Journal of Hydrology 543 (2016) 892–909

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Journal of Hydrology
journal homepage: www.elsevier.com/locate/jhydrol

Research papers

A comparison of numerical and machine-learning modeling of soil watercontent with limited input data
Fatemeh Karandish a,⇑, Jiří Šimůnek bab

Water Engineering Department, University of Zabol, Zabol, Iran
Department of Environmental Sciences, University of California Riverside, Riverside, CA 92521, USAa r t i c l e

i n f oArticle history:
Received 22 October 2016
Received in revised form 3 November 2016
Accepted 5 November 2016
Available online 8 November 2016
This manuscript was handled by P.
Kitanidis, Editor-in-Chief, with theassistance of Juan V. Giraldez, Associate
Editor
Keywords:
Machine-learning models
ANFIS
HYDRUS-2D
MLR
SVM
Soil water content
Water saving irrigationa b s t r a c t
Soil water content (SWC) is a key factor in optimizing the usage of water resources in agriculture since it
provides information to make an accurate estimation of crop water demand. Methods for predicting SWC
that have simple data requirements are needed to achieve an optimal irrigation schedule, especially forvarious water-saving irrigation strategies that are required to resolve both food and water security issues
under conditions of water shortages. Thus, a two-year field investigation was carried out to provide adataset to compare the effectiveness of HYDRUS-2D, a physically-based numerical model, with various
machine-learning models, including Multiple Linear Regressions (MLR), Adaptive Neuro-Fuzzy
Inference Systems (ANFIS), and Support Vector Machines (SVM), for simulating time series of SWC data
under water stress conditions. SWC was monitored using TDRs during the maize growing seasons of 2010
and 2011. Eight combinations of six, simple, independent parameters, including pan evaporation andaverage air temperature as atmospheric parameters, cumulative growth degree days (cGDD) and crop
coefficient (Kc) as crop factors, and water deficit (WD) and irrigation depth (In) as crop stress factors, wereadopted for the estimation of SWCs in the machine-learning models. Having Root Mean Square Errors(RMSE) in the range of 0.54–2.07 mm, HYDRUS-2D ranked first for the SWC estimation, while the
ANFIS and SVM models with input datasets of cGDD, Kc, WD and In ranked next with RMSEs ranging from1.27 to 1.9 mm and mean bias errors of 0.07 to 0.27 mm, respectively. However, the MLR models did not
perform well for SWC forecasting, mainly due to non-linear changes of SWCs under the irrigation process.
The results demonstrated that despite requiring only simple input data, the ANFIS and SVM models could
be favorably used for SWC predictions under water stress conditions, especially when there is a lack ofdata. However, process-based numerical models are undoubtedly a better choice for predicting SWCs
with lower uncertainties when required data are available, and thus for designing water saving strategies
for agriculture and for other environmental applications requiring estimates of SWCs.
Ó 2016 Elsevier B.V. All rights reserved.

1. Introduction
Because it uses about 70% of the world’s freshwater withdrawals (OECD, 2010), agriculture is a primary target for policies
that move the sector towards sustainable management of water.
Over past decades, irrigated agriculture has been widely developedaround the world as it significantly increases crop productivity.
However, irrigation consumes high volumes of water and may
induce off-site pollution of receiving water bodies (Billib et al.,2009). Thus, special attention should be paid to the water management of irrigated lands. Water should be applied in the right⇑ Corresponding author.
E-mailaddresses:(F. Karandish).

F.Karandish@uoz.ac.ir,

http://dx.doi.org/10.1016/j.jhydrol.2016.11.007
0022-1694/Ó 2016 Elsevier B.V. All rights reserved.

Karandish_h@yahoo.com

amount at the right time in order to achieve an optimal crop production and it should not be wasted as it is a valuable resource, andits applications should be environmentally sustainable (SAI, 2010).
An optimal irrigation schedule created by accurately estimating
the crop water demand (CWD) may help achieve these goals since
it prevents both yield and water losses (Payero et al., 2006; Klocke
et al., 2004; Stone, 2003). Among different developed methods,
estimating CWD based on the soil water balance is commonly
advised (Allen et al., 2011) since it considers real CWD, which
may produce significant water savings in irrigated agriculture.
Applying this method for irrigation scheduling requires knowledge
about temporal variations of soil water contents (SWCs) at the field
scale for crops both under optimal or water-stress conditions in asoil-plant continuum. This is because the plant water status, which
is a dominant factor determining the crop yield (Taiz and Zeiger,

F. Karandish, J. Šimůnek / Journal of Hydrology 543 (2016) 892–909

2006), is closely related to SWC (Dorji et al., 2005; Simon et al.,2009; Souza et al., 2009; Stoll et al., 2000; Xu, 2009). Thus, in orderto achieve the highest irrigation water use efficiency, the reductionof irrigation should be done only to the extent that SWC is kept
under optimal conditions to avoid a significant reduction in yield.
The need for suitable methods of estimating SWCs under water
stress conditions becomes more obvious when it is understoodthat the scarcity of fresh water is an increasingly global problem(Karandish et al., 2015), resulting from a gradual increase in fooddemand and a decrease in available fresh water resources, thatneeds to be urgently resolved (Sepaskhah and Ahmadi, 2010).
Under these circumstances, it is necessary to adopt water saving
irrigation strategies to cope with actual water availability. Partial
root zone drying (PRD) is a new deficit irrigation strategy, which
may produce water savings without a significant decrease in yields(Dry et al., 2000; Kang and Zhang, 2004; Kirda et al., 2004; Shaoet al., 2008; Tang et al., 2005; Karandish and Šimůnek, 2016).
Under the PRD strategy, which was introduced by Dry andLoveys (1998), one half of the root zone is irrigated while the otherhalf is dried out. Irrigated and dry sides of the root zone are periodically switched. Good knowledge of root zone SWCs greatly
influences the efficiency of PRD since the time of the irrigation shift
should be determined based on the soil water matric potential inorder to produce optimal results.
Despite of the importance of directly determining SWCs as atime series, the inability of doing so in the field at a low cost represents a restriction on the application of the soil water balance
equation for irrigation scheduling, especially in the planning andoptimizing stages of irrigation projects. Thus, further attempts
are required to find suitable methods for indirectly estimating
SWCs. In this regard, different models have been developed for
simulating SWCs in soil-crop systems during the past three decades. SWCs can be estimated using either simple soil water balance
models, which require only limited input information (Wessolek,
1989; Cameira et al., 2003; Panigrahi and Panda, 2003; Nishat
et al., 2007), or more complex process-based models such as
LEACHM (Wagenet and Hutson, 1987), GLEAMS (Leonard et al.,1987), WAVE 2.1 (Vanclooster et al., 1996; Fernández et al.,2002), EURO-ACCESS-II (Armstrong et al., 1996; Fernández et al.,2002), SWIM (Verburg et al., 1996), SWAP (van Dam et al., 1997),
SWAT (Neitsch et al., 2005), or HYDRUS (Šimůnek et al., 2008,2016), which require a large number of input parameters. However, since these parameters are directly related to soil, crop, andclimate properties, these models often provide superior predictions of SWCs than simpler soil water balance models.
HYDRUS-2D (Šimůnek et al., 2008, 2016) is one of the dynamic,
physically based models that is widely used to simulate soil water
dynamics (e.g., Cote et al., 2003; Skaggs et al., 2004; Ajdary et al.,2007; Rahil, 2007; Crevoisier et al., 2008; Lazarovitch et al.,2009; Siyal and Skaggs, 2009; Mubarak, 2009; Ramos et al.,2012; Tafteh and Sepaskhah, 2012). One of the advantages of thismodel is that its parameters are related to soil physical properties,
which may be measured either in-situ or in the lab. There are many
published studies in which HYDRUS-2D was applied to evaluate
temporal variations of soil-water-plant interactions, including
under the PRD conditions (Karandish and Šimůnek, 2016).
Karandish and Šimůnek (2016) demonstrated the high capabilityof the model to simulate temporal variations of various soil waterbalance components under full irrigation, deficit irrigation, and
PRD conditions. However, HYDRUS-2D requires the determinationof soil hydraulic parameters and exact initial and boundary conditions, which sometimes hampers its more widespread use.
Machine-learning approaches, such as artificial neural networks(ANN) and support vector machines (SVM), are another group ofmodels that have been applied during past decades for simulating
various hydrological processes including soil water dynamics

893(Jiang and Cotton, 2004; Ahmad and Simonovic, 2005;
Elshorbagy and Parasuraman, 2008; Zou et al., 2010; Dai et al.,2011; Asefa et al., 2006; Yu and Liong, 2007; Lin et al., 2009; Liuet al., 2010; Deng et al., 2011). These approaches provide a great
prediction capacity and do not require the knowledge of soil physical properties. On the other hand, they do require meteorologicalmeasurements and time-series of various soil variables such as soil
water contents or matric potentials as training data. Therefore,
their prediction capability is limited by information contained in
the data. Also, they do not have direct intuitive interpretation(Lamorski et al., 2013) with respect to evaluated processes. Literature reviews also revealed that another powerful artificial intelligence method, the adaptive neuro-fuzzy inference systems(ANFIS), has not yet been used for simulating SWCs.
Although machine-learning models have been previously used
to simulate time series of SWC variations, the emphasis usually
was on estimating future SWCs from their historic values (i.e., fromsoil water contents measured on previous days). However, this is arather restrictive use of these models, since it is hard to use thisapproach to determine irrigation scheduling, especially during
the planning stages. This problem becomes more severe when
the focus is on optimization problems and finding the optimal
water-saving irrigation schedule.
In addition, the earlier applications of machine-learning models
were all for non-stressed conditions under full irrigation. However,
it is becoming more difficult to irrigate crops to meet their fulldemand when water is in short supply, so models must be able
to accurately predict SWCs for stressed conditions as well. Thus,
the objectives of this study were to use both the machinelearning and physically-based models to predict time series of
SWCs in the rooting depth of maize under full irrigation as wellas under several different water saving irrigation strategies. The
physically-based HYDRUS-2D model was used to simulate temporal variations of SWCs under PRD treatments. Its predictions werecompared with predictions of the multiple linear regressions(MLR), ANFIS, and SVM models, which only require the knowledgeof simple and easily determined atmospheric and crop parameters.
Second, when only limited data is available, the possibility of substituting HYDRUS-2D with machine learning models for indirectly
estimating SWCs is evaluated.

2. Materials and methods
2.1. Field investigations
The experimental site of the Sari Agricultural Sciences and Natural Resources University is located on the coastal plain of northern Iran, (36.3°N, 53.04°E, 15 m below the sea level). Mean
annual rainfall is 616 mm, about 70% of which occurs during the
October-March period. The long-term average, minimum, andmaximum air temperatures are 17.3, 6, and 38.9 °C, respectively(Karandish and Šimůnek, 2016). Weather data were recorded daily
in the experimental area. The soil physical characteristics are summarized in Table 1.
The experiment was carried in a maize field using a randomized
complete block design with three surface drip irrigation treatments as described below. To measure retention curves, soil samples were taken every 20 cm to a depth of 80 cm for each treatmentin three replicates before crop sowing and using a 2-in ID augur.
SWCs at 11 different pressure heads were measured in the laboratory at each sample using a pressure plate apparatus. Thereafter,
the van Genuchten-Mualem model (Mualem, 1976; van
Genuchten, 1980) was used to describe the soil hydraulic properties. The soil hydraulic parameters for two soil horizons (i.e., 0–
20 cm and 20–80 cm soil depths) were obtained by fitting this894

F. Karandish, J. Šimůnek / Journal of Hydrology 543 (2016) 892–909

Table 1
Soil properties at the experimental site.
Depth (cm)Soil texture

Sand (%)

Silt (%)

Clay (%)

Field capacity (%)

Wilting point (%)

Bulk density (g cm3)

0–20
20–40
40–60
60–80
80–100

Sandy clay loam
Clay loam
Clay loam
Clay loam
Clay loam

49
40
30
37
36

22
25
36
30
28

27
35
34
33
34

30
32
32
32
32

15
14
14
14
14

1.40
1.38
1.35
1.37
1.37

model to the observed retention curve data of both layers using the
RETC program (van Genuchten et al., 1991). Fitted soil hydraulicparameters included the saturated SWC (hs), the residual SWC(hr), and the shape parameters a and n. Since no hydraulic conductivity data was available, the pedotransfer functions embedded in
the HYDRUS-2D software were used to predict the saturated soil

hydraulic conductivity (Ks) using measured soil physical properties
for each treatment.
The surface drip irrigation system was then installed prior tocrop sowing. Fig. 1a shows the locations of drip lines, drippers,
and plants in the experimental field. Five Time Domain Reflectometry (TDR) access tubes (TRIME FM, IMKO, Germany) for continuous monitoring the SWCs were installed in the maize root zoneof each treatment, as illustrated in Fig. 1b. TDR probes were used
at least two times a day to measure soil water contents (SWCs)
at depths of every 5 cm (i.e., at measuring points displayed inFig. 1b) during both growing seasons of 2010 and 2011 (i.e., fromsowing to harvest). Overall, SWCs were measured at each measuring time in 100 points for each treatment. Moreover, information
about the movement of the wetting front during irrigation eventswas collected at least 10 times in each treatment by measuring
SWCs one hour before, and immediately and 2, 6, 12, 24, 48 h after
the irrigation events in the 2010 growing season. For each treatment, such SWC information was collected for at least 10 irrigation
events during each growing season.
Thereafter, maize single-cross hybrid 704 was sown on May 26
of both 2010 and 2011, 5 cm deep, with 75  20 cm  cm crop row
and crop spacing, between and parallel to the drip lines (Fig. 1a,
Table 2). Crops were irrigated every other day using surface drip
irrigation during the entire growing season with a total 53 and55 irrigation events in the 2010 and 2011 growing seasons, respectively. Irrigation treatments included full irrigation (FI) and twopartial root-zone drying treatments, PRD75 and PRD55. The FI treatment was fully irrigated and the soil water content was kept close
to field capacity during the entire growing season. The irrigation
requirement (in mm) for the FI treatment was calculated using
Eq. (1) for each irrigation event:½In FI ¼

m
X
f½hFCi  ðhBIi ÞFI Di g

ð1Þ

i¼1

Fig. 1. Locations of drip line laterals, drippers, plants (a), and TDR probes (b) in theexperimental maize field.

where ½In FI is the net irrigation depth (mm) of the nth irrigation
event for the FI treatment, hFCi is the volumetric SWC at field capacity (FC, %) of the ith soil layer, ðhBIi ÞFI is the average volumetric SWC
before irrigation of the ith soil layer (%) in the FI treatment, Di is thesoil layer thickness (mm), i is the soil layer, and m refers to thenumber of soil layers down to a specific soil depth, for which [In]FIis calculated. (hBIi)FI was measured using TDR probes before eachirrigation event.
All treatments (including FI and both PRD treatments) received
the same amount of irrigation water during the first 55 days aftersowing (DAS) in 2010 and during 45 DAS in 2011. During the PRD
period (i.e., during 55–107 DAS in 2010 and during 45–110 DAS in

Table 2
Summary of agricultural and fertilization activities during the maize growing season.
Date

Agricultural activities and fertilization

Remarks

May 26 in 2010 and 2011
June 12, 2010 and June 5, 2011

Planting and fertilization
Fertilization

July 14, 2010 and July 4, 2011
July 19, 2010 and July 9, 2011
September 9, 2010 and September 12, 2011

Fertilization
Onset PRD treatments
Harvest150 kg ha1 triple superphosphate
65 kg ha1 urea and
50 kg ha1 potassium sulphate
135 kg ha1 urea and 100 kg ha1 potassium sulphate

F. Karandish, J. Šimůnek / Journal of Hydrology 543 (2016) 892–909

2011), the PRD treatments were scheduled to receive 55% (PRD55)
and 75% (PRD75) of the FI treatment’s irrigation amount at eachirrigation event. While in the FI treatment, both drip lines wereoperated simultaneously, in the PRD treatments during the PRD
period, to ensure partial root-zone drying, just one of the drip lines
was operated while the other was not during each irrigation event.
Only half of the root zone was thus irrigated during the PRD period,
while irrigation shifted between the two sides of the plants eachweek. The dripper discharge was two liters per hour.
Prior to implementing the PRD treatments, the information
about the movement of the wetting front was collected by measuring SWCs using TDR probes to determine the maximum lateraladvance of the wetting front for different irrigation water depths.
Such information was required to ensure that only about half ofthe rooting zone in the PRD75 and PRD55 treatments is irrigatedafter imposing the PRD treatments. A maximum observed lateraladvance of about 35 cm confirmed this assumption.
Plants were harvested on September 9, 2010 (107 DAS) and on
September 12, 2011 (110 DAS). Detailed description of all measurements during field investigation, especially those requiredfor calibration and validation of HYDRUS-2D, is presented in
Karandish and Šimůnek (2016).
2.2. HYDRUS (2D/3D)
2.2.1. Model description
HYDRUS-2D (Šimůnek et al., 2008) is a powerful software forsimulating the transient, two-dimensional movement of waterand nutrients in soils for a wide range of boundary conditions,and soil heterogeneities. Water flow in soils is described using
the Richards equation as follows:





@h
@
@h
@
@h
@k
þ

¼
Kx
Kz
 WUðh; x; zÞ
@t @x
@x
@z
@z
@z

ð2Þ

where h is the volumetric SWC (L3 L3), K is the unsaturated
hydraulic conductivity function (L T1), h is the soil water pressure
head (L), x is the lateral coordinate, z is the vertical coordinate (positive downwards), t is time (T), and WU(h, r, z) denotes root water
uptake (T1). WU is computed as follows:WUðh; x; zÞ ¼ cðhÞRDFðx; zÞWT pot

ð3Þ

where c(h) is the soil water stress function (dimensionless) of
Feddes et al. (1978), RDF is the normalized root water uptake distribution [L2], Tpot is the potential transpiration rate [L T1], and W is
the width of the soil surface [L] associated with the transpiration
process. In the present study, the root distribution was assumed
to be described using the Vrugt et al. (2001) function and to be constant in time (which is a restriction of HYDRUS-2D).
Crop evapotranspiration (ETc) under the FI treatment was
assumed to represent potential crop evapotranspiration sincecrops under this treatment were well-irrigated (i.e., there was nowater stress in the rooting zone since irrigation events were scheduled to always refill soil water content to field capacity), well fertilized, and were treated with pesticides to control weeds, aphids
and fungal diseases during both growing seasons. Moreover, thecrop achieved its full production when considering the maize yieldpotential for given climate conditions in the study area. Details
about how ETc was obtained is given in Karandish and Šimůnek(2016).
Measured leaf area index (LAI) was applied to divide ETc into
potential evaporation (Ep) and potential transpiration (Tp) as follows (Belmans et al., 1983):


Ep ¼ ET c eK gr LAI
T p ¼ ET c  Ep

ð4Þ

895

where Ep is potential evaporation [L T1], Tp is potential transpiration [L T1], ETc is crop evapotranspiration [L T1], and Kgr is an
extinction coefficient for global solar radiation [–]. Kgr was set to0.39 following the suggestions by Ritchie (1972) and Feddes et al.
(1978). Estimated values of Ep and Tp were used as input parametersin HYDRUS-2D. Temporal variations of ETc under different treatments can be found in Karandish and Šimůnek (2016).
The soil hydraulic properties were modeled using the van
Genuchten-Mualem constitutive relationships (van Genuchten,1980). The Galerkin finite element method was used for solving
Eq. (3) based on the iterative mass conservative scheme proposed
by Celia et al. (1990).
2.2.2. Geometry information and boundary conditions
The domain geometry was defined to represent a typical maize
field in which driplines are located between maize rows with a row
spacing of 75 cm (Fig. 1b). The two-dimensional transport domain
was a rectangle 75 cm wide between two neighboring emitters oneither side of one plant (Fig. 1b) and with a soil depth of 80 cm. The
soil depth was selected so that the observed maximum rooting
depth was situated above this depth. The driplines were considered to be line sources, since the emitter spacing along the driplines was relatively small (i.e., 20 cm in this field investigation).
The spatial domain was discretized using unstructured triangular
finite element mesh (FEM) defined using 2215 nodes. A nonuniform FEM was generated by HYDRUS-2D with finite element
sizes gradually increasing with distance from the emitters. A highnodal density is required in the immediate vicinity of the emittersto accurately model the large spatial gradients in soil water pressure heads caused by infiltrating water (Kandelous et al., 2011).
Two soil horizons with different soil hydraulic properties were
defined at the 0–20 cm and 20–80 cm soil depths.
Initial conditions were defined based on the field measurements. The time-variable and atmospheric boundary conditions
were specified at the soil surface to represent drip irrigation andto apply precipitation, evaporation, and transpiration fluxes,
respectively. A free drainage boundary condition was applied along
the bottom boundary, allowing for downward drainage. All otherremaining boundaries were assigned a no-flow boundary
condition.
2.2.3. Model calibration and validation
The sensitivity analysis was carried out to determine how thesoil hydraulic parameters affect the HYDRUS-2D simulated SWCs.
The sensitivity coefficients for individual parameters were calculated according to Šimůnek and van Genuchten (1996). Thereafter,
the most sensitive parameters were optimized in the calibration
process. Detailed information about the calibration and validation
process can be found in Karandish and Šimůnek (2016).
2.2.4. Multiple Linear Regressions (MLR)
Regression analysis is commonly used to describe quantitative
relationships between a response variable and one or more
explanatory variables (Tabari et al., 2012). The linear equation in
MLR is as follows:

Y ¼ a0 þ a1 X 1 þ a2 X 2 þ    þ an X nð5Þwhere Y is the dependent variable, a0  an are the MLR parameters,
and X 1  X n are the independent variables (Ozbayoglu andOzbayoglu, 2006).
For all evaluated MLR models, SWC was defined as a dependent
variable. First, a Pearson-correlation matrix was establishedbetween SWC as a dependent variable and different independent
variables. Thereafter, variables which had the highest correlation
coefficient with SWC were selected independent variables in the
MLR models. The set of independent variables included pan evap-

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F. Karandish, J. Šimůnek / Journal of Hydrology 543 (2016) 892–909

oration (E), average air temperature (T), crop coefficient (Kc), cumulative growth degree day (cGDD), net irrigation depth (In) andwater deficit (WD). These parameters were selected since they
are easy to obtain and affect SWCs. While T and E are considered
to be atmospheric factors, Kc and cGDD are considered to be crop
factors. WD and In represent stress conditions in the soil and affect
SWCs through root water uptake. WD is calculated as
WD ¼ ðhFC  hpwp Þ  a, where, hFC is the volumetric soil water content at field capacity, hpwp is the volumetric soil water content at
the permanent wilting point and a is a coefficient, which is set to1, 0.75, and 0.55 for the FI, PRD75, and PRD55 treatments, respectively. Eight combinations of input variables were evaluated in thisstudy: (i) E, Kc, and In; (ii) E, Kc, WD, and In; (iii) cGDD, E, Kc, and In;(iv) Kc, T, and In; (v) Kc, WD, T, and In; (vi) cGDD, Kc, T, and In; (vii)cGDD, Kc, and In, and (viii) cGDD, Kc, WD, and In.
2.3. Adaptive Neuro-Fuzzy Inference Systems (ANFIS)1 2bixci 1þ a i

ð6Þwhere x is input, and a, b, and c are comparative parameters and thenon-linear coefficients, which determine the shape of the MFs. The
set of the fuzzy variables is called the S1 set or the left-handed set(LHS). The output values of the first layer show the membership values of ach input for specific MFs.
Layer 2: This layer is the result of multiplying the input values
by the nodes, and then finally the firing strength (i.e., weight).
For example, for the first node, we have:wi ¼ lA1 ðx1 Þ lB1 ðx2 Þ

 1 ¼ Pw1
Wni¼1 wi;i ¼ 1; 2; . . . ; nð8Þwhere n is the number of nodes in each layer.
Layer 4: This is the terms layer, in which terms are achieved.
These terms are the results of operation on the input signals into
this layer:

 1f 1 ¼ w
 1 ðp1 x1 þ q1 x2 þ r 1 Þ
Z1 ¼ w

ð9Þ

where p1, q1, and r1 are the consequent parameters.
Layer 5: This is the last layer of the net, which includes only one
node, and is calculated by adding all input values into its total
output:nX
 if i
w

ð10Þ

i¼1

Jang (1993) introduced the Adaptive Neuro-Fuzzy Inference
Systems method (ANFIS), which combines neural networks andthe capabilities of the fuzzy theory (Tang et al., 2010). A popularteaching method in neuro-fuzzy systems is the fuzzy inference system, which uses the hybrid learning algorithms to identify the
fuzzy system parameters and to teach the model (Rehman andMohandes, 2008). The ANFIS model has a five-layer structure(Fig. 2), which is the result of adding the fuzzy logical models tothe artificial neural networks:
Layer 1 or Input Layer: In this layer, the membership degree ofthe nodes entering different fuzzy periods is determined by using
the membership functions (MFs). There are numerous kinds ofMFs, such as trapezoids, sigmoid, Gaussian, and bell-shaped functions. Two fuzzy sets are considered for each input. The shape ofthe MFs and the values of their overlapping are optional and determined by Eq. (5):lA ðxÞ ¼

Layer 3: Its nodes normalize the firing strength:

ð7Þ

In the structure of the ANFIS model, there are sets of comparative parameters S1 (the set of fuzzy variables), and consequentparameters S2. Simulation is correctly performed when bothparameters are estimated in a way that minimizes the values ofthe error function of the model in the training, examination, andvalidation procedures. There are usually two steps to calculatethese parameters. The first step is a forward pass, which considers
the static S1 and S2 parameters, by using the least square error
algorithm. The second step is called a backward pass, which considers that the static S1 and S2 are obtained using the Gradient Descent algorithm. These sets of functions, called Epoch, appear at
every stage of training (Tabari et al., 2012). By calculating theparameters of the model, the output values of the model are
obtained for each arranged couple, which is sent to the modelthrough the training data of the model. The estimated values arecompared to the real values in order to calculate the training error
function of the ANFIS model. In every step of the Epoch training
course, the values of the model parameters change; thus, the values of the training error and the test error of the model vary. The
maximum values are selected in a way that the model does not
require extra training (Dastirani et al., 2010).
One of the features of every ANFIS model is the type of function
considered for the model inputs. In this study, different MFs were
employed and different numbers of MFs were tried in each application. The combination of MFs and their numbers giving the minimum errors were selected. Different ANFIS architectures were
evaluated using a MATLAB code that included fuzzy logic. Efficientmodels were first determined for each combination of input variables. Then, the various ANFIS models were tested and obtained
results were evaluated using different criteria indices. For bothFig. 2. Schematic of the ANFIS architecture.

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F. Karandish, J. Šimůnek / Journal of Hydrology 543 (2016) 892–909

ANFIS and SVM models (described below), the same combinationsof independent input variables were used as were described above
for the MLR models (i.e., eight input combinations involving E, T, Kc,
WD, cGDD, and In).

nomial, radial basis function, and sigmoid kernels were used. The
best kernel was determined by a trial and error process. The coefficients ai a are obtained by maximizing the following form:Rðai a Þ ¼ eumNi¼1 ðai þ a Þ þ

2.4. Support Vector Machines (SVM) 12Introduced by Vapnik (1995), Support Vector Machines (SVM)
are a classifier derived from statistical learning theory. The SVM
can be used both for classification and regression problems. They
can be represented as a two-layer network where the weights
are non-linear in the first layer and linear in the second layer(Bray and Han, 2004). The Support Vector Regression (SVR) is usedin literature to describe regression with SVM. The regression estimation with SVR is to estimate a function according to a given dataset, {(xi,yi)}n, where xi denotes the input vector, yi denotes the output value, and n is the total number of data points (Tabari et al.,2012). The input vectors (xi) refer to independent variables,
whereas the target values (yi) refer to SWC. The linear regressionfunction uses the following function:f ðxÞ ¼ x/ðxÞ þ b

ð11Þ

where /(x) is a nonlinear function by which x is mapped into a feature space, b and x are a weight vector and a coefficient, respectively, that should be estimated from the data. A linear regression
is performed in a high dimensional feature space via non-linear
mapping. The coefficients b and x are estimated by minimizing
the sum of the empirical risks (the first term of Eq. (12)) and a complexity term (the second term of Eq. (12)).
nX
1
R ¼ c Le ðf ðxi Þ; yi Þ kxk2
2
i¼1


Le ðf ðxi Þ; yi Þ ¼

0
jf ðxÞ  yj  eð12Þ
forjf ðxi Þ  yi j < eotherwise

ð13Þ

where R is the sum of the empirical risks, c is a positive constant (anadditional capacity control parameter) that determines the tradeoff between the model complexity and the values up to which
errors larger than e are tolerated, x2 is the regularization term,
which denotes the Euclidean norm, y is the output value, and Le iscalled the e-insensitive loss function that measures the empiricalrisk and has the advantage of not requiring all input data fordescribing the regression vector x.
Eq. (12) shows that the loss function is equal to 0 if thedifference between the predicted f ðxÞ and the measured value yi
is less than e. The choice of e is easier than the choice of c and it
is often given as a desired percentage of the output values yi . So,a non-linear regression function is given by a function thatminimizes Eq. (11), subject to Eq. (12), as in the following
expression:f ðx; a; a Þ ¼

nXðai  a Þkðxi ; xÞ þ b

ð14Þ

i¼1

Coefficient ai a ¼ 0 or ai a P 0 for i ¼ 1; . . . ; N and the kernel function kðxi ; xÞ describe the inner products in the D-dimensional feature space:

kðx; yÞ ¼

DX/j ðxÞ/i ðyÞ

ð15Þ

i¼1

It should be mentioned that the features /j need not be computed; rather what is needed is the kernel function, which is very
simple and has a known analytical form. In this study, linear, poly-Xyi ða  ai ÞNXða þ ai Þ  ða þ ai Þkðxi ; xj Þ
j¼1

subject to

ð16Þ

N
X
ða  ai Þ ¼ 0 and 0 6 ai a 6 C
i¼1

Only a number of coefficients ai a will be different from zero,and the data points associated to them are called support vectors(Mohandes et al., 2004; Kisi and Cimen, 2009; Zhou et al., 2009).
2.5. Correspondence criteria indices
The root mean square error (RMSE), the mean bias error (MBE),
the model efficiency (EF) and the coefficient of determination (R)were calculated to provide the quantitative comparison of the correspondence between the predicted and observed data as follows(Parchami-Araghi et al., 2013):

sﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ
Pn2
i¼1 ðOi  P i Þ
RMSE ¼
n

ð17Þ

Pn
MBE ¼

i¼1 ðOi  P i ÞnPnðOi  Pi Þ2
EF ¼ 1  P i¼12
ni¼1 ðOi  Oi Þ

ð18Þ

ð19Þ

where Pi and Oi are the predicted and observed data, respectively, O are the averages of observed and simulated data, respectively,
and P
and n is the number of observations.
3. Results and discussion
The HYDRUS-2D, MLR, ANFIS, and SVM models were evaluatedto simulate temporal variations of SWC data during two years andfor the FI, PRD75, and PRD55 treatments. It should be noted that the
SWC results are presented for all methods, except for HYDRUS-2D,in terms of average values for the entire simulation domain in the
FI treatment, which is an 80 cm deep rectangle between two neighboring emitters on either side of one plant (as explained in Section 2); and in terms of average values for each sides of the plantin the PRD treatments. Although the results for HYDRUS-2D arepresented for every 20 cm soil depth in Table 4 and for different
positions of the wetting front in Fig. 3, the average RMSE andMBE values were calculated for the entire soil domain of 0–
80 cm by averaging these statistics from individual soil layers.
3.1. HYDRUS-2D
Soil hydraulic properties (i.e., Ks, hs, and hr) for the two horizonsof the soil profile (i.e., 0–20 cm and 20–80 cm soil depths) were
optimized for all treatments using the inverse option of HYDRUS2D that minimizes deviations between the observed and simulated
SWCs. Based on the results of the optimization, summarized inTable 3, hr was the most sensitive parameter, followed by Ks, forpredicting observed soil water contents in the 0–20 cm soil depth.
In contrast, for the 20–80 cm soil depth, Ks was the most sensitive
parameter. Soil water contents showed less sensitivity to hs during
the calibration process than to Ks and hr.

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F. Karandish, J. Šimůnek / Journal of Hydrology 543 (2016) 892–909

Fig. 3. A comparison of observed and simulated (HYDRUS-2D) soil water contents at 0 (a), 18.75 cm (b), 37.5 cm (c), 56.25 cm (d), and 75 cm (e) distance from the first
dripper in the FI treatment (BI and AI denote before and after irrigation, respectively).

Table 3
Measured and optimized soil hydraulic parameters (Ks is the saturated hydraulic conductivity, and hr and hs are the residual and saturated water contents, respectively).
Treatment

Soil depth (cm)Ks (cm day1)hs (cm3 cm3)*hr (cm3 cm3)

Optimized

Observed

Optimized

Observed

Optimized

Observed

FI

0–20
20–80

1.3
1.0

1.1
0.80

0.47
0.47

0.45
0.48

0.10
0.07

0.07
0.09

PRD75

0–20
20–80

1.22
0.95

1.05
0.83

0.47
0.47

0.47
0.48

0.10
0.08

0.08
0.08

PRD55

0–20
20–80

1.15
0.90

1.1
1.0

0.47
0.47

0.46
0.47

0.09
0.07

0.06
0.08

*
Observed refers to parameters fitted by RETC to retention curves determined in the laboratory. Only Ks, hs, and hr were optimized by HYDRUS-2D, while parameters a, l,and n were kept equal to values obtained by RETC.

As an example, a comparison between observed and simulated
SWCs during the calibration process for the FI treatment is illustrated in Fig. 3. Good agreement was obtained between observed
and simulated SWCs at the sampling points between two neighboring emitters when the optimized soil hydraulic parameters
were used during the calibration period. A lower R2 (0.96–0.97)
was obtained for SWCs observed in the vicinity of drippers where
SWC variations were higher. Small differences between observed
and simulated SWCs can be a consequence of the fact thatHYDRUS-2D provides point values of SWCs, while measurements
inherently average SWCs over a certain soil volume, in whichwater contents may vary, especially in the vicinity of emitters(Mguidiche et al., 2015). Similar calibration results were also
observed for the PRD75 and PRD55 treatments, which confirm thehigh potential of HYDRUS-2D to simulate SWCs.
Daily SWCs for 2011 growing seasons were used to validate
HYDRUS-2D (i.e., results are also presented for daily SWCs during
the 2010 growing season in Fig. 4 for the visual comparison of

F. Karandish, J. Šimůnek / Journal of Hydrology 543 (2016) 892–909

899

Fig. 4. A comparison of temporal variations of measured and simulated (HYDRUS-2D) average SWCs in the root zone (i.e., 0–80 cm) during the 2010 (calibration) and 2011
(validation) growing seasons for the FI (a and d), PRD75 (b and e), and PRD55 (c and f) treatments.

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F. Karandish, J. Šimůnek / Journal of Hydrology 543 (2016) 892–909

Table 4
A comparison of observed and simulated (with HYDRUS-2D) EWDi (an equivalent water depth over four horizons) using criteria indices.
Parameter

Depth
(cm)

2010

2011

FI

EWDi
(mm)

*

0–20
20–40
40–60
60–80

PRD55

PRD75

FI

PRD75

PRD55

RMSE*
(mm)

MBE
(mm)

RMSE
(mm)

MBE
(mm)

RMSE
(mm)

MBE
(mm)

RMSE
(mm)

MBE
(mm)

RMSE
(mm)

MBE
(mm)

RMSE
(mm)

MBE
(mm)

0.8
0.85
0.78
0.28

1.1
1.2
0.85
0.48

0.73
0.73
0.64
3.69

0.93
0.91
0.87
10

0.83
0.79
0.81
5.84

0.83
0.85
0.69
5.5

1.18
0.38
0.48
0.41

0.89
1.2
0.8
0.75

0.82
0.78
0.80
0.87

1.4
0.95
1.2
0.97

0.35
0.72
0.60
0.51

0.45
0.82
0.70
0.65

RMSE: Root mean square error, MBE: Mean bias error.

Table 5
Statistical performance evaluation criteria for the MLR models using measured and observed SWCs.
Model

Input variables

RMSE (mm)*

MBE (mm)EF

Equation**

MLR1
MLR2
MLR3
MLR4
MLR5
MLR6
MLR7
MLR8

E, Kc, In
E, Kc, WD, In
cGDD, E, Kc, In
Kc, T, In
Kc, WD, T, In
cGDD, Kc, T, In
cGDD, Kc, n
cGDD, Kc, WD, In

25.26
12.20
12.00
17.13
12.24
13.44
14.78
10.46

1.37
0.34
3.98
3.16
0.18
7.14
9.48
6.86

0.34
0.85
0.86
0.71
0.85
0.82
0.79
0.90

0.163 + 0.001E + 0.048Kc + 0.003In***0.19 + 0.000022E + 0.041Kc  0.03WD + 0.0018In
0.622–0.0028cGDD + 0.000022E  0.051Kc + 0.000041In
0.205 + 0.0511Kc + 0.0015T + 0.0028In
0.18 + 0.056Kc + 0.0000018T  0.019WD + 0.00185In
0.65–0.0031cGDD  0.0702Kc + 0.0000021T + 0.00000014In
0.721–0.0025cGDD  0.042Kc + 0.0018In
0.531–0.0028cGDD  0.0499Kc  0.065WD + 0.0015In*

RMSEs and MBEs are calculated for equivalent root zone water depth (EWDrz).
Equations are for SWCs.
DAS: days after sowing (d), Kc: crop coefficient (dimensionless), WD: water deficit (dimensionless), In: irrigation depth (mm), E: evapotranspiration (mm), GDD: growthdegree day (oCd).
*****

the observed and HYDRUS-2D simulated SWCs). Temporal variations of measured and simulated SWCs averaged over the rooting
depth (i.e., 0–80 cm) for both growing seasons, as well as therelated scatter plots, are displayed in Fig. 4 for all treatments.
HYDRUS-2D performed very well in simulating average root zone
SWCs and their seasonal trends. Similar results were found in
Pang et al. (2000) and Mguidiche et al. (2015), who concluded thatHYDRUS-2D is capable of simulating the general trend of SWCs forboth homogenous and heterogeneous soils.
The performance of the HYDRUS-2D model in simulating SWCs
in terms of RMSE and MBE is summarized in Table 4. RMSE values
characterizing differences between observed and simulated SWCs
varied between 0.28–1.18 mm for the FI treatment, 0.64–
3.69 mm for the PRD75 treatment, and 0.35–5.84 mm for the
PRD55 treatment (the RMSE and MBE values were calculated forequivalent water depths over four different horizons (EWDi)). In
general, higher RMSEs for the FI treatments were obtained forthe 0–40 cm surface soil depths, which experience higher variations of SWCs during the growing seasons, than for deeper depths.
On the other hand, the highest RMSEs for the PRD treatments wereobtained for the 60–80 cm soil depth. A small underestimation of
all components of the EWDi could be observed for all treatments,
with MBEs between 0.45–10 mm. Other studies also reported
small errors in the simulated components of the seasonal soil
water balance using HYDRUS-2D, with RMSEs less than 10 mm(Skaggs et al., 2004; Kandelous and Šimůnek, 2010; Mguidiche
et al., 2015).
3.2. The MLR models
Results for the MLR models in terms of predictions of equivalentwater depth over root zone (EWDrz) are summarized in Table 5.
With respect to RMSE, the MLR8 model with inputs of cGDD, Kc,
In, and WD had the best performance (RMSE = 10.46 mm,
MBE = 6.86, and EF = 0.0.9), followed by the MLR3 (RMSE = 12 mm,
MBE = 3.98, and EF = 0.86) and MLR2 (RMSE = 12.2 mm,

MBE = 0.34, and EF = 0.85) models. It seems that Kc, which represents the crop growth stage and consequently the crop’s waterrequirements, plays a major role in estimating EWDrz since italways has a higher multiplication coefficient than other input
variables. In fact, the higher the root water uptake, the lower the
EWDrz.
The input combination of E, Kc, and In (i.e., the MLR1 model)
yielded the lowest accuracy in estimating EWDrz. With respect toMBE, MLR7 is the worst model since neither atmospheric factors(i.e., T and E) nor WD are included in its input combination. Generally, underestimation of SWCs is observed for the MLR models,
except for MLR1 and MLR4. Overall, differences between observed
and simulated SWCs (averaged over the root zone), as illustrated
by scatter plots in Fig. 5, indicate a low accuracy of the MLR models
in simulating SWCs for different treatments. This low accuracy canbe ascribed to the non-linearity of the variably-saturated waterflow process and thus the inadequacy of linear relationships
between SWCs and selected input parameters.
3.3. The ANFIS and SVM models
Table 6 provides the final architectures of various ANFIS models
and the performance statistics for the training phase. Two membership functions (MFs) were found to be sufficient for simulating
EWDrz with the ANFIS models except for ANFIS1, for which five MFs
led to the best results. The increase in the number of MFs not only
does not provide any significant improvements in the results, butalso increases the number of parameters that need to be optimized.
Because of that, the number of MFs should be kept as low as possible (Ozger and Yıldırım, 2008). Except for ANFIS4, the generalized, bell-shaped built-in (Gb) and Gaussian combination (Gus2)
membership functions were the best MFs for all ANFIS models.
For ANFIS4, the Gaussian curve built-in (Gus1) function led to
the lowest error when simulating EWDrz.
With respect to RMSE, the ANFIS8 model, with inputs of cGDD,
Kc, WD, and In, had the best performance in simulating EWDrz

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F. Karandish, J. Šimůnek / Journal of Hydrology 543 (2016) 892–909

Fig. 5. Scatter plots between observed and simulated SWCs (averaged over the root zone) for the MLR models.

Table 6
The final architecture and performance statistics of the ANFIS models for the testing phase. The same combination of input variables were used in the ANFIS and MLR models (seeTable 5).
Model

Input variables

Best MF*NMF

RMSE (mm)**

MBE (mm)EF

ANFIS1ANFIS2ANFIS3ANFIS4ANFIS5ANFIS6ANFIS7ANFIS8E, Kc, In
E, Kc, WD, In
cGDD, E, Kc, In
Kc, T, In
Kc, WD, T, In
cGDD, Kc, T, In
cGDD, Kc, In
cGDD, Kc, WD, In

Gb
Gb
Gus2
Gus1
Gb
Gb
Gus2
Gus2

5
2
2
2
2
2
2
2

22.45
4.73
3.21
8.17
4.87
2.92
2.71
1.67

2.61
0.03
0.66
0.40
0.23
0.01
0.02
0.11

0.45
0.98
0.99
0.93
0.98
0.99
0.99
1.00

*
MF: Membership function, NMF: Number of membership function, Gb: Generalized bell-shaped built-in MF, Gus1: Gaussian curve built-in MF, Gus2: Gaussian combination MF.
**
RMSEs and MBEs are calculated for equivalent root zone water depth (EWDrz).

(RMSE = 1.67 mm, MBE = 0.11 mm, and EF = 1), and ANFIS7
ranked second (RMSE = 2.71 mm, MBE = 0.02 mm, and EF = 0.99).
ANFIS6, with inputs of cGDD, Kc, GDD, and In, ranked third. In contrast, ANFIS1, with input parameters Kc, E, and In, had the highest
error when simulating EWDrz (RMSE = 22.45, MBE = 2.61, andEF = 0.45). Based on the results summarized in Table 6, it seems
that crop parameters and factors representing soil stresses play amore important role in temporal variations of EWDrz, compared
to atmospheric parameters such as T and E. As discussed in the literature, crop factors are the most dominant factors, controllingroot water uptake under stress conditions, which consequently
controls SWCs (Allen et al., 1998).
As shown in Table 6, all ANFIS models underestimated EWDrz,
except for ANFIS5 and ANFIS8, which slightly overestimated EWDrz.
However, a visual inspection of scatter plots in Fig. 6, comparing
observed and ANFIS-estimated SWCs (averaged over the rootzone), clearly indicates the high potential of the ANFIS modeling.
Notice the high values of R2 in Fig. 6 for several ANFIS models.

The SVM models were similarly implemented using a MATLAB
code. First, different SVM architectures were tried. Once the appropriate model structures were determined for each combination of
input variables, the resulting SVM models were tested against
the experimental dataset, and the results were compared using
the performance statistics (Table 7). In the SVM modeling, anappropriate choice of kernels allows the data to become separable
in the feature space despite being non-separable in the original
space. This allows one to obtain non-linear algorithms from algorithms previously restricted to handling linearly separable data
sets (Bray and Han, 2004). Here, the radial basis function was thebest kernel for all SVM models.
With respect to RMSE, the SVM8 model, with RMSE = 1.9 mm,
MBE = 0.7 mm, and EF = 1, had the best performance in simulating
EWDrz, followed by SVM3 (RMSE = 2.44 mm, MBE = 0.28 mm, andEF = 0.99) and SVM7 (RMSE = 2.66 mm, MBE = 0.15 mm, andEF = 0.99). These three models had nearly the same performancestatistics and thus the selection of one of these models over the

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F. Karandish, J. Šimůnek / Journal of Hydrology 543 (2016) 892–909

Fig. 6. Scatter plots between observed and simulated SWCs (averaged over the root zone) for the ANFIS models.

Table 7
The final architecture and performance statistics of the SVM models for the testing phase. The same combination of input variables were used in the SVM and MLR models (seeTable 5).

*

Model

Input variables

Best Kernel

RMSE (mm)*

MBE (mm)

EF

SVM1
SVM2
SVM3
SVM4
SVM5
SVM6
SVM7
SVM8

E, Kc, In
E, Kc, WD, In
cGDD, E, Kc, In
Kc, T, In
Kc, WD, T, In
cGDD, Kc, T, In
cGDD, Kc, In
cGDD, Kc, WD, In

Radial Basis Function
Radial Basis Function
Radial Basis Function
Linear Function
Radial Basis Function
Radial Basis Function
Radial Basis Function
Radial Basis Function

21.33
3.96
2.44
8.45
4.75
3.05
2.66
1.90

2.39
0.43
0.28
0.45
0.37
0.21
0.15
0.27

0.51
0.98
0.99
0.93
0.98
0.99
0.99
1.00

RMSEs and MBEs are calculated for equivalent root zone water depth (EWDrz).

other should be dependent upon available data. Since pan evaporation (E) is not always available for all weather stations, the SVM8
and SVM7 models may be easier to use when limited data are
available. SVM6, with RMSE = 3.05 mm, MBE = 0.21 mm, andEF = 0.99, also had an acceptable accuracy for estimating EWDrz.
Similar to the MLR and ANFIS models, the combination of E, Kc,
and In for SVM1 led to the lowest accuracy in simulating EWDrz(RMSE = 21.33 mm, MBE = 2.39 mm, and EF = 0.51). However, thescatter plots between observed and simulated SWCs (again, averaged over the root zone) and their high R2 demonstrate the highpotential of the SVM modeling (Fig. 7).
3.4. Comparison of different models
A comparison of observed SWCs and SWCs estimated using theMLR8, ANFIS8, and SVM8 models for the FI, PRD75, and PRD55 treatments in 2010 is illustrated in Fig. 8 in the form of time series andscatter plots. It should be noted that these particular models had
the best performance in their groups. Regardless of the type ofmodel, a better performance was found for the stressed conditions(i.e., PRD75 and PRD55 treatments) than for the unstressed
conditions with full irrigation. This result is contrary to HYDRUS2D, which had better results for the FI treatment. Negative bias isalso obvious for almost all days during the simulation process. Furthermore, a visual inspection of Fig. 8 shows that the ANFIS and
SVM models had a better performance than the MLR models for
all treatments, which is also reflected by the R2 index. These models very well reproduced not only the temporal variations, but also
the actual daily values of SWCs. Especially, the response of the
MLR8 model to irrigation events is less dynamic than observed.
This is especially visible for the FI treatment and the wetted sidesof the PRD treatments. This documents that the MLR8 model could
not well describe non-linear changes in SWCs during irrigation
events. This also confirms the RMSE, MBE, and EF statistics given
in Tables 5–7.
The performance of different modeling approaches to predict
EWDrz is ranked in Table 8 based on the RMSE statistics. Being
physically based, the HYDRUS-2D model ranked first with respectF. Karandish, J. Šimůnek / Journal of Hydrology 543 (2016) 892–909

903

Fig. 7. Scatter plots between observed and simulated SWCs (averaged over the root zone) for the SVM models.

to RMSE. Previous studies have also demonstrated the high potential of HYDRUS-2D for simulating SWCs (e.g., Cote et al., 2003;
Ajdary et al., 2007; Rahil, 2007; Crevoisier et al., 2008; Siyal andSkaggs, 2009; Mubarak, 2009; Tafteh and Sepaskhah, 2012). A highaccuracy of HYDRUS-2D is to be expected since it is a physically
based model that solves numerically by means of the finite element method the Richards equation, which describes the highly
non-linear, variably-saturated water flow in soils. This approachrequires knowledge of the soil water retention curve and thehydraulic conductivity function, which are rarely known.
Table 8 shows that the ANFIS8 model ranked second(RMSE = 1.67 mm, MBE = 0.11 mm, and EF = 1.00) with respect
to the RMSE index, followed by the SVM8 (RMSE = 1.9 mm,
MBE = 0.27 mm, and EF = 1.00), SVM3 (RMSE = 2.44 mm,
MBE = 0.28 mm, and EF = 0.99) and SVM7 (RMSE = 2.66 mm,
MBE = 0.15 mm, and EF = 0.99) models. These models are then followed by ANFIS7 and ANFIS6, and further by SVM6, ANFIS3, SVM2,ANFIS2, SVM5, ANFIS5, ANFIS4 and SVM4. Overall, 14 best performing models for the estimation of EWDrz under stressed conditions were the ANFIS and SVM models, which performed far better
than the MLR models. Exceptions were the SVM1 and ANFIS1 models, which by ignoring the WD parameter had lower efficiency inestimating EWDrz, especially under stressed conditions. Such
results demonstrated that the performance of the ANFIS and SVM
models is highly dependent on the selected input datasets. Table 8shows that the MLR models occupy the last places in terms of theirperformance in simulating EWDrz. Using the MLR4 model with
input parameters of Kc, GDD, and In led to the worst results.
The ANFIS models combine the transparent linguistic
representation of a fuzzy system with the learning ability ofANN. Therefore, they can be trained to perform an input/output
mapping similar to ANN while providing an additional benefit ofbeing able to consider a set of rules on which the model is based.

This provides a further insight into the process being modeled(Sayed et al., 2003).
The main advantage of using the SVM models is their flexibility
and ability to model non-linear relationships. Furthermore, theSVM training process always seeks a global optimized solution
and avoids over-fitting, which eventually leads to a better general
performance than the ANN models. The SVM models are able toselect the key vectors in the training process, which includes support vectors and automatically removes the non-support vectors
from the model. This helps the model cope well with noisy conditions. The main disadvantage of the SVM technique is that it has no
physical basis. In addition, the SVM approach can only be used
when the training dataset is available (Bray and Han, 2004; Zhou
et al., 2009; Kisi and Cimen, 2009).
3.5. A further assessment of different models
The performance of different models was reevaluated using theobserved SWCs data from 2011. The results of the statistical comparisons between the HYDRUS-2D, MLR, ANFIS, and SVM models
are summarized in Table 9, where models are ranked with respect
to the RMSE index. Similar to 2010, the best performance criteria
for simulating EWDrz were obtained by HYDRUS-2D with
RMSE = 0.66 mm, MBE = 0.81 mm, and EF = 0.99. Thereafter, theSVM and ANFIS models generally provided more accurate EWDrz
estimations compared to the MLR models. However, the SVM1
and ANFIS1 models performed slightly worse than the MLR models. Overall, Table 9 indicates that the 15 best models for simulating EWDrz in 2011, apart from HYDRUS-2D, were the ANFIS and
SVM models, similar to 2010. A visual inspection of observed andsimulated SWC time series in Fig. 9 also demonstrated better performance of the ANFIS and SVM models compared to the MLR
models. This validation demonstrates that these models could be904

F. Karandish, J. Šimůnek / Journal of Hydrology 543 (2016) 892–909

Fig. 8. Average observed and estimated SWCs over the rooting depth (0–80 cm) using the MLR3, SVM3, and ANFIS8 models for different treatments in 2010.

905

F. Karandish, J. Šimůnek / Journal of Hydrology 543 (2016) 892–909
Table 8
Summary of the ranking of selected models (based on the testing phase in 2010).
Rank*

Model

Considered parameters**

RMSE (mm)***

MBE (mm)

EF

1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25

HYDRUS-2D
ANFIS8
SVM8
SVM3
SVM7
ANFIS7ANFIS6SVM6ANFIS3SVM2ANFIS2SVM5ANFIS5ANFIS4
SVM4
MLR8
MLR3
MLR2
MLR5
MLR6
MLR7
MLR4
SVM1
ANFIS1MLR1physically-based modelcGDD, Kc, WD, In
cGDD, Kc, WD, In
cGDD, E, Kc, In
cGDD, Kc, In
cGDD, Kc, In
cGDD, Kc, T, In
cGDD, Kc, T, In
cGDD, E, Kc, In
E, Kc, WD, In
E, Kc, WD, In
Kc, WD, T, In
Kc, WD, T, In
Kc, T, In
Kc, T, In
cGDD, Kc, WD, In
cGDD, E, Kc, In
E, Kc, WD, In
Kc, WD, T, In
cGDD, Kc, T, In
cGDD, Kc, In
Kc, T, In
E, Kc, In
E, Kc, In
E, Kc, In

1.02
1.67
1.90
2.44
2.66
2.71
2.92
3.05
3.21
3.96
4.73
4.75
4.87
8.17
8.45
10.46
12.00
12.20
12.24
13.44
14.78
17.13
21.33
22.45
25.26

1.45
0.11
0.27
0.28
0.15
0.02
0.01
0.21
0.66
0.43
0.03
0.37
0.23
0.40
0.45
6.86
3.98
0.34
0.18
7.14
9.48
3.16
2.39
2.61
1.37

0.99
1.00
1.00
0.99
0.99
0.99
0.99
0.99
0.99
0.98
0.98
0.98
0.98
0.93
0.93
0.90
0.86
0.85
0.85
0.82
0.79
0.71
0.51
0.45
0.34

*

Different models are ranked with respect to RMSE.
DAS: days after planting (d), Kc: crop coefficient (dimensionless), WD: water deficit (dimensionless), In: irrigation depth (mm), E: evapotranspiration (mm), GDD: growthdegree day (°C).
***
RMSEs and MBEs are calculated for equivalent root zone water depth (EWDrz). The RMSE values for HYDRUS-2D were obtained by averaging the values for all individual
soil layers which are given in Table 4**

Table 9RMSEs, MBEs and EFs for the selected models during the validation phase, i.e., the 2011 growing season.
Rank*

Model

Considered parameters**

RMSE (mm)**

MBE (mm)

EF

1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25

HYDRUS2DSVM8ANFIS8SVM3ANFIS6SVM7ANFIS7ANFIS3
SVM6
SVM2
SVM5
ANFIS2ANFIS5ANFIS4
SVM4
MLR2
MLR5
MLR3
MLR8
MLR6
MLR4
MLR7
SVM1
ANFIS1
MLR1

Physically-based modelcGDD, Kc, WD, In
cGDD, Kc, WD, In
cGDD, E, Kc, In
cGDD, Kc, T, In
cGDD, Kc, In
cGDD, Kc, In
cGDD, E, Kc, In
cGDD, Kc, T, In
E, Kc, WD, In
Kc, WD, T, In
E, Kc, WD, In
Kc, WD, T, In
Kc, T, In
Kc, T, In
E, Kc, WD, In
Kc, WD, T, In
cGDD, E, Kc, In
cGDD, Kc, WD, In
cGDD, Kc, T, In
Kc, T, In
cGDD, Kc, In
E, Kc, In
E, Kc, In
E, Kc, In

0.66
1.27
1.82
2.27
2.36
2.37
2.46
2.48
2.59
4.40
4.52
5.02
5.84
8.75
9.88
13.41
13.59
16.09
16.73
17.01
19.32
21.76
26.32
26.32
26.32

0.81
0.07
0.07
0.19
0.00
0.28
0.17
0.15
0.40
0.03
0.38
0.11
0.40
0.45
3.07
0.61
1.65
9.72
14.01
10.65
3.58
17.27
5.71
5.71
5.71

0.99
1.00
1.00
1.00
1.00
1.00
1.00
1.00
0.99
0.99
0.98
0.98
0.97
0.93
0.91
0.86
0.85
0.81
0.83
0.79
0.70
0.64
0.36
0.36
0.36

The RMSE values for HYDRUS-2D were obtained by averaging the values for all individual soil layers which are given inTable 4*
Different models are ranked with respect to RMSE.
**
RMSEs and MBEs are calculated for equivalent root zone water depth (EWDrz).

useful substitutes for HYDRUS-2D for optimizing the managementof water resources under stressed conditions and with limited data
availability in agriculture.
3.6. Uncertainty analysis
Predictions of SWCs immediately before irrigation events are
more important for the purpose of irrigation scheduling than ofthose at other times. Therefore, errors in SWCs and theirassociated uncertainties before and after irrigation events were
estimated with different models separately (Fig. 10). Absolute
errors in estimated SWCs immediately before irrigation eventsusing the MLR, ANFIS, SVM, and HYDRUS-2D models ranged
between
1.3–17.5%,
0.4–10,0.2–10.5%,
and
1.1–2.8%,
respectively. For times after irrigation events, the MLR, ANFIS,
SVM, and HYDRUS-2D models estimated SWCs with errors

906

F. Karandish, J. Šimůnek / Journal of Hydrology 543 (2016) 892–909

Fig. 9. SWC values observed and estimated using the MLR3, SVM8, and ANFIS6 models for different treatments in 2011.

F. Karandish, J. Šimůnek / Journal of Hydrology 543 (2016) 892–909

907

Fig. 10. Errors in estimating SWCs with different models for times immediately before (left) and after (right) irrigation events (Min, Max, and SD denote the minimum,
maximum, and standard deviation, respectively).

between
1.7–17.7%,
0.7–8.2%,
0.3–8.7%,
and
1.1–3.1%,
respectively.
Fig. 10 shows that the results simulated by HYDRUS-2D have alower uncertainty domain of estimated SWCs than those obtainedby the other models and are thus more reliable. Even though the
ANFIS and SVM models have absolute errors that are similar to
those obtained by HYDRUS-2D, they predict SWCs both before
and after irrigation events with higher uncertainties than
HYDRUS-2D. Nevertheless, the ANFIS and SVM models performedmuch better than the MLR models in estimating SWCs and associated uncertainties. It can be concluded that although HYDRUS-2D
predictions of temporal variations of SWCs in the root zone aresuperior to those made by the ANFIS and SVM models, these models can be reliably used for predicting SWCs when there is a lack ofdata required by HYDRUS-2D.

4. Conclusions
Since agriculture is known to be the biggest consumer of fresh
water, sustainable management of water resources for irrigation
use has become an important goal at the basin scale worldwide.
In this regard, modeling tools for a quick estimation of SWCs andfor determining the real crop water demand, especially under

new water-saving irrigation strategies, will help farmers improve
the water and crop productivity. In this regard, introducing efficientmethods for estimating SWCs when only limited information isavailable would be useful for irrigation scheduling, even in theplanning stage when the possible effects of new water-saving irrigation strategies are evaluated. Therefore, in this research, the possibility of indirect estimation of SWCs under water stress conditions
was investigated by means of both numerical and machinelearning models, including the HYDRUS-2D, MLR, ANFIS, and SVM
models, and using a dataset from a two-year field investigation.
Being a physically-based model, HYDRUS-2D ranked first in
simulating time series of SWCs at the field scale for selected irrigation strategies. However, both ANFIS and SVM models could be
successfully used for predicting SWCs when there is a lack of input
data required by HYDRUS-2D. When suitable input datasets
selected based on the correlation analysis were available, thesemodels estimated SWCs with an accuracy that approached thephysically based model (Tables 8 and 9). In addition, these models
performed especially well compared to HYDRUS-2D for water
stressed conditions, although less well for conditions with full
irrigation.
Using simple atmospheric data (E, T), crop parameters (cGDD,
Kc), and stress descriptor parameters (In, WD), the ANFIS, andSVM models showed a high potential for modeling SWCs with

908

F. Karandish, J. Šimůnek / Journal of Hydrology 543 (2016) 892–909

acceptable errors. Therefore, when suitable input datasets areavailable, the ANFIS and SVM models can be used for irrigation
scheduling and for management of agricultural water resources.
However, these models have to first be trained using a training
dataset that should include as many data and as many conditions
as possible so that they can take into account unusual events andobtain good accuracy in their predictions. The models introducedin this research could be used in a wide range of applications since
they were developed using only simple input parameters, which
could be easily determined even in projects with limited resources.

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