Schmucker-Weidelt Lecture Notes, Aarhus, 1975 - MTNet

- Only for P M, _A - is the M component unit matrix, admitting an
exact determination of - x. (A corresponds approximately to the resolution
function A (ro 1 r) in the Backus--Gilbert theory. But there
are differences: - is symmetrical, A(s[r) not, the norm of A - is
small if the resolution is poor, the norm of A(ro[r) is always 1. )
In generalized matrix inversion exists the same trade-off as in the
Backus-Gilbert method. Cbnsider the covariance matrix of the change:
of.< - x > due to random changes of x:
In particular the variance of is
P . Vk
var (x ) = a2 ZI (3)
Z ,
j=l j
k 0 k = 1,2, .:., M
showing that the variance of xk is largely due to the small eigenvalues
X By discarding small eigenvalues and the corresponding
j'
eigenvectors, the accuracy of x can be increased at the expense
k
of the resolution, since A - will deviate more from a (M x M) unit
matrix if instead of the required M eigenvectors a smaller number
is used. -
The model will in general not.reproduce the data. The repro-
duced data are
= G - - =GWy - - =Ex -
with the information density matrix
T - B=GH=LJQ. - - - -
(6.36)
. . Only in the case N = PI 2 is a unit matrix. In particular, B -
describes the linear dependence of the data in the overconstrained
case. High diagonal values will show that this date contains
specific information on the model which is not contained in other
data. On the other hand a large off-diagonal value shows that this
information is also contained in another data.
Valuable insight in the particular inverse problem can be obtained
by considering the parameter eigenvectors corresponding to high and