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206 I.-M.LEE AND D.-H.KIM
values utilizing the EBM (the posterior estimation). Unfortunately, the posterior
distribution is not a simple normal distribution with respect to θ due to the fact
that u k is a non-linear function of θ, which causes the covariance matrix to be
non-linear. In order to resolve this problem, with u k the estimated value, θ is
k
linearized. With the linearized u , it is possible to obtain the posterior covariance
matrix with the conventional Bayesian theory (Honjo, Wen-Tsung and Guha
1994: 709):
(7.18)
As shown in Equation (7.18), the covariance matrix of the posterior estimation
(Σ ) is composed of the prior estimation (V ) and the covariance matrix of
p
p
measurements (V ). It is interesting to observe that when the first term in
u
Equation (7.18) is added to the second term, it reduces the variance of the
posterior estimation since the covariance matrix of measurements is added to the
prior estimation in inverse matrix form.
Numerical method for back analysis
For problems involving geometries, external loadings, and engineering
properties, it is generally not possible to obtain an analytical solution. Hence, we
need to rely on numerical methods, such as the finite element method, to predict
the ground motion caused by underground excavation. For the finite element
program, we can either use conventional software or write our own program. The
existing elastoplastic finite element program, developed by Owen and Hinton
(1980), was modified by the authors to simulate a tunnel excavation and support
system. The Mohr-Coulomb failure criterion was used to represent the plastic
behavior.
Framework of the feedback system
General
This section introduces the implementation of the extended Bayesian method,
combined with the finite element method, for parameter estimation in
underground structures. The procedure for the proposed feedback analysis
technique is summarized as a flowchart in Figure 7.1.
The initially estimated values of ground parameters (the prior information) as
well as measured data are utilized for the feedback analysis. Geotechnical
parameters to be estimated in the current analysis are chosen and formulated
statistically. In situ measured values also have observation errors and may
contain bias terms due to initial ground movement that occurred before the