Page 193 - Numerical Analysis and Modelling in Geomechanics

174 ANNAMARIA CIVIDINI AND GIANCARLO GIODA
and that the error covariance matrix C , which depends on the accuracy of the
u
measuring device, is known,

(6.13)

If all measurements are statistically independent, C u is a diagonal matrix the
entries of which (variances) are related to the resolution of the instruments.
Also the unknown parameters p are regarded as random quantities and it is
assumed that the following expectations are known,

(6.14)

(6.15)

In the above equations, p 0 and C p,0 depend on the a priori information on the
unknown parameters. If the entries of vector p 0 are uncorrelated, C p,0 is a
diagonal matrix. Note that the values of the entries of this matrix increase with
decreasing accuracy of the initial information on the unknown parameters.
The Bayesian back analysis consists in combining a priori and experimental
information in order to achieve the best estimate of the unknown parameters.
Also in this case, as for the deterministic back analysis, a numerical model is set
up which allows us to calculate the quantities u, corresponding to the measured
ones u*, on the basis of the current parameter vector p.
Consider first the simple case in which u is linearly dependent on p through a
constant matrix L and constant vectors u′ and p′,

(6.16)

The best estimate of p can be obtained by minimising, with respect to p, the
following error function E ,
r
(6.17)

which consists of two parts: the first represents the discrepancy between
measured and calculated data, while the second is the discrepancy between
assumed and current parameters.
These discrepancies are weighted by means of the inverted covariance
matrices, which tend to vanish with decreasing accuracy of the a priori
information and of the experimental data.
By introducing eq. (6.16) into eq. (6.17), and by imposing that the derivatives
of E with respect to p vanish, the following system of linear equations is arrived
r
at, the solution of which leads to the optimal vector ,