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Infiltration is a vital phenomenon in the water cycle, and consequently, estimation of infiltration rate is important for many hydrologic studies. In the present paper, different data-driven models including Multiple Linear Regression (MLR), Generalized Reduced Gradient (GRG), two Artificial Intelligence (AI) techniques (Artificial Neural Network (ANN) and Multigene Genetic Programming (MGGP)), and the hybrid MGGP-GRG have been applied to estimate the infiltration rates. The estimated infiltration rates were compared with those obtained by empirical infiltration models (Horton’s model, Philip’s model, and modified Kostiakov’s model) for the published infiltration data. Among the conventional models considered, Philip’s model provided the best estimates of infiltration rate. It was observed that the application of the hybrid MGGP-GRG model and MGGP improved the estimates of infiltration rates as compared to conventional infiltration model, while ANN provided the best prediction of infiltration rates. To be more specific, the application of ANN and the hybrid MGGP-GRG reduced the sum of square of errors by 97.86% and 81.53%, respectively. Finally, based on the comparative analysis, implementation of AI-based models, as a more accurate alternative, is suggested for estimating infiltration rates in hydrological models.

Infiltration can be defined as the process by which water enters the surface of Earth [

Owing to the wide applications of the infiltration rate, its estimation has gained significant attention from researchers. Over the years, various infiltration models have been proposed by the researchers for the estimation of infiltration rates. They include models that have physical, semiempirical, and even empirical formulations. Despite the development of several models, no single model exists that outperforms other ones universally. The suitability of infiltration model for a particular site depends on the type of soil and field conditions [

Recent applications of computational techniques in water resource engineering have widened the scope further [

The present study aims to compare the performances of different infiltration methods. Additionally, it attempts to assess the capability MGGP and of the novel hybrid MGGP-GRG to model the infiltration process. In a bid to seek for a better time-dependent infiltration model, the performances of the MGGP-based models were compared with those of the conventional models, regression techniques, and commonly used neural network.

In the present study, the infiltration data reported by Sihag et al. [

Training and testing data used.

Data | Count | Infiltration rate (cm/min) | Sand (%) | Density (g/cm^{3}) |
---|---|---|---|---|

Training | 116 | 0.080–1.560 | 6.00–38 | 0.08–1.56 |

Testing | 38 | 0.080–1.480 | 6.00–38.00 | 0.08–1.48 |

Figure

Observed infiltration data.

There are a number of infiltration models available in the literature. Brief description of some of the commonly used infiltration models considered in the present study is as follows.

Horton [_{c} is the final or ultimate infiltration capacity occurring at _{c}; _{0} is the initial infiltration capacity at time

(2)

Philip [

By differentiating the above equation, the infiltration rate may be represented as

(3)

Kostiakov [_{i}. The modified version is shown in the following equation:

The parameters of different infiltration models were obtained by minimizing the sum of square of errors using a nonlinear optimization tool. Thus, the objective function becomes_{obs} is the observed infiltration rate and _{est} is the estimated infiltration rate at any time

MLR has been widely used in water resource engineering [^{3}.

GRG is a gradient-based nonlinear optimization technique [

In the present study, GRG solver embedded in Microsoft Excel was used to estimate the infiltration rate based on minimizing the sum of square of errors. Detailed explanation on working of GRG technique is available in the literature [

ANN is one well-documented AI model. It has been used for solving various problems in water resources and hydrological modelling [

MGGP is a modified version of genetic programming (GP), which is classified as an AI technique [

In this study, an open-access code of MGGP was exploited. This code was adopted form the literature [

The hybrid MGGP-GRG was first proposed for developing stage-discharge relationships in the literature [

Flowchart of the hybrid MGGP-GRG for estimating infiltration rates.

The performance of infiltration models and soft computing techniques was compared based on several criteria, which are presented in the following equations [_{obs} is the observed infiltration rate, _{est} is the estimated infiltration rate at any time, and

In a bid to determine how much the results achieved by a typical model are sensitive to each input parameter, a sensitivity analysis can be conducted [

The reliability analysis is basically conducted to investigate the overall consistency of a prediction model. For this analysis, the relative error for each data point is achieved by the estimation model and compared with a threshold. Then, the number of cases, which have an equal or lower relative error than the threshold specified, is divided by the total number of points. Finally, the aforementioned ratio in the percentage would be the reliability metric, which demonstrates how reliable the prediction model performs in accordance with the desirable threshold. In this study, the reliability analysis was carried out for all methods used for predicting the infiltration rate, while the threshold was selected to be 20% based on the literature [

Accurate estimation of infiltration rate plays a vital role in various aspects of watershed hydrology. The present work focuses on improving the estimates of the infiltration rate through application of different soft computing approaches. The infiltration rates estimated by these techniques were compared with those approximated by the conventional infiltration models (Horton’s model, modified Kostiakov’s model, and Philip’s model). In the conventional infiltration model, the observed infiltration rates and time were used as input data in accordance to model equations to obtain the estimated infiltration rate. On the other hand, in MLR, GRG, ANN, MGGP, and the hybrid MGGP-GRG models, the observed infiltration rates, time, sand percentage, and density were used as the input variables to obtain the estimated infiltration rates.

Table

Parameters obtained for the conventional infiltration models.

Models | Calibrated parameters | ||
---|---|---|---|

Horton | |||

Modified Kostiakov | |||

Philip | — |

The results of different approaches considered in the present study were compared with respect to four criteria for both train and test data. This comparative analysis is shown in Table

Comparative statistics for fit of model to the observed infiltration data.

Methods | Training phase | Testing phase | |||||||||
---|---|---|---|---|---|---|---|---|---|---|---|

SSE (cm^{2}/min^{2}) | NRMSE | WI | MAE (cm/min) | NSE | SSE (cm^{2}/min^{2}) | NRMSE | WI | MAE (cm/min) | NSE | ||

Conventional models | Horton | 4.645 | 0.135 | 0.610 | 0.123 | 0.236 | 1.650 | 0.149 | 0.636 | 0.127 | 0.319 |

Modified Kostiakov | 4.485 | 0.133 | 0.613 | 0.118 | 0.263 | 1.474 | 0.141 | 0.686 | 0.119 | 0.392 | |

Philip | 4.482 | 0.133 | 0.616 | 0.117 | 0.263 | 1.474 | 0.141 | 0.687 | 0.119 | 0.392 | |

MLR | 4.710 | 0.202 | 0.136 | 0.123 | 0.242 | 1.668 | 0.210 | 0.150 | 0.128 | 0.312 | |

GRG | 3.943 | 0.184 | 0.124 | 0.104 | 0.352 | 0.837 | 0.148 | 0.106 | 0.101 | 0.655 | |

AI-based models | ANN | 0.097 | 0.020 | 0.996 | 0.018 | 0.984 | 0.380 | 0.071 | 0.954 | 0.051 | 0.843 |

MGGP | 0.838 | 0.057 | 0.962 | 0.059 | 0.862 | 0.487 | 0.081 | 0.938 | 0.071 | 0.798 | |

MGGP-GRG | 0.836 | 0.057 | 0.962 | 0.059 | 0.862 | 0.483 | 0.081 | 0.938 | 0.070 | 0.801 |

A perusal of Table

Figures

Relative error plots for Horton’s model for the training and testing data.

Relative error plots for modified Kostiakov’s model for the training and testing data.

Relative error plots for Philip’s model for the training and testing data.

Relative error plots for the MLR model for the training and testing data.

Relative Error plots for the GRG model for the training and testing data.

Relative error plots for the ANN model for the training and testing data.

Relative error plots for the MGGP model for the training and testing data.

Relative error plots for the hybrid MGGP-GRG model for the training and testing data.

Figures

Estimated versus observed infiltration rates during the training phase.

Estimated versus observed infiltration rates during the testing phase.

Figure

Results of the sensitivity analysis based on (a) ANN and (b) MGGP.

The reliability analysis was carried out for the train and test data separately. The results of this analysis are presented in Figure

Results of the reliability analysis for train and test data.

The structure of the equations developed by the conventional infiltration models, MLR and GRG, are known in advance of applying these methods. On the other hand, ANN, MGGP, and the hybrid MGGP-GRG are highly nonlinear techniques with greater degrees of freedom and complexity and, therefore, provide better estimates of the infiltration rate. However, more precise results are obtained by ANN, MGGP, and the hybrid MGGP-GRG at the expense of higher computational efforts. These machine learning tools require a considerable number of runs, unlike the conventional models and MLR in which a single attempt is sufficient for determining the model output. Based on the comparative analysis conducted in this study, ANN certainly yielded to the best estimates of infiltration rates. However, the estimates obtained from the hybrid MGGP-GRG were also comparable, especially, for the test data. Furthermore, unlike ANN, the hybrid MGGP-GRG model provided explicit equations for predicting infiltration rates, which can be implemented in a typical hydrological modelling or preferred in practice by engineers, which may be counted as an advantage of this AI-based technique.

In the present study, published infiltration data was used to assess the performances of MGGP and the hybrid MGGP-GRG technique in modelling the infiltration rates of soil. The estimated infiltration rates were compared with those obtained by the conventional models (Horton’s model, Philip’s model, and modified Kostiakov’s model). It was observed that application of the hybrid MGGP-GRG and MGGP improved the estimates of infiltration rates as compared to the conventional infiltration model by over 80%. On the other hand, ANN provided the best estimates of infiltration rates. In addition to the accuracy improvement, the application of ANN, MGGP, and the hybrid MGGP-GRG increased the complexity of modelling equations. Future studies may focus on the comparison of the hybrid MGGP-based models with the other machine learning approaches, while applying the explicit infiltration models developed by either MGGP or the hybrid MGGP-GRG in hydrological models is anticipated in favor of assessing their performances in practice.

The data used in this study are available in the related literature.

The authors declare that they have no conflicts of interest regarding the publication of this paper.