# Estimation of Daily Stage–Discharge Relationship by Using Data-Driven Techniques of a Perennial River, India

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## Abstract

**:**

^{3}/s, NSE = 0.925, PCC = 0.964, WI = 0.979) outperformed the WANN and SVM-LF models with the combination of three inputs, i.e., current stage, one-day antecedent stage, and discharge, during the testing period. In addition, the SVM-RF model was found to be more reliable and robust than the other models and having important implications for water resources management at the study site.

## 1. Introduction

## 2. Materials and Methods

#### 2.1. Study Area and Data Collection

^{2}. Figure 1 illustrates the location map of the study area. The basin is strongly dominated by the south-west monsoon that starts in June and descends in mid-October. The average annual rainfall in the basin is about 1800 mm. The maximum temperature in the plains of the basin varies between 42 and 49 °C during May and goes down 8 to 14 °C during December–January. Geologically, the basin belongs mostly to Archean terrains. The rocks in the basin include Gneisses, Schist, Quartzite, and Amphibolite. Igneous rocks are also seen in the riverbed at some places.

^{3}/s) of 10 years (1st June 2004–31st October 2013) were obtained from the India-Water Resources Information System (WRIS) portal. The time series plot of the total available datasets of stage and discharge versus time is shown in Figure 2. The whole data were divided into two parts: (i) training dataset consisting of 70% (1st June 2004 to 31st October 2010) of the total data which were used for the development of the model, and (ii) remaining 30% (1st June 2011 to 31st October 2013) of the total data which were used for testing to check the prediction capability of the applied models (Figure 2). Figure 3 shows the relationship between stage and discharge through the rating curve at the study site. In contrast, Figure 4 illustrates the flowchart of the adopted methodology for discharge estimation at the Govindpur site.

#### 2.2. Wavelet Transforms

#### 2.3. Support Vector Machine (SVM)

^{n}are the training inputs and y ϵ, Y ⸦ R

^{n}are the training outputs. Assume a non-linear function $f\left(x\right)$ is given by Equation (4):

- Linear kernel function: the simplest type of kernel function and written by using Equation (8) [72]:$$K\left({x}_{i},\text{}{x}_{j}\right)=\left({x}_{i},\text{}{x}_{j}\right)$$
- Radial basis function (RBF): a mapping of RBF that is similar to Gaussian bell-shaped, and expressed by using Equation (9) [72]:$$K\left({x}_{i},\text{}{x}_{j}\right)=\mathrm{exp}\left(-\gamma \Vert {x}_{i}-\text{}{x}_{j}{\Vert}^{2}\right)$$

#### 2.4. Model Development and Performance Indicators

_{t}) at the study site.

^{2}) to the differences in observed and estimated means and variances [81,82]. It represents the ratio of the mean square error and the potential error [83]. The WI varies between zero and one (0 < WI ≤ 1), so near to 1 means a perfect agreement/fit, while approaching 0 means complete disagreements between the observed and estimated data (Equation (16)). The main disadvantages of WI are over-sensitivity to extremes values due to the squared differences. The high values of WI were reported even for poor model fits [81,82].

## 3. Results and Discussion

#### 3.1. Statistical Analysis

#### 3.2. Evaluation of Results from Various Trails

#### 3.3. Quantitative and Qualitative Evaluation of Results

^{3}/s), NSE, PCC, and WI were obtained as 104.426, 0.925, 0.964, and 0.979, respectively, for SVM-RF-1, 106.594, 0.922, 0.964, and 0.978 for SVM-RF-2, and 122.262, 0.897, 0.956, and 0.969 for SVM-RF-3. The order of model performance based on NSE from very good to unsatisfactory was attained as SVM-RF-1 (0.925) > SVM-RF-2 (0.922) > SVM-RF-3 (0.897) > SVM-LF-3 (0.893) > WANN-1 (0.888) > SVM-LF-1 (0.883) = SVM-LF-2 (0.883) = WANN-3 (0.883) > WANN-2 (0.866). The order of model performance on the basis of the RMSE from best to inferior was obtained as SVM-RF-1 (104.426) > SVM-RF-2 (106.594) > SVM-RF-3 (122.262) > SVM-LF-3 (124.954) > WANN-1 (127.349) > SVM-LF-1 (130.404) > WANN-3 (130.441) > SVM-LF-2 (130.556) > WANN-2 (139.559). The order of model performance based on the WI from best to inferior was found as SVM-RF-1 (0.979) > SVM-RF-2 (0.978) > WANN-3 (0.971) > SVM-LF-3 (0.970) > SVM-RF-3 (0.969) > WANN-1 (0.968) > SVM-LF-1 (0.967) = SVM-LF-2 (0.967) > WANN-2 (0.963). The comparison of results in Table 5 confirmed the superiority of the SVM-RF model with M-1 (inputs ${H}_{t},\text{}{H}_{t-1,\text{}}{Q}_{t-1}$) having the lowest value of RMSE = 104.426 m

^{3}/s, and the highest values of NSE = 0.925, PCC = 0.964, and WI = 0.979, closely followed by the SVM-RF-2 model.

^{3}/s), whereas low discharge (<180 m

^{3}/s) values are over-estimated by WANN, SVM-LF, and SVM-RF models during the testing period. The quantity of explained variation out of the total variation (R

^{2}: coefficient of determination) was obtained as excellent for SVM-RF-1 and SVM-RF-2 models. Based on R

^{2}values, the order of the model performance from very satisfactory to unsatisfactory [87] was found as SVM-RF-1 (0.930) = SVM-RF-2 (0.930) > SVM-RF-3 (0.914) > SVM-LF-3 (0.903) > WANN-3 (0.894) > WANN-1 (0.890) > SVM-LF-2 (0.887) > SVM-LF-1 (0.886) > WANN-2 (0.867).

^{3}/s, MAE = 13.1 m

^{3}/s, NSE = 0.94), and MARS at the Chakdara site (RMSE = 47.5 m

^{3}/s, MAE = 31.6 m

^{3}/s, NSE = 0.91). Ali and Shahbaz [96] evaluated the performance of an ANN for daily streamflow prediction in the Jhelum river basin, Pakistan. The results of the analysis revealed the better suitability of ANN in daily streamflow prediction with RMSE = 127.70 m

^{3}/s, PCC = 0.98, and NSE = 0.96 during the testing period. Mohammadi et al. [97] predicted the monthly streamflow of the Vu Gia Thu Bon river (Vietnam) using a standalone ANFIS and hybrid ANFIS coupled with the shuffled frog leaping algorithm (ANFIS-SFLA). The results of the perusal displayed the superior performance of the ANFIS-SFLA model with RMSE = 141.39 m

^{3}/s, NSE = 0.88, and PCC = 0.88 over the ANFIS model (RMSE = 167.81 m

^{3}/s, NSE = 0.83, PCC = 0.83). Mohammadi et al. [98] applied classical MLP and their hybrid integrated with particle swarm (MLP-PSO), PSO-multi-verse optimizer (MLP-PSO-MVO), and bi-linear (MLP-BL) to predict the daily streamflow at four stations, i.e., Brantford and Galt located in Grand River, Canada, and Macon and Elkton positioned in Ocmulgee and Umpqua rivers, United States. The results of the comparison revealed that the MLP-BL models (RMSE = 6.426/ 6.067/ 24.441/ 34.535 m

^{3}/s, MAE = 3.530/ 3.190/ 11.825/ 14.878 m

^{3}/s, and R

^{2}= 0.994/ 0.990/ 0.990/ 0.986) outperformed the other models at the Brantford, Galt, Macon, and Elkton stations, respectively. Tripura et al. [99] forecasted hourly streamflow of Barak riven basin, Assam (India) by employing the standalone co-active neuro-fuzzy inference system (CANFIS) and a hybrid of CANFIS optimized with the genetic algorithm (CANFIS-GA) and firefly algorithm (CANFIS-FA). They found that the CANFIS-FA model provides better results than the other models. The results of these studies support the application of artificial intelligence (AI) techniques in monthly and daily streamflow/discharge prediction. Likewise, the results of the current research are in fair agreement with the utility of the SVM-RF technique for daily discharge prediction at Govindpur station.

## 4. Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

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**Figure 5.**Partial Autocorrelation Function Values of (

**a**) Stage, and (

**b**) Discharge at the Study Site.

**Figure 6.**Observed Versus Estimated Discharge of Best (

**a**) WANN-1, (

**b**) SVM-LF-1, and (

**c**) SVM-RF-1 Models During the Testing Period at the Study Site.

**Figure 7.**Observed versus estimated discharge of best (

**a**) WANN-2, (

**b**) SVM-LF-2, and (

**c**) SVM-RF-2 models during the testing period at the study site.

**Figure 8.**Observed Versus Estimated Discharge of Best (

**a**) WANN-3, (

**b**) SVM-LF-3, and (

**c**) SVM-RF-3 Models During the Testing Period at the Study Site.

**Figure 9.**Taylor diagram of WANN, SVM-LF, and SVM-RF corresponding to (

**a**) scenario 1, (

**b**) scenario 2, and (

**c**) scenario 3 during the testing period at the study site.

**Table 1.**Statistics of Stage and Discharge Variables During Training, Testing, and Entire Periods at the Study Station.

Statistical Parameter | Training | Testing | Entire | |||
---|---|---|---|---|---|---|

H (m) | Q (m ^{3}/s) | H (m) | Q (m ^{3}/s) | H (m) | Q (m ^{3}/s) | |

Mean | 2.9461 | 243.50 | 2.7548 | 291.60 | 2.8887 | 257.93 |

Median | 2.5200 | 136.80 | 2.2000 | 157.12 | 2.4900 | 142.61 |

Minimum | 0.8600 | 1.3690 | 0.8600 | 3.5730 | 0.8600 | 1.3690 |

Maximum | 8.8400 | 2885.9 | 9.2400 | 2685.6 | 9.2400 | 2885.9 |

Std. Dev. | 1.5805 | 349.15 | 1.7028 | 381.48 | 1.6200 | 359.71 |

CV | 0.5364 | 1.4339 | 0.6181 | 1.3082 | 0.5608 | 1.3946 |

Skewness | 1.3012 | 3.6133 | 1.3441 | 2.8999 | 1.3013 | 3.3629 |

**Table 2.**Performance Indicators of WANN-1, SVM-LF-1, and SVM-RF-1 Models During Testing at the Study Station.

Model | Performance Indicators | |||
---|---|---|---|---|

RMSE | NSE | PCC | WI | |

WANN-1 | ||||

Trail-1 | 148.662 | 0.848 | 0.924 | 0.959 |

Trail-2 | 127.349 | 0.888 | 0.944 | 0.968 |

Trail-3 | 133.695 | 0.877 | 0.938 | 0.968 |

Trail-4 | 157.487 | 0.829 | 0.927 | 0.960 |

SVM-LF-1 | ||||

Trail-1 | 130.404 | 0.883 | 0.941 | 0.967 |

Trail-2 | 217.531 | 0.674 | 0.952 | 0.930 |

Trail-3 | 135.250 | 0.874 | 0.954 | 0.968 |

Trail-4 | 180.688 | 0.775 | 0.954 | 0.948 |

SVM-RF-1 | ||||

Trail-1 | 108.920 | 0.918 | 0.961 | 0.977 |

Trail-2 | 106.227 | 0.922 | 0.963 | 0.978 |

Trail-3 | 106.227 | 0.922 | 0.963 | 0.978 |

Trail-4 | 104.426 | 0.925 | 0.964 | 0.979 |

**Table 3.**Performance Indicators of WANN-2, SVM-LF-2, and SVM-RF-2 Models During Testing at the Study Station.

Model | Performance Indicators | |||
---|---|---|---|---|

RMSE | NSE | PCC | WI | |

WANN-2 | ||||

Trail-1 | 139.597 | 0.866 | 0.931 | 0.962 |

Trail-2 | 139.839 | 0.866 | 0.933 | 0.961 |

Trail-3 | 139.559 | 0.866 | 0.931 | 0.963 |

Trail-4 | 151.836 | 0.842 | 0.935 | 0.963 |

SVM-LF-2 | ||||

Trail-1 | 206.840 | 0.706 | 0.953 | 0.938 |

Trail-2 | 130.556 | 0.883 | 0.942 | 0.967 |

Trail-3 | 135.972 | 0.873 | 0.956 | 0.968 |

Trail-4 | 174.246 | 0.791 | 0.954 | 0.952 |

SVM-RF-2 | ||||

Trail-1 | 111.356 | 0.915 | 0.962 | 0.975 |

Trail-2 | 109.005 | 0.918 | 0.962 | 0.977 |

Trail-3 | 108.376 | 0.919 | 0.963 | 0.977 |

Trail-4 | 106.594 | 0.922 | 0.964 | 0.978 |

**Table 4.**Performance Indicators of WANN-3, SVM-LF-3, and SVM-RF-3 Models During Testing at the Study Station.

Model | Performance Indicators | |||
---|---|---|---|---|

RMSE | NSE | PCC | WI | |

WANN-3 | ||||

Trail-1 | 148.561 | 0.848 | 0.925 | 0.961 |

Trail-2 | 130.441 | 0.883 | 0.945 | 0.971 |

Trail-3 | 244.984 | 0.588 | 0.824 | 0.901 |

Trail-4 | 134.526 | 0.876 | 0.939 | 0.968 |

SVM-LF-3 | ||||

Trail-1 | 128.384 | 0.887 | 0.945 | 0.968 |

Trail-2 | 124.954 | 0.893 | 0.950 | 0.970 |

Trail-3 | 139.634 | 0.866 | 0.954 | 0.966 |

Trail-4 | 173.277 | 0.794 | 0.951 | 0.953 |

SVM-RF-3 | ||||

Trail-1 | 130.589 | 0.883 | 0.951 | 0.964 |

Trail-2 | 122.262 | 0.897 | 0.956 | 0.969 |

Trail-3 | 147.599 | 0.850 | 0.939 | 0.952 |

Trail-4 | 124.596 | 0.893 | 0.954 | 0.968 |

Model | Structure/Parameter | Performance Indicators | |||
---|---|---|---|---|---|

RMSE | NSE | PCC | WI | ||

WANN-1 | 12-5-1 | 127.349 | 0.888 | 0.944 | 0.968 |

SVM-LF-1 | $\gamma $ = 0.330, $\epsilon $ = 0.100, c = 10 | 130.404 | 0.883 | 0.941 | 0.967 |

SVM-RF-1 | $\gamma $ = 0.160, $\epsilon $ = 0.010, c = 10 | 104.426 | 0.925 | 0.964 | 0.979 |

WANN-2 | 20-9-1 | 139.559 | 0.866 | 0.931 | 0.963 |

SVM-LF-2 | $\gamma $ = 0.1428, $\epsilon $ = 0.010, c = 10 | 130.556 | 0.883 | 0.942 | 0.967 |

SVM-RF-2 | $\gamma $ = 0.120, $\epsilon $ = 0.010, c = 10 | 106.594 | 0.922 | 0.964 | 0.978 |

WANN-3 | 28-5-1 | 130.441 | 0.883 | 0.945 | 0.971 |

SVM-LF-3 | $\gamma $ = 0.143, $\epsilon $ = 0.010, c = 10 | 124.954 | 0.893 | 0.950 | 0.970 |

SVM-RF-3 | $\gamma $ = 0.160, $\epsilon $ = 0.100, c = 10 | 122.262 | 0.897 | 0.956 | 0.969 |

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## Share and Cite

**MDPI and ACS Style**

Kumar, M.; Kumari, A.; Kushwaha, D.P.; Kumar, P.; Malik, A.; Ali, R.; Kuriqi, A.
Estimation of Daily Stage–Discharge Relationship by Using Data-Driven Techniques of a Perennial River, India. *Sustainability* **2020**, *12*, 7877.
https://doi.org/10.3390/su12197877

**AMA Style**

Kumar M, Kumari A, Kushwaha DP, Kumar P, Malik A, Ali R, Kuriqi A.
Estimation of Daily Stage–Discharge Relationship by Using Data-Driven Techniques of a Perennial River, India. *Sustainability*. 2020; 12(19):7877.
https://doi.org/10.3390/su12197877

**Chicago/Turabian Style**

Kumar, Manish, Anuradha Kumari, Daniel Prakash Kushwaha, Pravendra Kumar, Anurag Malik, Rawshan Ali, and Alban Kuriqi.
2020. "Estimation of Daily Stage–Discharge Relationship by Using Data-Driven Techniques of a Perennial River, India" *Sustainability* 12, no. 19: 7877.
https://doi.org/10.3390/su12197877