# An Enhanced Feed-Forward Back Propagation Levenberg–Marquardt Algorithm for Suspended Sediment Yield Modeling

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## Abstract

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## 1. Introduction

^{®}, and catchment area (CA) as inputs. No attempt has been made to predict SSY at multiple gauge stations using a single model in the entire MRB using the ANN techniques with temporal data (WD, RF, and T) and spatial data (RT, R, and CA). Thus, in this study, a single generalized ANN model was developed using the combined data of 11 gauge stations to estimate SSY at each station of the 11 gauging stations of the entire MRB using the hydro-geo-climatical WD, RF, T, RT, R, and CA data as major controlling factors of SSY. The parameters for some AI methods are selected by a trial-and-error method to obtain a reasonably good result. However, this approach takes a significantly large amount of computational time to obtain the parameter value, and is also not guaranteed to be the optimal or near-optimal solution to the problems. In this study, the parameters for the ANN model were selected by the grid search technique to obtain a passably good result. After the development of a reliable ANN-based prediction model, the performance of the model was examined with the same test dataset. The results demonstrated that the proposed ANN-based model performed satisfactorily and had a greater capacity for generalization than other comparative MLR and SRC methods for SSY prediction. Moreover, the ANN model, which is developed using the combined data of 11 stations, provided better results at Tikarapara than the ANN models using the data of Tikarapara station only (ANN-1) and had more generalization capability. The ANN-1 model is developed using the RF, WD, T, RT, CA, and R of a single Tikarapara station only using the same method as the ANN model which is developed combined data of 11 gauging stations. Among all gauging stations, the proposed ANN prediction model provided the best accuracy at Tikarapara gauging station. It could be because Tikarapara is situated at the far downstream end of the MRB basin before meeting with the Bay of Bengal which has the maximum CA, RF, WD, and SSY among all the gauging stations. Many researchers have developed artificial intelligence (AI) models to predict sediment load by considering a set of temporal parameters, such as WD, RF, and T, for a specific geographical location. The ANN model prepared based on the data of 11 gauging stations performed better than the ANN-1 model which was developed based on the data of individual stations (Tikarapara) and has a greater generalization capability than the individual models of different gauging stations. The MRB case study focuses on the development of a highly generalized global single AI model using a huge amount of temporal as well as spatial data from 11 gauging stations and applied it at individual stations for the prediction of SSY in river systems which is our unique contribution.

## 2. Study Region

^{2}, accounting for 4.3% of India’s total geographical area [34]. The MRB is located between latitudes 19°20′ and 23°35′ north and between longitudes 80°30′ and 86°50′ east. At an altitude of roughly 442 m above sea level, the Mahanadi River starts in Raipur, Chhattisgarh, halfway between Pharsiya Village and Nagri Town. A total of 53 % of the river’s CA contribution is made in Chhattisgarh, 46% in Odisha, and the remaining amounts are split evenly between Maharashtra and Jharkhand [34,35]. According to the current sediment load, in terms of capacity to cause flooding and water potential, the MRB is ranked second among the Indian peninsular rivers [36,37]. From 1971 to 2004, the mean annual RF in the MRB basin ranged between twelve hundred and fourteen hundred mm [36]. According to daily statistics for the years 1969 to 2004, the two coldest months of the year are January and December with the lowest temperatures of 12°C, and the two warmest months of the year are May and April with the highest temperatures of 39°C to 40°C [36]. The river’s basin area contribution for the years 2005–2006 was 54.27% under agricultural land cover, 5.24% under wasteland, 32.74% under forest cover, 3.30% under built-up land, and 4.45% under aquatic bodies [36]. The Chilika Lake and Hirakud reservoir are two large sources of water in the MRB. A geographical location map of the MRB including gauging sites is shown in Figure 1. The different lithologies found in the basin area include 5% coastal alluvium, 7% khondalite, 15% charnockite, 17% shale and Lower Gondwana limestone, and 22% shale and Upper Gondwana sandstone [38]. Among the 11 measuring stations, Tikarapara has the lowest elevation, while Baronda has the highest. The maximum CA value (124,450 km

^{2}) is found at Tikarapara, which lies on the downward side of the MRB before it meets the Bay of Bengal, while the lowest CA value (2210 km

^{2}) is found in Andhiyarakhore, which lies in the upper part of the MRB. Table 1 summarizes the subbasin and tributary descriptions of the MRB. The highest CA is found in the Seonath tributary, while the lowest is found in the Jonk. There is an abundance of literature that has provided a full description of the MRB [24,25,29,37].

## 3. Methodology

## 4. Results and Discussion

^{9}, and the value was incremented and reduced by a factor of ten and 0.1, respectively. The model began with a random connection and bias weights values that were initialized and then updated every epoch to optimize performance. In this study, the maximum number of hidden neurons was restricted to 32, in view of computational time and model complexity [25,49,50]. The lower the complexity of the model, the easier it is to understand the interpretability of the artificial intelligence model [63]. Figure 6 shows the RMSE variation values with neurons an µ of the ANN using grid search techniques. This figure shows that when the optimum neurons in the hidden layer and µ are 31 and 0.06, respectively, then the ANN model produced the lowest RMSE value (0.00460) in the training phase. As a result, it was considered the best ANN model. To examine the effectiveness of the models, the error variance (VAR), RMSE, mean absolute error (MAE), coefficient of efficiency (CE), mean square error (MSE), and correlation coefficient ($r$) are often utilized as statistical performance measures.

## 5. Conclusions and Future Scope

## Author Contributions

## Funding

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 3.**The flowchart of the proposed ANN model with heuristic parameters selection processes for SSY prediction.

**Figure 5.**Monthly variations in WD, RF, T, and SSY at different locations (

**a**) Tikarapara (

**b**) Simga (

**c**) Andhiyarakhore (

**d**) Sundargarh (

**e**) Bamnidih (

**f**) Jondhra (

**g**) Kantamal (

**h**) Kurubhata (

**i**) Basantpur (

**j**) Baronda (

**k**) Rajim.

**Figure 7.**Hydrograph between predicted SSY and actual SSY of testing data of the ANN prediction model.

**Figure 8.**Scatter plots between predicted SSY and actual SSY of testing data of the ANN prediction model.

**Figure 9.**(

**a**) Hydrograph between the predicted and actual SSY at Tikarapara using the ANN-1 model; (

**b**) Scatter plot between the predicted SSY and actual at Tikarapara using the ANN model.

Name of the Sub-Basin and Tributary | Catchment Area (km^{2}) | Catchment Area of Mahanadi Basin (%) |
---|---|---|

Hasdeo | 9856 | 6.96 |

Jonk | 3484 | 2.47 |

Seonath | 30,761 | 21.72 |

Mand | 5200 | 3.67 |

Upper Mahanadi | 21,652 | 15.29 |

Ib | 12,447 | 8.79 |

Ong | 5128 | 3.62 |

Middle Mahanadi | 12,654 | 8.93 |

Tel | 22,818 | 16.12 |

Lower Mahanadi | 17,589 | 12.43 |

Total | 141,589 | 100 |

Abbreviations | Meaning |
---|---|

SSY | Suspended sediment yield |

ANN | Artificial neural network |

AI | Artificial intelligence |

WD | Water discharge |

RF | Rainfall |

T | Temperature |

R | Relief |

RT | Rock type |

CA | Catchment area |

MRB | Mahanadi river basin |

MSE | Mean square error |

RMSE | Root-mean-squared error |

MAE | Mean absolute error |

μ | Combinational coefficient of Levenberg–Marquardt algorithm |

LM | Levenberg–Marquardt |

CE | Coefficient of efficiency |

MLP | Multi-layer perceptron |

MAE | Mean absolute error |

r | Coefficient of correlation |

GDA | Gradient descending adaptive |

VAR | Error variance |

I | Input layer |

O | Output layer |

H | Hidden layer |

ARMA | Autoregressive moving average |

MLR | Multiple linear regression |

SRC | Sediment rating curve |

Source | SS | Df | MS | F | p > F |
---|---|---|---|---|---|

Columns | 138.938 | 3 | 46.3126 | 1694.08 | 0.001 |

Error | 288.579 | 10556 | 0.0273 | ||

Total | 427.517 | 10559 |

**Table 4.**Error statistics during the testing, training, and validation phases of the ANN model of 11 gauge stations of the MRB.

ANN | RMSE | MAE | r | VAR | MSE | CE |
---|---|---|---|---|---|---|

Validation | 0.011 | 0.004 | 0.741 | 1.000 × 10^{−4} | 1.000 × 10^{−4} | −1.255 |

Training | 0.005 | 0.002 | 0.976 | 2.090 × 10^{−5} | 2.090 × 10^{−5} | 0.952 |

Testing | 0.009 | 0.003 | 0.867 | 7.760 × 10^{−5} | 7.950 × 10^{−5} | −0.249 |

Tikarapara | 0.009 | 0.006 | 0.967 | 5.400 × 10^{−5} | 7.960 × 10^{−5} | 0.901 |

Simga | 0.003 | 0.002 | 0.780 | 7.540 × 10^{−6} | 7.820 × 10^{−6} | −0.196 |

Andhiyakore | 0.001 | 0.001 | 0.611 | 6.860 × 10^{−7} | 6.770 × 10^{−7} | −14.881 |

Sundargarh | 0.005 | 0.003 | 0.684 | 2.240 × 10^{−5} | 2.170 × 10^{−5} | 0.412 |

Bamnidih | 0.002 | 0.001 | 0.546 | 2.480 × 10^{−6} | 2.440 × 10^{−6} | −29.462 |

Jondhara | 0.004 | 0.002 | 0.922 | 1.620 × 10^{−5} | 1.760 × 10^{−5} | 0.778 |

Kantamal | 0.027 | 0.012 | 0.772 | 1.000 × 10^{−3} | 7.540 × 10^{−4} | 0.203 |

Kurubhata | 0.002 | 0.001 | 0.945 | 4.140 × 10^{−6} | 4.980 × 10^{−6} | 0.160 |

Basantpur | 0.006 | 0.004 | 0.943 | 4.540 × 10^{−5} | 4.910 × 10^{−5} | 0.785 |

Baronda | 0.001 | 0.001 | 0.849 | 1.400 × 10^{−6} | 1.290 × 10^{−6} | 0.667 |

Rajim | 0.002 | 0.001 | 0.716 | 2.520 × 10^{−5} | 2.540 × 10^{−6} | 0.455 |

**Table 5.**Performance comparisons in testing data error statistics of the ANN model and ANN-1 model of Tikarapara Station of the MRB.

Models | RMSE | MSE | MAE | Error Variance | r |
---|---|---|---|---|---|

ANN | 0.00892 | 7.95 × 10^{−5} | 0.002897 | 7.76 × 10^{−5} | 0.867 |

ANN-1 | 0.09847 | 0.00968 | 0.05694 | 0.006619 | 0.957 |

MLR | 8.960 × 10^{−3} | 8.03 × 10^{−5} | 0.004 | 7.92 × 10^{−5} | 0.843 |

SRC | 1.010 × 10^{−2} | 1.000 × 10^{−4} | 0.003 | 9.830 × 10^{−5} | 0.792 |

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**MDPI and ACS Style**

Yadav, A.; Chithaluru, P.; Singh, A.; Joshi, D.; Elkamchouchi, D.H.; Pérez-Oleaga, C.M.; Anand, D. An Enhanced Feed-Forward Back Propagation Levenberg–Marquardt Algorithm for Suspended Sediment Yield Modeling. *Water* **2022**, *14*, 3714.
https://doi.org/10.3390/w14223714

**AMA Style**

Yadav A, Chithaluru P, Singh A, Joshi D, Elkamchouchi DH, Pérez-Oleaga CM, Anand D. An Enhanced Feed-Forward Back Propagation Levenberg–Marquardt Algorithm for Suspended Sediment Yield Modeling. *Water*. 2022; 14(22):3714.
https://doi.org/10.3390/w14223714

**Chicago/Turabian Style**

Yadav, Arvind, Premkumar Chithaluru, Aman Singh, Devendra Joshi, Dalia H. Elkamchouchi, Cristina Mazas Pérez-Oleaga, and Divya Anand. 2022. "An Enhanced Feed-Forward Back Propagation Levenberg–Marquardt Algorithm for Suspended Sediment Yield Modeling" *Water* 14, no. 22: 3714.
https://doi.org/10.3390/w14223714