Geoscience Reference
In-Depth Information
10
−1
estimated spectrum
data variance
extrapolated spectrum
10
−2
10
−3
10
−4
10
−5
10
0
10
1
10
2
10
3
10
4
10
5
singular value index
Spectral Diagnostics from GCR. The estimated spectrum
O
.
R
/
Fig. 12.3
of the representer matrix
HBH
T
is shown by the
dark solid line
corresponding to the last GCR iterate (
R
D
)in
Fig.
12.2
.
Solid gray lines
show
O
.
R
/
based on iterates
p
D
10;20;
and
40
, for comparison. The
data variance is
,where
R
D
I
. The extrapolated spectrum is computed from a linear fit to
.log.i/;log.
i
.
R
///
in the range
p=4
i
3p=4
p
D
58
due to the background and observations
may be obtained as diagnostic information from the GCR iterates. Substituting
x
p
D
BH
T
x
p
in (
12.4
), one obtains
Finally, the two components of
J
.
x
p
/
J
.
x
p
/
D
J
B
.
x
p
/
C
J
R
.
x
p
/
D
x
p
HBH
T
x
p
(12.11)
HBH
T
x
p
y
/
T
R
1
.
HBH
T
x
p
y
C
.
/:
Because the GCR solver computes the residual
r
p
at each iterate, one has
HBH
T
C
R
.
/
x
p
D
y
r
p
:
(12.12)
Assuming that
Rx
p
can be computed on demand, then
HBH
T
x
p
D
y
r
p
Rx
p
;
(12.13)
and all terms in the expression for the objective function are computable. The
contribution from the background term is
J
B
.
x
p
/
D
.
x
p
/
T
.
y
r
p
Rx
p
/;
(12.14)
while the contribution from the observations is
J
R
.
r
p
C
Rx
p
/
T
R
1
.
x
p
/
D
.
r
p
C
Rx
p
/:
(12.15)
Search WWH ::
Custom Search