫺0.0233
URMSE
cm3 cm⫺3

H1m
H2m
H3m
H4m
H5m
RBC
RVG
COS1
COS2
VER
WOE

0.07130.06930.06960.04970.05190.07370.06870.07580.07540.06990.07

0.08090.07760.07890.04850.05160.08240.07980.08320.08290.08670.07950.0706
0.06660.06090.02190.02070.06740.06620.07090.07190.07300.0610than those found by Kern (1995) while the results for
the VER model are somewhat worse. Tietje and Tapkenhinrichs (1993) also found underestimated watercontents in their evaluation of the COS2, RBC, andVER models on a German dataset.
Models that did not include water retention points
showed weighted URMSE values ranging from 0.0551(WOE) to 0.0636 cm3 cm⫺3 (VER). Corresponding (biased) RMSE values ranged from 0.0575 to 0.0657 cm3
cm⫺3, respectively; however, in this case PTF H1m hada slightly higher error (0.0687 cm3 cm⫺3) than the VER
PTF. Excluding H4m and H5m, all models produced
the largest URMSE values at 100 cm. This pressure
head is located near the steepest decrease in water contents for most water retention curves. Relatively small
errors in estimated ␣ or hb can therefore cause large
URMSE errors. The weighted URMSE errors decreased
to 0.0396 and 0.0292 cm3 cm⫺3 when one (H4m) or two(H5m) retention points were included, respectively. Of
course, much of this decrease is realized at the included
retention points (at 330 and 15 000 cm), but the URMSE
values at most other pressures also decreased significantly. RMSE errors for the (modified) Models H1mthrough H5m were generally comparable to those for
Rosetta’s calibration database, presented in Table 2.

Estimation of Available Water Content
Knowledge of the available water content is useful inapplications that involve plants, such as regional or
global change soil–vegetation–atmosphere coupling studies, or for agricultural purposes. The available watercontent is loosely defined as the difference between
the water content at the pressure head where gravity
drainage becomes negligible (known as field capacity)and the permanent wilting point. The latter value isoften defined at 15 000-cm pressure. Since differentpressure heads for field capacity are commonly used, we
evaluated the models for three available water contents:
60–15 000 cm (AWC1, 1288 pairs of points), 100–15 000
cm (AWC2, 7309 pairs), and 330–15 000 cm (AWC3,
47 162 pairs).
The results in Table 4 show that, in general, most
Table 4. Available water content (AWC) for three pressure differences; negative mean errors (MEs) indicate underestimatedavailable water contents.
AWC1
60–15 000 cm

H1m
H2m
H3m
H4m
H5m
RBC
RVG
COS1
COS2
VER
WOE

0.00860.01020.00110.01730.01970.0009⫺0.0189⫺0.0043⫺0.0007⫺0.0114⫺0.0105

H1m
H2m
H3m
H4m
H5m
RBC
RVG
COS1
COS2
VER
WOE

0.07460.07450.0731
0.0637
0.05310.07280.06920.06890.06680.06770.0656

AWC2
100–15 000 cm
ME
cm3 cm⫺3⫺0.0089⫺0.0067⫺0.0053
0.01100.0175⫺0.0198⫺0.0310⫺0.0218⫺0.0177⫺0.0181⫺0.0115RMSE
cm3 cm⫺3
0.07650.07510.0761
0.06170.05120.07790.07850.07640.07390.07980.0735

AWC3
330–15 000 cm⫺0.0103⫺0.0084⫺0.0127
0.00630.0085⫺0.0313⫺0.0349⫺0.0253⫺0.0226⫺0.0328⫺0.02100.0626
0.06120.05780.03930.01740.06630.0671
0.06500.06290.06800.0575

Reproduced from Vadose Zone Journal. Published by Soil Science Society of America. All copyrights reserved.

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VADOSE ZONE J., VOL. 3, NOVEMBER 2004

PTFs underestimated available water content values.
Although Models H4m and H5m overestimate field capacity only slightly, these models are not likely to bepractical for estimating of available water contents sincein those situations the available water content can becalculated directly from the water content data used as
input to these models. Models H3m, RBC, and COS2
provided the lowest ME values for AWC1. Models H1mthrough H5m generally gave the best ME values for
AWC2 and AWC3. Root mean square residuals, however, are considerable. Except for H4m and H5m, theRMSE for AWC1 ranged from 0.0575 to 0.0798 cm3cm⫺3. It appears that Models H1m through H3m perform slightly worse for AWC1; for AWC2 and AWC3
little difference exists among the models. The WOE
model seems to slightly outperform the other PTFs forapplications involving available water.

SUMMARY AND CONCLUSION
In this study, we improved a sequence of hierarchical
PTFs by modifying parameter estimates from an optimum in terms of variance between fitted and estimatedparameters to an optimum in terms of variance between
estimated and observed water contents. The modification was accomplished by using simple linear equations
that modified the original estimates with an offset anda slope. The largest change was found for parameter n,
whereas smaller changes were necessary for the otherretention parameters. The procedure resulted in substantial reductions in the systematic errors. An average
underestimation of about 0.02 cm3 cm⫺3 was reduced toless than 0.0015 cm3 cm⫺3. The error in terms of rootmean-square residuals was reduced by about 0.005 to
0.008 cm3 cm⫺3. The modification procedure was foundto be simple, flexible, and can potentially be used forother models.
As we noted in a previous study (Schaap and Leij,
1998), the performance of a PTF depends strongly on
the databases being used for PTF calibration and testing.
The heuristic procedure outlined here makes it possible
to modify existing PTFs toward characteristics of particular datasets. A potential application is to take a calibrated PTF (e.g., any of the 11 models used in thisstudy) and modify its output using a smaller databaseof site-specific local data. This would be beneficial in
two ways. First, it is not necessary to invoke the complete PTF development procedure for site-specific data.
Pedotransfer function development for small data sets
is often problematic because the size of the available
database may limit the statistical reliability of the PTFs.
Second, insight obtained in the calibration of the original PTFs is largely retained since the parameter modification procedure makes only minor changes to the estimated parameters. Within this context we also mention
Bayesian modeling of PTFs (e.g., Meyer et al., 1999),
which offers a more formal way to accomplish the useof prior information (the existing PTFs) with new orsite-specific data.
The modified models were evaluated on selected records from the NRCS soil characterization database andcompared with six previously published PTFs (see Table 1). Given the range in complexity of the models thatwere evaluated in this study, we were somewhat surprised to find relatively small differences in performance
among the models. It is therefore difficult to identify asuperior PTF. Model complexity, required input variables, and overall model precision all play a role. The
COS1, COS2, and VER models are by far the most
simple in terms of model complexity, yet these models
do not necessarily perform much worse than other models. In terms of input data, Model H1m may be the most
attractive since it requires only the USDA textural class,which is readily available or may be estimated in thefield by experienced soil surveyors. Next are Models
H2m, COS1, and COS2, which require sand, silt, andclay percentages. For this group of PTFs, Model H2mprovides the lowest URMSE, but a higher ME. Models
H3m, RVG, RBC, VER, and WOE additionally requirebulk density and, in the case of VER and WOE, alsoorganic C or organic matter content. With the possible
exception of the WOE model, this group does not necessarily provide smaller errors than the PTFs based on
only sand, silt, and clay. We previously showed (Schaap
et al., 2001) that PTFs that use bulk density probably
perform better near saturation. Unfortunately, no water
retention points below a pressure head of 60 cm wereavailable in the NRCS database to test this assertion.
Models H4m and H5m require measured water contents
and, although more accurate, they may generally be themost difficult to implement in practice. However, therelatively good performance of these models indicatesthat they can be used to generate complete water retention curves for a large part of the NRCS database. Other
PTFs may be used for records where the required input
water contents at pressure heads of 330 and 15 000 cm
are missing. In addition, the hierarchical Models H1mthrough H5m are also able to estimate saturated andunsaturated conductivities (Schaap and Leij, 2000; Schaap
et al., 2001). The modified models can thus be used topopulate the NRCS database with a complete set of
hydraulic parameters, and consequently increase theusefulness of this database for a large number of applications.
ACKNOWLEDGMENTS
The authors gratefully acknowledge the support by NSF
and NASA (EAR-9804902), the ARO (39153-EV), and the
SAHRA science and technology center under a grant from
NSF (EAR-9876800).

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