Schmucker-Weidelt Lecture Notes, Aarhus, 1975 - MTNet

should be small. Hence it required to minimize simultaneously the
quadratic forms
N N
s= C ai% sik 7 c2= C a a E .
i k lk
i,k=l
i , k=l
subject to the condition (6.3), i-e.
There does not ex'ist a set of a. which minimizes s and E' separately.
L
As a compromise only a combination
can be minimized. In (6.15) , c is an arbitrary positive scaling
factor which accounts for the different dimensions of s and E' and
W is a parameter
which weighs the particular importance of s and E ~ For . W == 1 the
spread is minimized without regarding the error of the spatially
averaged quantity . Conversely .for W = 0 the spread s is
large and the error E' is a minimum. Hence, in general there is a
trade-off between resoluti.on - and accurac~, which for a particular
is shown in the following figure.
ro
EZ max
E'
2 -- - -L ---min
I
W=O
I --
s
S
min max
Near s = s . the trade-off curve i.s rather steep. Hence, a small
m m
sacrifice of resolving power will largely reduce the error of the
.average . This uncertainty relation between resolution and
accuracy is the central point 05 the Backus-Gilbert procedure.
, S