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and
0
is the
p
p
zero matrix. The matrices in (
7.17
) are defined as:
P
f
D
D
E
©
f
©
f
;
(7.51a)
T
f
D
D
E
©
f
©
2
f
;
(7.51b)
D
E
©
f
©
2
f
F
f
D
;
(7.51c)
R
D
˝
©
o
©
o
˛
;
(7.52a)
R
4
D
˝
©
o
©
2
o
˛
;
(7.52b)
D
E
H
2
C
H
2
D
E
©
o
©
2
f
©
f
©
2
o
A
D
;
(7.53)
B
D
R
˝
HP
f
H
T
C
HP
f
H
T
˝
R
;
(7.54)
D
H
T
©
f
©
o
E
D
H
H
T
E
˝
H
©
f
©
o
˝
©
o
©
f
©
o
©
f
C
D
C
;
(7.55)
N
2
vector into the
p
2
predictor
and
H
2
D
H
˝
H
is the matrix operator that takes an
space and copious use of the identity,
.
H
©
f
/
˝
.
H
©
f
/
D
.
H
˝
H
/.
©
f
˝ ©
f
/
,has
been made.
In (
7.51a
),
P
f
is a square matrix listing the necessary second moments of the
prior distribution. In (
7.51b
),
T
f
is a rectangular matrix listing the necessary third
moments of the prior distribution. In (
7.51c
),
F
f
is a square matrix listing the
necessary fourth moments of the prior distribution. In (
7.52a
),
R
4
is a square matrix
listing the necessary fourth moments of the observation likelihood. Note that even
when the observation error covariance matrix,
R
, is diagonal
R
4
is not diagonal. The
matrices
A
,
B
,and
C
are sparse, square matrices that represent various combinations
of observation error covariances and forecast error covariances.
The covariance matrix of squared innovations is
H
T
HP
f
H
T
C
R
1
HT
f
H
2
˝
v
2
˛˝
v
2T
˛
:
(7.56)
H
2
F
f
H
2
C
H
2
T
f
… D
A
C
B
C
C
C
R
4
Appendix 2: Non-Gaussian Phase Uncertainty from Variable
Shear Flows
To isolate the effects of phase uncertainty we focus on the one-dimensional
advection equation:
@p
@t
C
c.x/
@p
@x
D
0;
(7.57)
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