Schmucker-Weidelt Lecture Notes, Aarhus, 1975 - MTNet

small eigenvalues. The parameter vector of a high eigenvalue shows
the parameter combination which can be resolved well, the parameter
eigenvector of a small eigenvalue gives the combination of para-
meters for which only a poor resolution can be obtained.
The generalized inverse is used both to invert a given data set and
to estimate the information contents of this data set when the
final solution is reached.
hring the inversion procedure one has two too1.s to stabilize the
notable unstable process of linearization:
a) application of only a fraction of the con~puted parameter correc--
tion, leading to a trade-off between convergence rate and sta-
bility;
b) decrease of number of eigenvalues taken into account.
In the final estimation of the data contents one might prescribe
for exh parameter a maximum variance. Then one has to determine
from (6.35) the number of eigenvalues leading to a value nearest
to the prescribed one. Finally the row of the resolution matrix
for this particular parameter is calculated.
6.3. 'mri.vation of the kernels for the linearized inverse problem
of electromagnetic induction
6.3.1. The one-dimensional case
Both for the Backus-Gilbert procedure and for the generalized linear
inversion a knowledge of the change of the data due to a small
change in the conductivity structure is required.
In the one-dimensional case the pertinent differential equation and
datum are
, flr(z,w) = I K ~ + iwp0o(z)}f (zrw)
Consider two conductivity profiles o and o with correspondicg
1 2
fields fl and f2. Multiplying the equation for fl with f2 and the
equation for f2 with flr subtracting the resnltinq equations and
-
integrating the difference over z from zero to infinity, we obtain
.I (f;'f2 - f:fl)dz = iwp o 1 (ol-cr2)flf2 dZ
0 0