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Water Resources Research
RESEARCH ARTICLE
10.1002/2016WR019533
Key Points:
 The main wetting branch of the soil
water retention curve is estimatedbased on its main drying branch The machine learning method isused for analysis
 Results are compared with classicalmethods of estimating the main
wetting branch
Supporting Information:
 Supporting Information S1
 Data Set S1Correspondence to:K. Lamorski,
k.lamorski@ipan.lublin.pl

Citation:
Lamorski, K., J.r Simůnek, C. Sławinski,
and J. Lamorska (2017), An estimationof the main wetting branch of the soil
water retention curve based on its
main drying branch using the machinelearning method, Water Resour. Res.,53, doi:10.1002/2016WR019533.
Received 19 JUL 2016
Accepted 28 JAN 2017
Accepted article online 3 FEB 2017

An estimation of the main wetting branch of the soil water
retention curve based on its main drying branch using themachine learning method
2, Cezary Sławin
 ski1, and Joanna Lamorska3
Krzysztof Lamorski1, Jirı Simůnek

1Institute of Agrophysics, Polish Academy of Sciences, Lublin, Poland, 2Department of Environmental Sciences, Universityof California Riverside, Riverside, California, USA, 3Institute of Agricultural Sciences, State School of Higher Education inChełm, Chełm, Poland

Abstract In this paper, we estimated using the machine learning methodology the main wetting branchof the soil water retention curve based on the knowledge of the main drying branch and other, optional,
basic soil characteristics (particle size distribution, bulk density, organic matter content, or soil speciﬁc
surface). The support vector machine algorithm was used for the models’ development. The data neededby this algorithm for model training and validation consisted of 104 different undisturbed soil core samples
collected from the topsoil layer (A horizon) of different soil proﬁles in Poland. The main wetting and drying
branches of SWRC, as well as other basic soil physical characteristics, were determined for all soil samples.
Models relying on different sets of input parameters were developed and validated. The analysis showedthat taking into account other input parameters (i.e., particle size distribution, bulk density, organic matter
content, or soil speciﬁc surface) than information about the drying branch of the SWRC has essentially no
impact on the models’ estimations. Developed models are validated and compared with well-known
models that can be used for the same purpose, such as the Mualem (1977) (M77) and Kool and Parker(1987) (KP87) models. The developed models estimate the main wetting SWRC branch with estimation
errors (RMSE 5 0.018 m3/m3) that are signiﬁcantly lower than those for the M77 (RMSE 5 0.025 m3/m3) or
KP87 (RMSE 5 0. 047 m3/m3) models.

1. Introduction
Soil water retention curves (SWRCs) are one of the most important soil hydrological characteristics requiredfor both agricultural and environmental research related to the vadose zone. SWRCs link the soil water content with the soil water potential and represent indispensable information for the modeling of soil waterﬂow processes. Although laboratory measurements are the ultimate source of information about retention
curves, for many reasons SWRCs are commonly estimated using various statistical models, such as the socalled pedotransfer functions (PTF) [e.g., Vereecken et al., 1989, 2016; Schaap et al., 2001]. The main reason
for using PTF estimations of SWRCs instead of direct measurements is their long duration and high cost.
Typical steady state equilibrium measurements of the retention curve for a full range of soil water potentials
can last several months.
PTFs estimate SWRCs based on various physical and chemical soil characteristics. For example, particle size distribution and dry bulk density are commonly used predictor variables in PTFs. Additional soil variables such asorganic carbon content, soil-speciﬁc surface area, and/or cation exchange capacity can also be used. There
were numerous PTF models developed utilizing different statistical and/or soft computing methods for SWRC
estimation. Early PTFs were often developed using statistical regression [Rawls et al., 1982; Vereecken et al.,€sten et al., 1999; Walczak et al., 2006], leading to some still often used models. Various soft computing
1989; Wo
methods of statistical inference such as artiﬁcial neural networks (ANN) [Schaap et al., 2001; Jana et al., 2008],
th et al.,the k-nearest neighbors algorithm (k-NN) [Nemes et al., 2006; Botula et al., 2013], regression trees [To2012, 2015], or support vector machines (SVM) [Lamorski et al., 2008] were used later on.
C 2017. American Geophysical Union.
V

All Rights Reserved.

LAMORSKI ET AL.

Soil water retention curves of many soil materials exhibit hysteretic behavior, which means that the dependence between the soil water potential (h) and the soil water content (h) is not unique and depends on the

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history of soil water content changes. As a result, soil can exhibit an interval of soil water contents for a single soil water potential. Under natural conditions, available states in the h-h space are limited by the socalled main drying and main wetting curves. The main wetting and drying curves are obtained when the
SWRC is measured starting at zero or full saturations, respectively. Since most laboratory methods for SWRC
measurements determine the main drying branch, existing PTFs usually produce values for the main drying
branch as well. However, for many modeling applications, it is advisable to consider hysteresis, and thus
necessary to also know the main wetting branch of the SWRC. Only when both wetting and drying branchesof the SWRC are known can numerical models consider soil hysteresis and determine actual values of soil
water content for given values of soil water potential, taking into account historical values of soil water
potential.
Hysteresis in the SWRC affects many vadose zone ﬂow processes, often with practical implications. For
example, the inﬂuence of hysteresis on runoff predictions in catchments was demonstrated by Mirus [2015].
Other studies showed that hysteresis inﬂuences slope stability, and neglecting it can lead to an underestimation of landslide conditions [Ebel et al., 2010; Bordoni et al., 2015]. Consideration of SWRC hysteresis in and Mls, 2010; Ma et al.,vadose zone transport modeling has an inﬂuence on soil water ﬂow [Trpkosova
2011] and solute transport [Vereecken et al., 1995] modeling results, especially in the upper soil layer and in
the rhizosphere [Elmaloglou and Diamantopoulos, 2009; Carminati et al., 2010]. On the other hand,the effects of soil heterogeneity may overshadow the effects of hysteresis on soil water ﬂow processes[Vereecken et al., 1995].
Physical phenomena which result in SWRC hysteresis [Bachmann and van der Ploeg, 2002; Albers, 2014]
include (a) an ink-bottle effect due to irregular shapes of pores, (b) hysteresis of the interface contact angle,
i.e., the difference between ascending and descending contact angles, (c) a rain drop effect, i.e., air bubble
entrapment or Haines jumps, and (d) temporal nonequilibrium due to fast pore ﬁlling events. The research
into SWRC hysteresis is still under active development, and different modeling approaches have beendeveloped to understand or predict the SWRC hysteretic behavior. Two main modeling methodologies canbe distinguished: physically based models and phenomenological models.
Physically based models take into account soil water interactions. These models focus on real physical phenomena that occur at the level of soil pores and that together with soil structure inﬂuence unsaturated soilhydrological properties [Mahmoodlu et al., 2016]. Real representations of pore spaces obtained from computed microtomography scanning (mCT) [Ahrenholz et al., 2008] or equivalent pore network models [JoekarNiasar et al., 2013; Arroyo et al., 2015; Rostami et al., 2015] and individual phenomena leading to the hysteresis effect such as contact angle hysteresis [Zhou, 2013] or soil-water interactions and pore morphology[Chan and Govindaraju, 2011; Gan et al., 2013] are used and analyzed in these models. While these models
provide insight into various physical factors leading to hysteresis, their practical applications are very limited
since they are very complicated and often dependent on hard-to-get input data.
Phenomenological models are based, on the other hand, on simpliﬁed assumptions regarding the soil
medium and/or on the general Everett’s domain hysteresis theory. The ﬁrst attempt to apply the domain
theory to soil moisture hysteresis was the independent domain model of Poulovassilis [1962]. The similarity
hypothesis for the distribution of geometrical menisci curvatures for drying/wetting relationships was introduced by Philip [1964], which allowed for the reduction of information needed to describe the hysteretic
process to only the main wetting and drying branches. The similarity concept was further developed by
Mualem [1973, 1984], who stated that the domain model for the water distribution function can be
expressed as a simple product of two functions, one solely dependent on wetting and the other only on
drying soil water potential. Another approach, which is based on a scaling hypothesis that assumes a simple
dependence between main and scanning curves, was proposed by Scott et al. [1983]. This approach isequivalent to the Mualem model based on the similarity hypothesis. The Scott et al. [1983] model was thenadopted for SWRCs described using the van Genuchten [1980] analytical function [Kool and Parker, 1987;
Parker and Lenhard, 1987]. The van Genuchten [1980] model has been recently extended to take into
account deformation of soil pores due to swelling effects [Gallipoli, 2012]. Another approach to describing
the main and scanning curves is to deﬁne them indirectly, such as using different tangent inclinations
dependent on suction [Pedroso and Williams, 2010]. The scanning curves can then be determined by integrating a differential equation introduced to describe the shapes of the curves. A comprehensive review ofthese types of models of hysteresis was recently provided by Pham et al. [2005] and Albers [2014].

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Phenomenological models are still being further improved by taking into account additional multiple
effects or phenomena [Rudiyanto et al., 2013, 2015]. Phenomenological models are very important for thesoil water modeling practice because they allow for simple and easy-to-incorporate approaches for describing the hysteretic behavior of soils based on standard soil water retention information.
Most hysteretic retention models are based on information about the main drying and wetting branches of
SWRCs. However, laboratory measurements of the main wetting branch of the SWRC are rarely made during
soil surveys, and usually, only the main drying branch of SWRC is measured. It is thus necessary to somehow
estimate the main wetting branch when the soil hysteretic behavior is to be considered; some classicalmodels of the soil water hysteresis allow for that. For example, the universal domain model estimates hysteretic SWRCs, including the main wetting branch, based only on information about the main drying branch[Mualem, 1977]. Kool and Parker [1987] suggested an approximate method for the estimation of the main
wetting branch of the SWRC parameterized using the van Genuchten [1980] function based on parametersof the main drying branch. Other models need additional information regarding the main wetting branch,such as the water entry value [Hogarth et al., 1988] or two points measured on the wetting branch [Phamet al., 2003]. Despite the fact that these models give a more accurate estimation [Pham et al., 2005] of thewetting branch of a SWRC, their practical applications are cumbersome due to prohibitive requirements on
additional input data characterizing the main wetting branch of a SWRC.
The aim of this work is to develop a statistical model that will allow for the estimation of the main wetting
branch of a SWRC based on knowledge of the main drying branch of the SWRC, while also providing theoption to take additional information about soil physical characteristics into account. The technique of support vector machines will be used to develop this model.

2. Materials and Methods
2.1. Soil Material
Measured soil retention data are needed for the development and validation of SWRC estimating models.
In this study, 104 different undisturbed soil core samples collected from the top soil layer (A horizon) of different soil proﬁles in Poland were analyzed (Figure 1). Standard 100 cm3 (a 5 cm diameter) sampling rings
were used in the analysis. Soils were selected to be representative of the entire country (Poland) and were asubset of the Soil Databank of Arable Mineral Soils of Poland [Bieganowski et al., 2013a]. Soils were mainlyclassiﬁed as Haplic Luvisols and Eutric Cambisols, i.e., the two main soil types in Poland. Mollic Gleysols,
Eutric Fluvisols, Haplic Phaeozems, DistricFluvisols, Rendzic Leptosols, and TerricHistosols were also represented.

Figure 1. Textural distribution (PSD was determined by LDM) of the soil
samples used in this work.

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Drying and wetting branches of the SWRC
were determined during a 3 year long
campaign of laboratory measurements.
Soil cores were initially saturated withwater and then subjected to drying; thedrying branch of the SWRC was measured
ﬁrst and the wetting branch was determined afterward starting from a soil water
potential of 21554.78 kPa, which wasreached at the end of the drying phase.
Soil water contents were determined forseven values of a soil water potential of
23.1, 29.81, 215.54, 249.16, 2155.47,
2491.66, and 21554.78 kPa for the drying branch and for six values of a soil
water potential of 21554.78, 2310.21,
298.1, 215.54, 29.81 and 23.90 kPa for
the wetting branch. Sandboxes and thehanging water column method were3

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Figure 2. Schematic of the measurement cell: (a) a side view and (b) a top view.

used for soil water potentials higher than or equal to 29.81 kPa, while pressure chambers were utilized forlower potentials. The drying branch of SWRC was measured using the standard ceramic plate extractors (bySoilmoisture Equipment Corp.), which allowed for the soil water potential to be only reduced between subsequent measurements, as water could only be released from the measurement system.
For the determination of the wetting branches of the retention curves, different pressure chambers
were used, which allowed for wetting of measured soil cores through the chamber ceramics between
subsequent measurements. For that purpose, modiﬁed Soilmoisture Equipment volumetric pressure plate
extractors (Product #1250) were used. The system was originally designed to measure changes in thesoil water contents of the samples based on the extracted/applied soil water volume measured byburette. This system was adapted so that TDR probes [Skierucha et al., 2012a, 2012b] could be used todirectly measure the soil water content in the samples (see Figure 2) rather than derive it from burette
measurements. In the new setup, the TDR probes were vertically installed in the soil samples from thetop, and such a TDR placement and the validity of their measurements were experimentally veriﬁed[Pastuszka et al., 2014]. In addition to making measurements easier, this modiﬁcation also allowed us toplace three soil samples in each chamber (the original system allowed only one soil core in a pressurechamber), which allowed for the simultaneous determination of the drying branches of 21 soil cores
using seven pressure chambers.
In addition to SWRCs, other soil physical characteristics were determined including dry bulk density (BD),particle size distribution (PSD), which was measured using the laser diffraction method (LDM), the totalorganic carbon (OC), which was measured using high-temperature catalytic oxidation using TOC-VPCH(Shimadzu), and soil speciﬁc surface (SSS), which was determined using the water vapor adsorption
method. The soil samples were initially sieved through the 2 mm sieve for PSD, OC, and SSS
determination.
A Malvern Mastersizer 2000 with a measurement range of 0.02 mm to 2 mm was used for PSD determination. For obtaining homogeneity in the measured soil suspension, a Hydro G dispersion unit was used
[Sochan et al., 2012]. The pump speed was 1750 rpm, and the stirrer speed was 700 rpm. The soil sample
was dispersed by ultrasonication, and the power of the probe was 35 W with the time interval equal to 4
min. Light intensity measured on the detectors was recalculated into PSD according to the Mie theory (ISO13320:2009, 2009). The Mie model parameters were an absorption coefﬁcient of 0.1 and a refractive coefﬁcient of 1.52 [Bieganowski et al., 2013b]. For each soil, the PSD curve was determined as an average value of
three replications. Soil samples were not pretreated in any way before the LDM measurement.
The soil speciﬁc surface (SSS) area was determined for samples initially dried at 1058C for 24 h. The adsorption and desorption isotherms of water vapor were determined using the Dynamic Vapor Sorption-Intrinsic

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Table 1. Statistical Properties of the Entire Soil Data Seta
Statistic3

BD (g/cm )Sand (%)
Silt (%)
Clay (%)
OC (%)SSS (m2/g)

Mean

St. Dev.

Min

Max

1.395
55.80239.6954.504
1.87
17.6170.143
25.45
23.1192.638
0.799
12.911

1.02
11.28
7.615
0.518
0.722
3.14

1.76
91.57278.81812.713
5.547
80.46(Surface Measurement Systems Ltd., UK)
for a relative humidity from 0 to 100% at
208C. The BET model was used for thedetermination of SSS based on desorption isotherms [Brunauer et al., 1938].

While the data set collected for this study
is relatively large (104 soil samples), allameasurements were carried out with theBD, bulk density; OC, organic carbon; SSS, speciﬁc soil surface; Sand, Silt,
and Clay are percentages of particular particle sizes.
same experimental methodology. Such adata set, which could be used for the
analysis of hysteresis, is rather unique, as most similar studies usually rely on much smaller data sets or data
sets collected from different sources. An overview of descriptive statistics for the soil data set is presented
in Table 1.
2.2. SWRC Data Pretreatment
Laboratory measurements provided information about the drying and wetting branches of SWRCs in theform of measured pairs of soil water contents and soil water potentials. The soil hydraulic parameters of thevan Genuchten (vG) function were ﬁtted to both drying and wetting branches of SWRCs for all soil samples.
An assumption was made that both branches had the same residual and saturated water contents. For ﬁtting of the vG model’s parameters, a custom script was written in R language that was based on the NelderMead minimization of RMSEs (root-mean-square errors) between measured and vG model predicted soil
water contents for given soil water potentials.
2.3. SVM SWRC Modeling
The support vector machines approach (SVM), one of many algorithms among machine learning methodologies, was selected for the SWRC model development. The SVM approach has been previously used in subsurface hydrology for facies delineation [Tartakovsky and Wohlberg, 2004; Wohlberg et al., 2006], SWRC
modelling [e.g., Lamorski et al., 2008; Twarakavi et al., 2009], or for predictions of time-variable soil watercontents [e.g., Lamorski et al., 2013; Karandish and Simůnek, 2016]. While SVM was originally developed forsolving classiﬁcation problems, its usage has later been extended to regression-type problems [Vapnik,
1995]. When used for regression modeling, SVM estimates one output variable based on a set of independent input variables. Since SVM is a supervised learning method, it requires a training data set for the model’s development. The resulting model reproduces input-output relationships present in the training data
set and is able to make estimations for any values of input variables. A model based on a training data set issubject to validation based on a testing data set. SVM models, similarly like ANN or k-NN-based models,
have to be developed and used by means of software because their ﬁnal form is not a simple, explicit relationship. In this study, the R statistical computing environment was used for that purpose.
Support vectors (SV), a kernel function, and model parameters are selected during the SVM model development and together constitute the model. The core of the SVM model are so-called support vectors (SV),which are a subset of a training data set, thoroughly selected during the model development procedure.
There is a strict dependence between the number of SVs and the generalization capability of the model. An
optimal number of SVs is about half of a training data set. Incorporating more SVs in the model will lead toovertraining, i.e., model estimations will be very good for a training data set, while estimation errors will bevery high for a testing data set [Lamorski et al., 2014].
The SVM methodology used in this paper was previously applied for the development of pedotransfer functions (PTF) for the estimation of SWRCs by Lamorski et al. [2014], in which all details can be found. The mSVM algorithm was used for the estimation of SWRCs. SVM rely on a selectable model component, the socalled kernel function; a radial basis kernel function was used in the study of Lamorski et al. [2014]. The
model parameters were searched for using the genetic algorithm optimization technique. The objectivefunction that was used in this process was based on RMSE between measured and estimated SWRC, while
also explicitly taking into account the number of support vectors used in the developed model. The use ofsuch an objective function instead of one only based on RMSE avoids overtraining during the model
development.

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2.4. SVM Model Development
The soil data set was randomly divided into
Model
Input
Estimation
the training (78 soils) and testing (26 soils)
Name
Variables
Methoddata sets for the SVM model development.
SWRC
DirectM0d
Parametric
M0p
Five different types of models (M0 through
SWRC, BD
DirectM1dM4) were developed based on the regression
ParametricM1panalysis of available soil data. While all ﬁve
SWRC, BD, Sand, Silt, Clay
DirectM2d
ParametricM2p
models allowed for the determination of theSWRC, BD, Sand, Silt, Clay, OC
DirectM3d
wetting branch of SWRC, they used different
ParametricM3p
sets of input parameters. All models used
SWRC, BD, Sand, Silt, Clay, OC, SSS
DirectM4d
ParametricM4p
information about the drying branches ofaSWRCs. All models, except for M0, used addiA list of variables considered by the estimation methods ofdifferent SVM models.
tional soil characteristics as input parameters(i.e., PSD, BD, OC, and SSS; see Table 2). Soilphysical parameters were included as input for the developed models since they were correlated with thewetting branch of SWRC, and thus their inclusion into the analysis could have a positive impact on model
estimations.
Table 2. Names and Deﬁnitions of SVM Modelsa

SWRC data were either directly (subscript d) or parametrically (subscript p) considered in each SVM model.
In both cases, the ﬁrst step was to ﬁt the van Genuchten model to measured retention data points. In the
direct approach, 15 soil water contents evaluated from the van Genuchten functions ﬁtted to the drying
branches of SWRCs for the following soil water potentials were considered: 20.00981, 20.031, 20.0981,
20.310, 20.981, 23.102, 29.81, 215.547, 231.021, 249.166, 298.1, 2155.478, 2310.21, 2491.664, and21554.780 kPa. The direct models estimated soil water contents for the wetting branches of SWRCs for thesame set of soil water potentials. In the parametric models, SWRC input information was represented using
the van Genuchten parameters ﬁtted to the drying branches of SWRCs. Similarly, the parametric models
estimated the vG parameters of the wetting branches of SWRCs. Altogether, 10 different SVM models wereconstructed and validated (see Table 2).
Both direct and parametric approaches for the model development were considered here since there is still
an ongoing discussion regarding the best SWRC modeling methodology. While some researchers reportbetter accuracy of the direct approach [Schaap, 2004; Børgesen and Schaap, 2005], others reported no differences between direct and parametric approaches [Merdun, 2006].
2.5. Models’ Cross Validation
Different cross-validation techniques can be utilized to assess the quality of the developed models
and their selected parameters when the data learning concept is used [Hastie et al., 2009]. The k-foldcross validation was used for that purpose in this study. The training data set, which was used formodel development and consists of 78 soil data records, was randomly divided into ﬁve disjunctive
subsets (i.e., folds), with 15 or 16 soil data records in each subset (fold 1–16, fold 2–16, fold 3–16, fold
4–15, fold 5–15). The folds used for data preparation were sequentially rotated ﬁve times, whichallowed for the preparation of ﬁve sets of data, which were then used for model development. Five
training data sets were set up, each with four joined folds (62 or 63 samples). One SVM submodel
was then created for each of the ﬁve different training data sets. The resulting ﬁnal models, whichwere developed using the k-fold method, estimate water contents as an average outcome from theﬁve submodels.
2.6. Classical Models for Estimating Wetting Branches of SWRCs
The wetting branch of the SWRC may be estimated using several approaches suggested in the literature[Mualem, 1977; Kool and Parker, 1987; Hogarth et al., 1988; Pham et al., 2003]. However, if only the knowledge of the drying branch of the SWRC is to be used as input information for this estimate, only two models
are currently available [Mualem, 1977; Kool and Parker, 1987].
Mualem’s [1977] model, which is based on the dependent-domain theory of hysteresis, is described byequation (1). This model directly relates soil water contents of the wetting branch of the SWRC with soil
water contents of the drying branch of the SWRC for a given soil water potential:

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qﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ
Sew ðhÞ512 12Sed ðhÞ(1)where h is the pressure head (m), and Sew ðhÞ and Sed ðhÞ are effective degrees of saturation for the wetting
and drying branches, respectively, described by equations:Sew ðhÞ5hw ðhÞ2hres
hsat 2hres(2)

Sed ðhÞ5hd ðhÞ2hres
hsat 2hres

(3)

and

where hres and hsat are the residual and saturated water contents, respectively, and hd(h) and hw(h) are thewater contents on the draining and wetting branches of the SWRC, respectively.
The model of Kool and Parker [1987], contrary to the model of Mualem [1977] that operates directly with thewater contents of the drying branch of the SWRC, relates parameters of the van Genuchten parametric representation of the drying and wetting branches of the SWRC. The model of Kool and Parker [1987] assumes
that the n parameters for the wetting and drying branches of the SWRC are the same and that the a parameters of the two branches are related as follows:aw 52ad(4)

where aw and ad are vG function parameters for the wetting and drying branches of the SWRC, respectively.
The other vG function parameters have the same values for both wetting and drying branches of SWRC.
The Kool and Parker [1987] model of hysteresis is, for example, implemented into the widely used vadosezone HYDRUS [Simůnek et al., 2008] and SWAP [van Dam et al., 2008] models.

3. Results and Discussion
The developed SVM models were validated against the testing data set, which is a basic source of information about a model’s quality of estimation. Additionally, estimates made by the SVM models were comparedwith estimates made by the alternative—literature models that allow for the estimation of the SWRC wetting branch based only on information about the SWRC drying branch. These models will be further referred
to as M77 [Mualem, 1977] and KP87 [Kool and Parker, 1987].
The results presented in Table 3 indicate that there are no substantial differences between estimations
made by different SVM models and that improvements obtained by the more complex SVM models withrespect to R2 or RMSE as a comparison criterion are not signiﬁcant. Models M0d and M0p, which only use
information about the drying branches of SWRCs, have the same statistical indices as models M1d and M1p,
Table 3. Statistics of Various SVM and Classical Modelsa
R2

RMSE

Model Name

Value

SD

Group

Value (m3/m3)

SD (m3/m3)

Group

M0d
M0p
M1d
M1p
M2d
M2p
M3d
M3p
M4d
M4p
M77 [Mualem, 1977]
KP87 [Kool and Parker, 1987]

0.984
0.985
0.984
0.984
0.986
0.986
0.986
0.985
0.985
0.986
0.97
0.939

0.02
0.02
0.02
0.021
0.017
0.016
0.017
0.015
0.017
0.015
0.036
0.061

A
A
A
A
A
A
A
A
A
A
AB
B

0.018
0.017
0.018
0.018
0.018
0.018
0.018
0.02
0.019
0.019
0.025
0.047

0.014
0.016
0.015
0.017
0.017
0.017
0.018
0.021
0.018
0.02
0.014
0.022

A
A
A
A
A
A
A
A
A
A
AB
B

a 2

R , coefﬁcient of determination; RMSE, root-mean-square error; SD, standard deviation; Group, grouping based on the Tukey test forgroup mean differences. The RMSE and R2 values for the SVM models average results obtained by ﬁve submodels generated for eachfold. SDs are related to the spread of results achieved for different folds.

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which additionally consider the bulk density (BD) as an independent variable for estimations. In fact, theinclusion of BD has no discernible impact on estimations. A similar observation was made by Schaap et al.
[2001], who showed that consideration of BD in their pedotransfer functions produced better estimates
only for the saturated water content, which in our study, we kept the same as for the SWRC drying branch.
No improvements in estimations were also observed when PSD information was considered among input
variables. Consideration of additional soil variables, such as OC or SSS, produces only a limited increase inprediction quality (Table 3).
Since consideration of other soil information, in addition to the drying branches of SWRCs, has not substantially improved SVM models’ predictive capabilities, from a practical point of view, it is not necessary to use
such information for the estimation of the SWRC wetting branch. This result can be interpreted in the following way. In the statistical sense, information about soil physical parameters is already taken into accountand thus ‘‘hidden’’ in information about the drying branches of SWRCs. As a result, an explicit use of theseparameters as additional input variables for the estimation of the wetting branches has no discernible
impact on the results.
The developed SVM models performed better than the M77 model (Table 3), which also estimates the wetting branch of the SWRC solely based on the drying branch. Although, the differences in predictions by the
SVM models and the M77 model were not signiﬁcant in a statistical sense (using the Tukey test for groupmean differences with a conﬁdence level of 0.95), the RMSE for the M77 model was higher (0.025) than for
the SVM models (0.018).
Signiﬁcant differences (the Tukey test) in the quality of estimations of the wetting branches of SWRCs wereobserved (Table 3) between the SVM models and the KP87 model [Kool and Parker, 1987]. For the randomly
selected testing data set, the RMSE for the KP87 model was about 3 times larger than for the SVM models.
While Kool and Parker [1987] suggested the use of aw 5 2ad, other authors proposed different relations
depending on the soil type. For example, Likos et al. [2014], for a soil data set involving 25 samples, determined the average relation of aw 5 2.24ad, while for subsets of cohesive soils, aw 5 1.74ad, and cohesionlesssoils, aw 5 3.14ad. A comparison of the KP87 model with our experimental data shows that this model significantly underestimates hysteresis of SWRCs and that both the SVM models and the M77 model provide better predictions. Our data further indicate that the constrains of the KP77 model, requiring the same nparameter for both the drying and wetting branches, are too restrictive and prevent successful applicationof this model to our data set.
One of the reasons for these differences may be the fact that the KP87 model [Kool and Parker, 1987] wasproposed based on SWRC data that were only measured for a narrow range of soil water potentials
between 0 cm and about 2100 cm, while the measurement range for our data set was signiﬁcantly larger.
The other reason for relatively high KP87 model estimation errors may be the fact that for our data set, theFigure 3. Comparisons between the vG parameters (a) a and (b) n ﬁtted to the measured wetting and drying branches of SWRCs for theentire data set. Dotted lines represent 1:1 lines and solid lines represent linear regression lines ﬁtted to measured and estimated watercontents.

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Figure 4. Comparison of measured (points) and ﬁtted (lines) main drying (‘‘dry. measurement’’) and wetting (‘‘wet. measurement’’) SWRC branches for six selected soils from the testingdata set. Estimated main wetting branches by direct and parametric PTFs are shown as ‘‘wet. M0d’’ and ‘‘wet. M0p’’ lines, respectively. Main wetting branches estimated using the methodof Mualem [1977] and Kool and Parker [1987] are displayed as ‘‘wet. M77’’ and ‘‘wet. KP87’’ lines, respectively. Soil_id is the soil identiﬁcation number in the database, and PSD 5 (sand/silt/clay) are percentages of particular particle sizes.

n parameters for wetting and related drying branches are not the same, which is an assumption of the KP87
model. In this study, both of the vG parameters a and n were independently optimized for the wetting and drying SWRC branches. Their dependence is displayed in Figure 3. The n parameter for the wetting branch is usuallylarger than for the drying branch, and there is a correlation between these two values (R2 5 0.45). On the other
hand, values of the wetting and drying a parameters are not correlated (R2 5 0.03; Figure 3). The awet values arein most cases, although not all, higher than the adry values. The mean values of a and n parameters for the entire
data set are: <adry> 5 0.0789, <awet> 5 0.233, <ndry> 5 1.23, and <nwet> 5 1.35, i.e., <awet> 5 2.95<adry>.
However, this ratio of a values cannot be used for estimating the wetting branch from the drying branch as isdone in the KP87 model because there is no correlation between awet and adry, and thus this ratio does not
express any trend between the awet and adry values.
Figure 4 shows a comparison of measured and estimated wetting and drying SWRC branches for six
soils selected from the testing data set. It presents experimentally measured values of soil water contents for the drying branch of SWRCs and their ﬁt with the vG function. Figure 4 additionally shows
experimentally measured values of soil water contents for the wetting branch (dots), vG functions

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estimated using the SVM parametricmodels or ﬁtted to water contents estimated using the SVM direct models
for the wetting branch of SWRCs, aswell as estimations made using the
KP87 and M77 models. These examples provide visual conﬁrmations ofconclusions discussed above related toTable 3.

Figure 5. Mean residuals between the vG functions ﬁtted to the measured soil
water contents for the wetting SWRC branches and those estimated by different
models for the testing data set.

Residuals, i.e., mean absolute valuesof the estimation error for the testingdata set, between the vG functions ﬁtted to the measured soil water contents of the SWRC wetting branches
and vG functions estimated using the

Figure 6. Scatterplots of residuals as a function of texture for different models (a) M0d, (b) M0p, (c) M77, and (d) KP87, used for the estimation of the wetting branches of SWRCs.

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Figure 7. Comparison between measured and estimated (i.e., derived from estimated vG functions by the M0d model) soil water contents for selected soil water potentials. Dotted lines
represent the 1:1 line, solid lines represent a linear regression line ﬁtted to measured and estimated water contents.

Figure 8. Comparison between measured and estimated (i.e., derived from estimated vG functions by the M77 model) soil water contents for selected soil water potentials. Dotted lines
represent the 1:1 line, solid lines represent a linear regression line ﬁtted to measured and estimated water contents.

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M0d, M0p, M77, and KP87 models, are presented in Figure 5. The estimation errors for the KP87 model arehighest almost throughout the entire interval of soil water pressure heads, except below the pressure headof 25 cm where the estimation errors for the M77 model are larger. Interestingly, the M77 model has adifferent trend of estimation errors than the other models, especially near saturation (see subplot in Figure5). Its estimation errors almost up to the saturation point have large values when compared to the othermodels. The errors of the parametric SVM model (M0p) are larger than those of the direct SVM model (M0d)in the range of the pressure heads from 23 to 2300 cm.
Residuals for each soil from the testing data set were also individually analyzed using scatter plots for any
possible dependency on the particle size distribution (Figure 6). One can observe a similar dependency of
the M0p, M0d, and M77 models on texture with the estimation errors being smaller for soils with higher
sand (lower silt) contents than for ﬁne textured soils. The KP87 model does not display such dependency,
and residuals are approximately in the same wide range regardless of soil texture.
The analyses of estimation errors reported so far were based on differences between the vG functions of
the wetting branches of SWRCs estimated by different models and the vG ﬁts to measured points of thewetting branches. Figure 7 shows the comparisons between measured and estimated (i.e., derived fromestimated vG functions) soil water contents for selected soil water potentials. The coefﬁcients of determination show a fairly good agreement with R2 2 ð0:7; 0:88Þ between measured and estimated soil water contents. All ﬁtted linear models have slopes less than 1 and small values of positive intercepts (see Figure 7),
which indicates that the models underestimated water contents. The comparisons between measured andestimated (using the M77 model) soil water contents for selected soil water potentials are presented on Figure 8. The coefﬁcients of determination between measured and estimated soil water contents R2
2 ð0:56; 0:80Þ are slightly lower than for the SVM models, but still relatively high considering that the M77
model was not calibrated on the basis of this soil data set. In this case the results are also biased and the
M77 model underestimates water contents compared to measured values of the wetting branch of SWRC.

4. Conclusions
Statistical models were developed in this study that estimate wetting branches of the soil water retention
curve using the information about the routinely determined drying branch and other optional soil physicalcharacteristics (PSD, BD, OM, or SSS). Developed models for the estimation of the wetting branch of SWRC
utilize soft computing methodology that is based on the SVM algorithm. The developed models were validated against testing data, producing a good agreement between measured and estimated wetting
branches of SWRCs. The RMSE was about 0.018 m3/m3, which compares well with typical estimation errors
obtained by statistical methods used for SWRC modeling [e.g., Schaap et al., 2001].
It was observed that taking soil information other than about the drying branches of SWRCs into accounthas essentially no impact on the models’ estimations. It can be concluded that developed SVM models canestimate the wetting branches of SWRCs solely based on information about the drying branches of SWRCs.
Other than the drying branches of SWRCs, consideration of any soil physical properties as model input
parameters is of no use and leads only to a needless increase in model complexity. The developed M0d or
M0p models may be used for most practical applications.
Two types of models were developed and validated in this study: the direct and parametric models, which
either directly use SWRC data points or the van Genuchten parameters of the drying branches of SWRCs,
respectively. These two approaches are alternatively used in many PTF modeling studies, and different
authors advocate one of these two modeling approaches. Our results (Table 3 and Figures 5 and 6) showthat there are only minor differences between the PTF models that were developed directly using SWRC
data points or the van Genuchten parameters as input variables. There are no substantial differences
between the parametric and direct approaches for hysteretic SWRC modeling.
The estimation errors of developed models were compared with the alternative classical models (Mualem[1977] or Kool and Parker [1987]) that can be used for the same purpose. In both cases, the new SVM-basedmodels produced much lower estimation errors. The RMSE for the M77 model was about 1.4 times larger
than for the SVM model. Much worse results were obtained using the KP87 model, which had a RMSE 2.6

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larger than the SVM model. This was mainly due to the fact that the assumptions of the KP87 model were
not valid for the underlying database.
However, taking into account the complexity of the SVM-based models (M0d and M0p), which cannot be
described explicitly, and the simplicity of the M77 model, which is described by a single equation, one may
still prefer using the M77 model for practical applications.

Acknowledgments
This work has been partially supportedby the National Science Centre,
research grant 2011/01/B/ST10/07544.
The data set used for this study isavailable as a supporting information
or upon request from thecorresponding author.

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