Modelling of the impact of water quality on the infiltration rate of the soil
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Abstract
The concept behind of this paper is to check the potential of the three regression-based techniques, i.e. M5P tree, support vector machine (SVM) and Gaussian process (GP), to estimate the infiltration rate of the soil and to compare with two empirical models, i.e. Kostiakov model and multi-linear regression (MLR). Totally, 132 observations were obtained from the laboratory experiments, out of which 92 observations were used for training and residual 40 for testing the models. A double-ring infiltrometer was used for experimentation with different concentrations (1%, 5%, 10% and 15%) of impurities and different types of water quality (ash and organic manure). Cumulative time (T_{f}), type of impurities (I_{t}), concentration of impurities (C_{i}) and moisture content (W_{c}) were the input variables, whereas infiltration rate was considered as target. For SVM and GP regression, two kernel functions (radial based kernel and Pearson VII kernel function) were used. The results from this investigation suggest that M5P tree technique is more precise as compared to the GP, SVR, MLR approach and Kostiakov model. Among GP, SVR, MLR approach and Kostiakov model, MLR is more accurate for estimating the infiltration rate of the soil. Thus, M5P tree is a technique which could be used for modelling the infiltration rate for the given data set. Sensitivity analyses suggest that the cumulative time (T_{f}) is the major influencing parameter on which infiltration rate of the soil depends.
Keywords
Double-ring infiltrometer Gaussian process Support vector regression M5P tree modelIntroduction
The process in which water moves into the soil through the top surface soil is called the infiltration, and the rate by which it enters into the soil is called the infiltration rate (Haghighi et al. 2010). It plays the important role in the hydrologic cycle. There are many factors which influence the infiltration rate, that is, rainfall intensity, suction head, humidity, water content, types of impurities, field density and humidity. It is associated with the surface runoff and groundwater recharge (Uloma et al. 2014) and also helpful in water supply system, landslides, design of irrigation, flood control system and drainage (Igbadun and Idris 2007). With the help of infiltration rate, we can easily find out sorptivity and unsaturated hydraulic conductivity of the soil (Chow et al. 1988; Scotter et al. 1982). Hydraulic properties of soil are necessary for design of drainage system (Brooks and Corey 1964). At catchment level, infiltration characteristic is one of the dominant factors in determining the flooding condition (Bhave and Sreeja 2013). The soil capacity of infiltration affects the amount of surface flow (Diamond and Shanley 2003). Infiltration rate in soil is inversely proportion to the water-holding capacity of soil (Singh et al. 2014). Physical changes of soil also affect the infiltration rate (Gupta and Gupta 2008; Smith 2006: Micheal 1978).
Water quality of soil is also affected the infiltration rate and ultimately affected the natural and artificial ground water recharge. Generally, there are many impurities present in the earth surface which can easily mix with the water and changes the quality of the water. Many people studied about the concept of the water quality and infiltration. Singh et al. (2017) used the two types of impurities (ash and organic manure) in his study with three soft computing techniques (M5P model tree, artificial neural network and random forest) and found that random forest predicts the infiltration rate well as compared to the other methods. Sihag (2018) studied the infiltration rate by mixing different proportions fly ash and rice husk ash in sand with fuzzy logic and artificial neural network and found that artificial neural network outperforms the fuzzy logic. Singh et al. (2017a, 2018) and Sihag and Singh 2018 utilised various infiltration models (empirical model) in his study to calculate the infiltration rate of the soil for the given study area. Tiwari et al. (2017) used the generalised regression neural network, MLR, M5P model tree and SVM to predict the cumulative infiltration of soil and found that SVM works well than the other techniques. Various researchers have been used various soft computing techniques in hydraulics and environmental engineering applications (Sihag et al. 2017b, c, 2018a; Haghiabi et al. 2018; Nain et al. 2018a; Tiwari et al. 2018; Parsaie et al. 2017a, b; Shiri et al. 2016, 2017; Parsaie and Haghiabi 2015, 2017; Parsaie 2016; Azamathulla et al. 2016; Baba et al., 2013). These researchers found that these techniques work exceptionally well. Keeping it in the view, the focus of this investigation is on the prediction of the infiltration rate by using M5P tree, GP, MLR and SVM. Furthermore, the results were also compared with the empirical model (Kostiakov 1932) and sensitivity analysis was performed to find out the most important influencing parameter for predicting the infiltration rate of the soil.
Soft computing techniques
The soft computing technique is one of the most relevant and modern techniques used in the civil engineering problems (Sihag et al. 2018b, c; Nain et al. 2018b, Haghiabi et al. 2017; Kisi et al. 2017; Parsaie et al. 2017c; Kisi et al. 2015; Parsaie and Haghiabi 2014; Shiri and Kisi 2012). In this investigation, GP, SVM and M5P tree models were used. The description of the GP, SVM and M5P tree is given below.
Gaussian process (GP) regression
GP regression relies upon the postulation that nearby observation must share the information mutually and it is an approach for mentioning earlier straight over the function space. The simplification of Gaussian distribution is known as Gaussian regression. The matrix and vector of Gaussian distribution are expressed as covariance and mean in GP regression. Due to having earlier knowledge of function reliance and data, the validation for generalisation is not essential. The GP regression models are capable of recognising the foresee distribution consequent to the input test data (Rasmussen and Williams 2006).
A GP is the collection of numbers of random variable, and any finite number of them has a collective multivariate Gaussian distribution. Assuming u and v stand for input and output domain accordingly, thereupon × pairs (g_{i}, h_{i}) are drawn freely and equivalently distribution. For regression, it is assumed that h ⊆ R_{e}; then, a GP on p is expressed by the mean function v_{0}: u R_{e} and covariance function µ: u × u R_{e}. Readers are requested to follow the Kuss (2006a, b) to get the exhaustive details of GP.
Support vector machine (SVM)
This method was first proposed by Vapnik (1998) and based on statistical learning theory. Main principle of SVM is optimal separation of classes. From the separable classes, SVM selects the one which have lowest generalisation error from infinite number of linear classifier or set upper limit to error which is generated by structural risk minimisation. In this way, the maximum margin between two classes can be found from the selected hyperplane and sum of distances of the hyperplane from the nearby point of two classes will set highest margin between two classes. Readers are requested to follow the Smola (1996) to get the exhaustive details of SVM. Cortes and Vapnik (1995) gave the idea of kernel function for nonlinear support vector regression.
M5P tree
M5P tree (Quinlan 1992) is a binary decision tree that uses linear regression function at the leaf (terminal node) which helps in predicting continuous numerical attributes. This method involves two stages for generation of model tree. First stage consists of splitting criteria to generate a decision tree. Splitting criteria for this method are based on treating the standard deviation of class value. Splitting process causes less standard deviation in child node as compared to parent node and thus considered as pure (Quinlan 1992). Out of all possible splits, M5P tree chooses the one that maximises the error reductions. This process of splitting the data may overgrow the tree which may cause over fitting. So, the next stage involves in removing over fitting using pruning method. It trims overgrown trees by substituting the subtrees with linear regression function. In this technique of tree generation, parameter space is split into surfaces and building a linear regression model in each of them.
Conventional models
In this investigation, two conventional models were used. The description of the conventional models was listed below.
Multi-linear regression (MLR)
Kostiakov model
Materials and methodology
Details of the experimental procedure along with the range of infiltration rate
Time (min.) | Type of impurities | Concentration of impurity (%) | Water content (%) | Range of the infiltration rate (mm/h) |
---|---|---|---|---|
5 | Organic manure, ash | 1, 5, 10, 15 | 3.83, 8.43, 10.16, 11.51, 13.65 | 24–84 |
10 | Organic manure, ash | 1, 5, 10, 15 | 3.83, 8.43, 10.16, 11.51, 13.66 | 12–48 |
15 | Organic manure, ash | 1, 5, 10, 15 | 3.83, 8.43, 10.16, 11.51, 13.67 | 12–36 |
20 | Organic manure, ash | 1, 5, 10, 15 | 3.83, 8.43, 10.16, 11.51, 13.68 | 6–24 |
30 | Organic manure, ash | 1, 5, 10, 15 | 3.83, 8.43, 10.16, 11.51, 13.69 | 6–21 |
40 | Organic manure, ash | 1, 5, 10, 15 | 3.83, 8.43, 10.16, 11.51, 13.70 | 3–18 |
50 | Organic manure, ash | 1, 5, 10, 15 | 3.83, 8.43, 10.16, 11.51, 13.71 | 3–18 |
60 | Organic manure, ash | 1, 5, 10, 15 | 3.83, 8.43, 10.16, 11.51, 13.72 | 3–18 |
90 | Organic manure, ash | 1, 5, 10, 15 | 3.83, 8.43, 10.16, 11.51, 13.73 | 2–28 |
120 | Organic manure, ash | 1, 5, 10, 15 | 3.83, 8.43, 10.16, 11.51, 13.74 | 2–24 |
180 | Organic manure, ash | 1, 5, 10, 25 | 3.83, 8.43, 10.16, 11.51, 13.75 | 2.21 |
Data set
Features of the data used
Variables | Minimum | Maximum | Mean | SD | Kurtosis | Skewness |
---|---|---|---|---|---|---|
Training data set | ||||||
T_{f} (min.) | 5 | 180 | 57.6630 | 53.3414 | 0.2810 | 1.1566 |
C_{i} (%) | 1 | 15 | 7.2174 | 5.2011 | − 1.3709 | 0.1210 |
W_{c} (%) | 3.83 | 13.65 | 8.7021 | 3.6694 | − 1.3878 | − 0.2521 |
f(t) (mm/h) | 1 | 84 | 15.1793 | 15.9358 | 6.1863 | 2.3142 |
Testing data set | ||||||
T_{f} (min.) | 5 | 180 | 53.3750 | 48.6271 | 1.3901 | 1.3840 |
C_{i} (%) | 1 | 15 | 6.5000 | 5.0637 | − 1.3367 | 0.2401 |
W_{c} (%) | 3.83 | 13.65 | 8.2635 | 3.8315 | − 1.5351 | − 0.0002 |
f(t) (mm/h) | 2 | 96 | 15.3875 | 16.7426 | 13.6287 | 3.2757 |
All data set | ||||||
T_{f} (min.) | 5 | 180 | 56.3636 | 51.8111 | 0.4986 | 1.2096 |
C_{i} (%) | 1 | 15 | 7.0000 | 5.1512 | − 1.3555 | 0.1572 |
W_{c} (%) | 3.83 | 13.65 | 8.5692 | 3.7101 | − 1.4396 | − 0.1741 |
f(t) (mm/h) | 1 | 96 | 15.2424 | 16.1204 | 8.2289 | 2.5971 |
Correlation matrix of input data set
Variables | T_{f} (min.) | C_{i} (%) | W_{c} (%) | f(t) (mm/h) |
---|---|---|---|---|
T_{f} (min.) | 1.0000 | 0.0000 | 0.0000 | − 0.4322 |
C_{i} (%) | 0.0000 | 1.0000 | 0.1796 | 0.1077 |
W_{c} (%) | 0.0000 | 0.1796 | 1.0000 | − 0.1033 |
f(t) (mm/h) | − 0.4322 | 0.1077 | − 0.1033 | 1.0000 |
Detail of kernel functions
- 1.
Radial basis kernel (RBF) = \(e^{{ - \gamma \left| {a - b} \right|^{2} }}\)
- 2.
Pearson VII kernel function (PUK) = \(\left( {{1 \mathord{\left/ {\vphantom {1 {\left[ {1 + \,\left( {{{2\sqrt {\left\| {a\, - \,b} \right\|}^{2} \sqrt {2^{{\left( {{1 \mathord{\left/ {\vphantom {1 \omega }} \right. \kern-0pt} \omega }} \right)}} - \,1} \,} \mathord{\left/ {\vphantom {{2\sqrt {\left\| {a\, - \,b} \right\|}^{2} \sqrt {2^{{\left( {{1 \mathord{\left/ {\vphantom {1 \omega }} \right. \kern-0pt} \omega }} \right)}} - \,1} \,} \sigma }} \right. \kern-0pt} \sigma }} \right)^{2} } \right]}}} \right. \kern-0pt} {\left[ {1 + \,\left( {{{2\sqrt {\left\| {a\, - \,b} \right\|}^{2} \sqrt {2^{{\left( {{1 \mathord{\left/ {\vphantom {1 \omega }} \right. \kern-0pt} \omega }} \right)}} - \,1} \,} \mathord{\left/ {\vphantom {{2\sqrt {\left\| {a\, - \,b} \right\|}^{2} \sqrt {2^{{\left( {{1 \mathord{\left/ {\vphantom {1 \omega }} \right. \kern-0pt} \omega }} \right)}} - \,1} \,} \sigma }} \right. \kern-0pt} \sigma }} \right)^{2} } \right]}}^{\omega } } \right)\)
Primary parameters using GP, SVM and M5P tree
Approaches | Primary parameters |
---|---|
M5P tree | m = 4 |
GP with RBF | Gaussian noise = 0.80, γ = 3.5 |
GP with PUK | Gaussian noise = 0.80, ω = 0.02, σ = 0.5 |
SVM with RBF | C = 2, γ = 3.5 |
SVM with PUK | C = 2, ω = 0.02, σ = 0.5 |
Statistical performance evaluation criteria
Correlation coefficient (CC) and root-mean-square error (RMSE) values were calculated to investigate the performance of GP, SVM and M5P tree modelling approaches.
Coefficient of correlation (CC)
Root-mean-square error (RMSE)
Results and discussion
Results of the different modelling approaches and empirical models for training and testing data set
Approaches | Training data set | Testing data set | ||
---|---|---|---|---|
CC | RMSE | CC | RMSE | |
GP_RBF | 0.7753 | 10.4193 | 0.4374 | 14.9329 |
GP_PUK | 0.9937 | 8.3337 | 0.4115 | 15.8927 |
SVM_RBF | 0.8162 | 9.8525 | 0.5278 | 14.1891 |
SVM_PUK | 1.0000 | 0.0933 | 0.4289 | 15.4325 |
M5P | 0.9072 | 7.7038 | 0.8490 | 9.4356 |
MLR | 0.6262 | 12.6476 | 0.4405 | 15.9657 |
Kostiakov model | 0.7112 | 11.3162 | 0.4608 | 15.0521 |
Comparison of the results
Statistical information of the infiltration rate with different soft computing techniques and empirical models
Approaches | Minimum | Maximum | Mean | SD | Kurtosis | Skewness |
---|---|---|---|---|---|---|
Training data set | ||||||
Actual | 1.00 | 84.00 | 15.1793 | 15.9358 | 6.1863 | 2.3142 |
GP_RBF | 0.77 | 37.17 | 15.0424 | 9.4499 | − 0.7493 | 0.3505 |
GP_PUK | 7.20 | 45.74 | 15.1815 | 7.6481 | 4.3109 | 1.9056 |
SVM_RBF | 0.91 | 43.24 | 13.3878 | 9.8245 | 0.1587 | 0.8043 |
SVM_PUK | 1.02 | 83.94 | 15.2025 | 15.8756 | 6.2540 | 2.3261 |
M5P tree | 0.50 | 46.69 | 14.5036 | 10.6377 | 0.9470 | 1.1090 |
MLR | 1.83 | 63.04 | 12.8963 | 11.4244 | 4.5405 | 1.9507 |
Kostiakov model | 3.35 | 38.36 | 13.4477 | 10.3690 | 0.7651 | 1.2985 |
Testing data set | ||||||
Actual | 2 | 96 | 15.3875 | 16.7426 | 13.6287 | 3.2757 |
GP_RBF | − 1.61 | 36.17 | 14.7312 | 8.5849 | − 0.0222 | 0.2881 |
GP_PUK | 12.36 | 20.75 | 15.1809 | 1.7746 | 1.1680 | 0.8328 |
SVM_RBF | 1.00 | 39.52 | 13.3489 | 8.7057 | 0.6250 | 0.7803 |
SVM_PUK | 10.11 | 25.85 | 15.1678 | 3.2476 | 1.6642 | 0.9674 |
M5P tree | 0.25 | 46.69 | 15.1861 | 10.6086 | 2.2856 | 1.5043 |
MLR | 1.90 | 62.03 | 12.5197 | 12.5803 | 6.1334 | 2.3440 |
Kostiakov model | 3.35 | 38.36 | 12.3710 | 9.2193 | 2.7331 | 1.6930 |
Sensitivity analysis (SA)
Sensitivity analysis using M5P tree model
Combinations of the variables | Parameter removed | M5P tree model | |
---|---|---|---|
CC | RMSE (mm/h) | ||
T_{f}, I_{t}, C_{i}, W_{c} | 0.9072 | 7.7038 | |
I_{t}, C_{i}, W_{c} | T _{f} | 0.1971 | 15.5368 |
T_{f}, C_{i}, W_{c} | I _{t} | 0.8557 | 9.0473 |
T_{f}, I_{t}, W_{c} | C _{i} | 0.8822 | 8.2711 |
T_{f}, I_{t}, C_{i} | W _{c} | 0.8418 | 9.0481 |
Conclusions
Knowledge of infiltration process is essential for agriculture, hydrologic study, watershed management, irrigation system design and drainage design. In this investigation, three soft computing techniques (SVR, GP and M5P tree) and two empirical models (MLR and Kostiakov model) were used to estimate the infiltration rate of the soil with different water qualities. The obtained results concluded that the M5P tree model is the most efficient model to predict the infiltration rate of the soil with different water qualities than the SVR, GP, MLR and Kostiakov model, whereas the results of SVM were more suitable as compared to the GP and MLR and also gave better prediction than Kostiakov model. Thus, M5P tree model was the most suitable model for predicting the infiltration rate of the soil. Finally, SA suggests that cumulative time is an essential parameter which affects the infiltration rate of the soil with different water qualities using M5P model tree for this data set.
Notes
References
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