Accurate estimation of infiltration losses is very important in many hydrologic projects. However, the lack of available data often forces hydrologists to resort to the application of empirical and semi-empirical models for estimating the infiltration rates. Estimation of infiltration rate from these empirical and semi-empirical models is highly dependent on the method adopted to determine the parameters of models. One of the most common infiltration models known as the Kostiakov model was modified to rectify zero infiltration at the start of the infiltration process. Recently, a simplified approach for determining the parameters of the modified Kostiakov model was proposed. The present article focuses on analyzing the reliability of the simplified approach for determining the parameters of the modified Kostiakov model. In this regard, six observed datasets were analyzed from different parts of the world to arrive at some generalized results on the reliability of the simplified approach. Cumulative infiltration obtained from simplified approaches was compared with that obtained through the application of an excel spreadsheet-based nonlinear optimization solver. It was found that on average Sum of Square of Error reduced by 72% on application of nonlinear optimization as compared to the simplified approach.

  • Recently, a simplified approach was proposed to determine the modified Kostiakov infiltration model.

  • Comparison of simplified approach to nonlinear optimization.

  • Nonlinear optimization was found more accurate.

Infiltration is defined as the process by which water enters the surface of Earth (Mishra et al. 2003; Chahinian et al. 2005; Machiwal et al. 2006; Zakwan 2019). Determination of infiltration losses is a critical input in many of the water resource projects (Chahinian et al. 2005; Angelaki et al. 2021; Khasraei et al. 2021). Infiltration is the source of groundwater recharge and is, therefore, an important component of the hydrologic cycle (Bayabil et al. 2019; Zakwan 2019; Sihag et al. 2021).

Several infiltration models have been developed by scientists across the world to determine the infiltration rates (Haghighi et al. 2010; Zakwan et al. 2016). Empirical infiltration models are based on field and laboratory data (Mishra et al. 2003). Mishra et al. (2003) performed a comprehensive analysis of the suitability of various infiltration models across the globe. A similar attempt was made by Zakwan (2019), Ebrahimian et al. (2020), Thomas et al. (2020) and Khasraei et al. (2021) by considering the infiltration data corresponding to different soil types scattered across the globe, but, no single infiltration model was found suitable for all type of soils.

However, the Kostiakov model is a prominent and one of the most used infiltration model for determining the infiltration characteristics of soil. Parhi et al. (2007) realized that the Kostiakov model requires modification for the initial infiltration rate corresponding to time (t = 0) and they modified the Kostiakov model, popularly known as the modified Kostiakov model. Thereafter, various approaches have been proposed for determining the parameters of Kostiakov and modified Kostiakov model. Conventionally, the model parameters were determined through the graphical method by plotting cumulative infiltration (Z) against time (t) on logarithmic scale and determining the slope and intercept of the best fit straight line. Haghiabi et al. (2011) proposed the application of the dimensionless form of the Kostiakov model to attain higher accuracy in the estimated infiltration rate. However, Zakwan (2017) observed that the accuracy of the infiltration rate estimated by the Kostiakov model does not increase by turning it into dimensionless form. Sihag et al. (2017) proposed a novel infiltration model, derived from the Kostiakov model and claimed that infiltration rates estimated by the novel model are most accurate amongst the empirical infiltration models. Zakwan (2019) tested the novel infiltration models to various infiltration datasets across the globe, and found that the novel infiltration model was not the most accurate infiltration model for all the places and soil.

In general, the suitability of the infiltration model for a particular site depends on the type of soil and field conditions and hence, different models have performed better for different places and type of soil. However, many researchers have also asserted that the accuracy in determining the model parameters also affects the performance of infiltration models. Recently, Gebul (2022) claimed that a simplified approach for determining the parameters of the modified Kostiakov model provides a better estimate of infiltration rates. In this regard, the present articles explores the efficacy of a simplified approach by comparing its performance with a nonlinear optimization model for estimating the infiltration rates for soils from two different datasets.

Infiltration datasets from different testing sites were used to test the efficacy of a simplified approach. The infiltration tests were conducted using a double-ring infiltrometer at all the data collection sites. The average cumulative infiltration along with soil type and climatic conditions are presented in Table 1. Five infiltration datasets correspond to Ethiopia and one dataset is from Bangladesh. The texture of the soil ranged from clay, clay loam to sandy loam.

Table 1

Characteristics of datasets used in the present study

SourceLocationClimatic conditionSoil typeDatasetAverage cumulative infiltration (cm)Standard deviation cumulative infiltration (cm)
Gebul (2022)  Wonji Showa irrigation field, Ethiopia Arid and semi-arid climatic condition Clay 11.07 6.91 
Silty clay 6.24 2.90 
Silty clay 5.59 1.72 
Arba Minch University, Ethiopia Tropical climate Sandy clay 6.66 4.01 
Haramaya University, Ethiopia Tropical climate Silty clay Loam 18.81 12.38 
Hasan et al. (2015)  Agricultural field, Gazipur, Bangladesh Tropical wet and dry climate Clay loam 8.49 5.90 
SourceLocationClimatic conditionSoil typeDatasetAverage cumulative infiltration (cm)Standard deviation cumulative infiltration (cm)
Gebul (2022)  Wonji Showa irrigation field, Ethiopia Arid and semi-arid climatic condition Clay 11.07 6.91 
Silty clay 6.24 2.90 
Silty clay 5.59 1.72 
Arba Minch University, Ethiopia Tropical climate Sandy clay 6.66 4.01 
Haramaya University, Ethiopia Tropical climate Silty clay Loam 18.81 12.38 
Hasan et al. (2015)  Agricultural field, Gazipur, Bangladesh Tropical wet and dry climate Clay loam 8.49 5.90 

The present study focuses on the accuracy of estimation of infiltration through the modified Kostiakov model. The Kostiakov and modified Kostiakov model are presented in Equations (1) and (2).

The Kostiakov model

Kostiakov (1932) proposed an equation to calculate cumulative infiltration capacity as
formula
(1)
where Z refers to cumulative infiltration, t refers to time from starting of infiltration, k and a are Kostiakov model constants.

However, this equation was modified to yield a non-zero infiltration at time (t = 0).

The modified Kostiakov model

Smith (1972) modified the Kostiakov equation to include the constant term b.
formula
(2)

Simple approach for estimating the parameters

Simplified approach as proposed by Gebul (2022) includes following steps:

Step 1: Preparation of cumulative infiltration versus time data.

Step 2: Selecting the governing equation as shown in Equation (2).

Step 3: Select two data points, near the beginning (t1, Z1) and end (t2, Z2) of the infiltration test data series, respectively. These points can also be obtained by interpolation from the cumulative infiltration versus time curve.

Step 4: Estimate the value, t3, based on Equation (3).
formula
(3)
Step 5: Based on t3, value find the corresponding cumulative infiltration depth (Z3) from cumulative infiltration versus time curve. The constant b in Equation (2) can now be calculated using Equation (4).
formula
(4)
Once b has been determined, Equation (2) can be linearized using logarithmic transformation as shown in Equation (5):
formula
(5)

Step 6: Values of k and a in Equation (2) can be determined by rewriting Equation (5) for each time step and then solving the set of linear equations obtained.

Nonlinear optimization model for estimating the parameters

Many soft computing tools are available today to solve complex nonlinear problems. Amongst them, spreadsheet-based Excel solver is the simplest equation solving tool available which can be easily used even without programming skills. Furthermore, most of the hydrologic office staff, practitioners, researchers and field engineers are aware about the use of spreadsheet and over the years the Excel solver has become a reliable source of nonlinear modelling in water resource engineering (Zakwan 2022).

In this regard, the present study explores its applicability in determining the parameters of the Kostiakov model presented by Equation (2). Minimization of the sum of squares of the error between estimated and observed cumulative infiltration was set as the objective function for determining the parameters (k, a, and b) in the modified Kostiakov model. The objective function is presented by Equation (6).
formula
(6)
where Zobs is the observed cumulative infiltration and Zest is the estimated cumulative infiltration rate at any time t.
As a part of the optimization model k, a and b were set as the decision variable and a non-negative constrain was put on the decision variable b. Excel solver was run with the set up as shown in Figure 1, to obtain the values of decision variables. Once the values of decision variables were determined, the cumulative infiltration was calculated using Equation (2). Detailed explanation of the working of the generalized reduced gradient (GRG) technique has been discussed in detail by several authors (Lasdon et al. 1978; Lasdon & Smith 1992; Fylstra et al. 1998; Weiss & Gulliver 2001; Zakwan et al. 2016).
Figure 1

Flow chart for the proposed methodology.

Figure 1

Flow chart for the proposed methodology.

Close modal
The performance of different infiltration models was compared based on the following criteria:
formula
(7)
formula
(8)
formula
(9)
formula
(10)
where Zobs is the observed cumulative infiltration, Zest is the cumulative infiltration rate at any time and is average cumulative infiltration capacity. Nash criteria is an indicator of goodness-of-fit, its value ranges from 0.0 to 1.0 with values closer to unity indicating a better agreement between measured and estimated data.
Cumulative infiltration estimated by three methods at six testing sites was compared based on performance evaluation criterion as shown in Table 2. Analysis of Table 2 reveals that application of Excel solver resulted in the least errors (SSE, MAE, and MXAE) and best goodness-of-fit (NE) for all the six datasets considered. The results obtained from the three approaches were also assessed qualitatively by plotting the graph for estimated and observed cumulative infiltration and time as shown in Figure 2. Figure 2 also reveals that the cumulative infiltration obtained through the solver is much closer to observed data as compared to the simplified approach proposed by Gebul (2022). This indicates that irrespective of testing site and soil type, Excel solver provides the most reliable estimate of modified Kostiakov model parameters and in turn cumulative infiltration. On the other hand, the cumulative infiltration obtained from the simplified approach (Gebul 2022) and the graphical method are almost comparable.
Table 2

Comparative statistics for fit of the three approaches

LocationGraphical model
Simplified approach
Proposed approach
SSEMAEMXAENESSEMAEMXAENESSEMAEMXAENE
7.291 0.694 1.623 0.987 6.224 0.512 2.132 0.989 2.193 0.314 1.059 0.996 
3.806 0.562 1.302 0.989 32.903 1.676 3.092 0.910 2.791 0.546 0.915 0.992 
0.409 3.116 5.432 0.643 32.141 1.602 3.846 0.884 0.197 0.128 0.271 0.999 
1.690 0.434 0.744 0.996 2.337 0.491 0.808 0.995 1.056 0.309 0.649 0.998 
9.690 0.851 2.638 0.997 2.399 0.533 0.858 0.999 0.368 0.187 0.389 0.999 
1.462 0.313 0.669 0.999 4.408 0.587 1.025 0.996 1.360 0.309 0.615 0.999 
LocationGraphical model
Simplified approach
Proposed approach
SSEMAEMXAENESSEMAEMXAENESSEMAEMXAENE
7.291 0.694 1.623 0.987 6.224 0.512 2.132 0.989 2.193 0.314 1.059 0.996 
3.806 0.562 1.302 0.989 32.903 1.676 3.092 0.910 2.791 0.546 0.915 0.992 
0.409 3.116 5.432 0.643 32.141 1.602 3.846 0.884 0.197 0.128 0.271 0.999 
1.690 0.434 0.744 0.996 2.337 0.491 0.808 0.995 1.056 0.309 0.649 0.998 
9.690 0.851 2.638 0.997 2.399 0.533 0.858 0.999 0.368 0.187 0.389 0.999 
1.462 0.313 0.669 0.999 4.408 0.587 1.025 0.996 1.360 0.309 0.615 0.999 
Figure 2

Cumulative infiltration as depicted by various approaches at different sites.

Figure 2

Cumulative infiltration as depicted by various approaches at different sites.

Close modal

These results suggest that the simplified approach presented by Gebul (2022) does not lead to significant improvement from the conventional graphical method and lacks on the ground of accuracy.

The simplified approach proposed by Gebul (2022) was tested against the graphical and solver-based approach to determine the model parameters and the cumulative infiltration rate was compared based on performance evaluation criteria as mentioned in Equations (7)–(10). Both quantitative (Table 2) and qualitative (Figure 2) comparison clearly demonstrate that the solver-based parameter determination is more reliable and the use of the simplified approach does not significantly improve the estimates of infiltration rates as compared to the conventional graphical method. In fact, the SSE values obtained from an excel solver are more than 50% lesser than the SSE values obtained from the simplified approach for all the six gauging sites. Similarly, observations can also be made for other performance indices as presented in Table 2.

The results were further confirmed by validation through the jack-knifing technique in which one data point is left for validation during the calibration process. The cumulative infiltration as obtained from different approaches for the validation data point at different location is presented in Table 3. It may be observed that the cumulative infiltration obtained from the proposed solver approach is closest to the observed cumulative infiltration which again suggests that the proposed approach is most accurate in the prediction of infiltration.

Table 3

Cumulative infiltration for validation data point obtained from different approaches

LocationObserved infiltration (cm)Graphical method (cm)Simplified approach (cm)Proposed approach (cm)
13.5 14.335 13.217 13.641 
6.335 4.731 6.444 
6.2 6.501 7.980 6.321 
6.4 5.789 5.586 5.809 
15 15.630 15.554 14.666 
8.2 8.684 9.084 8.669 
LocationObserved infiltration (cm)Graphical method (cm)Simplified approach (cm)Proposed approach (cm)
13.5 14.335 13.217 13.641 
6.335 4.731 6.444 
6.2 6.501 7.980 6.321 
6.4 5.789 5.586 5.809 
15 15.630 15.554 14.666 
8.2 8.684 9.084 8.669 

Basically, the graphical method is inferior to the nonlinear optimization mainly because of the logarithmic transformation required for linearizing the graphs. Ferguson (1986) found that logarithmic transformation of data to linearize power curve fitting and re-transformation of data for estimation, introduces a bias in the estimate causing under/over estimation. Earlier, similar observations were made by Miller (1984). It is because of this bias that the conventional graphical method fails to provide an accurate estimation of cumulative infiltration. The deviation from observed data becomes more pronounced during validation as the curve fitting was performed on log-transformed data rather than actual data (Zakwan 2022).

A perusal of the simplified approach presented by Gebul (2022) also reveals that in Step 5, equations have been made linearized through logarithmic transformation. This appears to be the main cause of inaccuracy in the estimates obtained through Gebul (2022) method. Furthermore, the selection of data points in step 3 is very subjective. The calculation of time (t3) is entirely dependent on the selection of these two points and Z3 is calculated based on t3. Finally, Z3 plays a crucial role in determining b and further k and a. Different selection points (Z1 and Z2) will result in different values of k, a, and b. Finally, for site 4, the b value according to the simplified approach comes out to be −0.37. Practically, a negative value of b is not possible as for time (t = 0), it will represent water coming out of the ground rather than infiltrating into the ground. In Excel, solver negative value of b can easily be avoided by providing a non-negativity constraint on b.

Based on the above discussion it may be concluded that the excel solver approach is the best suited approach for determining the parameters of modified Kostiakov. Furthermore, the estimation of the Kostiakov model can be done in a single step using solver, as compared to the simplified approach that involve several steps such as curve fitting, selecting two random points on fitted curve, and finally the estimation of parameters. The Kostaikov model is widely used to determine the infiltration rate in various watershed models, evaluating the abstraction for the preparation of hydrographs, determination of soil properties, and runoff modelling. The simplified approach for determining the parameters of the Kostiakov model could have found a wide application, but, unfortunately, testing of the modified approach against the solver and graphical method by considering six datasets revealed that its application may not lead to accurate estimation of infiltration. Timely evaluation of such approaches would limit their injudicious application, and at the same time, highlight the need to develop methodologies which may lead to better estimation of infiltration model parameters.

In quest for gaining higher accuracy in estimating the infiltration rates, recently some researchers have proposed a simplified approach for determining the parameters of the modified Kostiakov model. The present study analyses the proposed simplified approach against the nonlinear optimization technique for the estimation of infiltration rates by considering six infiltration datasets pertaining to different experimental sites. It was observed that infiltration rates estimated through application of nonlinear optimization are far better than the infiltration rates obtained through a simplified approach. Furthermore, the proposed simplified approach is calculation intensive and very subjective. Therefore, it may be concluded that the simplified approach does not provide a suitable alternative for determining the modified Kostiakov model.

Data cannot be made publicly available; readers should contact the corresponding author for details.

The authors declare there is no conflict.

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