Page 195 - Numerical Analysis and Modelling in Geomechanics

176 ANNAMARIA CIVIDINI AND GIANCARLO GIODA
2) The quantities , corresponding to the in situ measurements, are determined
by means of a finite element stress analysis based on the current parameter
values.
3) The current “sensitivity” matrix L(p′) is evaluated numerically as a finite
difference approximation. This requires the solution of n stress analysis
problems (n being the number of unknown parameters p ). The vector of
i
parameters used in each analysis coincides with vector p′ but the i-th
component is perturbed by a small quantity ≥ p . i Denoting by ≥ u , the
i
difference between the quantities obtained at step (2) and those derived from
the i-th stress analysis, the sensitivity matrix can be expressed as,

(6.25)

4) Vector is evaluated by solving the equation system (6.18), where the current
values of L, u′ and p′ are introduced.
5) The iterative procedure ends when the difference between p′ and is smaller
than a pre-assigned tolerance, otherwise p′ is set equal to and the process is
continued from step (2). The main diagonal of the covariance matrix
calculated at the end of the iterative process through eq.(6.23) represents the
variances of the estimated values of the parameters.

It is worthwhile observing that the Bayesian approach is applicable also when the
number of unknown parameters exceeds the number of in situ measurements, if a
reliable initial guess on the parameters can be formulated.
Consider in fact the limit case in which no experimental information is available.
This case is equivalent to the situation in which the accuracy of the experimental
data is so poor that the entries of the corresponding inverted covariance matrix
C u vanish. Consequently, eq.(6.18) reduces to a trivial form expressing the
equivalence between the optimal values of the parameters and their initial
estimate P .
0
Another limit case is when no a priori information is available, or when its
reliability is so low that the corresponding inverted covariance matrix vanishes.
In this case eq. (6.18) becomes,

(6.26)

Furthermore, if all the (uncorrelated) in situ measurements have the same
accuracy, matrix C u can be eliminated from eq. (6.26), thus obtaining the
following least square expression for the best estimate of the unknown
parameters,
(6.27)