INTRODUCTION
Infiltration has an important role in landsurface and subsurface hydrology,
runoff generation, soil erosion, irrigation rate. In addition, the infiltration
rate of the soil is influence by various factors depending on the condition
of soil surface, its chemical and physical properties (Siyal
et al., 2002). Also water infiltration is an index of soil compaction
(Younesi Alamouti and Navabzadeh, 2007). Hence, infiltration
process modeling have been considered through the past century (Kostiakov,
1932; Smith, 1972; Philip, 1957;
Mein and Larson, 1973; Kao and Hunt,
1996; Argyrokastritis and Kerkides, 2003) and there
are a large number of models for its computation. An accurate infiltration model,
predicting the real infiltration correctly, is required to estimate the runoff
initiation time, planning of irrigation systems and management of water resources.
Through the past century, several infiltration models have been developed and
categorized as physicallybased, semiempirical and empirical (Mishra
et al., 1999).
Philip model: Philip (1957) developed an infiniteseries
solution to solve the nonlinear partial differential Richards’ equation
which describes transient fluid flow in a porous medium. For cumulative infiltration,
the general form of the Philip model is expressed in powers of the squareroot
of time as:
where, I is the cumulative infiltration (L), S is the sorptivity (LT^{1/2}), t is the time of infiltration (T), and A is a parameter with dimension of the saturated hydraulic conductivity (LT^{1}).
Swartzendruber model: Swartzendruber (1987)
presented an infiltration model:
where, f_{c} is the final infiltration rate (LT^{1}), c and d are empirical constants.
Horton model: Horton (1940) presented a threeparameter
semiempirical infiltration model expressed as:
where, C is the final infiltration rate (LT^{1}). Parameters c, m (LT^{1}), and a (T^{1}) must be evaluated using observed infiltration data.
Kostiakov model: A simple and general form of infiltration model presented
by Kostiakov (1932) is:
where, α_{1} and β_{1} are constants and evaluated using the observed infiltration data.
Modified Kostiakov model (MK): Kostiakov was modified by adding the
term of ultimate infiltration capacity (α_{3}) by Smith
(1972), as follow:
where, α_{3} is the final infiltration rate (LT^{1}), and α_{2} and β_{2} are the same as α_{1} and β_{1} at the Kostiakov model.
Revised modified Kostiakov model (RMK): Recently, Parhi
et al. (2007) revised Modified Kostiakov model and obtained a four
parameters model as:
where, α_{4}, β_{3}, α_{5} and β_{4} are parameters to be determined empirically, using measured infiltration data.
SCS model: Experts of US Department of Agriculture,
Natural Resources and Conservation Service (1974) found that the Kostiakov
model did not apply at long time. They have done many experiments and concluded
that coefficient of 0.6985 to be added to this model which would work well at
all times:
where, a and b are constants and evaluated using the observed infiltration data.
There are several approaches for selection of a suitable model. One of the
simplest approaches is minimizing the difference between observed and predicted
data to find the best model. For example, a model with higher R^{2}
may be preferred more than one with smaller R^{2}. Gifford
(1976) and Machiwal et al. (2006) used the
coefficient of determination (R^{2}) to compare infiltration models.
Mishra et al. (2003) examined the suitability
of the infiltration models with coefficient of efficiency. Turner
(2006) and Dashtaki et al. (2009) used both
the coefficient of determination (R^{2}) and the Mean Root Mean Square
Error (MRMSE) to select the best infiltration model. AbdelNasser
et al. (2007) used correlation coefficient to compared Kostiakov
and Philip models.
The objectives of this study were to test a variety of models with different
underlying assumptions to determine which model represents best the soil infiltration.
To achieve these objectives, we compared seven models described above. Three
comparison techniques were considered to define the best models: the coefficient
of determination (R^{2}), Mean Root Mean Square Error (MRMSE), Root
Mean Square Error (RMSE statistic).
MATERIALS AND METHODS
The infiltration data were obtained by Double Rings method from 95 locations
in 5 different provinces with different climates in Iran. The soil of these
regions are classified as Mollisols, Aridisols, Inceptisols and Entisols in
soil taxonomy (Soil Survey Staff, 2010). The infiltration
experiments were conducted until infiltration rate reached a constant value
for each soil. However, the minimum required time for infiltration measurement
was 270 min. Each infiltration measurement was replicated three times, using
double ring apparatus with outer and inner diameters of 70 and 30 cm, respectively.
The soil texture class of surface horizons were determined as clay loam, silty
loam, loam and silty clay loam.
In this study, the RMK model with four fitting parameters was used as a referenced model to compare with results from the other six models (i.e., Philip, SCS and Kostiakov with two parameters; Horton, Swartzendruber and MK with three parameters).
The RMSE statistic is an index of the correspondence between measured and predicted data and has frequently been used as a means of evaluating the accuracy of models. The mean RMSE (MRMSE) values and mean R^{2} values of all soils for each model were calculated. The model having the smallest MRMSE value and highest mean R^{2} was selected as best model. RMSE and R^{2} statistics were calculated as follow:
where, I_{o}, I_{p} and N are observed, predicted and number of observed data values, respectively.
All models were fitted to experimental infiltration data using an iterative nonlinear regression procedure which finds the values of the fitting parameters that give the best fit between the model and the data. This procedure was done using the Matlab 7.11 software.
RESULTS AND DISCUSSION
A statistic of the estimated parameter values (minimum, maximum and mean) of the infiltration models is given in Table 1.
R^{2} ranged from 0.77 to 1.00 among all of the soils and all of the
models (Table 1, Fig. 1). The greatest amounts
of R^{2} values were obtained with RMK, MK and Swartzendruber models.

Fig. 1: 
Box plot for R^{2} percentiles as the goodness of
fit of seven models for all soils. PH = Philip, KO = Kostiakov, HO = Horton,
SW = Swartzendruber, MK = Modified Kostiakov and RMK = Revised modified
Kostiakov models 
Table 1: 
Statistics of optimized parameters of the infiltration models
for all soils 

The SCS model with two parameters yielded the lowest R^{2}. Among the
models with three parameters, MK and Swartzendruber, had mean R^{2}
higher than that of Horton model. Among the models with two parameters, mean
R^{2} was higher for the Kostiakov model than that for Philip and SCS
models. The range of R^{2} values of each model obtained by curvefitting
is presented in Fig. 1. Considering mean R^{2}, RMK,
MK and Swartzendruber models were selected as the best models for all soils.
According to the results obtained from mean of RMSE (MRMSE) values (Table
1), the MK model provided the lowest values, indicating that infiltration
was well described by this model. Parlange and Haverkamp
(1989), Dashtaki et al. (2009) and Araghi
et al. (2010) reported that the MK model is the best model for quantifying
the infiltration process compared to the other infiltration models.
Table 2: 
Ranks of infiltration models using the results criteria 

A comparison model was accepted for the soil if its RMSE was smaller than RMSE of reference model. Considering RMSE criterion, the RMK model was the best in 67 out of 95 soils (70.6%) compared with the MK and higher number of soils for other models.
The results of ranking models according to MRMSE and R^{2} statistics Table 1 given in Table 2.
According to the R^{2} statistic the goodness of cumulative infiltration can be estimated by the RMK, MK, Swartzendruber, Kostiakov, Horton, Philip, SCS models, respectively.
Based on the results of ranking model given in Table 2 the
CSC model obtained the lowest ranking between the all of the models and all
of the criteria. Dashtaki et al. (2009) reported
a better performance for Horton model than Kostiakov and Philip models. This
finding is different with that obtained by Dashtaki et
al. (2009) and Parhi et al. (2007) reported
a better performance for RMK model than MK and Kostiakov models. These results
are concordant with that obtained by Parhi et al.
(2007). The mean RMSE indicated different pattern in term of model ranking.
The mean RMSE reveals that the correspondence between measured and predicted
infiltration is highest for MK model and lowest for SCS model. The results of
present study indicated that the empirical models had best fit on the double
ring data, because they are on the basis of data derived from field experiments
without any preassumptions. However, Swartzendruber model showed better performance
than Kostiakov, Horton and SCS empirical models. Based on the results of ranking
model given in Table 2 the CSC model obtained the lowest ranking
between the all of the models.
As a result of paired ttests on the RMSE, we found that several pairs of models performed identically based on RMSE statistics. PhilipSwartzendruber, PhilipHorton and SwartzendruberMK pairs performed identically based on RMSE statistics which showed that pairs of models made no great differences in predicted values at measurement points of cumulative infiltration in most of soils. This result indicated that models within each pair (Philip Swartzendruber, PhilipHorton and SwartzendruberMK) might have statistically identical performance even though the equations are different.
Effect of soil texture on performance of models: The R^{2 }percentiles are shown for various soil textures in Fig. 2. The R^{2} percentiles were higher for the RMK, MK, Swartzendruber, Kostiakov models than other models for loam soils. The R^{2} percentiles of RMK models were very close to 1.00 for majority of silty clay loam soils. In addition, for silt loam and clay loam soils RMK and MK models had a higher R^{2}. Using R^{2} analysis, the RMK and MK were the best models for silt loam and clay loam soils and RMK, MK, Swartzendruber and Kostiakov were the best models for loam and silty clay loam soils. Other models were smaller number of soils better than RMK model.

Fig. 2: 
Box plot for R^{2} percentiles as the goodness of
fit of seven models for soil textural classes. PH = Philip, KO = Kostiakov,
HO = Horton, SW = Swartzendruber, MK = Modified Kostiakov and RMK = Revised
modified Kostiakov models 
CONCLUSIONS
The results of this study indicated that all seven models account for >90% of the variance (R^{2}) in cumulative infiltration of majority of soils. Based on the mean RMSE and R^{2} the MK model was the best model for prediction cumulative infiltration and SCS with two parameters was the worst model.
Using paired ttests for RMSE, we found that PhilipSwartzendruber, PhilipHorton and SwartzendruberMK model pairs could be considered to perform identically at the 95% significance level.
Texture of soil could affect the performance of cumulative infiltration models. Among four soil classes, the RMK model with four parameters showed better fits for loam, clay loam and silty clay loam soils and worse fit than MK model for silt loam soils.