Bayesian analysis for uncertainty quantification of in situ stress data

Bayesian inference of rock strength anisotropy: Uncertainty analysis of the Hoek–Brown failure criterion

“…The large uncertainty of this last rock sample is related to the small number of peak strength observations (more of which later). These findings agree with results of Feng et al, 58 which have indicated that the more in situ stress data implicates in more reliable stress estimates. However, as in engineering practice only limited strength data are available, estimates of the parameter uncertainty ranges are of paramount importance for geotechnical reliability analysis.…”

“…74 For example, in geotechnical engineering, there is a steadily growing body of literature on the application of Bayesian analysis to inverse modeling and quantification of model parameter and output uncertainty. 51,[55][56][57][58]60,70 Bayesian inference allows for an exact description of parameter uncertainty by treating the parameters (and nuisance variables) as probabilistic variables with joint posterior probability density function, 𝑝(θ|σ 1 ). This multivariate distribution, the so-called posterior parameter distribution, is the consequence of two antecedents, a prior distribution, 𝑝(θ), which captures our initial degree of beliefs in the values of the model parameters, θ, and a likelihood function, 𝐿(θ|σ 1 ), which quantifies, by the rules of probability theory, the level of confidence (= conditional belief) in the parameter values in light of the observed peak strength data, σ1 = (σ 1,1 , σ1,2 , … , σ1,𝑛 ), where the subscripts of σ1,2 refer to sample 2 of the peak strength σ1 , etc.…”

“…In Bayesian analysis, it is good practice to access if these assumptions are violated as the posterior distribution and the corresponding prediction uncertainty intervals might be subject to misinterpretation. 58,79,80 To assess the validity of these assumptions, Fig. 3 shows diagnostic tests of the residuals as a function of the peak stress, 𝜎 1 (top), the quantile-quantile plot (middle) and the autocorrelation function (bottom) for the (8) gneiss A (a) and (1) Green river shale II (b).…”

“…There is a growing awareness in the geotechnical community about the importance of uncertainty quantification as researchers are trying to quantify design reliability and inform geotechnical engineers, contractors and other professionals about soil and rock conditions and decision makers seek to better determine the overall technical and financial risks of engineering structures and mitigate geotechnicalrelated risks. 51,53,[55][56][57][58][59] This paper builds on our recent work on slope stability analysis, which embraced uncertainty analysis of safety factors to improve risk analysis and decision-making. 60,61 Recent publications in the rock engineering literature have introduced methods for assessing the uncertainty of in situ stress data, 58,62 rock mass properties [63][64][65] and slope stability, 56,66 including its application to the design of foundations, 67 tunnels 68 and gas reservoirs.…”

“…51,53,[55][56][57][58][59] This paper builds on our recent work on slope stability analysis, which embraced uncertainty analysis of safety factors to improve risk analysis and decision-making. 60,61 Recent publications in the rock engineering literature have introduced methods for assessing the uncertainty of in situ stress data, 58,62 rock mass properties [63][64][65] and slope stability, 56,66 including its application to the design of foundations, 67 tunnels 68 and gas reservoirs. 69 Many of these contributions adopted a Bayesian approach to reconciling rock engineering models with field and/or laboratory data.…”