# 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

- Hasnain, S.I.; Thayyen, R.J. Discharge and suspended-sediment concentration of meltwaters, draining from the Dokriani glacier, Garhwal Himalaya, India. J. Hydrol.
**1999**, 218, 191–198. [Google Scholar] [CrossRef] - Kisi, O.; Yaseen, Z.M. The potential of hybrid evolutionary fuzzy intelligence model for suspended sediment concentration prediction. Catena
**2019**, 174, 11–23. [Google Scholar] [CrossRef] - Chang, H.H. Case Study of Fluvial Modeling of River Responses to Dam Removal. J. Hydraul. Eng.
**2008**, 134, 295–302. [Google Scholar] [CrossRef] - Kisi, O. Constructing neural network sediment estimation models using a data-driven algorithm. Math. Comput. Simul.
**2008**, 79, 94–103. [Google Scholar] [CrossRef] - Yadav, A.; Prasad, B.B.V.S.V.; Mojjada, R.K.; Kothamasu, K.K.; Joshi, D. Application of Artificial Neural Network and Genetic Algorithm Based Artificial Neural Network Models for River Flow Prediction. Rev. Intell. Artif.
**2020**, 34, 745–751. [Google Scholar] [CrossRef] - Safari, M.J.S.; Mohammadi, B.; Kargar, K. Invasive weed optimization-based adaptive neuro-fuzzy inference system hybrid model for sediment transport with a bed deposit. J. Clean. Prod.
**2020**, 276, 124267. [Google Scholar] [CrossRef] - ASCE. Task Committee on Application of Artificial neural networks in Hydrology, Artificial neural networks in hydrology. I: Preliminary concepts. J. Hydrol. Eng.
**2020**, 5, 115–123. [Google Scholar] - Maier, H.R.; Dandy, G.C. Neural network for the prediction and forecasting of water resources variables: A review of modeling issues and applications. Environ. Model. Softw.
**2000**, 15, 101–124. [Google Scholar] [CrossRef] - Sahoo, S.; Jha, M.K. Groundwater-level prediction using multiple linear regression and artificial neural network techniques: A comparative assessment. Hydrogeol. J.
**2013**, 21, 1865–1887. [Google Scholar] [CrossRef] - Shamseldin, A.Y. Application of a neural network technique to RF-runoff modeling. J. Hydrol.
**1997**, 199, 272–294. [Google Scholar] [CrossRef] - Zealand, C.; Burn, D.H.; Simonovic, S.P. Short term streamflow forecasting using artificial neural networks. J. Hydrol.
**1999**, 214, 32–48. [Google Scholar] [CrossRef] - Imrie, C.E.; Durucan, S.; Korre, A. River flow prediction using artificial neural networks: Generalization beyond the calibration range. J. Hydrol.
**2000**, 233, 138–153. [Google Scholar] [CrossRef] - Melesse, A.M.; Wang, X. Multi-temporal scale hydrograph prediction using artificial neural networks. J. Am. Water Resour. Assoc.
**2006**, 42, 1647–1657. [Google Scholar] [CrossRef] - Ahmad, S.; Simonovic, S.P. An artificial neural network model for generating hydrograph from hydro-meteorological parameters. J. Hydrol.
**2005**, 315, 236–251. [Google Scholar] [CrossRef] - Zhang, Q.; Stanley, S.J. Forecasting raw-water quality parameters for the North Saskatchewan River by neural network modeling. Water Res.
**1997**, 31, 2340–2350. [Google Scholar] [CrossRef] - Melesse, A.; Jayachandran, K.; Zhang, K. Modeling coastal eutrophication at Florida Bay using neural networks. J. Coast. Res.
**2008**, 24, 190–196. [Google Scholar] [CrossRef] - Solomatine, D.P.; Torres, L.A. Neural network approximation of a hydrodynamic model in optimizing reservoir operation. In Proceedings of the 2nd International Conference on Hydroinformatics, Zurich, Switzerland, 9–13 September 1996; pp. 201–206. [Google Scholar]
- Wen, C.G.; Lee, C.S. A neural network approach to multi-objective optimization for water quality management in a river basin. Water Resour. Res.
**1998**, 34, 427–436. [Google Scholar] [CrossRef] - Brion, G.M.; Lingireddy, S. A neural network approach to identifying non-point sources of microbial contamination. Water Res.
**1999**, 33, 3099–3106. [Google Scholar] [CrossRef] - Zhu, Y.M.; Lu, X.X.; Zhou, Y. Suspended sediment flux modeling with artificial neural network: An example of the Longchuanjiang River in the Upper Yangtze Catchment, China. Geomorphology
**2007**, 84, 111–125. [Google Scholar] [CrossRef] - Rajaee, T.; Mirbagheri, S.A.; Zounemat-Kermani, M.; Nourani, V. Daily suspended sediment concentration simulation using artificial neural network and neuro-fuzzy models. Sci. Total Environ.
**2009**, 407, 4916–4927. [Google Scholar] [CrossRef] - Melesse, A.M.; Ahmad, S.; McClain, M.E.; Wang, X.; Lim, Y.H. Suspended Sediment Load Prediction of River Systems: An artificial neural network approach. Agric. Water Manag.
**2011**, 98, 855–866. [Google Scholar] [CrossRef] - Sharma, N.; Zakaullah, M.; Tiwari, H.; Kumar, D. Runoff and sediment yield modeling using artificial neural network and support vector machines: A case study from Nepal watershed, Model. Earth Syst. Environ.
**2015**, 1, 23. [Google Scholar] [CrossRef] - Yadav, A.; Chatterjee, S.; Equeenuddin, S.M. Suspended sediment yield Estimation using Genetic Algorithm-based Artificial Intelligence Models in Mahanadi River basin. Hydrol. Sci. J.
**2018**, 63, 1162–1182. [Google Scholar] [CrossRef] - Yadav, A.; Chatterjee, S.; Equeenuddin, S.M. Prediction of Suspended Sediment Yield by ANN and Traditional Mathematical Model in MRB, India. Sustain. Water Resour. Manag.
**2017**, 4, 745–759. [Google Scholar] [CrossRef] - Yadav, A.; Satyannarayana, P. Multi-objective genetic algorithm optimization of artificial neural network for estimating suspended sediment yield in Mahanadi River basin, India. Int. J. River Basin Manag.
**2020**, 18, 207–215. [Google Scholar] [CrossRef] - Yadav, A. Estimation and Forecasting of Suspended Sediment Yield in Mahanadi River Basin: Application of Artificial Intelligence Algorithms; NIT Rourkela: Orissa, India, 2019. [Google Scholar]
- Yadav, A.; Joshi, D.; Kumar, V.; Mohapatra, H.; Iwendi, C.; Gadekallu, T.R. Capability and Robustness of Novel Hybridized Artificial Intelligence Technique for Sediment Yield Modeling in Godavari River, India. Water
**2022**, 14, 1917. [Google Scholar] [CrossRef] - Yadav, A.; Chatterjee, S.; Equeenuddin, S.M. Suspended sediment yield modeling in Mahanadi River basin, India by multi-objective optimization hybridizing artificial intelligence algorithms. Int. J. Sediment Res.
**2021**, 36, 76–91. [Google Scholar] [CrossRef] - Pramanik, N.; Panda, R.K. Application of neural network and adaptive neuro-fuzzy inference systems for river flow prediction. Hydrol. Sci. J.
**2009**, 54, 247–260. [Google Scholar] [CrossRef] - Kar, A.K.; Lohani, A.K.; Goel, N.K.; Roy, G.P. Development of Flood Forecasting System Using Statistical and artificial neural network Techniques in the Downstream Catchment of Mahanadi Basin, India. Water Resour. Prot.
**2010**, 2, 880–887. [Google Scholar] - Meher, J. RF and Runoff Estimation Using Hydrological Models and ANN Techniques. Ph.D. Thesis, National Institute of Technology, Rourkela, India, 2014. [Google Scholar]
- Samantaray, S.; Ghose, D.K. Evaluation of suspended sediment concentration using descent neural networks. Procedia Comput. Sci.
**2018**, 132, 1824–1831. [Google Scholar] [CrossRef] - Kant, A.; Suman, P.K.; Giri, B.K.; Tiwari, M.K.; Chatterjee, C.; Nayak, P.C.; Kumar, S. Comparison of multi-objective evolutionary neural network, adaptive neuro-inference system and bootstrap-based neural network for flood forecasting. Neural. Comput. Appl.
**2013**, 23, 231–246. [Google Scholar] [CrossRef] - Asokan, S.; Dutta, D. Analysis of water resources in the Mahanadi River basin, India under projected climate conditions. Hydrol. Process
**2008**, 22, 3589–3603. [Google Scholar] [CrossRef] - India-WRIS, Water Resources Information System of India. 2015. Available online: http://india-wris.nrsc.gov.in/wrpinfo/index.php?title=Mahanadi (accessed on 8 August 2016).
- Bastia, F.; Equeenuddin, S.M. Spatio-temporal variation of water flow and sediment discharge in the MRB, India. Glob. Planet. Chang.
**2016**, 144, 51–66. [Google Scholar] [CrossRef] - Chakrapani, G.J.; Subramanian, V. Heavy metals distribution and fractionation in sediments of the Mahanadi River basin, India. Environ. Geol.
**1993**, 22, 80–87. [Google Scholar] [CrossRef] - Bhattacharya, B.; Lobbrecht, A.H.; Solomatine, D.P. Neural networks and reinforcement learning in control of water systems. J. Water Resour.
**2003**, 129, 458–465. [Google Scholar] [CrossRef] - Kumar, V. Evaluation of computationally intelligent techniques for breast cancer diagnosis. Neural Comput. Applic.
**2021**, 33, 3195–3208. [Google Scholar] [CrossRef] - Kumar, V.; Lalotra, G.S. Predictive Model Based on Supervised Machine Learning for Heart Disease Diagnosis. In Proceedings of the IEEE International Conference on Technology, Research, and Innovation for Betterment of Society (TRIBES), Raipur, India, 17–19 December 2021; pp. 1–6. [Google Scholar] [CrossRef]
- Wang, Y.M.; Traore, S. Time-lagged recurrent network for forecasting episodic event suspended sediment load in typhoon prone area. Int. J. Phys. Sci.
**2009**, 4, 519–528. [Google Scholar] - Haghizade, A.; Shui, L.T.; Goudarzi, E. Estimation of yield sediment using artificial neural network at basin scale. Aust. J. Basic Appl. Sci
**2010**, 4, 1668–1675. [Google Scholar] - Singh, G.; Panda, R. Daily sediment yield modeling with artificial neural network using 10-fold cross validation method: A small agricultural watershed, Kapgari, India. Int. J. Earth Sci. Eng.
**2011**, 6, 443–450. [Google Scholar] - Karayiannis, N.B.; Venetsanopoulos, A.N. ALADIN: Algorithms for learning and architecture determination. In Artificial Neural Networks; Springer: Berlin/Heidelberg, Germany, 1993; pp. 195–218. [Google Scholar]
- Cybenko, G. Approximation by superpositions of a sigmoidal function. Math. Control Signals Syst.
**1989**, 2, 303–314. [Google Scholar] [CrossRef] - Hornik, K.; Stinchcombe, M.; White, H. Multilayer feed forward networks are universal approximators. Neural Netw.
**1989**, 2, 359–366. [Google Scholar] [CrossRef] - Tang, Z.; Almeida, D.C.; Fishwick, P.A. Time series forecasting using neural networks vs. Box-Jenkins methodology. J. Simul.
**1991**, 57, 303–310. [Google Scholar] - Chatterjee, S.; Bandopadhyay, S. Global neural network learning using genetic algorithm for ore grade prediction of iron ore deposit. Min. Resour. Eng.
**2007**, 12, 258–269. [Google Scholar] - Chatterjee, S.; Bandopadhyay, S. Reliability estimation using a genetic algorithm-based artificial neural network: An application to a laud-haul-dump machine. Expert Syst. Appl.
**2012**, 39, 10943–10951. [Google Scholar] [CrossRef] - Jin, L.; Whitehead, P.G.; Rodda, H.; Macadam, I.; Sarkar, S. Simulating climate change and socio-economic change impacts on flows and water quality in the Mahanadi River basin system, India. Sci. Total Environ.
**2018**, 637, 907–917. [Google Scholar] [CrossRef] [PubMed] - Boukhrissa, Z.A.; Khanchoul, K.; le Bissonnais, Y.; Tourki, M. Compare the Ann and Sediment rating curve model for prediction of suspended sediment load in EI Kebir catchment, Algeria. J. Earth Syst. Sci.
**2013**, 122, 1303–1312. [Google Scholar] [CrossRef] - Samarasinghe, S. Neural Networks for Applied Sciences and Engineering: From Fundamentals to Complex Pattern Recognition; CRC Press: Boca Raton, FL, USA, 2016; p. 555. [Google Scholar]
- Demuth, H.B.; Beale, M. Neural Network Toolbox for Use with MATLAB, Users Guide; The Mathworks Inc.: Natick, MA, USA, 1998. [Google Scholar]
- Bachiller, A.R.; Rodríguez, J.L.G.; Sánchez, J.C.R.; Gómez, D.L. Specific sediment yield model for reservoirs with medium-sized basins in Spain: An empirical and statistical approach. Sci. Total Environ.
**2019**, 681, 82–101. [Google Scholar] [CrossRef] - Faraway, J. Practical Regression and ANOVA in R. 2002. Available online: http://cran.r-project.org/doc/contrib/Faraway-PRA.pdf (accessed on 18 October 2022).
- Patel, A.K.; Chatterjee, S. Computer vision-based limestone rock-type classification using probabilistic neural network. Geosci. Front.
**2016**, 7, 53–60. [Google Scholar] [CrossRef] - Ghose, D.K.; Swain, D.P.C.; Panda, D.S.S. Sediment load analysis using ANN and GA. In Applied Mechanics and Materials; Trans Tech Publications Ltd.: Bach, Switzerland, 2012; Volume 110–116, pp. 2693–2698. [Google Scholar]
- Zhu, Y.M.; Lu, X.X.; Zhou, Y. Sediment flux sensitivity to climate change: A case study in the Longchuanjiang catchment of the upper Yangtze River, China. Glob. Planet. Change
**2008**, 60, 429–442. [Google Scholar] [CrossRef] - Harrison, C.G.A. What factors control mechanical erosion rates? Int. J. Earth Sci.
**2000**, 88, 752–763. [Google Scholar] [CrossRef] - Syvitski, J.P.M.; Peckham, S.D.; Hilberman, R.; Mulder, T. Predicting the Terrestrial Flux of Sediment to the Global Ocean: A Planetary Perspective. Sediment. Geol.
**2003**, 162, 5–24. [Google Scholar] [CrossRef] - Altun, H.; Bilgil, A.; Fidan, B.C. Treatment of multidimensional data to enhance neural network estimators in regression problems. Expert Syst. Appl.
**2007**, 32, 599–605. [Google Scholar] [CrossRef] - Jin, Y.; Sendhoff, B.; Korner, E. Evolutionary multi-objective optimization for simultaneous generation of signal-type and symbol-type representations. In Proceedings of the 3rd International Conference on Evolutionary Multi-Criterion Optimization, Guanajuato, Mexico, 9–11 March 2005; pp. 752–766. [Google Scholar]
- Legates, D.R.; McCabe, G.J. Evaluating the use of goodness-of-fit measures in hydrology and hydroclimatic model validation. Water Resour. Res.
**1999**, 35, 233–241. [Google Scholar] [CrossRef] - Legates, D.R.; McCabe, G.J. A refined index of model performance: A rejoinder. Int. J. Climatol.
**2013**, 33, 1053–1056. [Google Scholar] [CrossRef] - Chakrapani, G.J.; Subramanian, V. Factors controlling sediment discharge in the Mahanadi River Basin. India. J. Hydrol.
**1990**, 117, 169–185. [Google Scholar] [CrossRef]

**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