Schmucker-Weidelt Lecture Notes, Aarhus, 1975 - MTNet

Agai-n, from a finite erroneous data set we can extract only averaged
estimates with statistical uncertainties, i.e.
As in the linear case A(r Ir) is built up from a linear combination
0
of the data kernels
N
A(rolr) = C a. (rorm)Gi(ro,m).
i= 1 I
Introducing (6.1 9) into (6.20) we obtain
which for nonlinear kernels is different from (6.8) , since i.11 this
nonlinear case gi (m) qi (m) .
In the linear case, two models m and m' which both sa.tisfy the
data lead to the same average model in1 (r ) >. In the nonlinear case,
0
the average models are different; the difference, however, is of
the second order in (m' -m) . (Exercise! )
The Backus-Gilbert procedure in the nonlinear case requires a model
which already nearly fits the data. Then it can give an appraisal
of the inforn~ation contents of a given data set.
6.2. Generalized - matrix inversion
The generalized matrix inversion is an alternative procedure to
the Backus-Gilbert method. It is strictly applicable only to linear
problems, where the model under consideration consists of a set of
discrete unknown parameters. Nonlinear problems are generally
linearized to get in the range of this method. Assume that we :mnt
T
to determine the M component parameter vector p with - p = (plr...,pb5:
and that we have' N functidnalr, (rules) g. i = 1, . . . , N which
1'
assign to any model E a number, which when measured has the average
value y . and variance var (y . ) :
1 1