Navin K. C. Twarakavi*
Auburn Univ.
201 Funchess Hall
Auburn, AL 36849

Jirka Šimůnek

Dep. of Environmental Sciences
Univ. of California
Riverside, CA 92521

M. G. Schaap
Dep. of Soil, Water, and Environmental Science
Univ. of Arizona
Shantz Bldg.
Tucson, AZ 85721

SOIL PHYSICS

Development of Pedotransfer Functions for
Estimation of Soil Hydraulic Parameters using
Support Vector Machines
Modeling flow in variably saturated porous media requires reliable estimates of the hydraulicparameters describing the soil water retention and hydraulic conductivity. These soil hydraulicproperties can be measured using a wide variety of laboratory and field methods. Frequently,
this proves to be an arduous task because of the high spatial and temporal variability of soil
properties. In the last decade, researchers have shown a keen interest in developing a classof indirect approaches, called pedotransfer functions (PTFs), to overcome this problem.
Pedotransfer functions predict soil hydraulic parameters using easily obtainable soil properties
such as textural information, bulk density and/or few retention points. In this paper, we use
a new methodology called Support Vector Machines (SVMs) to derive a new set of PTFs.
Support vector machines represent a pattern recognition approach where the overall prediction
error and complexity of the SVM structure are minimized simultaneously. We used the same
database that was utilized to develop ROSETTA to generate the SVM-based PTFs. The
performance of the SVM-based PTFs was analyzed using the coefficient of determination, root
mean square error (RMSE) and mean error (ME). All soil hydraulic parameters estimated using
the SVM-based PTFs showed improved confidence in the estimates when compared with the
ROSETTA PTF program. Estimates of water contents and saturated hydraulic conductivities
using the hydraulic parameters predicted by the SVM-based PTFs mostly improved comparedwith those obtained using the artificial neural network (ANN)-based ROSETTA. The RMSE
for water contents decreased from 0.062 to 0.034 as more predictors were used, while the RMSE
for the saturated hydraulic conductivity decreased from 0.716 to 0.552 (dimensionless log10
units). Similarly, the bias in the water contents estimated using the SVM-based PTF was reduced
significantly compared with ROSETTA.
Abbreviations: ANN, Artificial neural networks; HYPRES, Database of hydraulic properties of European
soils; ME, Mean Error; PTFs, Pedotransfer functions; RMSE, Root mean squared error; SLT, Statisticallearning theory; SVMs, Support vector machines; UNSODA, Unsaturated soil hydraulic properties
database; WISE, World inventory of soil emission potentials.

A

number of hydrological models used to simulate situations
ranging from field-scale water flow to global climate change
rely on numerical techniques that simulate heat, water, and solute fluxes in the vadose zone. The use of flow models for variably
saturated conditions requires accurate estimates of the hydrauliccharacteristics that govern water retention and water flow in soils(Wösten et al., 2001). The hydraulic characteristics of soils varyspatially from one location to another, and are also scale-dependent (Hopmans et al., 2002). The temporal variability can also
occur as a result of various biological and human activities, suchas root-growth, soil management and agricultural practices, orSoil Sci. Soc. Am. J. 73:1443-1452
doi:10.2136/sssaj2008.0021
Received 23 Jan. 2008.
*Corresponding author (twarakavi@auburn.edu).
© Soil Science Society of America
677 S. Segoe Rd. Madison WI 53711 USA
All rights reserved. No part of this periodical may be reproduced ortransmitted in any form or by any means, electronic or mechanical,
including photocopying, recording, or any information storage andretrieval system, without permission in writing from the publisher.
Permission for printing and for reprinting the material contained
herein has been obtained by the publisher.
SSSAJ: Volume 73: Number 5 • September–October 2009

physical processes, such as soil shrinking-swelling, soil crusting,and/or water repellence (Wösten et al., 2001). Therefore, it isnecessary to characterize the hydraulic characteristics of soils while
keeping their spatial and temporal variability in perspective.
Ideally, it would be best to measure soil hydraulic parametersin the laboratory or field so that the variability in space and time canbe sufficiently characterized. However, this difficult task is rarely accomplished because of the significant financial and time investments
required for such measurements. Additionally, the spatial variability of soil hydraulic properties, their scale-dependency and possibly
large modeling domains can make such characterization difficult.
To circumvent these practical difficulties with direct methods, researchers have shown keen interest in developing indirect
methods beginning as early as 1912 (Briggs and Shantz, 1912).
The basic idea behind these indirect methods, often called pedotransfer functions (PTFs after Bouma and van Lanen, 1987),
is to predict hard-to-measure soil hydraulic properties (such asretention parameters and hydraulic conductivities) using easily
obtainable input information (such as soil texture, bulk density,and particle-size distribution). In essence, PTFs represent predictive functions that translate the data we have (input) into thedata we need (output) (Wösten et al., 2001)
1443

The complexity of existing PTF models varies from simple
lookup tables that provide hydraulic parameters for particular textural classes (e.g., Carsel and Parrish, 1988; Wösten etal., 1995), to linear/nonlinear regression-based approaches (cf.
Rawls and Brakensiek, 1985; Minasny et al., 1999), to models
that incorporate physical relationships in soil water flow processes (e.g., Haverkamp and Parlange, 1986; Arya and Paris, 1981;
Tyler and Wheatcraft, 1989; Nimmo et al., 2007). More sophisticated methods use various kinds of ANNs (Pachepsky et al.,
1996; Tamari et al., 1996; Minasny et al., 1999), Group Methodof Data Handling (Pachepsky and Rawls, 1999), or make use ofmulti-dimensional nearest neighbor techniques (Nemes et al.,2006a, 2006b). Wösten et al. (2001) provided a detailed reviewof the various PTFs that had been developed.
Pedotransfer functions are usually developed by examining the relationships between input data (textural properties)
and soil hydraulic parameters (such as retention curve and/or
hydraulic conductivity function parameters) from existing soil
databases. Several large databases such as UNSODA (Leij etal., 1996), HYPRES (Wösten et al., 1999), and WISE (Batjes,
1996) are available for the development of PTFs. The accuracy
and reliability of a given PTF approach is determined by how
well the various relationships between the input data (i.e., information such as texture or bulk density) and the output data (i.e.,soil hydraulic parameters) are represented in the PTF structure.
These relationships often tend to be highly nonlinear, and are
therefore not well established. Consequently, regression-based
PTFs (cf. Rawls and Brakensiek, 1985; Minasny et al., 1999)
have often performed poorly because they require adequate priorknowledge of these relationships.
Even though regression-based PTFs have been widely usedin the past due to their simplicity, PTFs based on pattern recognition approaches (cf. Pachepsky et al., 1996; Tamari et al., 1996;
Schaap et al., 2001) seem to have become more popular.
Pattern recognition approaches help describe the underlying relationships between the given inputs and outputs by ‘learning’ from a training data set. After training, pattern recognition
methods lead to high-dimensional nonlinear functions. It canbe said that pattern recognition approaches are mathematicalmodels obtained in an experimental way. If there were no data
(examples, patterns, and observations), there would be no learning, and consequently no pattern recognition tools. One suchcommonly used pattern recognition tool belongs to a class of
methods called Artificial Neural Networks (ANNs).
An ANN is a classical pattern recognition paradigm inspiredby the way biological nervous systems, such as the brain, process
information (Hastie et al., 2001). Artificial Neural Networks arecomposed of a large number of highly interconnected processing elements (called neurons) working in unison to solve specificproblems. The key element of ANNs is user-defined, that is, theircomplicated inter-woven structure. Because of this, an ANN’s
optimality for a specific problem is heavily influenced by user
expertise, though this can be mitigated somewhat by exploring a
number of different ANN topologies (e.g., Ye et al., 2007). Oncethe neural network structure is selected, the objective of learning from the training data in ANNs is to calculate the optimal
weight for each of the links in the neural net by minimizing theoverall prediction error. The optimal high-dimensional relation
between the input data (such as particle size or bulk density) and
1444

output data (hydraulic parameters) is learned from a given set oftraining data (input-output response patterns).
While ANN-based PTFs have been relatively successful,there are a number of weaknesses that need to be considered intheir development and application. Key weaknesses include: (i)ANNs have a number of coefficients (weights) that do not permit
easy physical interpretation (Schaap et al., 2001), (ii) the ANNs’s
structure has to be selected a priori and therefore may not be optimal since there are many types of neurons and many types ofpossible connections (Wösten et al., 2001), (iii) a higher numberof neurons and connections than required can result in overfitting and over parameterization (Hastie et al., 2001) and (iv) due
to the complexity of the ANNs structure and the large number
of weights that are being “trained” as the network “learns”, there
is no assurance that the learning algorithm will find optimum
weights that minimize prediction errors. The procedure can getstuck in a local minimum, even though it can be overcome mostly. In lieu of the problems associated with ANN-based PTFs,
there is need for a better pattern recognition tool to improve the
PTFs accuracy and reliability. This has been duly recognized in areview by Wösten et al. (2001). Tamari et al. (1996) used radialbasis functions to develop PTFs. Radial basis functions are an
improvement over traditional ANNs but they suffer from manyof the same weaknesses as ANNs.
Recently, a number of new pattern recognition tools havebeen proposed that aim to improve on the weaknesses of ANNs.
For example, over-fitting can be mitigated by constraining the
ANN optimization using independent data (Schaap, 2004) orby using bayesian regularization (MacKay, 1992; Ye et al., 2007).
Support vector machines (SVMs) provide another promising
approach which, unlike ANNs where the complexity of thestructure is fixed a priori and only the prediction error can be
minimized, represent a pattern recognition approach where theoverall prediction error and complexity of the SVM structure are
simultaneously minimized (Vapnik, 1995; 1998).
The SVM regression methodology (Vapnik, 1995, 1998)
is based on the Statistical Learning Theory (SLT), which is aunique philosophy for addressing the problems and techniques ofpattern recognition. The SLT proved that a pattern recognition
method would generalize well (i.e., provide good performanceon independent data) when the structural complexity of the pattern recognition method and the prediction error in the training sets were simultaneously minimized (Vapnik, 1995, 1998).
Good performance on a training data set is a necessary but insufficient condition for a robust pattern recognition method.
Support vector machines have been used successfully for awide array of classification and regression problems. Recently,
a number of SVM applications have been introduced in hydrological sciences (Kanevski and Maignan, 2004; Tartakovsky and
Wohlberg, 2004; Wohlberg et al., 2006). Dibike et al. (2001) applied SVM to remotely sensed image-processing problems andreported a superior performance over the traditional ANNs.
Liong and Sivapragasam (2000) showed that SVM performance
was superior to ANNs in forecasting flood stage. Wohlberg etal. (2006) showed that SVM methodology tends to improve thepredictions in some case studies, especially in regression-based
problems such as kriging. Asefa et al. (2004, 2005) successfully
used the SVM methodology for a host of pattern recognition
problems in hydrology, ranging from ground water modeling, to
SSSAJ: Volume 73: Number 5 • September–October 2009

stream flow predictions. In this paper, we analyze the performance of thepattern recognition approach basedon the SVMs for generating PTFs.

MATERIALS AND
METHODS
Soil Hydraulic Parameters
In this study, we consider the retention curve and hydraulic conductivity function described by the van
Genuchten model (van Genuchten,1980). Of all available models for representing soil hydraulic properties, the van
Genuchten model is perhaps the most Fig. 1. Textural distribution of soil samples containing (a) retention data, and (b) saturated
widely used model for characterizing hy- hydraulic conductivity.
draulic conductivity and water content
dependence on the capillary pressure head. The van Genuchten model
where K(Se) is the unsaturated hydraulic conductivity (LT−1) as a funcis described as follows.
tion of the soil water saturation (or pressure head), Ks is the saturated
hydraulic conductivity (LT−1), and L is an empirical parameter, often
θ s -θ rìassumed to be 0.5 (Mualem, 1976). In this research, we develop PTFs
h< 0
n m
ïθr +
for the following soil hydraulic parameters: (a) retention curve param(1+ α h )[1a]θ(h) = íeters (θs, θr, α and n) and (b) saturated hydraulic conductivity Ks.
ïî θs

Se (h) =h³0

Data Setθ(h)- θ r
θ s -θ r

m =1-1/ n,

[1b]
n >1

[1c]

where θ(h) is the volumetric water content, (L3 L−3) as a function of thepressure head h (L), θs and θr are the saturated and residual volumetricwater contents, Se(h) is the effective soil water saturation (-) for the pressure head h (L), and α (L−1) and n (-) are van Genuchten shape parameters. A combination of Eq. [1] with the Mualem pore-size distribution
model (Mualem, 1976) yields the following expression for unsaturated
hydraulic conductivity (van Genuchten, 1980):
1éù
K (Se ) = K S ëê1-(1- Sem )m ûú

2

[2]

L
s e

For developing the PTFs, we used the same data set used by Schaapand Leij (1998) and Schaap et al. (2001) for calibrating the ROSETTA
model. Schaap and Leij (1998) assembled this set from a variety of available data sets. The assembled data set consists of data that are heterogeneous and diverse with respect to the measurement procedures, sources
and scales of measurements. Figure 1 shows the textural distribution of
the data set used in this study. Table 1 also lists the summary statistics
for the retention parameters and the saturated hydraulic conductivity
for different textural classes in the dataset. The dataset contained 2134
soil samples with water retention data. As each of these soil samples had
at least six retention curve points, the entire data set included 20,574
θ(h) points. Measurements for saturated hydraulic conductivity (Ks) wereavailable for 1306 of these soils samples.
The retention curve parameters were estimated for the 2134 soil
samples with the retention data using a curve-fitting approach. The water

Table 1. Average water retention parameters, saturated hydraulic conductivities, and bulk densities for each textural class in the database.
The values in parentheses represent the standard deviation (Schaap and Leij, 1998).
Class

Water retention

Saturated hydraulic conductivity

Db†g cm−3θr
cm3 cm−3θs
cm3 cm−3log(cm−1)1.53(0.12)
1.52(0.19)
1.37(0.25)
1.46(0.26)
1.28(0.27)
1.57(0.18)
1.32(0.18)
1.42(0.19)
1.33(0.09)
1.39(0.20)
1.59(0.10)

0.053(0.029)
0.049(0.042)
0.061(0.073)
0.039(0.054)
0.065(0.073)
0.063(0.078)
0.090(0.082)
0.079(0.076)
0.050(0.041)
0.098(0.107)
0.117(0.114)

0.375(0.055)
0.390(0.070)
0.399(0.098)
0.387(0.085)
0.439(0.093)
0.384(0.061)
0.482(0.086)
0.442(0.079)
0.489(0.078)
0.459(0.079)
0.385(0.046)−1.45(0.25)−1.46(0.47)−1.95(0.73)−1.57(0.56)−2.30(0.57)−1.68(0.71)−2.08(0.59)−1.80(0.69)−2.18(0.30)−1.82(0.68)−1.48(0.57)

Silty Clay29
1.36(0.15)†Bulk density.
‡Number of soils per textural class.

0.111(0.119)

0.481(0.080)−1.79(0.64)

Sand
Loamy Sand
Loam
Sandy Loam
Silt Loam
Sandy Clay Loam
Silty Clay Loam
Clay Loam
Silt
Clay
Sandy Clay

Na
308
205
249
481
332
181
89
150
6
92
12

SSSAJ: Volume 73: Number 5 • September–October 2009log(α)N‡

Db

log(Ks)0.50(0.18)
0.24(0.16)
0.17(0.13)
0.16(0.11)
0.22(0.14)
0.12(0.12)
0.18(0.13)
0.15(0.12)
0.22(0.13)
0.10(0.07)
0.08(0.06)

253
167
113
314
135
135
40
62
3
60
10

g cm−3
1.53(0.13)
1.53(0.19)
1.42(0.22)
1.55(0.18)
1.42(0.14)
1.59(0.18)
1.36(0.12)
1.44(0.23)
1.39(0.03)
1.40(0.23)
1.60(0.08)log(cm d−1)2.81(0.59)
2.02(0.64)
1.08(0.92)
1.58(0.66)
1.26(0.74)
1.12(0.85)
1.05(0.76)
0.91(1.09)
1.64(0.27)
1.17(0.92)
1.06(0.89)

0.12(0.10)14

1.33(0.16)

0.98(0.57)log(n)

1445

retention data for each soil sample were fitted using an algorithm utilized by
Schaap and Leij (1998) to estimate the following soil hydraulic parameters(θs, θr, α and n). Fitting was performed with the simplex or amoeba algorithm using the following constraints: 0.0 ≤ θr ≤ 0.3 cm3 cm−3, 0.6φ ≤ θs ≤ φ
cm3 cm−3 (where φ is the total porosity), 0.0001 ≤ α ≤ 1.000 cm−1 and1.001 ≤ n ≤ 10. Measured saturated hydraulic conductivity was directly
used as it was available for 1306 of the soil samples. The soil hydraulicparameters for each soil sample (that is, fitted retention curve parameters and the measured saturated hydraulic conductivity, where available)
were then juxtaposed to other soil properties (such as textural information, bulk density, and water contents at particular pressure heads) tocreate the input data for the training and testing of PTFs. For improving
the statistical robustness of the PTFs, logarithmically transformed values of the measured saturated hydraulic conductivities [log(Ks)] and thefitted retention curve shape parameters [log(α) and log(n)] were used
(Schaap et al., 2001).
It is important to note that some textural classes (such as silt) do
not have a sufficient number of samples in the data set, and thus calibrated PTFs for these textural classes are likely to be highly uncertain. It
can be expected that the uneven distribution of soil samples in different
textural classes will impact the reliability of the PTF predictions.

Theory of Support Vector Regression
The goal of a pattern recognition method is to estimate an unknown real-value variable using the mathematical functional relationship as follows:y = β( x )+ δ

[3]where δ is an independent and identically distributed zero mean random
error (noise), x is a multivariate input of m-dimensions, y is a scalar output and β(x) is a function, likely nonlinear. The estimation of the β(x)function is based on a finite number (nt) of samples, the so-called training data set: (xi , yi ), (i = 1,..., nt ) .
The SVM regression is based on the generalized regression formulation. In the SVM regression, the input x is first mapped onto an
m-dimensional space using some fixed (nonlinear) mapping, and thena linear regression model relating the input-output data is constructed.
Mathematically, the SVM regression is formulated as:mf ( x , w ) = å w j g j ( x )+ b

[4]

j =1

where g j(x), j = 1..m denotes a set of nonlinear transformations, wj areassociated weights and b is the bias term.
Similar to traditional regression methods, an SVM regression
model is developed by minimizing the error in predictions during thetraining stage. The quality of each prediction for a noisy data set is measured by a novel error function known as the ε-insensitive loss function(Vapnik, 1995; 1998):ìL( y, f ( x , w)) = íî0
if y - f ( x , w) £ ε
y - f ( x , w) - ε otherwise[5]

The value of ε in the loss function dictates the sensitivity of the transformation to noise in the data. A value of ε equal to zero essentially convertsthe ε-insensitive loss function into a least-modulus loss function and Eq.
[5] into the mean absolute error. It has been shown in many studies thata non-zero ε results in a good SVM regression (Vapnik, 1995; 1998).
Figure 2 shows a schematic of the ε-insensitive loss function. Once theresiduals are estimated using the ε-insensitive loss function, the cumulative training error can be described as:

R(w) =1 nt
å L( yi , f ( x i ,w))
nt i=1[6]

The SVM regression performs a linear regression using ε-insensitive
loss function while trying to reduce the SVM’s structural complexity
2
by minimizing w (Vapnik, 1995, 1998). This is achieved by introducing the non-negative slack variables ξi and ξi*, i = 1..nt to measure
the deviation of training samples outside the ε-insensitive zone (Fig.
22). Minimization of w influences the structural complexity of the
SVM regression model because it influences the final estimate of hyperparameters which govern the structure of SVM model. Thus, the SVM
regression is formulated as a minimization of the following function:nt
1| w | 2 +C å (x i + x i* )
2
i =1

[7]

Subject toì y i - f (x i , w) - b £ e + x i*ï
í f (x i , w) + b - yi £ e + x i
ï x , x * ³ 0, i = 1,..., nt
î i iThis optimization problem can be transformed into a dual problem (Vapnik,
1995), and its solution is given as:
nt

f (x) = å (a i - a i* ) K (x i , x) + b
i =1

such that0 £ a i* £ C , 0 £ a i £ C[8]where nt is the number of points in the training data set and K (x i , x) is a kernel function.
A number of kernels are available(Vapnik, 1995, 1998). The kernel funcFig. 2. Illustration of the SVM regression with the •-insensitive loss function. All the errors contributed tion is a symmetrical function that satisfies
by points within the •-tube are not considered during the SVM regression.
the so-called Mercer’s conditions (Vapnik,
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SSSAJ: Volume 73: Number 5 • September–October 2009

1995, 1998). The performance of SVMs depends at least as strongly on
the choice of a kernel as on a kernel’s parameterization (Wohlberg et al.,2006). In fact, the lack of a principled way to identify the optimal kernel isconsidered a main weakness of SVMs. In this study, we consider one of themost commonly used kernels, namely the Radial Basis kernel. The radialbasis function is mathematically expressed in Eq. [9].

æ x - xi 2 öK ( x i , x ) = exp ç ÷ç
2 γ 2 ÷ø
è

[9]where γ is the kernel parameter called as the kernel width. One may
define the radial basis kernel by specifying the kernel width γ a priori.
Readers are referred to Vapnik, (1995, 1998) for a detailed description.
During the training of SVM regression models, only certain data
points in the training data set have non-zero coefficients in Eq. [8]. These
points are also called as support vectors. The number of support vectors(nsv) is a good indicator of how well the SVM model may perform on an
unseen dataset. A larger percentage of support vectors will lead to overfitting of the training data set and poor predictions in the testing dataset, and a smaller percentage will lead to under-fitting. For optimallyfitting SVM using a training data set, the percentage of support vectors
should be around 50% (Hastie et al., 2001).
Support vector machine-based regression has been found to show
an improved performance for some cases in the past (Wohlberg et al.,2006). The performance of the SVM regression depends on a good selection of the following so-called hyper parameters: cost (C), insensitivity value (ε) and the radial basis kernel width (γ). The cost parameter C
determines the tradeoff between the complexity of the SVM structure
and the prediction error of the training set. For example, if the cost C
is set too large, the resulting SVM regression would give minimal importance to the necessity of minimizing the SVM structure complexity.

This could result in over-fitting of the training data and poor generalization. The insensitivity parameter, ε, controls the width of the insensitive zone. Larger values of ε lead to smaller numbers of support vectors,
which can result in poor generalization.
Figure 3 shows a schematic of a trained SVM structure, and thesteps involved in estimating the final output value, f(xt,w), given an input
vector xt. The input vector xt and the support vectors Xi (i = 1,.., n SV )are nonlinearly transformed using the pre-defined kernel function andthen linearly combined using weights wi and the bias b obtained during
training to estimate the final output value.
In this research, the SVM regression was performed in two stages:(i) training and (ii) testing. Before training, the training data was standardized for zero mean and unit variance. The standardized training
data was used to develop the SVM regression models. The mean andstandard deviation values used for standardizing, was used in the laterstages for de-standardizing the predictions. The training stage aims at
finding the optimal estimates of cost, C, insensitivity value, ε and the radialbasis kernel width, γ, to achieve the best generalization. During the testingstage, the ability of the trained SVM to predict final values is evaluated.
In the past, optimal estimations of the SVM hyper-parameters
were performed using the following three procedures: (i) based on apriori knowledge and user expertise, (ii) using a thorough grid-based
search approach, and (iii) using an analytical estimation based on thestatistical properties of the training data. In this study, we have opted
to use the grid-based search approach. The objective of the grid-searchmethod is to obtain the optimal hyper-parameters by estimating theerror in the predictions for a training data set for every possible combination of the hyper-parameters within a feasible hyper-parameter
space. The hyper-parameter which results in a minimum training error
was chosen as optimal. To estimate the training error, we used a five-foldcross-validation approach. Readers are referred to Hastie et al. (2001)

Fig. 3. Flowchart showing the steps involved during estimation of output when input data is fed to a trained SVM regression model. Note that theSteps (2), (3) and (4) are performed at the same instance when the appropriate kernel such as the radial basis kernel is used.
SSSAJ: Volume 73: Number 5 • September–October 20091447

for a detailed explanation of the five-fold cross-validation approach. The
optimal hyper-parameters for the SVMs were estimated by searching
within the feasible parameter space. The feasible parameter space foreach of the hyper-parameters was constructed using a set of minimum
and maximum possible values that were assumed to be the following apriori (0.0001 < C < 1000, 0.0001 < γ < 100, 0 < ε < 1). We assumed
the aforementioned range of hyper-parameters to be appropriate basedon number of previous studies (e.g., Asefa et al., 2004; 2005). A meshincrement of 0.01 was selected for the grid search to ensure optimality ofhyper-parameters and the computational efficiency.

Performance Criteria
At different stages of the PTF development, it is required to quantify the amount by which an estimated value differs from the ‘true’ valueof the quantity being estimated. Such quantification describes how wellthe estimator describes the ‘true’ values. In this research, such differences
between estimated values and the ‘true’ values are quantified using thefollowing performance criterion: (i) root mean square error (RMSE),(ii) mean error (ME), and (iii) coefficient of determination (R2).

RMSE =1 N
å(ζ i -ζ'i )2
N i=1

[10]

ME =

1 N
å(ζ i -ζ'i )
N i=1

[11]

N

å(ζ -ζ )R =1- iN=1
2

i

' 2
i

å(ζ -ζ )
i

[12]

2

i

i=1

where N is the number of values, ζ i (i = 1..N) is the ‘true’ value of thequantity, z is the mean of the ‘true’ values and ζ’i (i = 1..N) are the corresponding estimated values.

Support Vector Machine Methodology
We developed new PTFs based on the SVM methodology for thesaturated hydraulic conductivity [log(Ks)] and the four parameters of
the water retention function [θs, θr, log(α), and log(n)]. The R statistical language was used for developing the SVM-based PTFs. Four SVMbased PTFs were developed for each set of soil hydraulic parameters byvarying the input data predictors. The first set of PTFs (called S1) used
sand, silt, and clay percentages as input predictors. The second set of
PTFs (called S2) used bulk density, and sand, silt, and clay percentages
as predictors, while the input data of the third set of PTFs (called S3)
also included the water content at the pressure head of −330 cm. The
last model (called S4) included the water content at the pressure headof −15,000 cm in addition to those predictors used for the third set of
PTFs (S3). Pedotransfer functions for retention parameters were developed by training with the corresponding parameter data that were estimated by fitting the van Genuchten retention model to retention dataof the soils (2134 soil samples). On the other hand, PTF for saturated
hydraulic conductivity was trained using the measured saturated hydraulic conductivity values (1306 soil samples). Therefore, performance
analysis of the PTFs using RMSE, R2 and ME (Equations 10–12) was
done by comparing the predicted values with fitted estimates [in case ofθs, θr, log(α), and log(n)] and measured values [in case of log(Ks)].
Pedotransfer functions were developed by combining the SVM
approach with the bootstrap method (Efron and Tibshirani, 1993). In
1448

the bootstrap method, a single dataset of size ‘N’ is randomly resampledwith replacement to create ‘B’, statistically similar subsets also of size‘N’, while each of the ‘B’ subsets contain about 63% of the parent dataset (Schaap et al., 2001). Using each of these ‘B’ subsets (also called astraining data set), PTFs were first trained. The trained PTFs were thentested using the 37% of the parent dataset that was not represented in
the subset. Incorporating the bootstrapping method into the generationof PTFs has several advantages: (i) it allows for estimating uncertainty
in the PTF prediction, and (ii) it creates complementary testing andtraining data sets. A ‘B’ value of 60 was used in our calibration, following Efron and Tibshirani (1993) who suggested a B value between 50
and 100 for a robust model. Each of the SVM-based PTFs comprised of
60 SVMs that were calibrated using the bootstrapped subsets of the original
dataset. Predictions of the soil hydraulic parameters from 60 SVMs werefirst averaged to report the mean parameter estimates and their standard deviations. The mean value of the soil parameters from the 60 bootstraps werereported as the predictions from the PTFs.
For each of the PTFs used to predict the water retention parameters, we also compared the predicted values in the training and testingdata sets with the corresponding fitted values for all of the 60 bootstraps
using the RMSE criterion. In case of the PTF developed to predict thesaturated hydraulic conductivity, we compared predicted values in the
training and testing data sets with the corresponding measured estimates
for all of the 60 bootstraps using the RMSE criterion. The mean of the
RMSE in the training and testing data sets of the 60 bootstraps for each
PTF was reported as the training and testing error.
It is of interest to analyze how the errors in the predicted soil
hydraulic parameters translated in the accuracy of the water contents
that may be estimated using these predicted parameters (Eq. [1a–1c]).
For this purpose, we calculated RMSE and ME values between watercontents estimated using parameters predicted by PTFs (S1-S4) andthe measured water contents [20,574 θ(h) points]. To understand how
these RMSE and ME values varied for water contents (θ(h)) as a function of pressure head (h), we also calculated these values for water contents grouped using 10 suction classes between 0, 3.2, 10, 32, 100, 320,
1000, 3200, 10 000, 32000, and 10000 cm. It is important to note thatwater contents estimated by using parameters predicted by PTFs cannot
be more accurate than those estimated by using fitted parameters that wereused to develop the PTFs. The RMSE and ME were also estimated between
the water contents estimated using the fitted parameters and the measured
water contents for comparison purposes.

RESULTS AND DISCUSSION
Development of Support Vector Machine-based
Pedotransfer Functions
The four SVM-based PTFs (S1, S2, S3, and S4) were developed using the procedure described above. Table 2 lists thesummary statistics for the hyper-parameters of all PTFs, alongwith the percentage of training data set that represents support
vectors. One may view support vectors as those records in the
training data set providing the maximum value to the SVM
method. It may be seen from Table 2 that all developed PTFs
have an optimal number of support vectors around 50% of the
training dataset which is also the recommended value (Hastie etal., 2001). The optimal percentage of support vectors was also
reflected in the performance of SVMs on the training and testingdata sets.

SSSAJ: Volume 73: Number 5 • September–October 2009

Table 2. Summary statistics of estimates of hyperparameters and percentage of support vectors (SV) for the SV machines (SVM)-based
Pedotransfer functions (PTFs) that predict water retention parameters and saturated hydraulic conductivity using different sets of input data.
The estimated means and standard deviations (in parentheses) of all bootstraps in a PTF are reported. SSC– sand, silt, and clay percentages,
Db– bulk density, θ330 and θ15000– water contents at pressure heads of −330 and −15,000 cm, respectively.
Parameterθrθslog10(α)
log10(n)log10(Ks)θrθslog10(α)
log10(n)log10(Ks)θrθslog10(α)
log10(n)log10(Ks)θrθslog10(α)
log10(n)log10(Ks)

PTF Model

Input data

S1

SSC

S2

SSS+DbS3

SSC+ Db + θ330

S4

SSC+ Db + θ330 +θ15000Cost, C
9.2 (3.35)
23 (2.1)
9.4 (2.97)
22.6 (1.67)
16.6 (2.44)
6.2 (3.35)
4.4 (2.1)
4.2 (2.97)
13.6 (1.67)
16.6 (2.44)
16.6 (3.35)
9.4 (2.79)
20 (3.56)
8.8 (1.51)
9 (1.34)
13.3 (3.29)
8 (4.83)
9 (2.26)
20 (3.9)
6.4 (1.41)

As described in the methodology section, we estimated the
RMSE of the predictions in the training and testing data sets for
all the PTFs (Table 3). Table 3 shows that the performance of the
PTFs for the training and testing data is very close, suggesting anear optimal training of the PTFs
To enable direct comparisons with Schaap et al. (2001), we
used the developed PTFs to predict soil hydraulic parameters forthe entire data set. Comparisons were made using RMSE andR2. Table 4 lists the RMSE and R2 values of the soil hydraulicpredictions by the SVM-based PTFs for the entire dataset. One
obvious observation is that R2 values between fitted and PTFpredicted soil hydraulic parameters increase and RMSE decreases as more input predictors are used in the PTFs (from S1 to S4).
This observation suggests that all predictors in the input data set
have some information that is useful for estimating the hydraulicparameters. In all of the PTFs, correlations between predictionsof θr and corresponding fitted estimates is the weakest among thepredicted parameters. Bulk density seems to considerably improve predictions of θs (compare S2 and S1). On the other hand,the accuracy of predictions also seems to increase significantly
Table 3. Root mean squared error (RMSE) of the water retention
parameters and saturated hydraulic conductivities predicted by different support vector machine (SVM)-based pedotransfer functions(PTFs) for the training and testing data sets.
Parameterθrθslog(α)
log(n)
log(Ks)Data
Training
Testing
Training
Testing
Training
Testing
Training
Testing
Training
Testing

S1
0.063
0.072
0.075
0.081
0.558
0.567
0.132
0.132
0.714
0.72

S2
0.061
0.07
0.052
0.055
0.531
0.537
0.127
0.13
0.662
0.662

S3
0.052
0.064
0.049
0.052
0.47
0.476
0.11
0.113
0.561
0.57

S4
0.039
0.045
0.047
0.05
0.342
0.348
0.08
0.083
0.55
0.556

SSSAJ: Volume 73: Number 5 • September–October 2009

Kernel width γ

Insensitivity ε

Number of SV(-)
0.28 (0.1)
1.01 (0.05)
0.45 (0.35)
0.27 (0.09)
0.82 (0.32)
0.4 (0.1)
0.27 (0.05)
0.67 (0.35)
0.26 (0.09)
0.16 (0.32)
0.16 (0.18)
0.16 (0.3)
0.16 (0.45)
0.28 (0.12)
0.28 (0.11)
0.17 (0.11)
0.22 (0.13)
0.34 (0.07)
0.27 (0.3)
0.21 (0.15)(-)
0.48 (0.14)
0.62 (0.2)
0.68 (0.3)
0.5 (0.09)
0.41 (0.25)
0.7 (0.14)
0.52 (0.2)
0.74 (0.3)
0.5 (0.09)
0.51 (0.25)
0.51 (0.15)
0.41 (0.11)
0.4 (0.2)
0.4 (0.07)
0.43 (0.22)
0.51 (0.22)
0.42 (0.08)
0.29 (0.11)
0.19 (0.09)
0.37 (0.2)%
57.55 (8.76)
42.53 (16.05)
38.74 (21.61)
41.61 (8.76)
57.17 (23.03)
41.3 (8.76)
34.96 (16.05)
33.02 (21.61)
40.35 (8.76)
43.83 (23.03)
43.83 (9.5)
42.95 (8.5)
48.86 (10.92)
43.31 (4.53)
43.2 (19.12)
43.53 (19.12)
40.41 (9.55)
56.86 (11.05)
58.41 (7.76)
45.13 (21.05)(especially for θr, log (α) and log(n)) when retention curve data
are used as additional predictors (compare S2 to S3 and S4).
We also compared the R2 values for the predicted soil hydraulic parameters with the corresponding values for the ANNbased ROSETTA which is given in Schaap et al. (2001). It wasnoted that the RMSE and R2 values suggest a significant improvement in predictions obtained by all the SVM-based PTFs
when compared with the ANN-based ROSETTA. Comparedwith ROSETTA, SVM-based PTFs, using SSC as the input,
shows the most improvement. Also, the extent of improvementin the soil hydraulic parameters predicted by SVM seems to belesser and lesser with the number of inputs used in the PTFs.
Table 5 shows RMSE between the observed water contents
and those estimated by using parameters predicted by ROSETTA
and SVM-based PTFs. The RMSE estimates in Table 4 also indicate SVM-based PTFs are more accurate in predicting watercontents. One may attribute this improvement in predictions
due to one or both of the following reasons: (i) the improvementof SVM over ANNs or (ii) the superior structure of SVM-based
Table 4. Root mean squared error (RMSE) and R2 estimates for thewater retention parameters and saturated hydraulic conductivities
predicted by different support vector machine (SVM)-based pedotransfer functions (PTFs) for the entire dataset.
Parameterθrθslog(α)
log(n)
log(Ks)Performancemeasure
R2
RMSE
R2
RMSE
R2
RMSE
R2
RMSE
R2
RMSE

S1

S2

S3

S4

0.282
0.066
0.44
0.077
0.497
0.561
0.686
0.132
0.681
0.716

0.38
0.064
0.787
0.053
0.567
0.533
0.705
0.128
0.737
0.662

0.48
0.056
0.816
0.05
0.684
0.472
0.791
0.111
0.817
0.564

0.79
0.041
0.831
0.048
0.848
0.344
0.897
0.081
0.826
0.552

1449

PTFs over ROSETTA. The ANN-based ROSETTA predicts
all four VG parameters using ONE single model (i.e., the ANN
has four outputs corresponding to different parameters). In thecase of the SVM-based PTFs developed here, there are four different models, each predicting one output (multi-model). It is
our opinion that the multi-model approach provides SVM-based
PTFs more flexibility, and could be one of the reasons why the
SVM-based PTFs have lower prediction errors. To check if theobserved improvement was due to multi-model nature of SVMbased PTFs or due to the superior structure of SVM comparedwith ANN, we need to do further research comparing singlemodel ANN, multi-model ANNs, single-model SVM and multimodel SVMs. This is a topic for future research.

Uncertainty
Estimations of uncertainty (such as standard deviation) in
predictions by PTFs can be very useful, especially for further
statistical and modeling studies. The user may view the uncertainty estimate as a measure of PTF’s reliability (e.g., Schaap,2004). Since we used bootstrapping for the SVM-based PTFs,
we can easily estimate standard deviations for the predicted soil
hydraulic parameters. Figure 4 shows the distribution of mean(μ) soil hydraulic parameters predicted by the S2 PTFs and theassociated uncertainty (σ, standard deviation) as a function ofsoil texture for a specified bulk density of 1.3 g cm−3. Here, we
use the standard deviation to represent the uncertainty under theassumption that the errors are distributed normally, which may
not necessarily be true. As already observed for ROSETTA by
Schaap et al. (2001), the textural dependence of predicted parameter values concur with established knowledge. For example,
θr increases for finer textures, while log(Ks) increases for coarser
textures. Standard deviations for the predicted values of soil
hydraulic parameters show similarity. They are especially highfor textures that are less well represented in the input data set(compare Fig. 1 with Fig. 4). This was consistently observed forpredictions by all PTFs (S1-S4), indicating that gathering more
information for textures that are less represented in the training
dataset (such as silt) is a necessity.

Water Contents and Saturated
Hydraulic Conductivity
The soil hydraulic parameters predicted using different
SVM-based PTFs were used to estimate water contents at different pressure heads with the van Genuchten model. These preTable 5. Root mean square error (RMSE) estimates between themeasured and corresponding predicted water contents and saturated hydraulic conductivities by support vector machine (SVM)based pedotransfer functions (PTFs) and ROSETTA.
PTF
Model
S1
S2
S3
S4

Input data used†

PTF

SVM-based
ROSETTASVM-based
SSC+DbROSETTASVM-based
SSC+ Db + θ33
ROSETTA
SSC+ Db + θ33+θ15000 SVM-based
ROSETTA
SSC

Direct fit†SSC, sand silt and clay; Db, bulk density.
1450

θ(h)

log(Ks)

0.062
0.076
0.053
0.068
0.038
0.047
0.034
0.044
0.012

0.716
0.717
0.662
0.666
0.564
0.586
0.552
0.581
NA

dicted water contents were then compared with the corresponding measured values using RMSE. Table 5 summarizes RMSE ofwater contents and saturated hydraulic conductivities predictedusing ROSETTA and the SVM-based PTFs for different input
predictors. For saturated hydraulic conductivities, RMSE of
predictions by ROSETTA and SVM are comparable for lowerorder PTFs which use only textural information and bulk density. However, when retention data points are also included into
the PTFs training (as in S3 and S4), the SVM-based PTFs show
considerable improvement. Compared with ROSETTA, RMSE
for water contents predicted by the SVM-based PTFs decreased by
10 to 25% for different PTFs. Compared with other PTFs, Table
5 indicates that the hydraulic conductivities predictions by SVMbased PTFs show only a marginal improvement over ROSETTA.
Figure 5 shows RMSE between water contents estimatedusing parameters predicted by PTFs (S1-S4) and the measured
water contents as a function of the pressure head. A similar analysis was performed by Schaap et al. (2001). The figure also shows
the RMSE for the water contents estimated by using the fitted
parameters. Naturally, one may expect that the water contents
predicted by the PTFs cannot be more accurate than those estimated by using the fitted parameters. Compared with the results of ROSETTA (Schaap et al., 2001), the SVM-based PTFs
show improvements for different pressure head intervals. Outof all the PTFs, only S4 performs better at lower pressure heads(<−32,000 cm), while all other PTFs seem to have poor performance at these pressure heads. It is important to note that we
had only 28 points below the pressure head of −32,000 cm. Thus,
the poor performance of PTFs at lower pressure heads could bea statistical problem.
Figure 6 depicts the ME for water contents estimated using
the parameters predicted by the SVM-based PTFs as a functionof pressure head. The ME for the water contents estimated using
the RETC-fitted parameters is also shown. Clearly, the RETCfitted parameters show minimal ME indicating minimal bias in
the estimated water contents. The water contents estimated using
the parameters predicted by the SVM-based PTFs show relatively
more bias. The water contents are overestimated near saturation(especially when log(h) < 0.5 cm) and underestimated at lesser
saturations. It was observed that this trend is very similar to ME
of the water contents estimated by using ROSETTA (Schaap etal., 2001). However, the absolute ME value of the water contents
estimated using parameters predicted by SVM-based PTFs indicate lesser bias than those estimated using parameters estimatedby ROSETTA. Similarly in Fig. 6, the ME of the estimated watercontents seems to be reduced as more predictors are used in thecorresponding PTF.

SUMMARY AND CONCLUSIONSThe SVM methodology was successfully applied to develop
PTFs that used different input predictors to estimate soil hydraulic parameters. These PTFs utilize some or all of the following
predictors: sand, silt, and clay percentages, bulk density and retention data at one or two points (at the field capacity and thewilting point). Bootstrapping was performed to allow for theestimation of uncertainty in the predictions. It was observed thatan increase in the number of predictors resulted in improved predictions by PTFs. It was also observed that the predictions by the SVMbased PTFs showed considerable improvement over ROSETTA.
SSSAJ: Volume 73: Number 5 • September–October 2009

Fig. 4. Distribution of estimated parameter values (a, c, e, g, and i)and the associated uncertainties (b, d, f, h, and j) for the SVM based
PTF S2 that uses sand, silt and clay percentages and the bulk densityof 1.3 g cm-3.

However, we note that large uncertainties in PTFs’ estimatesremain due to the lack of data for some textural classes (such as silt)
in the training. This limitation underscores the need for gathering
more information, especially for fine-textured classes (see Table 1).
One may also expect that the performance of PTFs would improve
if additional information, such as the organic matter content, soil
structure and chemical properties, was available. However, as noted by Schaap et al. (2001), care should be taken to avoid predictors that are difficult to measure or are not commonly measured
because this may put the very idea of PTFs in jeopardy.
SSSAJ: Volume 73: Number 5 • September–October 2009

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