Estimation of Infiltration Rate using a Nonlinear Regression Model
Abstract
The primary abstraction of precipitation is infiltration. Accurate estimation of the infiltration rate is helpful in determining the direct runoff, groundwater recharge, water availability, and degree of saturation of soil. In this study, infiltration rate was estimated for the soils in Vishakhapatnam, Andhra Pradesh, India. The agriculture sector in the study region contributes to about 12% of the total GDP of the state. The study region faces an acute shortage of water and hence, accurate estimation of infiltration rates is vital. In this regard, the application of an Excel Solver was explored in estimating the infiltration rates. The results of the infiltration rates obtained through the Excel Solver were compared with the conventional Graphical method. In the present study, the best infiltration model was selected using the sum of square error (SSE), Maximum Absolute Error (MAE), Wilmot Index (WI), Mean Absolute Relative Error (MARE), and NashSutcliffe Efficiency (NSE). Horton’s model was found to be the bestfit model for the Samantha Hostel, NCC Building, Assembly Hall, and CSE building. For the dispensary, the Kostiakov model was found to be the bestfit model of the study area. Upon application of the Excel Solver, the SSE was reduced by 70.48%, 39.55%, and 87.39% for the Kostiakov, Horton, and SCS models, respectively, as compared to the Graphical method. The WI improved by 5.68%, 36.39%, and 17.85% for the Kostiakov, Horton and SCS models, respectively, for the Excel Solver model as compared to the Graphical method. Based on this analysis, the Excel Solver was found to be a reliable tool for determining the parameters of nonlinear equations, and therefore its application should be explored in different fields of engineering.
1 Introduction
Infiltration is the process by which water enters the soil surface of earth (Zakwan et al. 2016a; Hillel and Baker 1988; Patle et al. 2019). Infiltration rate during rainfall plays a vital role to determine the quantity of runoff over the top layer of soil, for designing irrigation structures, estimation of groundwater recharge, and estimation of contaminant transport in surface and subsurface water (Vieira et al. 1981; Liu et al. 2001).
Various parameters like type of soil, temperature, vegetation on the soil surface, rainfall intensity and duration are factors that influence the infiltration rate. SakellariouMakrantonaki et al. (2016) observed that hydraulic conductivity of soil is one of the most important parameters controlling the movement of water into the ground, thereby influencing the irrigation demand and groundwater potential. Angelaki et al. (2021) asserted that sorptivity is an important soil property that influences infiltration and its effect on cumulative infiltration increases as the infiltration process is prolonged.
Due to these reasons, modeling and prediction of infiltration rates are an inevitable part of hydrological modeling. Considering this, a number of infiltration models have been proposed and developed by various scientists for determination of the infiltration rate (Turner 2006; Haghighi et al. 2010; Hasan et al. 2015; Zakwan 2017; Singh et al. 2018; Gundalia 2018; Rajasekhar et al. 2018; Dahak et al. 2022; Zakwan 2019; Muneer et al. 2022). Various infiltration models have been developed to calculate the model parameters, efficiencies, and validation of models for different soil conditions (Ogbe et al. 2011; Igbadun and Idris 2007; Adindu et al. 2015; Mishra et al. 2003).
A number of studies have been conducted to develop, validate, and compare model parameters and applicability for different soil conditions. Three infiltration models, Kostiakov, Modified Kostiakov, and SCS models, were used to evaluate the infiltration rates of the NIT campus Haryana by Sihag et al. (2017). Three statistical comparison criteria like MAE, Bias, and RMSE were applied to determine the best performing infiltration model. A novel infiltration model was also developed using nonlinear regression modeling, and a comparison of the novel model with the other three models revealed an improvement in the estimation of infiltration rate.
Recently, soft computing techniques like Artificial Neural Network (ANN), Random Forest Regression (RF), and M5P Tree, were used by Singh et al. (2021) for the prediction of infiltration process of soil in the Kurukshetra district of Haryana. Performance of the models was compared with the results of the empirical model by Kostiakov and a Multi Linear Regression model (MLR) and suggests that the cumulative infiltration values of the RF method are better than the Kostiakov method, and other soft computing techniques.
To predict cumulative infiltration depth, double ring infiltrometer experiments were carried out at the agricultural research farm of Bahauddin Zakaria University to determine the model parameters in the study area such as empirical constant c, infiltration decay constant k using the Kostiakov model, sorptivity S, conductivity constant A for Philip’s model, parameter k, and infiltration capacity for Horton’s model were evaluated to estimate the bestfit model of the study area. Measured cumulative infiltration depths in the field using the estimated model parameters of three models showed that there was good agreement on the study area. Based on performance indicators such as Root Mean Square Error (RMSE), Mean Percent Difference (MPD), and Model Efficiency (ME), the most suitable model for the chosen locations was selected. Horton’s model was indicated to be the best model to simulate the infiltration data in the study area (Farid et al. 2019).
Nontraditional optimization techniques were developed to estimate the optimum parameter values of two infiltration models using the Kostiakov and Modified Kostiakov models by Deep and Das (2008). The dimensionless form of Kostiakov’s equation was developed to estimate the accurate parameters of the Kostiakov model (Haghiabi et al. 2011). Accuracy of the dimensionless model proposed by Haghiabi et al. (2011), was assessed by Zakwan (2017) and found that to estimate the cumulative infiltration nonlinear optimization technique is more accurate than the dimensionless form of Kostiakov’s equation and the Graphical method. Experiments were carried out to evaluate the infiltration characteristics of the soil situated at Andhra University Campus, Visakhapatnam. The parameters were obtained by using Kostiakov, Philip, Horton, and Green–Ampt models and compared with the observed field data. (Sreejani et al. 2017).
Thomas et al.’s (2020) study of the suitability of infiltration models has been evaluated using Kostiakov, Philip, Horton, and Green–Ampt models. The parameters estimated by these four models were compared with the field data obtained from the wetland soils using linear regression analysis. To check the performance of the simulated models, a comparison of statistical parameters was estimated using the Correlation Coefficient, Root Mean Square Error, Mean Absolute Error, and the Mean Bias Error, and found good agreement with the Philip models compared to the other models.
Fadadu et al. (2018) applied a Horton infiltration model to estimate the parameters of the soils at the College of Agricultural Engineering and Technology, Gujarat using the infiltration data obtained from the different locations using a double ring infiltrometer. The Decay constant was estimated using the Graphical method and semilog plot of time (t) vs. (ffc). The performance of the Horton model was determined using the least square fitting with the field data and was found suitable for the study area.
To estimate the infiltration rates, two artificial intelligence techniques (AI), ANN and Multigene Genetic Programming (MGGP), and Hybrid MGGP GRG, and different data driven models like Multiple Linear Regression (MLR), Generalized Reduced Gradient (GRG), were applied by Zakwan and Niazkar (2021), and Singh et. al. (2021). The infiltration rates obtained from the data driven models were compared with empirical models like the Horton model, Philip model, and Modified Kostiakov model. The study reveals that the best estimates of infiltration rate were obtained from the Philip model. Results also indicated that as compared to the conventional model’s application of a hybrid MGGPGRG model, an MGGP model gave the best infiltration rate. The study suggested that for estimation of infiltration rates in hydrological models, application of AI based models are more accurate as compared to empirical models. However, they stated that gradientbased optimization techniques are sufficient to handle the nonlinearity of infiltration models. In this regard, this paper explores the application of gradient based Excel Solver techniques in estimating the parameters of various infiltration models of the soils in Vishakhapatnam, Andhra Pradesh, India and compare the results with the Graphical method.
2 Materials and methods
2.1 Study area
Infiltration data of various locations in Andhra University, Visakhapatnam, India has been obtained from the literature (Sreejani et. al. 2017). The infiltration tests were conducted using a doublering infiltrometer with an inner and outer diameter of 30 cm and 60 cm, respectively. The infiltration characteristics, along with the location name, are provided in Table 1.
Table 1 Infiltration characteristics and location name.
S. No  Location  Average Infiltration rate (cm/hr)  Standard deviation (cm/hr)  Number of observations till constant infiltration 
1  Samantha Hostel  51.53  11.54  27 
2  Dispensary  13.38  1.94  25 
3  NCC Building  5.45  0.71  23 
4  Assembly hall  3.12  0.86  14 
5  CSE Building  4.56  0.89  24 
Vishakhapatnam is situated in the Eastern Ghats on the coast of the Bay of Bengal in Andhra Pradesh, India. Andhra University, which is located at Visakhapatnam, lies between latitude 17⁰43′ 5.38″ N and latitude 18⁰ 19′ 17.61″ E, covering an area of 460 acres. The elevation of the region varies from 10 MSL to 62.5 MSL (Sreejani et al. 2017). Climatic conditions at Visakhapatnam are tropical humid, characterized by warm weather in summer (February to May) and moderate winters (November to January). The average temperature in the region is around 23.7°C, with the maximum temperature reaching 43°C during the summer. The yearly precipitation is around 100 cm (Sreejani et al. 2017). Sugarcane and paddy are the most important crops cultivated in the region. The agriculture sector of the region contributes to about 12.46% of the total GDP of the state. With sugarcane and paddy being the most water intensive crops, and with the study region facing a shortage of water, it becomes imperative to study the infiltration characteristics of the soil with high accuracy. In this regard, this article aims to explore the accuracy of the Excel Solver in determining the infiltration parameters and in turn, infiltration rates.
2.2 A brief description of infiltration models chosen for the present study
Infiltration models are mainly classified into three categories (Mishra et al. 2003) such as physical models, (Green and Ampt 1911; Philip 1957), semiempirical models (Singh and Yu 1990), and empirical models (Kostiakov 1932; Horton 1939).
Horton model
Horton (1939) developed an empirical equation for infiltration rate which is given as:
(1) 
Where:
f  =  Infiltration capacity (cm/hr) at any time t, from the start of rainfall, 
f_{0}  =  Initial infiltration capacity (cm/hr) at time t = 0, 
f_{c}  =  final steady state infiltration capacity (cm/hr) at t = t_{c}, 
t  =  time in hours, and 
k  =  Hortons decay coefficient representing the rate of decrease in infiltration capacity, which depends upon soil characteristics and vegetation cover. 
The parameters f_{0}, k, and f_{c}_{ }are evaluated from measured infiltration data.
When rearranging the above terms, the equation becomes:
(2) 
In the Graphical method, the parameters in the Horton model are calculated by drawing ln ( f_{0 }– f_{c }) plot against time. The slope of the plot represents Horton’s decay coefficient.
Kostiakov model
Kostiakov (1932) proposed an equation for cumulative infiltration:
(3) 
Where:
F  =  Cumulative infiltration capacity (cm), and 
a, b  are  the constants with a > 0, and 0 < b < 1. 
Graphically, the parameters of the Kostiakov model can be obtained by taking the slope and intercept of the double log graph of Equation 3.
Philip model
Philip (1957) proposed an infinite series solution of the Richard’s equation to drive a relationship between cumulative infiltration and soil properties, represented as (Fan et al. 2019):
(4) 
Where:
F  =  cumulative infiltration (cm), 
s  =  soil suction potential function known as sorptivity, 
K  =  Darcy’s hydraulic conductivity (cm/hr), and 
t  =  time after start of infiltration (hr). 
Differentiating the above equation, infiltration capacity may be expressed as:
(5) 
Where:
f  =  infiltration capacity (cm/hr) at any time t from the start. 
Graphically, the parameters of the Kostiakov model can be obtained by taking the slope and intercept of the graph between f and 1/√t.
Green–Ampt model
Green–Ampt (1911) developed a model for infiltration capacity based on Darcy’s law and expresses the physical model as:
(6) 
Where:
Ƞ  =  Effective porosity of soil, 
S  =  Capillary suction at the wetting front (cm/hr^{0.5}), and 
K  =  Darcy’s hydraulic conductivity (cm/hr). 
The difference between total porosity and the fraction of the pore space occupied by clay or shale is effective porosity. For clean sands, total porosity is the same as effective porosity. Equation 6 (above) is rewritten as:
(7) 
Where:
m  and  n are Green–Ampt parameters of infiltration. 
Values of infiltration capacity f are plotted against 1/F on an arithmetic graph. The intercept on the ordinate axis is m, and n serves as slope when the bestfit straight line is drawn through the plotted points.
Soil Conservation Service (SCS) model
The Soil Conservation Service (SCS) model is expressed as follows (Jury et al. 1991):
(8) 
Where:
a, b  are  the constants, 
F  =  cumulative infiltration (cm), and 
t  =  time in hours. 
Graphically, the parameters of the SCS model can be obtained by taking the slope and intercept of the double log graph from the above equation.
2.3 Model development using Excel Solver and model selection
To determine the model parameters of the infiltration model defined in the previous subsection using an Excel Solver, the objective function was defined as minimizing the sum of the squares of errors between the observed and estimated infiltration rate.
(9) 
Where:
=  observed infiltration rate,  
=  estimated infiltration rate at any time t,  
i  =  index of observed/estimated rate, from 1 to N, 
N  =  total number of observations 
The accuracy of the infiltration models depends on the type of soil. Various methods have been used by different researchers for the selection of suitable models for the experimental data. Minimizing the difference between the observed and simulated values is one of the simplest methods to determine the bestfit infiltration model. The parameters of the abovementioned infiltration model were treated as the decision variables and constraints, as applicable to the various infiltration models applied in Excel. The solver was then run to obtain the various infiltration model parameters. In this study, the best infiltration model was selected using the SSE, MAE, WI, MARE, and NSE.
NashSutcliffe Efficiency
The NSE (Nash and Sutcliffe 1970) has a value between ∞ and 1.
Its value is defined by:
(10)  
(11)  
(12)  
(13) 
2.4 Excel Solver
Excel Solver is an optimization tool built into Microsoft Excel for solving optimization problems related to large engineering problems. Excel Solver can be used to find the optimal values of the objective function for a given target cell on an Excel work sheet. The optimal parameters of linear and nonlinear equations of the given problem can be obtained by the Excel Solver. For optimization of linear equations, Linear Programming Solver, GRG Solver (Generalized Reduced Gradient), and Evolutionary Solver can be used for optimization of nonlinear equations (Barati 2013). GRG Solver uses a nonlinear optimization code developed by Lasdon et al. (1978). The GRG Solver optimization technique is a gradientbased nonlinear optimization technique (Zakwan et al. 2016b; Muzzammil et al. 2018). GRG Solver is a proven robust tool and a reliable method for solving highly difficult, nonlinear programming problems (Lasdon and Smith 1992).
For estimating infiltration parameters and rating curve parameters, the GRG technique is superior to the conventional Graphical method (Zakwan et al. 2021). To determine the optimization of objective function, GRG Solver uses two techniques, i.e., QuasiNewton method and Conjugate Gradient method. The default choice to solve the optimization problems is the QuasiNewton method. GRG Solver automatically switches between QuasiNewton and Conjugate Gradient method based on the computer storage available (Zakwan and Muzzammil 2016). GRG Solver is a deterministic method and requires assumed initial values for parameter estimation. In this study, the Excel Solver is used for optimizing the objective function.
3 Results and discussion
The present work analyses the infiltration characteristics of five sites at Andhra University Campus, Visakhapatnam, India based on the five commonly used infiltration models by using the Graphical method as well as the Excel Solver. The results of infiltration rates obtained from the two approaches were compared based on performance indices. The performance indices of the two approaches for various sites are mentioned in Tables 2 to 6. Analysis of these tables reveals that infiltration rates estimated by Excel Solver are more accurate than using the Graphical method. However, the estimates obtained for the Green–Ampt and Philip models were similar, as can be observed from Tables 2 and 3. These observations could be explained from the findings of Ferguson (1986), who stated that logarithmic transformation of data to obtain straightline results in the introduction of bias in the estimates. This bias may lead to underestimation or overestimation, depending upon the nature and number of data points. Such observations were also made by Zakwan and Niazkar (2021) and Zakwan (2020) in the case of sediment rating curves and rainfall envelope curves, respectively. Therefore, in the case of Horton, Kostiakov, and the SCS method where logarithmic transformation was used for curve fitting, the Graphical method led to under/over estimation of the infiltration rate. But, in the case of the Philip and Green–Ampt models, the equations are themselves in the form of linear equations, and therefore no logarithmic transformation was required to obtain the parameters, and hence there was no difference observed in the infiltration rates obtained from the Graphical and Solver methods. However, the application of the Excel Solver is advantageous in the Philip and Green–Ampt models as well, because the values of sorptivity, hydraulic conductivity, effective porosity, and capillary suction at the wetting front can be determined separately using the Excel Solver model.
Among the infiltration models used in the present study, no single model was found to be suitable for all locations, indicating that the suitability of infiltration models to estimate the infiltration rate depends upon the soil type and soil conditions. Such observations were also made by Mishra et al. (2003), Machiwal et al. (2006), Haghighi et al. (2010), and Mirzaee et al. (2014).
In the present study area, the Horton model was found to be the bestsuited model for the Samantha Hostel, NCC Building, Assembly Hall, and CSE building, while the Kostiakov model was the bestfit model for the Dispensary. Figure 1 shows the best infiltration model at different sites along with the observed infiltration data. It may be observed from Figure 1 that the curve of the estimated infiltration rate satisfies the infiltration data at all the locations.
Figure 1 Bestfit models against the observed data for different locations (a) Samantha, (b) Dispensary Building, (c) NCC Building, (d) Assembly Building, and (e) CSE Building.
The Green–Ampt model failed to mimic the infiltration process at all the locations considered in the present study, as is evident from Table 3. Basically, the Green–Ampt model is suitable under rainy conditions. Further, the assumption of the existence of a distinct wetting front is rarely possible, leading to an inaccurate estimation of infiltration rate from the Green–Ampt model (Mirzaee et al. 2014; Zakwan 2019).
Table 2 Errors in the Philip Solver model.
Location  SSE  NSE  WI  MAE  MARE  MXARE  
Approach  Graphical  Solver  Graphical  Solver  Graphical  Solver  Graphical  Solver  Graphical  Solver  Graphical  Solver 
Samantha Hostel  2566  2566  0.83  0.83  0.95  0.95  8.89  8.89  0.26  0.26  0.92  0.92 
Dispensary  79.05  79.05  0.84  0.84  0.95  0.95  1.49  1.49  0.13  0.13  0.33  0.33 
NCC Building  96.203  96.20  0.36  0.36  0.73  0.73  1.57  1.57  0.41  0.41  1.14  1.14 
Assembly Hall  1.22  1.22  0.96  0.96  0.99  0.99  0.23  0.23  0.08  0.08  0.27  0.27 
CSE Building  31.16  31.16  0.79  0.79  0.94  0.94  0.97  0.97  0.26  0.26  0.46  0.46 
Table 3 Errors in the Green–Ampt model.
Location  SSE  NSE  WI  MAE  MARE  MXARE  
Approach  Graphical  Solver  Graphical  Solver  Graphical  Solver  Graphical  Solver  Graphical  Solver  Graphical  Solver 
Samantha Hostel  5185.70  5185.70  0.66  0.66  0.89  0.89  12.70  12.70  0.38  0.38  1.31  1.31 
Dispensary  172.92  172.92  0.65  0.65  0.88  0.88  2.30  2.30  0.20  0.20  0.47  0.47 
NCC Building  106.72  106.72  0.29  0.29  0.66  0.66  1.80  1.80  0.47  0.47  1.33  1.33 
Assembly Hall  1.95  1.95  0.93  0.93  0.98  0.98  0.29  0.29  0.47  0.47  0.33  0.33 
CSE Building  70.4  70.49  0.52  0.52  0.829  0.829  1.531  1.531  0.423  0.423  0.775  0.775 
Table 4 Errors in the Kostiakov Solver model.
Location  SSE  NSE  WI  MAE  MARE  MXARE  
Approach  Graphical  Solver  Graphical  Solver  Graphical  Solver  Graphical  Solver  Graphical  Solver  Graphical  Solver 
Samantha Hostel  4338.41  1234.01  0.726  0.92  0.88  0.98  9.51  6.05  0.35  0.19  1.41  0.69 
Dispensary  173.03  29.64  0.69  0.939  0.91  0.98  2.36  0.92  0.21  0.08  0.55  0.23 
NCC Building  113.0  102.51  0.25  0.32  0.73  0.72  1.69  1.50  0.46  0.40  0.19  1.02 
Assembly hall  11.86  0.70  0.58  0.975  0.99  0.99  0.89  0.18  0.38  0.07  0.87  0.23 
CSE Building  51.69  17.01  0.66  0.889  0.92  0.97  1.12  0.73  0.40  0.20  0.87  0.39 
Table 5 Errors in the Horton Solver model.
Location  SSE  NSE  WI  MAE  MARE  MXARE  
Approach  Graphical  Solver  Graphical  Solver  Graphical  Solver  Graphical  Solver  Graphical  Solver  Graphical  Solver 
Samantha Hostel  1089.8  987.32  0.93  0.94  0.82  0.98  4.99  4.69  0.09  0.08  0.19  0.19 
Dispensary  43.02  73.14  0.89  0.82  0.97  0.96  0.96  1.52  0.06  0.115  0.17  0.27 
NCC Building  299.02  61.03  0.57  0.91  0.79  0.98  1.21  1.31  0.1  0.3  0.59  0.53 
Assembly hall  41.389  0.71  0.64  0.96  0.05  0.99  5.32  0.18  1.893  0.06  2.47  0.15 
CSE Building  13.03  1.36  0.89  0.99  0.96  0.997  0.61  0.20  0.12  0.05  0.29  0.19 
Table 6 Errors in the SCS Solver model.
Location  SSE  NSE  WI  MAE  MARE  MXARE  
Approach  Graphical  Solver  Graphical  Solver  Graphical  Solver  Graphical  Solver  Graphical  Solver  Graphical  Solver 
Samantha Hostel  9943.9  1234.2  0.35  0.92  0.81  0.99  17.09  6.06  0.58  0.19  1.94  0.69 
Dispensary  804.3  29.65  0.01  0.93  0.98  0.999  5.45  0.92  0.45  0.08  0.85  0.23 
NCC Building  108.58  102.52  0.28  0.32  0.67  0.975  1.69  1.49  0.41  0.39  0.99  1.02 
Assembly Hall  15.21  0.70  0.46  0.98  0.875  0.999  1.02  0.18  0.43  0.07  0.925  0.23 
CSE Building  80.38  17.05  0.48  0.889  0.88  0.999  1.59  0.75  0.499  0.197  0.999  0.39 
The parameters of infiltration models of five locations were estimated using Graphical and Microsoft Excel Solvers. The statistical performance of these models was analyzed using Nash Sutcliffe Criterion (NSE), Wilmot Index (WI), Mean Absolute Error (MAE), Mean Absolute Relative Error (MARE), and Maximum Absolute Relative Error (MXARE), and criteria were estimated for all five locations of the study area (Samantha Hostel, Dispensary Building, NCC Building, Assembly Hall, and CSE Building), and for all models (Philip, Green–Ampt, Kostiakov, Horton, and SCS) using the Solver and Graphical approach. Results have shown that the statistical performance of the Philip and Green–Ampt models are the same for all locations for both approaches (Graphical and Excel Solver), where in the other models, are shown as maximum deviations. Upon application of the Excel Solver, the Sum of Square of Error (SSE) reduced by 70.48%, 39.55%, and 87.39% for Kostiakov, Horton, and SCS models, respectively, as compared to the Graphical method; while the Wilmot Index (WI) improved by 5.68%, 36.39%, and 17.85% for Kostiakov, Horton and SCS models, respectively, for the Excel Solver model, as compared to the Graphical method.
The infiltration rate estimated by the Horton model was the most successful in predicting the bestfit measured experimental data, while the infiltration rates predicted by the Philip model were in least agreement with the observed data. On average, the SSE, MAE, MARE, and MXARE were reduced by 59.32%, 39.45%, 47.54% and 56.45%, respectively, for the Horton model, as compared to the Philip models. Similarly, NSE and WI improved by 21.23% and 7.89%, respectively, for the Horton model as compared to the Philip model. The infiltration rates predicted by the SCS model were found to be in poor agreement with the observed data at most of the locations.
4 Conclusions
In this study, application of the Excel Solver was explored to estimate infiltration rates. Published infiltration data was used to determine the infiltration rates of the soils. Commonly used infiltration models such as Horton, Philip, Kostiakov, Green–Ampt, and SCS were employed to estimate the infiltration rates. The parameters of these models were determined by using Graphical and Excel Solver models. It was observed that the Horton model was the bestfit model, and the Green–Ampt model was the worstfit model for most of the study sites. The infiltration rates obtained through the Excel Solver were found to be in close agreement with the observed data and were far better than the estimates obtained from Graphical method. For the Dispensary, the Kostiakov model is found to be the bestfit model of the study area. Based on the present analysis, the Excel Solver was found to be a reliable tool for finding the parameters of nonlinear equation and therefore, its application should be explored in different fields of engineering.
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