Abstract
The failure of rock slopes leads to disastrous consequences and thus necessitates their reliability analysis. There are various methods to perform the reliability analysis of a rock slope, viz., conventional, numerical, and Soft Computing (SoCom). However, due to different environmental and loading conditions, there is a need to explore SoCom methods to evaluate the probability of failure of rock slopes. Therefore, in this paper, three soft computing techniques viz., Monte Carlo Simulation (MCS), Generalized Regression Neural Networks (GRNN), and Gaussian Process Regression (GPR) have been proposed to analyze the rock slope stability under various conditions. The variables used in this study were c, ϕ, and σt, and the output was the factor of safety (Fs), which was further utilized for the training and testing of the models. The training and testing of the models were performed on different sets of datapoints (viz., 50, 100, 300, 500, 1000, 5000, 10,000). The MCS algorithm was used to generate various sets of datapoints. Furthermore, statistical parameters were used to assess the performance of the proposed SoCom. A comparative study has been performed to check the adaptability of MCS, GRNN, and GPR models for performing the reliability analysis of rock slopes. Training vs. testing datasets has been plotted to realize the fitting of these models. Furthermore, the observations from Taylor’s plot indicate that the MCS, GRNN, and GPR models are capable of predicting the reliability of slope in terms of reliability index; however, GPR outperformed the other two models. The findings imply that the performance assessment of MCS, GRNN, and GPR should also be tried for reliability and risk analysis of other geotechnical and rock engineering problems.
Similar content being viewed by others
References
Armstrong JS, Collopy F (1992) Error measures for generalizing about forecasting methods: empirical comparisons. Int J Forecast 8:69–80. https://doi.org/10.1016/0169-2070(92)90008-W
Aslam B, Zafar A, Khalil U (2023) Comparative analysis of multiple conventional neural networks for landslide susceptibility mapping. Nat Hazards 115:673–707. https://doi.org/10.1007/S11069-022-05570-X/FIGURES/16
Baecher GB, Christian JT (2003) Reliability and statistics in geotechnical engineering. Technometrics. https://doi.org/10.1198/tech.2005.s838
Behar O, Khellaf A, Mohammedi K (2015) Comparison of solar radiation models and their validation under Algerian climate: the case of direct irradiance. Energy Convers Manag 98:236–251. https://doi.org/10.1016/j.enconman.2015.03.067
Bucher CG (1988) Adaptive sampling: an iterative fast Monte Carlo procedure. Struct Saf 5:119–126. https://doi.org/10.1016/0167-4730(88)90020-3
Chen Z, Hu X, Bu X (2021) Effect of weak intercalation on failure mode of rock slopes under seismic excitation. Nat Hazards 105:363–381. https://doi.org/10.1007/S11069-020-04314-Z/FIGURES/14
Chu X, Li L, Wang Y (2015) Slope reliability analysis using length-based representative slip surfaces. Arab J Geosci 8:9065–9078. https://doi.org/10.1007/s12517-015-1905-5
Duzgun HSB, Yucemen MS, Karpuz C (2003) A methodology for reliability-based design of rock slopes. Rock Mech Rock Eng 36:95–120. https://doi.org/10.1007/s00603-002-0034-0
Fujimoto Y, Iwata M, Zheng Y (1991) Fitting-adaptive importance sampling in reliability analysis. Computational stochastic mechanics. Springer, Dordrecht, pp 15–26
Ganji A, Jowkarshorijeh L (2012) Advance first order second moment (AFOSM) method for single reservoir operation reliability analysis: a case study. Stoch Environ Res Risk Assess 26:33–42. https://doi.org/10.1007/s00477-011-0517-1
Ge H, Tu J, Qin F (2011) Analysis of slope stability with first order second moment method. Int J Digit Content Technol Its Appl 5:445–451. https://doi.org/10.4156/jdcta.vol5.issue12.54
Gravanis E, Pantelidis L, Griffiths DV (2014) An analytical solution in probabilistic rock slope stability assessment based on random fields. Int J Rock Mech Min Sci 71:19–24. https://doi.org/10.1016/j.ijrmms.2014.06.018
Gueymard C (2014) A review of validation methodologies and statistical performance indicators for modeled solar radiation data: towards a better bankability of solar projects. Renew Sustain Energy Rev 39:1024–1034
Harr EM (1987) Reliability based design in civil engineering
Huang X, Jin H (2018) An earthquake casualty prediction model based on modified partial Gaussian curve. Nat Hazards 94:999–1021. https://doi.org/10.1007/S11069-018-3452-3/TABLES/11
Ji ZM, Chen TL, Wu FQ et al (2022) Assessment and prevention on the potential rockfall hazard of high-steep rock slope: a case study of Zhongyuntai mountain in Lianyungang, China. Nat Hazards 115:2117–2139. https://doi.org/10.1007/S11069-022-05630-2/FIGURES/13
Jiang SH, Li DQ, Zhang LM, Zhou CB (2014) Slope reliability analysis considering spatially variable shear strength parameters using a non-intrusive stochastic finite element method. Eng Geol 168:120–128. https://doi.org/10.1016/j.enggeo.2013.11.006
Jimenez-Rodriguez R, Sitar N (2007) Rock wedge stability analysis using system reliability methods. Rock Mech Rock Eng 40:419–427. https://doi.org/10.1007/s00603-005-0088-x
Karamchandani A, Allin Cornell C (1991) Adaptive hybrid conditional expectation approaches for reliability estimation. Struct Saf 11:59–74. https://doi.org/10.1016/0167-4730(91)90027-7
Khalokakaie R, Zare Naghadehi M (2012) Ranking the rock slope instability potential using the Interaction Matrix (IM) technique; a case study in Iran. Arab J Geosci 5:263–273. https://doi.org/10.1007/s12517-010-0150-1
Kourosh MA, Mosrafa S, Heydari SM (2011) Uncertainty and reliability analysis applied to slope stability: a case study from sungun copper mine. Geotech Geol Eng 29:581–596. https://doi.org/10.1007/s10706-011-9405-1
Kung GT, Juang CH, Hsiao EC, Hashash YM (2007) Simplified model for wall deflection and ground-surface settlement caused by braced excavation in clays. J Geotech Geoenvironmental Eng 133:731–747. https://doi.org/10.1061/(ASCE)1090-0241(2007)133:6(731)
Legates DR, Davis RE (1997) The continuing search for an anthropogenic climate change signal: limitations of correlation-based approaches. Geophys Res Lett 24:2319–2322. https://doi.org/10.1029/97GL02207
Legates DR, McCabe GJ (1999) Evaluating the use of “goodness-of-fit” measures in hydrologic and hydroclimatic model validation. Water Resour Res 35:233–241. https://doi.org/10.1029/1998WR900018
Legates DR, Mccabe GJ (2013) A refined index of model performance: a rejoinder. Int J Climatol 33:1053–1056. https://doi.org/10.1002/joc.3487
Li DQ, Jiang SH, Chen YF, Zhou CB (2011) System reliability analysis of rock slope stability involving correlated failure modes. KSCE J Civ Eng 15:1349–1359. https://doi.org/10.1007/s12205-011-1250-5
Li S, Zhao HB, Ru Z (2013) Slope reliability analysis by updated support vector machine and Monte Carlo simulation. Nat Hazards 65:707–722. https://doi.org/10.1007/s11069-012-0396-x
Li L, Wang Y, Cao Z (2014) Probabilistic slope stability analysis by risk aggregation. Eng Geol 176:57–65. https://doi.org/10.1016/j.enggeo.2014.04.010
Li DQ, Tang XS, Phoon KK (2015) Bootstrap method for characterizing the effect of uncertainty in shear strength parameters on slope reliability. Reliab Eng Syst Saf 140:99–106. https://doi.org/10.1016/j.ress.2015.03.034
Liang L, Xue-song C (2012) The location of critical reliability slip surface in soil slope stability analysis. Procedia Earth Planet Sci 5:146–149. https://doi.org/10.1016/j.proeps.2012.01.025
Mahmoodzadeh A, Mohammadi M, Farid Hama Ali H et al (2022) Prediction of safety factors for slope stability: comparison of machine learning techniques. Nat Hazards 111:1771–1799. https://doi.org/10.1007/S11069-021-05115-8/FIGURES/20
Mehta AK, Kumar D, Singh P, Samui P (2021) Modelling of seismic liquefaction using classification techniques. Int J Geotech Earthq Eng 12:12–21. https://doi.org/10.4018/IJGEE.2021010102
Mockus J (2005) The Bayesian approach to global optimization. In: System modeling and optimization. Kluwer academic publishers, pp 473–481
Moriasi DN, Arnold JG, Van Liew MW et al (2007) Model evaluation guidelines for systematic quantification of accuracy in watershed simulations. Trans ASABE 50:885–900. https://doi.org/10.13031/2013.23153
Nash JE, Sutcliffe JV (1970) River flow forecasting through conceptual models part I: a discussion of principles. J Hydrol 10:282–290. https://doi.org/10.1016/0022-1694(70)90255-6
Ni W, Tang H, Liu X et al (2014) Dynamic stability analysis of wedge in rock slope based on kinetic vector method. J Earth Sci 25:749–756. https://doi.org/10.1007/s12583-014-0462-2
Pariseau WG, Puri S, Schmelter SC (2008) A new model for effects of impersistent joint sets on rock slope stability. Int J Rock Mech Min Sci 45:122–131. https://doi.org/10.1016/j.ijrmms.2007.05.001
Park H, West TR (2001) Development of a probabilistic approach for rock wedge failure. Eng Geol 59:233–251. https://doi.org/10.1016/S0013-7952(00)00076-4
Park HJ, West TR, Woo I (2005) Probabilistic analysis of rock slope stability and random properties of discontinuity parameters, Interstate Highway 40, Western North Carolina, USA. Eng Geol 79:230–250. https://doi.org/10.1016/j.enggeo.2005.02.001
Park HJ, Um JG, Woo I, Kim JW (2012) The evaluation of the probability of rock wedge failure using the point estimate method. Environ Earth Sci 65:353–361. https://doi.org/10.1007/s12665-011-1096-7
Park HJ, Lee JH, Kim KM, Um JG (2016) Assessment of rock slope stability using GIS-based probabilistic kinematic analysis. Eng Geol 203:56–69. https://doi.org/10.1016/j.enggeo.2015.08.021
Pathak S, Nilsen B (2004) Probabilistic rock slope stability analysis for Himalayan condition. Bull Eng Geol Environ 63:25–32. https://doi.org/10.1007/s10064-003-0226-1
Pinheiro M, Sanches S, Miranda T et al (2015) A new empirical system for rock slope stability analysis in exploitation stage. Int J Rock Mech Min Sci 76:182–191. https://doi.org/10.1016/j.ijrmms.2015.03.015
Prasomphan S, Machine SM (2013) Generating prediction map for geostatistical data based on an adaptive neural network using only nearest neighbors. Int J Mach Learn Comput 3:98
Rackwitz R (1987) Structural reliability: analysis and prediction. Struct Saf 23:194–195. https://doi.org/10.1016/s0167-4730(01)00007-8
Rasmussen CE (2004) Gaussian processes in machine learning. Springer, Berlin, pp 63–71
Ray R, Kumar D, Samui P et al (2021) Application of soft computing techniques for shallow foundation reliability in geotechnical engineering. Geosci Front 12:375–383. https://doi.org/10.1016/j.gsf.2020.05.003
Reale C, Xue J, Pan Z, Gavin K (2015) Deterministic and probabilistic multi-modal analysis of slope stability. Comput Geotech 66:172–179. https://doi.org/10.1016/j.compgeo.2015.01.017
Rubinstein RY, Kroese P (1981) Simulation and the monte carlo method. Wiley, Hoboke
Sardana S, Verma AK, Verma R, Singh TN (2019) Rock slope stability along road cut of Kulikawn to Saikhamakawn of Aizawl, Mizoram, India. Nat Hazards 99:753–767. https://doi.org/10.1007/S11069-019-03772-4/FIGURES/6
Siddique T, Mondal MEA, Pradhan SP et al (2020) Geotechnical assessment of cut slopes in the landslide-prone Himalayas: rock mass characterization and simulation approach. Nat Hazards 104:413–435. https://doi.org/10.1007/S11069-020-04175-6/FIGURES/9
Singh P, Kumar D, Samui P (2020) Reliability analysis of rock slope using soft computing techniques. Jordan J Civ Eng 14:27–42
Singh P, Bardhan A, Han F et al (2022) A critical review of conventional and soft computing methods for slope stability analysis. Model Earth Syst Environ. https://doi.org/10.1007/s40808-022-01489-1
Singh P, Mehta A, Kumar D, Samui P (2019) Rock slope reliability analysis using genetic programming. In: ICGGE-2019, MNNIT Allahabad. Prayagraj, p 7
Specht DF (1991) A general regression neural network. IEEE Trans Neural Netw 6:568–576. https://doi.org/10.1109/72.97934
Srinivasulu S, Jain A (2006) A comparative analysis of training methods for artificial neural network rainfall-runoff models. Appl Soft Comput J 6:295–306. https://doi.org/10.1016/j.asoc.2005.02.002
Stone RJ (1993) Improved statistical procedure for the evaluation of solar radiation estimation models. Sol Energy 51:289–291. https://doi.org/10.1016/0038-092X(93)90124-7
Tan X, hui, Shen M fen, Hou X liang, et al (2013) Response surface method of reliability analysis and its application in slope stability analysis. Geotech Geol Eng 31:1011–1025. https://doi.org/10.1007/s10706-013-9628-4
Taylor KE (2001) Summarizing multiple aspects of model performance in a single diagram. J Geophys Res Atmos 106:7183–7192. https://doi.org/10.1029/2000JD900719
Topal T (2007) Discussion of “A new approach for application of rock mass classification on rock slope stability assessment” by Liu and Chen, Engineering Geology, 89:129–143 (2007). Eng Geol 3:99–100. https://doi.org/10.1016/j.enggeo.2007.07.001
Viscarra Rossel RA, McGlynn RN, McBratney AB (2006) Determining the composition of mineral-organic mixes using UV-vis-NIR diffuse reflectance spectroscopy. Geoderma 137:70–82. https://doi.org/10.1016/j.geoderma.2006.07.004
Wang L, Hwang JH, Juang CH, Atamturktur S (2013) Reliability-based design of rock slopes: a new perspective on design robustness. Eng Geol 154:56–63. https://doi.org/10.1016/j.enggeo.2012.12.004
Wei LW, Chen H, Lee CF et al (2014) The mechanism of rockfall disaster: a case study from Badouzih, Keelung, in northern Taiwan. Eng Geol 183:116–126. https://doi.org/10.1016/j.enggeo.2014.10.008
Williams CKI (1998) Prediction with Gaussian processes: from linear regression to linear prediction and beyond. Learning in graphical models. Springer, Dordrecht, pp 599–621
Willmott CJ (1981) On the validation of models. Phys Geogr 2:184–194. https://doi.org/10.1080/02723646.1981.10642213
Willmott CJ (1982) Some comments on the evaluation of model performance. Bull Am Meteorol Soc 63:1309–1313. https://doi.org/10.1175/1520-0477(1982)063%3c1309:SCOTEO%3e2.0.CO;2
Willmott CJ (1984) On the evaluation of model performance in physical geography. In: Spatial statistics and models, pp 443–460
Xiaoyan H, Li H, Huasheng Z et al (2020) Objective approach for rainstorm based on dual-factor feature extraction and generalized regression neural network. Nat Hazards 104:1987–2002. https://doi.org/10.1007/S11069-020-04258-4/FIGURES/6
Yan J, Chen J, Li Y et al (2022) Kinematic-based failure angle analysis for discontinuity-controlled rock slope instabilities: a case study of Ren Yi Peak Cluster in Fusong County, China. Nat Hazards 111:2281–2296. https://doi.org/10.1007/S11069-021-05137-2/TABLES/6
Yang ZG, Li TC, Dai ML (2009) Reliability analysis method for slope stability based on sample weight. Water Sci Eng 2(3):78–86
Yang Y, Xing H, Yang X et al (2015) Two-dimensional stability analysis of a soil slope using the finite element method and the limit equilibrium principle. IES J Part A Civ Struct Eng 8:251–264. https://doi.org/10.1080/19373260.2015.1072299
Youssef AM, Pradhan B, Al-Harthi SG (2015) Assessment of rock slope stability and structurally controlled failures along Samma escarpment road, Asir Region (Saudi Arabia). Arab J Geosci 8:6835–6852. https://doi.org/10.1007/s12517-014-1719-x
Zeng P, Jimenez R, Jurado-Piña R (2015) System reliability analysis of layered soil slopes using fully specified slip surfaces and genetic algorithms. Eng Geol 193:106–117. https://doi.org/10.1016/j.enggeo.2015.04.026
Zhou JW, Cui P, Yang XG (2013) Dynamic process analysis for the initiation and movement of the Donghekou landslide-debris flow triggered by the Wenchuan earthquake. J Asian Earth Sci 76:70–84. https://doi.org/10.1016/j.jseaes.2013.08.007
Zhou JW, Jiao MY, Xing HG et al (2017) A reliability analysis method for rock slope controlled by weak structural surface. Geosci J 21:453–467. https://doi.org/10.1007/s12303-016-0058-1
Funding
The authors declare that no funds, grants, or other support were received during the preparation of this manuscript.
Author information
Authors and Affiliations
Contributions
Prithvendra Singh: Conceptualization, Data curation, Formal analysis, Methodology, Software, Validation, Writing-original draft, Writing-review & editing. Pijush Samui: Conceptualization, Resources, Validation, Review & editing, Supervision. Edy Tonnizam Mohamad: Review & editing, Supervision. Ramesh Murlidhar Bhatawdekar: Review & editing, Supervision. Wengang Zhang: Review & editing, Supervision.
Corresponding author
Ethics declarations
Competing interest
The authors have no relevant financial or non-financial interests to disclose.
Additional information
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Rights and permissions
Springer Nature or its licensor (e.g. a society or other partner) holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.
About this article
Cite this article
Singh, P., Samui, P., Mohamad, E.T. et al. Application of MCS, GRNN, and GPR for performing the reliability analysis of rock slope. Nat Hazards (2024). https://doi.org/10.1007/s11069-024-06472-w
Received:
Accepted:
Published:
DOI: https://doi.org/10.1007/s11069-024-06472-w