Schmucker-Weidelt Lecture Notes, Aarhus, 1975 - MTNet

6.1.2. The llnear inverse problem
The data may have two defects:
a) insufficient
b) inaccurate
Certainly in the above problem the two data M and 0 are insufficient
to determine the continuous function p(r). Generally, the properties
a) and b) are inherent to all real data sets when a continuous func-
tion is sought. The lack of data smoothes out details and only
some average quantities are available, the observational error in-
troduces statistical uncertainties in the model. (If we were looking
for a descrete model with fewer parameters than data, then incon-
sistency can arise as a third defect.) Because of the lack of data,
instead of m at r we can obta.in only an averaged quantity ,
0
which is still. subject to statistical incertainties due to errors
in the data. Let
a
< m(ro) > = I A(ro[r)m(r)dr, (6.2)
0
where a is subject to
The latter condition ensures that agrees with m, if m is a con-
stant. A(r ( r) is the window, througlil which the real but unknown
0
function m(r) can be seen. It is the - averaqinq - or resolution Punctio -
The more A at ro resembles a &-function the better is the resolution
at ro. Resolvable are only the projections of m(r1 into the
space of the data kernels Gi. The part of m, which is orthogonal
to the data kernels cannot be resolved. Hence, it is reasonable
to write A as a linear combination of the data kernels
where the coefficients a. (ro) have to be determined in such a way
I
that A is as peaked as possible at ro. The aost obvious choice
would be to minimize
subject to (6.3). For'c~rn~utational ease Backus and Gilbert prefer