Page 194 - Numerical Analysis and Modelling in Geomechanics

BACK ANALYSIS OF GEOTECHNICAL PROBLEMS 175

(6.18)

To obtain the covariance matrix associated with vector is necessary to recall that
if a vector a is linearly dependent on a vector b of random variables through
matrix A,

(6.19)

the following relationship exists between the covariance matrices, C a and C ,
b
associated with the two vectors
(620)

On the basis of eq. (6.18), a linear relation can be established between vectors , p 0
and u*,

(6.21)

where I is the identity matrix, and matrix M has the following expression,
0
(6.22)

Since p 0 and u* are statistically independent, eq. (6.20) allows expressing the
covariance matrix associated to vector in the following form,

(6.23)

Eqs.(6.18) and (6.23) cannot be directly applied to the majority of calibration
problems in the field of geomechanics, due to the fact that u is in general a non-
linear function of p (even in the simple case of linear elastic behaviour of the
soil/ rock mass). In this case an iterative procedure can be adopted, by linearising
the relationship between u and p, in the neighbourhood of the current parameter
vector p′, through a Taylor’s series expansion truncated at the linear terms (cf. eq.
(6.16)),

(6.24)

The main steps of the iterative solution procedure can be summarised as follows:

1) At the beginning of iterations the current parameter vector p′ is set equal to
the initial estimate.