Engineering with Computers

, Volume 35, Issue 1, pp 337–350 | Cite as

Unsaturated soil slope characterization with Karhunen–Loève and polynomial chaos via Bayesian approach

  • Hao-Qing Yang
  • Lulu ZhangEmail author
  • Jianfeng Xue
  • Jie Zhang
  • Xu Li
Original Article


Field measured data reflect real response of soil slopes under rainfall infiltration and can provide representative estimates of in situ soil properties. In this study, an efficient probabilistic back analysis method for characterization of spatial variability of soil properties is used to investigate the effects of field responses with various monitoring schemes on characterization of spatial variability in unsaturated soil slope. A hypothetical heterogeneous slope of spatially varied saturated hydraulic conductivity subjecting to steady-state rainfall infiltration is analyzed as a numerical example. The spatially varied soil saturated hydraulic conductivity is parameterized by the Karhunen–Loève expansion (KLE) with a given covariance. The random variables corresponding to the truncated KLE terms are considered as variables to be estimated with Bayesian inverse method. Synthetic pore water pressure data corrupted with artificial noise are utilized as measurement data. Nine schemes with various locations, spacings and depths of monitoring sections are discussed. The results show that the local variability can be reduced substantially around the monitoring points of pore pressure. The spatial variability can be estimated more accurately with a smaller spacing of measurement points. When measurement points are installed with a spacing of 16.5 m, the posterior average COV of ks field is around 2% and the RMSE of the MAP field is only 5.90 × 10− 7 m/s. For schemes with different depths, the RMSEs of the MAP field does not change much but the posterior uncertainty of the estimated field is reduced with the increase of borehole depth.


Spatial variability Karhunen–Loève Polynomial chaos Markov Chain Monte Carlo Monitoring 



The work in this paper was substantially supported by the National Basic Research Program of China (973 Program, Project No. 2014CB049100) and the Natural Science Foundation of China (Project Nos. 51679135 and 51422905). The authors are grateful for the support from the National Program for support of Top-notch Young Professionals, and Shanghai Science and Technology Committee (Project No. 16DZ1200503).


  1. 1.
    Armaghani DJ, Mohamad ET, Hajihassani M, Yagiz S, Motaghedi H (2016) Application of several non-linear prediction tools for estimating uniaxial compressive strength of granitic rocks and comparison of their performances. Eng Comput 32(2):189–206Google Scholar
  2. 2.
    Atkinson KE (1967) The numerical solution of Fredholm integral equations of the second kind. SIAM J Numer Anal 4(3):337–348MathSciNetzbMATHGoogle Scholar
  3. 3.
    Babuška I, Nobile F, Tempone R (2007) A stochastic collocation method for elliptic partial differential equations with random input data. SIAM J Numer Anal 45(3):1005–1034MathSciNetzbMATHGoogle Scholar
  4. 4.
    Bagarello V, Sferlazza S, Sgroi A (2009) Testing laboratory methods to determine the anisotropy of saturated hydraulic conductivity in a sandy-loam soil. Geoderma 154(1–2):52–58Google Scholar
  5. 5.
    Baum RL, Godt JW, Savage WZ (2010) Estimating the timing and location of shallow rainfall-induced landslides using a model for transient, unsaturated infiltration. J Geophys Res-Earth Surf 115:F03013Google Scholar
  6. 6.
    Baum RL, Savage WZ, Godt JW (2008) TRIGRS-A fortran program for transient rainfall infiltration and grid-based regional slope-stability analysis. In: Version 2.0. Open-File Report 2008–1159. U.S. Geological Survey, Denver, COGoogle Scholar
  7. 7.
    Box GE, Tiao GC (2011) Bayesian inference in statistical analysis. John Wiley & SonsGoogle Scholar
  8. 8.
    Cao ZJ, Wang Y (2013) Bayesian approach for probabilistic site characterization using cone penetration tests. J Geotech Geoenviron Eng 139(2):267–276Google Scholar
  9. 9.
    Cao ZJ, Wang Y, Li DQ (2016) Site-specific characterization of soil properties using multiple measurements from different test procedures at different locations—a Bayesian sequential updating approach. Eng Geol 211:150–161Google Scholar
  10. 10.
    Chen X (2000) Measurement of streambed hydraulic conductivity and its anisotropy. Environ Geol 39(12):1317–1324Google Scholar
  11. 11.
    Cho SE (2009) Probabilistic assessment of slope stability that considers the spatial variability of soil properties. J Geotech Geoenviron Eng 136(7):975–984Google Scholar
  12. 12.
    Deng JH, Lee CF (2001) Displacement back analysis for a steep slope at the Three Gorges Project site. Int J Rock Mech Min Sci 38(2):259–268Google Scholar
  13. 13.
    Ering P, Babu GS (2016) Probabilistic back analysis of rainfall induced landslide-A case study of Malin landslide, India. Eng Geol 208:154–164Google Scholar
  14. 14.
    Franck BM, Krauthammer T (1988) Development of an expert system for preliminary risk assessment of existing concrete dams. Eng Comput 3(3):137–148Google Scholar
  15. 15.
    Fredlund DG, Xing A (1994) Equations for the soil-water characteristic curve. Can Geotech J 31(4):521–532Google Scholar
  16. 16.
    Ghanem RG, Spanos PD (1991) Spectral stochastic finite-element formulation for reliability analysis. J Eng Mech 117(10):2351–2372Google Scholar
  17. 17.
    Ghanem RG, Spanos PD (2003) Stochastic finite elements: a spectral approach. Courier CorporationGoogle Scholar
  18. 18.
    Harris SJ, Orense RP, Itoh K (2012) Back analyses of rainfall-induced slope failure in Northland Allochthon formation. Landslides 9(3):349–356Google Scholar
  19. 19.
    Hess KM, Wolf SH, Celia MA (1992) Large-scale natural gradient tracer test in sand and gravel, Cape Cod, Massachusetts: 3. Hydraulic conductivity variability and calculated macrodispersivities. Water Resour Res 28(8):2011–2027Google Scholar
  20. 20.
    Hu BX, He C (2006) Using sequential self-calibration method to estimate a correlation length of a log-conductivity field conditioned upon a tracer test and limited measured data. Stoch Environ Res Risk Assess 21(1):89–96MathSciNetGoogle Scholar
  21. 21.
    Hughson DL, Yeh TCJ (2000) An inverse model for three-dimensional flow in variably saturated porous media. Water Resour Res 36(4):829–839Google Scholar
  22. 22.
    Jardani A, Dupont JP, Revil A, Massei N, Fournier M, Laignel B (2012) Geostatistical inverse modeling of the transmissivity field of a heterogeneous alluvial aquifer under tidal influence. J Hydrol 472:287–300Google Scholar
  23. 23.
    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 168:120–128Google Scholar
  24. 24.
    Jiang SH, Papaioannou I, Straub D (2018) Bayesian updating of slope reliability in spatially variable soils with in-situ measurements. Eng Geol In pressGoogle Scholar
  25. 25.
    Juang CH (2001) Three-dimensional site characterisation: neural network approach. Geotechnique 51(9):799–809Google Scholar
  26. 26.
    Karhunen K (1947) Über lineare Methoden in der Wahrscheinlichkeitsrechnung. Math-Phys. Universitat Helsinki, Annales Academiae Scientiarum FennicaezbMATHGoogle Scholar
  27. 27.
    Ledesma A, Gens A, Alonso EE (1996) Parameter and variance estimation in geotechnical back analysis using prior information. Int J Numer Anal Methods Geomech 20(2):119–141zbMATHGoogle Scholar
  28. 28.
    Leong EC, Rahardjo H (1997) Permeability functions for unsaturated soils. J Geotech Geoenviron Eng 123(12):1118–1126Google Scholar
  29. 29.
    Li DQ, Jiang SH, Cao ZJ, Zhou W, Zhou CB, Zhang LM (2015) A multiple response-surface method for slope reliability analysis considering spatial variability of soil properties. Eng Geol 187:60–72Google Scholar
  30. 30.
    Li DQ, Jiang SH, Cheng YG, Zhou CB (2013) A comparative study of three collocation point methods for odd order stochastic response surface method. Struct Eng Mech 45(5):595–611Google Scholar
  31. 31.
    Li S, Zhao H, Ru Z, Sun Q (2016) Probabilistic back analysis based on Bayesian and multi-output support vector machine for a high cut rock slope. Eng Geol 203:178–190Google Scholar
  32. 32.
    Lloret-Cabot M, Fenton GA, Hicks MA (2014) On the estimation of scale of fluctuation in geostatistics. Georisk 8(2):129–140Google Scholar
  33. 33.
    Loève M (1948) Fonctions aléatoires de second ordre. Supplement to P Levy Proces stochastiques et mouvement Brownien Gauthier-Villars, PariszbMATHGoogle Scholar
  34. 34.
    Lv Q, Liu Y, Yang Q (2017) Stability analysis of earthquake-induced rock slope based on back analysis of shear strength parameters of rock mass. Eng Geol 228:39–49Google Scholar
  35. 35.
    Mahdiyar A, Hasanipanah M, Armaghani DJ, Gordan B, Abdullah A, Arab H, Majid MZA (2017) A Monte Carlo technique in safety assessment of slope under seismic condition. Eng Comput 33:807–817Google Scholar
  36. 36.
    Mantoglou A (2005) On optimal model complexity in inverse modeling of heterogeneous aquifers. J Hydraul Res 43(5):574–583Google Scholar
  37. 37.
    Murakami H, Chen X, Hahn MS, Liu Y, Rockhol ML, Vermeul VR, Zachara JM, Rubin Y (2010) Bayesian approach for three-dimensional aquifer characterization at the Hanford 300 Area. Hydrol Earth Syst Sci 14(10):1989–2001Google Scholar
  38. 38.
    Nobile F, Tempone R, Webster CG (2008) A sparse grid stochastic collocation method for partial differential equations with random input data. SIAM J Numer Anal 46(5):2309–2345MathSciNetzbMATHGoogle Scholar
  39. 39.
    Peng XY, Zhang LL, Jeng DS, Chen LH, Liao CC, Yang HQ (2017) Effects of cross-correlated multiple spatially random soil properties on wave-induced oscillatory seabed response. Appl Ocean Res 62:57–69Google Scholar
  40. 40.
    Phoon KK, Huang HW, Quek ST (2005) Simulation of strongly non-Gaussian processes using Karhunen–Loeve expansion. Prob Eng Mech 20(2):188–198Google Scholar
  41. 41.
    Phoon KK, Kulhawy FH (1999) Characterization of geotechnical variability. Can Geotech J 36(4):612–624Google Scholar
  42. 42.
    Rehfeldt KR, Boggs JM, Gelhar LW (1992) Field study of dispersion in a heterogeneous aquifer: 3. Geostatistical analysis of hydraulic conductivity. Water Resour Res 28(12):3309–3324Google Scholar
  43. 43.
    Sharma LK, Singh R, Umrao RK, Sharma KM, Singh TN (2017) Evaluating the modulus of elasticity of soil using soft computing system. Eng Comput 33(3):497–507Google Scholar
  44. 44.
    Smolyak S (1963) Quadrature and interpolation formulas for tensor products of certain classes of functions. Soviet Math Dokl 4:240–243zbMATHGoogle Scholar
  45. 45.
    Soize C, Ghanem R (2004) Physical systems with random uncertainties: chaos representations with arbitrary probability measure. SIAM J Sci Comput 26(2):395–410MathSciNetzbMATHGoogle Scholar
  46. 46.
    Srivastava A, Babu GS, Haldar S (2010) Influence of spatial variability of permeability property on steady state seepage flow and slope stability analysis. Eng Geol 110(3–4):93–101Google Scholar
  47. 47.
    Sudret B (2008) Global sensitivity analysis using polynomial chaos expansions. Reliab Eng Syst Saf 93(7):964–979Google Scholar
  48. 48.
    Sudret B (2014) Polynomial chaos expansions and stochastic finite-element methods. In Phoon KK, Ching J (eds), Risk and Reliability in Geotechnical Engineering (pp 265–300) CRC PressGoogle Scholar
  49. 49.
    Sudret B, Berveiller M, Lemaire M (2006) A stochastic finite element procedure for moment and reliability analysis. Eur J of Comput Mech 15(7–8):825–866zbMATHGoogle Scholar
  50. 50.
    Thompson GR, Long LG (1989) Hibernia geotechnical investigation and site characterization. Can Geotech J 26(4):653–678Google Scholar
  51. 51.
    Tian M, Li DQ, Cao ZJ, Phoon KK, Wang Y (2016) Bayesian identification of random field model using indirect test data. Eng Geol 210:197–211Google Scholar
  52. 52.
    Trandafir AC, Sidle RC, Gomi T, Kamai T (2008) Monitored and simulated variations in matric suction during rainfall in a residual soil slope. Environ Geol 55(5):951–961Google Scholar
  53. 53.
    Turcke MA, Kueper BH (1996) Geostatistical analysis of the Borden aquifer hydraulic conductivity field. J Hydrol 178(1–4):223–240Google Scholar
  54. 54.
    Vanmarcke E (2010) Random Fields: Analysis and Synthesis. World ScientificGoogle Scholar
  55. 55.
    Vardon PJ, Liu K, Hicks MA (2016) Reduction of slope stability uncertainty based on hydraulic measurement via inverse analysis. Georisk 10(3):223–240Google Scholar
  56. 56.
    Vrugt JA, Ter Braak CJ, Gupta HV, Robinson BA (2009) Equifinality of formal (DREAM) and informal (GLUE) Bayesian approaches in hydrologic modeling? Stoch Environ Res Risk Assess 23(7):1011–1026Google Scholar
  57. 57.
    Vrugt JA, ter Braak CJF, Clark MP, Hyman JM, Robinson BA (2008) Treatment of input uncertainty in hydrologic modeling: Doing hydrology backward with Markov Chain Monte Carlo simulation. Water Resour Res 45(12):W00B09Google Scholar
  58. 58.
    Wang L, Hwang JH, Luo Z, Juang CH, Xiao J (2013) Probabilistic back analysis of slope failure—A case study in Taiwan. Comput Geotech 51:12–23Google Scholar
  59. 59.
    Wang Y, Au SK, Cao ZJ (2010) Bayesian approach for probabilistic characterization of sand friction angles. Eng Geol 114(3):354–363Google Scholar
  60. 60.
    Wang Y, Huang K, Cao ZJ (2014) Bayesian identification of soil strata in London clay. Géotechnique 64(3):239Google Scholar
  61. 61.
    Wang Y, Zhao T (2017) Statistical interpretation of soil property profiles from sparse data using Bayesian compressive sampling. Géotechnique 67(6):523–536Google Scholar
  62. 62.
    Wang Y, Zhao T, Phoon KK (2017) Direct simulation of random field samples from sparsely measured geotechnical data with consideration of uncertainty in interpretation. Can Geotech J. Google Scholar
  63. 63.
    Whitman RV (2000) Organizing and evaluating uncertainty in geotechnical engineering. J Geotech Geoenviron Eng 126(7):583–593Google Scholar
  64. 64.
    Xiu D (2007) Efficient collocational approach for parametric uncertainty analysis. Commun Comput Phys 2(2):293–309MathSciNetzbMATHGoogle Scholar
  65. 65.
    Xiu D (2009) Fast numerical methods for stochastic computations: a review. Commun Computatl Phys 5(2–4):242–272MathSciNetzbMATHGoogle Scholar
  66. 66.
    Xiu D, Karniadakis GE (2002) The Wiener–Askey polynomial chaos for stochastic differential equations. SIAM J Sci Comput 24(2):619–644MathSciNetzbMATHGoogle Scholar
  67. 67.
    Yang HQ, Zhang LL, Li DQ (2018) Efficient method for probabilistic estimation of spatially varied hydraulic properties in a soil slope based on field responses: A Bayesian approach. Comp Geotech.
  68. 68.
    Yu FW, Peng XZ, Su LJ (2017) A back-propagation neural-network-based displacement back analysis for the identification of the geomechanical parameters of the Yonglang landslide in China. J Mt Sci 14(9):1739–1750Google Scholar
  69. 69.
    Zeng P, Li T, Jimenez R, Feng X, Chen Y (2018) Extension of quasi-Newton approximation-based SORM for series system reliability analysis of geotechnical problems. Eng Comput 34(2):215–224Google Scholar
  70. 70.
    Zhang D, Lu Z (2004) An efficient, high-order perturbation approach for flow in random porous media via Karhunen–Loève and polynomial expansions. J Comput Phys 194(2):773–794zbMATHGoogle Scholar
  71. 71.
    Zhang J, Tang WH, Zhang L (2010) Efficient Probabilistic Back-Analysis of Slope Stability Model Parameters. J Geotech Geoenviron Eng 136(1):99–109Google Scholar
  72. 72.
    Zhang J, Zhang LM, Tang WH (2010) Slope reliability analysis considering site-specific performance information. J Geotech Geoenviron Eng 137(3):227–238Google Scholar
  73. 73.
    Zhang LL, Li J, Li X, Zhang J, Zhu H (2016) Rainfall-Induced Soil Slope Failure: Stability Analysis and Probabilistic Assessment. Taylor & Francis CRC Press, Boca RatonGoogle Scholar
  74. 74.
    Zhang LL, Zhang J, Zhang LM, Tang WH (2010) Back analysis of slope failure with Markov chain Monte Carlo simulation. Comput Geotech 37(7–8):905–912Google Scholar
  75. 75.
    Zhang LL, Zheng YF, Zhang LM, Li X, Wang JH (2014) Probabilistic model calibration for soil slope under rainfall: effects of measurement duration and frequency in field monitoring. Geotechnique 64(5):365–378Google Scholar
  76. 76.
    Zhang LL, Zuo ZB, Ye GL, Jeng DS, Wang JH (2013) Probabilistic parameter estimation and predictive uncertainty based on field measurements for unsaturated soil slope. Comput Geotech 48(4):72–81Google Scholar
  77. 77.
    Zhang LM, Dasaka SM (2010) Uncertainties in site-specific profiles versus variability in pile founding depth. J Geotech Geoenviron Eng 136(11):1475–1488Google Scholar
  78. 78.
    Zhu H, Zhang LM, Zhang LL, Zhou CB (2013) Two-dimensional probabilistic infiltration analysis with a spatially varying permeability function. Comput Geotech 48(4):249–259Google Scholar
  79. 79.
    Zieher T, Rutzinger M, Schneider-Muntau B, Perzl F, Leidinger D, Formayer H, Geitner C (2017) Sensitivity analysis and calibration of a dynamic physically based slope stability model. Nat Hazards Earth Syst Sci 17(6):971–992Google Scholar

Copyright information

© Springer-Verlag London Ltd., part of Springer Nature 2018

Authors and Affiliations

  • Hao-Qing Yang
    • 1
    • 2
    • 3
  • Lulu Zhang
    • 1
    • 2
    • 3
    Email author
  • Jianfeng Xue
    • 4
  • Jie Zhang
    • 5
  • Xu Li
    • 6
  1. 1.State Key Laboratory of Ocean EngineeringShanghai Jiao Tong UniversityShanghaiChina
  2. 2.Collaborative Innovation Center for Advanced Ship and Deep-Sea Exploration (CISSE)ShanghaiChina
  3. 3.Department of Civil EngineeringShanghai Jiao Tong UniversityShanghaiChina
  4. 4.School of Engineering and ITUniversity of New South Wales, CanberraCampbellAustralia
  5. 5.Key Laboratory of Geotechnical and Underground Engineering of Ministry of Education and Department of Geotechnical EngineeringTongji UniversityShanghaiChina
  6. 6.Department of Geotechnical and Geoenvironmental Engineering, School of Civil EngineeringBeijing Jiaotong UniversityBeijingChina

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