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the singular values. This singular spectrum is useful when assessing the observing
array or covariance model, since it establishes a criterion for counting the number
of degrees of freedom effectively constrained by the data (
Bennett 1985
,
1992
).
When the observation error is not a constant it is advantageous to transform with
the change of variables,
v
D
R
1=2
x
.
The singular spectrum can be used to develop a stopping criterion for the iterative
solver in terms of the predicted percent of variance explained. Recall that the
representer matrix
can be interpreted as a covariance matrix, the trace of which is
the total amount of variance expected in the observations exclusive of measurement
noise (
Bennett 2002
). Recall also, that the degrees of freedom associated with
singular vectors may be classified as either smoothed or interpolated by the data
assimilation, according to whether
R
i
.
R
/<
or
i
.
R
/>
, respectively (
Bennett
2002
). Let
denote the mode number with the singular value comparable to the
measurement error, e.g.,
k
k
.
R
/>
k
C
1
.
R
/
,then
k
X
S
D
i
.
R
/
(12.9)
i
D
1
is the expected total observed variance explainable by the given data assimilation
system. In practice
O
.
R
/
D
.
R
/
is not known exactly, but its approximation
/
1
.
U
is available from the orthogonal decomposition of
U
. An approximation
O
.
R
/
O
e
.
R
/
to
S
can be made by extrapolating
out to
i
D
k
. Letting
denote this
approximate spectrum, then the fraction of
S
explained by stopping at iterate
p
may
be estimated as
p
X
!
k
X
!
1
O
i
.
R
/
O
i
.
R
/
f
D
:
(12.10)
i
D
1
i
D
1
Figure
12.3
shows an application of these ideas with the data-assimilative ocean
model described in
Zaron et al.
(
2009
). The estimated spectrum
O
.
R
/
is computed
for iterates
p
D
10;20;40
(gray) and for the final iterate
p
D
58
(black). The
O
e
.
R
/
extrapolated spectrum
is computed from a power-law fit to the middle 50 %
of the singular values, and one sees that the extrapolated spectrum and data error
variance intersect at approximately
k
D
200
; thus, one expects approximately 142
additional iterates would be necessary to minimize
J
.
/
x
. Applying (
12.10
)to
f
D
88
compute the fraction of variance explained, one finds
%. In other words,
p
D
58
88
the solution obtained by stopping the solver at
accounts for
of the
p>k
is un-explainable with the covariance model
B
, and it is ascribed to observation
error. While the details are certainly problem-dependent, we have found that
O
.
R
/
adequately approximates the true spectrum when judged against the uncertainty in
B
. Experience with idealized, low-dimensional, data assimilation problems suggests
that these methods are applicable in realistic systems, where complete knowledge
of the spectra cannot be obtained.
explainable observed variance. Note that the variance associated with modes
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