Comparison and Estimation of Four Infiltration Models

Infiltration is an important component of the hydrological cycle. It provides soil moisture in the vadose zone to support plant growth. This study was conducted to compare the validity of four infiltration models with measured values from the double ring infiltrometer. The parameters of the four models compared were estimated using the linear regression analysis. The C.C was used to show the performance of the predictability of the models. The RMSE, MAE and MBE were employed to check the anomalies between the predicted and the observed values. The results showed that, average values of the C.C ranged from 0.9294 - 0.9852. The average values of the RMSE were 4.0033, 17.489, 11.2400 and 49.8448; MAE were 3.1341, 15.9802, 10.6525, and 61.4736; and MBE were 0.0786, 9.5755, −0.0007 and 47.0204 for Philip, Horton, Green Ampt and Kostiakov respectively for the wetland soils. Statistical results also from the Fisher’s multiple comparison test show that the mean infiltration rate estimated from the Green Ampt’s, Philip’s and Horton’s model was not significantly soils that exhibit similar characteristic as Besease wetland soils.


Introduction
Infiltration is the process by which water on the ground surface enters the soil.
Infiltration plays a vital role in soil and water conservation as it determines the amount of runoff over the soil surface during irrigation and precipitation. The infiltration rate of a soil, thus ability of the soil to accept heavy rainfall or irrigation depends on the characteristics of the soil [1] [2]. Substantial reduction in time and cost of field measurement of infiltration can be achieved by using infiltration models [3]. Poor infiltration rate indicates potential of high runoff and erosion which affects the amount of water stored in the plant root zone [4]. This makes it difficult for the soil to meet the required water demand for crop production.
Design, operation and management of surface irrigation system rely greatly on the infiltration behaviour or characteristics of the soil, because the infiltration behaviour of the soil directly determines the essential variables such as inflow rate, length of run, application time and depth of percolation [5]. These infiltration characteristics of the soil are determined when fitted mathematically into infiltration models. But not all infiltration models can be applied to all soils [1].
Many researchers have compared the accuracy of the various models by comparing the computed and observed infiltration rates. Under different conditions, a particular model shows better predictions than others. But till date, it is not specifically mentioned which model gives the best prediction [6]. [7] estimated and compared Kostiakov, Novel and the Modified Kostiakov infiltration models in the Kurukshetra district of India. They concluded that, the Novel model was more accurate in predicting infiltration rate. [8] investigated the capability of the novel infiltration model in estimating the infiltration rate from actual field data in comparison to Philip, Kostiakov and Modified Kostiakov models in similar conditions. Findings from their research indicated that the novel model was the most suitable among three other models used for the estimation of infiltration rate of the study area. [9] also compared Philip's model, modified Philip's model, Horton's model, and Green Ampt's model in NIT Kurukshetra campus in India at ten different locations to predict infiltration rates and found out that, the infiltration rate versus time plot for the field data and predicted data did not accurately match; but the Modified Philip's model was much closer to the observed field data. [10]

Study Area
Besease is a predominantly farming area in the Ejisu Municipal District of the  (Figure 1). The climate of the study area is mostly related to the semi-humid type. The region is characterised with two distinct seasons, the wet season which begins from April and ends in October while the dry season extends from the month of November to March. The wet seasons can be categorised less than two rainy seasons. The major rainy season which ranges from mid-March to July and the minor rainy season starts from September to mid-November. The mean annual rainfall is 1420 mm; mean monthly temperature is 26.5˚C, the relative humidity ranges from 64% in January to 84% in August. The average monthly maximum and minimum evapotranspiration (ETo) for the study area were 127.5 mm and 64.7 mm and has an annual ETo of 1230 mm. The area is drained by the Oda River which is seasonal and whose basin is about 143 km 2 [12].
The study area is located in the moist semi-deciduous forest zone. Grass spe-

Sample Collection
Soil samples were collected with core samplers of height 10 cm to an average depth of 100 cm ( Figure 1). Disturbed soil samples were taken from the field at site P1 -P2, P1 -P9, P6 -P9, P7 -P8, P11 -P14, and P13 -P4 and air dried, ground and passed through the 2 mm sieve to obtain the soil fractions for the determination of soil texture.

Measurement of Infiltration Rates
Double ring infiltrometers, consisting of two concentric rings, were used to measure the infiltration rate. Rings were 250 mm deep and were made from 12-guage steel with sharpened bottom edges. They were driven into the ground to 50 mm depth. Grass was cut to near soil level and a pad was placed inside the inner ring to prevent puddling. The inner and outer edges were tamped to seal possible cracking. Generally, the water level was kept at or above 50 mm depth. The difference in height between the inner and outer rings was kept to a minimum. The rate of fall of water was measured in the inner ring while a pool of water was maintained at approximately the same level in the outer ring to reduce the amount of lateral flow from the inner ring. The rate of fall of the water level in the inner cylinder was measured at 2, 3, 5, 10, 15, 20, 30, 45 and 60 minutes and at 30-minute intervals thereafter. The accumulated volume of water entering the soil was converted to the infiltration rate (mm/h) and was plotted against elapsed time whereby a declining slope was obtained. Fifty-five (55) samples (replicates) were used for the measurement of soil infiltration rate. The field infiltration rate measurement was considered as observed. The aim of the measurements was to obtain a steady-state infiltration rate. This is achieved when the amount of infiltrated water was constant in time, i.e. when the infiltration curve (instantaneous infiltration against time) levels out. To estimate the infiltration rate at steady state, the terminal infiltration rate (i.e. the infiltration rate obtained at the end of the experiment in about 2 h), was used as an approximation of the steady state infiltration rate.

Infiltration Models and Parameters
In this study, Kostiakov's, Philip's, Horton's and Green Ampt's infiltration models were fitted to the infiltration data.

Kostiakov's Model [13]
Kostiakov's model, an empirical model expresses cumulative infiltration equa- where p F = cumulative infiltration (cm), t = time from start of infiltration (min), and a and b are constants that depends on the soil initial conditions.
The parameters in the Kostiakov equation are obtained from the plot of ( ) ln t and the best fit straight line through the plotted points gives as the intercept and b as the slope.

Philip's Model [14]
Philip's physical based model expresses infiltration rate as

Horton's Model [15]
Horton's semi-empirical model expressed the decay of infiltration capacity with time as an exponential decay given by

Green Ampt's Model [16]
Green Ampt proposed a model for infiltration capacity based on Darcy's law and expresses the physical model as

Coefficient of Correlation (C.C)
Coefficient of correlation is a statistical measure that calculates the strength of the relationship between the relative movements of two variables. The coefficient of correlation is calculated as

Root Mean square Error (RMSE)
The root mean square error exaggerates the prediction error, thus the difference between the predicted value and the actual value. This is evaluated by where a is the calculated and b is observed values of the infiltration rate and N is the number of observations.

Mean Bias Error (MBE)
This is the average difference between the predicted values and the observed values of the infiltration models. The mean bias error is estimated by where a is the calculated and b is observed values of the infiltration rate and N is the number of observations.

Mean Absolute Error (MAE)
The absolute error is the absolute value of the difference between the predicted value and the observed value. The absolute error is estimated by where a is the calculated and b is observed values of the infiltration rate and N is the number of observations.

Results and Discussion
Results from Table 1 shows that the final infiltration rate for the studied site Comparing the predicted and measured infiltration rate, the average values of C.C shown in Table 3  performance. This may be due to the fact that their parameters lack a consistent physical interpretation and also the process involved in the evaluation of the parameters might be very sensitive to approximation errors and errors due to parallax while determining the initial and steady state infiltration rates from the graph as inputs for the prediction of cumulative infiltration [17]. However, the    [20]. One or few of the infiltration models are better and for a specific site condition [21], [1] which presupposes that not all models are applicable in all soils. Consequently, the application of these models under verified field conditions leads to the determination of the appropriate infiltration characteristics for the equations that would optimize infiltration simulation, irrigation performance and minimize water wastage [12].

Conclusions
The prediction accuracy of four infiltration models was validated with measured values using the double ring infiltrometer. Comparison of the field and predicted infiltration rate indicated that the infiltration rate predicted by the Philip's model was much closer to the observed data.