Environmental Engineering Reference
In-Depth Information
conclude with some final remarks on the potential and challenges faced by Bayesian updat-
ing in the context of geotechnical risk and reliability applications.
5.2 baYeSIan analYSIS
Bayesian analysis allows one to consistently and effectively combine new information (mea-
surement, data) with existing models. As illustrated in Figure 5.1 , Bayesian updating can be
applied to learn model parameters (a) from direct sampling and measurements of soil and
loading parameters, and (b) from measurements and observations of system parameters.
Task (a) corresponds to classical statistical inference, where the probability distribution of
the model parameters is learned based on samples; task (b) corresponds to a probabilistic
solution of an inverse analysis or parameter identification. In this chapter, we will consider
both applications of Bayesian updating.
For illustration purposes, consider the following situation: A construction is planned
with a geotechnical model, using initial estimates of soil and hydraulic parameters. The
model predicts stability, deformations, and groundwater flow at the site. Such a model and
its parameters are subject to uncertainty, which in the classical geotechnical design are
addressed through the use of safety factors, characteristic values, and conservative assump-
tions. Alternatively, it is possible to quantify the uncertainty explicitly through a probabilis-
tic analysis. Thereby, the main uncertain or random parameters of the model are represented
by random variables with corresponding probability distributions. We denote the set of
these random variables with X . Through the geotechnical model, all events of interest can
be expressed as a function of X . As an example, the event F = loss of stability” is expressed
through a limit state function (LSF) g ( X ), so that Fg
=
{( )
X
0
},
that is, failure corre-
sponds to the LSF taking negative values.
During the construction, additional measurements become available, for example, addi-
tional soil probes are taken and tested, deformations are measured, or water ingress is moni-
tored. These measurements provide information on X , either directly or indirectly. The
information is direct if one or more of the parameters in X are measured, for example,
measurements of soil parameters. The information is indirect if the measurement outcomes
are related to the parameters X through some model, as is, for example, the case if defor-
mations are measured at the construction site. Mathematically, these measurements are
events, which we here denote with Z. In the above example, let the measurement outcome
be = “deformation between 20 and 30 mm.”
Sampling, measurements
of soil parameters
Statistical inference
(Classical Bayesian updating)
Parameters
X,Θ
Model
(analytical, FEM, ...)
System performance
(model output)
Reliability and risk
assessment
Measurements
deformation, performance, ...
Bayesian model updating
(Inverse analysis, parameter identification)
Figure 5.1 Overview on Bayesian updating in the context of geotechnical models.
 
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