A Hybrid CNN–BiLSTM–Attention Deep Learning Framework for Sediment Transport in an Agricultural Gully System

Gully erosion threatens soil and water resources across agricultural landscapes, yet most predictive tools for small watercourses rely on simplistic empirical or linear models. This study develops a hybrid Conv1D-BiLSTM-Attention deep learning architecture to predict sediment transport rate in a gully system actively rehabilitated with Morning Glory (Ipomoea carnea) vegetative cover. A six-variable dataset (flow depth, channel slope, soil shear stress, flow velocity, channel breadth and average depth; 100 observations across five field scenarios) was compiled and linked to governing sediment-transport equations (continuity, Straub shear stress, Duboys and Meyer-Peter-Müller bed-load relations) and the Wu-Waldron root-reinforcement model. The hybrid network, trained with Adam under a weighted Huber loss with L2 regularisation and dropout using 10x-repeated 5-fold cross-validation, achieved a hold-out R² of 0.90 (RMSE = 0.45 kg/s/m, MAPE = 20.1%; bootstrap 95% CI: 0.72-0.97) and a cross-validated R² of 0.68 ± 0.36, outperforming a benchmark multilayer perceptron (R² = 0.62 ± 0.37) though not significantly different from either the MLP or multivariate linear regression (R² = 0.78 ± 0.28; Mann-Whitney U, p = 0.06 and 0.08). Permutation importance and Kernel SHAP identified flow depth and soil shear stress as dominant predictors, while Monte Carlo dropout produced 95% prediction intervals with 73% empirical coverage, somewhat under the nominal target at this sample size. Morning Glory reduced mean sediment transport by 25.5% at 1.0 m ponding depth but only 1.4% at 1.5 m, indicating diminishing vegetative control once hydraulic loading exceeds root-reinforcing capacity. This integrated, physically grounded and uncertainty-aware framework provides a reproducible basis for scaling hybrid deep-learning sediment transport prediction to larger agricultural landscapes.

Hybrid deep learning; CNN-BiLSTM-Attention; Sediment transport; Gully erosion

1. Abdar, M., Pourpanah, F., Hussain, S., Rezazadegan, D., Liu, L., Ghavamzadeh, M., Fieguth, P., Cao, X., Khosravi, A., Acharya, U.R., Makarenkov, V., Nahavandi, S., 2021. A review of uncertainty quantification in deep learning: Techniques, applications and challenges. Information Fusion, 76, 243-297. https://doi.org/10.1016/j.inffus.2021.05.008 [Google Scholar] [Crossref]

2. AlDahoul, N., Essam, Y., Kumar, P., Ahmed, A.N., Sherif, M., Sefelnasr, A., Elshafie, A., 2021. Suspended sediment load prediction using long short term memory neural network. Scientific Reports, 11, 7826. https://doi.org/10.1038/s41598-021-87415-4 [Google Scholar] [Crossref]

3. Amiri, M., Pourghasemi, H.R., Ghanbarian, G.A., Afzali, S.F., 2019. Assessment of the importance of gully erosion effective factors using Boruta algorithm and its spatial modeling and mapping using three machine learning algorithms. Geoderma, 340, 55-69. https://doi.org/10.1016/j.geoderma.2018.12.042 [Google Scholar] [Crossref]

4. Andualem, T. G., Hewa, G. A., Myers, B. R., Peters, S., Boland, J., 2023. Erosion and sediment transport modeling: A systematic review. Land, 12(7), 1396. https://doi.org/10.3390/land12071396 [Google Scholar] [Crossref]

5. Arabameri, A., Chandra Pal, S., Costache, R., Saha, A., Rezaie, F., Seyed Danesh, A., Pradhan, B., Lee, S., Hoang, N.-D., 2021. Prediction of gully erosion susceptibility mapping using novel ensemble machine learning algorithms. Geomatics, Natural Hazards and Risk, 12, 469-498. https://doi.org/10.1080/19475705.2021.1880977 [Google Scholar] [Crossref]

6. ASCE Task Committee on Application of Artificial Neural Networks in Hydrology, 2000. Artificial neural networks in hydrology. I: Preliminary concepts. Journal of Hydrologic Engineering, 5(2), 115-123. https://doi.org/10.1061/(ASCE)1084-0699(2000)5:2(115) [Google Scholar] [Crossref]

7. Bahdanau, D., Cho, K., Bengio, Y., 2014. Neural machine translation by jointly learning to align and translate. arXiv preprint arXiv:1409.0473. https://doi.org/10.48550/arXiv.1409.0473 [Google Scholar] [Crossref]

8. Bhattacharya, B., Price, R. K., Solomatine, D. P., 2007. Machine learning approach to modeling sediment transport. Journal of Hydraulic Engineering, 133(4), 440-450. https://doi.org/10.1061/(ASCE)0733-9429(2007)133:4(440) [Google Scholar] [Crossref]

9. Bishop, C.M., 2006. Pattern Recognition and Machine Learning. Springer, New York. https://doi.org/10.1007/978-0-387-45528-0 [Google Scholar] [Crossref]

10. Bottou, L., Curtis, F.E., Nocedal, J., 2018. Optimization methods for large-scale machine learning. SIAM Review, 60(2), 223-311. https://doi.org/10.1137/16M1080173 [Google Scholar] [Crossref]

11. Breiman, L., 2001. Random forests. Machine Learning, 45(1), 5-32. https://doi.org/10.1023/A:1010933404324 [Google Scholar] [Crossref]

12. Brunton, S.L., Noack, B.R., Koumoutsakos, P., 2020. Machine learning for fluid mechanics. Annual Review of Fluid Mechanics, 52(1), 477-508. https://doi.org/10.1146/annurev-fluid-010719-060214 [Google Scholar] [Crossref]

13. Cybenko, G., 1989. Approximation by superpositions of a sigmoidal function. Mathematics of Control, Signals and Systems, 2(4), 303-314. https://doi.org/10.1007/BF02551274 [Google Scholar] [Crossref]

14. Duboys, P., 1879. Le Rhône et les rivières à lit affouillable. Annales des Ponts et Chaussées, 18(5), 141-195. [Google Scholar] [Crossref]

15. Efron, B., 1979. Bootstrap methods: Another look at the jackknife. Annals of Statistics, 7(1), 1-26. https://doi.org/10.1214/aos/1176344552 [Google Scholar] [Crossref]

16. Fisher, A., Rudin, C., Dominici, F., 2019. All models are wrong, but many are useful: Learning a variable\'s importance by studying an entire class of prediction models simultaneously. Journal of Machine Learning Research, 20(177), 1-81. https://jmlr.org/papers/v20/18-760.html [Google Scholar] [Crossref]

17. Francisca, O.O., Nelson, C.A., Collins, C.A., Opeyemi, S.O., 2023. Review of gully erosion in Anambra State: Geology, causes, effects, control measures and challenges associated with its mitigation. Journal of Geography, Environment and Earth Science International, 27(9), 102-116. https://doi.org/10.9734/jgeesi/2023/v27i9689 [Google Scholar] [Crossref]

18. Gal, Y., Ghahramani, Z., 2016. Dropout as a Bayesian approximation: Representing model uncertainty in deep learning. Proceedings of the 33rd International Conference on Machine Learning (ICML), 1050-1059. https://proceedings.mlr.press/v48/gal16.html [Google Scholar] [Crossref]

19. Geman, S., Bienenstock, E., Doursat, R., 1992. Neural networks and the bias/variance dilemma. Neural Computation, 4(1), 1-58. https://doi.org/10.1162/neco.1992.4.1.1 [Google Scholar] [Crossref]

20. Glorot, X., Bengio, Y., 2010. Understanding the difficulty of training deep feedforward neural networks. Proceedings of the 13th International Conference on Artificial Intelligence and Statistics (AISTATS), 249-256. https://proceedings.mlr.press/v9/glorot10a.html [Google Scholar] [Crossref]

21. Goodfellow, I., Bengio, Y., Courville, A., 2016. Deep Learning. MIT Press, Cambridge, Massachusetts. https://www.deeplearningbook.org/ [Google Scholar] [Crossref]

22. Gray, D.H., Sotir, R.B., 1996. Biotechnical and Soil Bioengineering Slope Stabilization: A Practical Guide for Erosion Control. John Wiley & Sons, New York. [Google Scholar] [Crossref]

23. Hastie, T., Tibshirani, R., Friedman, J., 2009. The Elements of Statistical Learning: Data Mining, Inference, and Prediction (2nd ed.). Springer, New York. https://doi.org/10.1007/978-0-387-84858-7 [Google Scholar] [Crossref]

24. Hochreiter, S., Schmidhuber, J., 1997. Long short-term memory. Neural Computation, 9(8), 1735-1780. https://doi.org/10.1162/neco.1997.9.8.1735 [Google Scholar] [Crossref]

25. Hornik, K., Stinchcombe, M., White, H., 1989. Multilayer feedforward networks are universal approximators. Neural Networks, 2(5), 359-366. https://doi.org/10.1016/0893-6080(89)90020-8 [Google Scholar] [Crossref]

26. Huang, F., Zhang, X., 2024. A new interpretable streamflow prediction approach based on SWAT-BiLSTM and SHAP. Environmental Science and Pollution Research, 31(16), 23896-23908. https://doi.org/10.1007/s11356-024-32725-z [Google Scholar] [Crossref]

27. Huber, P.J., 1964. Robust estimation of a location parameter. Annals of Mathematical Statistics, 35(1), 73-101. https://doi.org/10.1214/aoms/1177703732 [Google Scholar] [Crossref]

28. Igwe, C.A., 2003. Erodibility of soils of the upper rainforest zone, Southeastern Nigeria. Land Degradation & Development, 14(4), 323-334. https://doi.org/10.1002/ldr.541 [Google Scholar] [Crossref]

29. Igwe, O., Egbueri, J.C., 2018. The characteristics and the erodibility potentials of soils from different geologic formations in Anambra State, Southeastern Nigeria. Journal of the Geological Society of India, 92(4), 471-478. https://doi.org/10.1007/s12594-018-1044-1 [Google Scholar] [Crossref]

30. Jozefowicz, R., Zaremba, W., Sutskever, I., 2015. An empirical exploration of recurrent network architectures. Proceedings of the 32nd International Conference on Machine Learning (ICML), 2342-2350. https://proceedings.mlr.press/v37/jozefowicz15.html [Google Scholar] [Crossref]

31. Kaloop, M. R., Elsayed, M., Eldessouki, M., Hu, J. W., Lee, S.-J., ELRashidy, N., 2025. Enhancement framework for modeling sediment discharge in rivers using novel supervised deep learning approaches. Journal of Soils and Sediments, 25(9), 2777-2796. https://doi.org/10.1007/s11368-025-04111-w [Google Scholar] [Crossref]

32. Kamal, A.M., Shakour, Z.T.A., Ali, S.A., Sleem, A.A., Haggag, E.G., 2017. Phytochemical and biological investigation of Ipomoea carnea Jacq. grown in Egypt. International Journal of Pharmacognosy and Phytochemical Research, 9(2), 143-153. https://doi.org/10.25258/phyto.v9i2.8073 [Google Scholar] [Crossref]

33. Kingma, D.P., Ba, J., 2015. Adam: A method for stochastic optimization. Proceedings of the International Conference on Learning Representations (ICLR). https://doi.org/10.48550/arXiv.1412.6980 [Google Scholar] [Crossref]

34. Kirkby, M. J., Bracken, L. J., 2009. Gully processes and gully dynamics. Earth Surface Processes and Landforms, 34(14), 1841-1851. https://doi.org/10.1002/esp.1849 [Google Scholar] [Crossref]

35. Kohavi, R., 1995. A study of cross-validation and bootstrap for accuracy estimation and model selection. Proceedings of the 14th International Joint Conference on Artificial Intelligence (IJCAI), 1137-1143. https://www.ijcai.org/Proceedings/95-2/Papers/016.pdf [Google Scholar] [Crossref]

36. Kratzert, F., Klotz, D., Brenner, C., Schulz, K., Herrnegger, M., 2018. Rainfall-runoff modelling using long short-term memory (LSTM) networks. Hydrology and Earth System Sciences, 22(11), 6005-6022. https://doi.org/10.5194/hess-22-6005-2018 [Google Scholar] [Crossref]

37. Lakshminarayanan, B., Pritzel, A., Blundell, C., 2017. Simple and scalable predictive uncertainty estimation using deep ensembles. Advances in Neural Information Processing Systems (NeurIPS), 30, 6402-6413. https://papers.nips.cc/paper/2017 [Google Scholar] [Crossref]

38. Lal, R., 2001. Soil degradation by erosion. Land Degradation & Development, 12(6), 519-539. https://doi.org/10.1002/ldr.472 [Google Scholar] [Crossref]

39. Latif, S. D., Chong, K. L., Ahmed, A. N., Sherif, M., Sefelnasr, A., El-Shafie, A., 2023. Sediment load prediction in Johor River: Deep learning versus machine learning models. Applied Water Science, 13, 79. https://doi.org/10.1007/s13201-023-01874-w [Google Scholar] [Crossref]

40. Lawrence, J.E.R., 1983. Morning glory. Fibrous Root Plants of the Tropics: Ecology and Utilisation. Tropical Agronomy Series, No. 4. [Google Scholar] [Crossref]

41. LeCun, Y., Bengio, Y., Hinton, G., 2015. Deep learning. Nature, 521, 436-444. https://doi.org/10.1038/nature14539 [Google Scholar] [Crossref]

42. Legates, D.R., McCabe, G.J., 1999. Evaluating the use of goodness-of-fit measures in hydrologic and hydroclimatic model validation. Water Resources Research, 35(1), 233-241. https://doi.org/10.1029/1998WR900018 [Google Scholar] [Crossref]

43. Liu, D.C., Nocedal, J., 1989. On the limited memory BFGS method for large scale optimization. Mathematical Programming, 45(1-3), 503-528. https://doi.org/10.1007/BF01589116 [Google Scholar] [Crossref]

44. Lundberg, S.M., Lee, S.-I., 2017. A unified approach to interpreting model predictions. Advances in Neural Information Processing Systems (NeurIPS), 30, 4765-4774. https://papers.nips.cc/paper/2017 [Google Scholar] [Crossref]

45. Maier, H.R., Taghikhah, F.R., Nabavi, E., Razavi, S., Gupta, H., Wu, W., Radford, D.A., Huang, J., 2024. How much X is in XAI: Responsible use of \"explainable\" artificial intelligence in hydrology and water resources. Journal of Hydrology X, 25, 100185. https://doi.org/10.1016/j.hydroa.2024.100185 [Google Scholar] [Crossref]

46. Meyer-Peter, E., Müller, R., 1948. Formulas for bed-load transport. Proceedings of the 2nd Meeting of the International Association for Hydraulic Structures Research, 39-64. [Google Scholar] [Crossref]

47. Moayedi, H., Ahmadi Dehrashid, A., Nguyen Le, B., 2024. A novel problem-solving method by multi-computational optimization of artificial neural network for modelling and prediction of the flow erosion processes. Engineering Applications of Computational Fluid Mechanics, 18(1), 2300456. https://doi.org/10.1080/19942060.2023.2300456 [Google Scholar] [Crossref]

48. Mohebzadeh, H., Biswas, A., Rudra, R., Daggupati, P., 2022. Machine learning techniques for gully erosion susceptibility mapping: A review. Geosciences, 12(12), 429. https://doi.org/10.3390/geosciences12120429 [Google Scholar] [Crossref]

49. Molnar, C., 2022. Interpretable Machine Learning: A Guide for Making Black Box Models Explainable (2nd ed.). Independently published. https://christophm.github.io/interpretable-ml-book/ [Google Scholar] [Crossref]

50. Morgan, R.P.C., 2005. Soil Erosion and Conservation (3rd ed.). Blackwell Publishing, Oxford. [Google Scholar] [Crossref]

51. Nearing, G.S., Kratzert, F., Sampson, A.K., Pelissier, C.S., Klotz, D., Frame, J.M., Prieto, C., Gupta, H.V., 2021. What role does hydrological science play in the age of machine learning? Water Resources Research, 57, e2020WR028091. https://doi.org/10.1029/2020WR028091 [Google Scholar] [Crossref]

52. Nearing, M.A., Pruski, F.F., O\'Neal, M.R., 2004. Expected climate change impacts on soil erosion rates: A review. Journal of Soil and Water Conservation, 59(1), 43-50. [Google Scholar] [Crossref]

53. Ng, K.W., Huang, Y.F., Koo, C.H., Chong, K.L., El-Shafie, A., Ahmed, A.N., 2023. A review of hybrid deep learning applications for streamflow forecasting. Journal of Hydrology, 625, 130141. https://doi.org/10.1016/j.jhydrol.2023.130141 [Google Scholar] [Crossref]

54. Nocedal, J., 1980. Updating quasi-Newton matrices with limited storage. Mathematics of Computation, 35(151), 773-782. https://doi.org/10.1090/S0025-5718-1980-0572855-7 [Google Scholar] [Crossref]

55. Ola, A., Dodd, I.C., Quinton, J.N., 2015. Can we manipulate root system architecture to control soil erosion? SOIL, 1, 603-612. https://doi.org/10.5194/soil-1-603-2015 [Google Scholar] [Crossref]

56. Pedregosa, F., Varoquaux, G., Gramfort, A., Michel, V., Thirion, B., Grisel, O., Blondel, M., Prettenhofer, P., Weiss, R., Dubourg, V., Vanderplas, J., Passos, A., Cournapeau, D., Brucher, M., Perrot, M., Duchesnay, E., 2011. Scikit-learn: Machine learning in Python. Journal of Machine Learning Research, 12, 2825-2830. https://jmlr.org/papers/v12/pedregosa11a.html [Google Scholar] [Crossref]

57. Pimentel, D., 2006. Soil erosion: A food and environmental threat. Environment, Development and Sustainability, 8(1), 119-137. https://doi.org/10.1007/s10668-005-1262-8 [Google Scholar] [Crossref]

58. Poesen, J., Nachtergaele, J., Verstraeten, G., Valentin, C., 2003. Gully erosion and environmental change: Importance and research needs. Catena, 50(2-4), 91-133. https://doi.org/10.1016/S0341-8162(02)00143-1 [Google Scholar] [Crossref]

59. Rahmati, O., Tahmasebipour, N., Haghizadeh, A., Pourghasemi, H.R., Feizizadeh, B., 2017. Evaluation of different machine learning models for predicting and mapping the susceptibility of gully erosion. Geomorphology, 298, 118-137. https://doi.org/10.1016/j.geomorph.2017.09.006 [Google Scholar] [Crossref]

60. Robbins, H., Monro, S., 1951. A stochastic approximation method. Annals of Mathematical Statistics, 22(3), 400-407. https://doi.org/10.1214/aoms/1177729586 [Google Scholar] [Crossref]

61. Rumelhart, D.E., Hinton, G.E., Williams, R.J., 1986. Learning representations by back-propagating errors. Nature, 323, 533-536. https://doi.org/10.1038/323533a0 [Google Scholar] [Crossref]

62. Schoklitsch, A., 1934. Der Geschiebetrieb und die Geschiebefracht. Wasserkraft und Wasserwirtschaft, 29(4), 37-43. [Google Scholar] [Crossref]

63. Schuster, M., Paliwal, K.K., 1997. Bidirectional recurrent neural networks. IEEE Transactions on Signal Processing, 45(11), 2673-2681. https://doi.org/10.1109/78.650093 [Google Scholar] [Crossref]

64. Setargie, T.A., Tsunekawa, A., Haregeweyn, N., Tsubo, M., Fenta, A.A., Berihun, M.L., Sultan, D., Yibeltal, M., Ebabu, K., 2023. Random forest-based gully erosion susceptibility assessment across different agro-ecologies of the Upper Blue Nile Basin, Ethiopia. Geomorphology, 431, 108671. https://doi.org/10.1016/j.geomorph.2023.108671 [Google Scholar] [Crossref]

65. Shah, S.M.R., Hameed, M., Ahmad, M.S.A., 2022. Contribution of environmental and soil physicochemical attributes in invasiveness of Ipomoea carnea Jacq. in diverse habitats along elevation and salinity gradients. Pakistan Journal of Botany, 54(6), 2013-2024. https://doi.org/10.30848/PJB2022-6(4) [Google Scholar] [Crossref]

66. Shakya, D., Deshpande, V., Kumar, B., 2023. Predicting total sediment load transport in rivers using regression techniques, extreme learning and deep learning models. Artificial Intelligence Review, 56, 10067-10098. https://doi.org/10.1007/s10462-023-10422-6 [Google Scholar] [Crossref]

67. Shen, C., 2018. A transdisciplinary review of deep learning research and its relevance for water resources scientists. Water Resources Research, 54, 8558-8593. https://doi.org/10.1029/2018WR022643 [Google Scholar] [Crossref]

68. Shields, A., 1936. Anwendung der ähnlichkeitsmechanik und der Turbulenzforschung auf die Geschiebebewegung. Mitteilungen der Preussischen Versuchsanstalt für Wasserbau und Schiffbau, Berlin. [Google Scholar] [Crossref]

69. Solomatine, D.P., Ostfeld, A., 2008. Data-driven modelling: Some past experiences and new approaches. Journal of Hydroinformatics, 10(1), 3-22. https://doi.org/10.2166/hydro.2008.015 [Google Scholar] [Crossref]

70. Srivastava, N., Hinton, G., Krizhevsky, A., Sutskever, I., Salakhutdinov, R., 2014. Dropout: A simple way to prevent neural networks from overfitting. Journal of Machine Learning Research, 15(1), 1929-1958. https://jmlr.org/papers/v15/srivastava14a.html [Google Scholar] [Crossref]

71. Straub, L. G., 1935. Some observations on hydraulic and hydrologic engineering. Proceedings of the American Society of Civil Engineers. [Google Scholar] [Crossref]

72. Tao, H., Al-Khafaji, Z. S., Qi, C., Zounemat-Kermani, M., Kisi, O., Tiyasha, T., Chau, K. W., Nourani, V., Melesse, A. M., Elhakeem, M., Yaseen, Z. M., 2021. Artificial intelligence models for suspended river sediment prediction: State-of-the-art, modeling framework appraisal, and proposed future research directions. Engineering Applications of Computational Fluid Mechanics, 15(1), 1585-1612. https://doi.org/10.1080/19942060.2021.1984992 [Google Scholar] [Crossref]

73. Taylor, K.E., 2001. Summarizing multiple aspects of model performance in a single diagram. Journal of Geophysical Research, 106(D7), 7183-7192. https://doi.org/10.1029/2000JD900719 [Google Scholar] [Crossref]

74. Truong, P., Loch, R., 2004. Vetiver system for erosion and sediment control. Proceedings of the 13th International Soil Conservation Organisation Conference, Brisbane, Australia. [Google Scholar] [Crossref]

75. Truong, P., Van, T.T., Pinners, E., 2008. Vetiver System Applications: A Technical Reference Manual. The Vetiver Network International, Bangkok. [Google Scholar] [Crossref]

76. Valentin, C., Poesen, J., Li, Y., 2005. Gully erosion: Impacts, factors and control. Catena, 63(2-3), 132-153. https://doi.org/10.1016/j.catena.2005.06.001 [Google Scholar] [Crossref]

77. Van Rijn, L. C., 1984a. Sediment transport, part I: Bed load transport. Journal of Hydraulic Engineering, 110(10), 1431-1456. https://doi.org/10.1061/(ASCE)0733-9429(1984)110:10(1431) [Google Scholar] [Crossref]

78. Van Rijn, L. C., 1984b. Sediment transport, part II: Suspended load transport. Journal of Hydraulic Engineering, 110(11), 1613-1641. https://doi.org/10.1061/(ASCE)0733-9429(1984)110:11(1613) [Google Scholar] [Crossref]

79. Vapnik, V.N., 1998. Statistical Learning Theory. Wiley, New York. [Google Scholar] [Crossref]

80. Vaswani, A., Shazeer, N., Parmar, N., Uszkoreit, J., Jones, L., Gomez, A.N., Kaiser, L., Polosukhin, I., 2017. Attention is all you need. Advances in Neural Information Processing Systems (NeurIPS), 30, 5998-6008. https://doi.org/10.48550/arXiv.1706.03762 [Google Scholar] [Crossref]

81. Waldron, L.J., 1977. The shear resistance of root-permeated homogeneous and stratified soil. Soil Science Society of America Journal, 41(5), 843-849. https://doi.org/10.2136/sssaj1977.03615995004100050005x [Google Scholar] [Crossref]

82. Willmott, C.J., Matsuura, K., 2005. Advantages of the mean absolute error (MAE) over the root mean square error (RMSE) in assessing average model performance. Climate Research, 30(1), 79-82. https://doi.org/10.3354/cr030079 [Google Scholar] [Crossref]

83. Wu, T.H., McKinnell, W.P., Swanston, D.N., 1979. Strength of tree roots and landslides on Prince of Wales Island, Alaska. Canadian Geotechnical Journal, 16(1), 19-33. https://doi.org/10.1139/t79-003 [Google Scholar] [Crossref]

84. Wu, T.H., 2013. Root reinforcement of soil: Review of analytical models, test results, and applications to design. Canadian Geotechnical Journal, 50(3), 259-274. https://doi.org/10.1139/cgj-2012-0160 [Google Scholar] [Crossref]

85. Zounemat-Kermani, M., Kheimi, M., 2026. Explainable artificial intelligence in hydrology: A review. Water Resources Management, 40(3), 1-28. https://doi.org/10.1007/s11269-025-04435-9 [Google Scholar] [Crossref]

86. Additional references (advanced sediment-transport and gully-trapping equations, Section 2.2.2) [Google Scholar] [Crossref]

87. Ackers, P., White, W.R., 1973. Sediment transport: New approach and analysis. Journal of the Hydraulics Division, ASCE, 99(HY11), 2041-2060. [Google Scholar] [Crossref]

88. Bagnold, R.A., 1966. An Approach to the Sediment Transport Problem from General Physics. U.S. Geological Survey Professional Paper 422-I, Washington, D.C. [Google Scholar] [Crossref]

89. Brune, G.M., 1953. Trap efficiency of reservoirs. Transactions, American Geophysical Union, 34(3), 407-418. https://doi.org/10.1029/TR034i003p00407 [Google Scholar] [Crossref]

90. Churchill, M.A., 1948. Discussion of “Analysis and use of reservoir sedimentation data” by L.C. Gottschalk. In: Proceedings of the Federal Inter-Agency Sedimentation Conference, Denver, CO. [Google Scholar] [Crossref]

91. Einstein, H.A., 1950. The Bed-Load Function for Sediment Transportation in Open Channel Flows. USDA Soil Conservation Service Technical Bulletin No. 1026, Washington, D.C. [Google Scholar] [Crossref]

92. Engelund, F., Hansen, E., 1967. A Monograph on Sediment Transport in Alluvial Streams. Teknisk Forlag, Copenhagen. [Google Scholar] [Crossref]

93. Exner, F.M., 1925. Über die Wechselwirkung zwischen Wasser und Geschiebe in Flüssen. Sitzungsberichte der Akademie der Wissenschaften Wien, Mathematisch-Naturwissenschaftliche Klasse, 134, 165-204. [Google Scholar] [Crossref]

94. Rouse, H., 1937. Modern conceptions of the mechanics of fluid turbulence. Transactions of the ASCE, 102, 463-505. [Google Scholar] [Crossref]