Geoscience Reference
In-Depth Information
b
m
i.e. the peak amplitudes are inversely proportional to
and to the square of the
a
1
m
mode scale
. Expressions (
8.36
)-(
8.38
) can be useful in generating the first guess
values for
z
m
to initialize an iterative procedure of approximating experimental data.
After the model parameters are established, the action of
B
1
can be computed
recursively (cf. (
8.21
)and(
8.22
)):
B
1
D
Y
m
I
j
z
2
m
j
2
D
/
.2
h
z
2
m
i
I
D
(8.39)
The inverse BEC model (
8.39
) can then be employed to compute either the action
of
B
with an iterative inversion algorithm or to directly compute the gradient of
a 3dVar cost function involving the quadratic form
x
T
B
1
x
,where
x
is the state
vector.
The above analysis gives an insight on the shape of the local CFs and provides a
direct connection between the scales described by
B
and the polynomial coefficients
of the considered BEC models (
8.9
), (
8.10
), (
8.25
)or(
8.39
). The second important
ingredient in constructing the BEC operator
C
(
8.3
) is estimating the diagonal
elements of
B
, which is a more technical but equally important problem.
8.3
Diagonal Estimation
8.3.1
Stochastic Methods
In the last few decades a large family of stochastic algorithms were developed for
estimating elements and traces of extra-large matrices emerging from numerical
soluitons of the PDEs in applied physics (e.g.,
Girard 1987
;
Dong and Liu 1994
;
Hutchison 1989
).
Weaver and Courtier
(
2001
) were among the first to use this
approach in geophysical applications for estimating the diagonal of the operator
(
8.9
).
The underlying idea is to define an ensemble of
random vectors
s
k
on
the model grid and perform componentwise averaging of the products
K
s
Bs
D
according to the formula:
d
.
/
D
s
ˇ
s
˛
s
ˇ
s
;
x
(8.40)
where the overline denotes averaging over the ensemble and ˇ, ˛ stand for the
componentwise multiplication and division of the vectors respectively. Simple
considerations show that when all the components of
s
have identical
ı
-correlated
d
from the off-diagonal elements
distributions with zero mean, the contributions to
d
converges to
d
D diag
B
as
K
!1. More accurately, the
tend to cancel out, and
squared relative approximation error
/
D
.
d
d
"
2
.
/
2
=
d
2
x
(8.41)
Search WWH ::
Custom Search