Geoscience Reference
In-Depth Information
where
z
D
y
T
x
b
T
is
;
X
D
H
T
I
n
T
is
.m
C
n/
1
.m
C
n/
n
© D
Œ
©
o
©
b
T
and
.m
C
n/
1
is
with zero mean and variances matrix
R
0
D
0
B
The generalized LS solution for
™
is BLUE and is given by
™ D
.
X
T
1
X
/
1
X
T
1
z
(4.34)
see
Talag r an d
(
1997
). After some algebra this equation equals (
4.11
). Thus
z
D
X
™ D
H
T
x
a
x
a
T
D
X
X
T
1
X
/
1
X
T
1
z
.
and by (
4.5
) the influence matrix becomes
S
yy
S
yb
S
by
S
bb
R
1
HAH
T
R
1
HA
B
1
AH
T
z
D
@
™
S
zz
D
@
z
@
z
D
D
B
1
A
@
where
S
yy
D
@
Hx
a
@
y
I
S
yb
D
@
x
a
@
y
I
S
by
D
@
Hx
a
D
@
x
a
@
x
b
@
x
b
I
S
bb
. Note that
S
yy
D
S
as defined
in (
4.4
).
Generalized LS regression is different from ordinary LS because the influence
matrix is not symmetric anymore. For idempotence, using (
4.33
)iteasytoshow
that
S
zz
S
zz
D
S
zz
:
Finally,
D
B
1
A
D
I
n
H
T
R
1
HA
S
bb
hence,
H
T
R
1
HA
tr
.
S
bb
/
D
n
tr
.
/
D
n
tr
.
S
yy
/
it follows that
S
bb
/
D
n
The trace of the influence matrix is still equal to the parameter's dimension.
.
S
zz
/
D
tr
.
S
yy
/
C
tr
.
tr
References
Bauer P, Buizza R, Cardinali C, Thepaut J-l (2011) Impact of singular vector based satellite data
thinning on NWP. Q J R Meteorol Soc 137:286-302
Bormann N, Saarinen S, Kelly G, Thepaut J-N (2003) The spatial structure of observation errors
in atmospheric motion vectors from geostationary satellite data. Mon Wea Rev 131:706-718
Cardinali C (2009) Monitoring the forecast impact on the short-range forecast. Q J R Meteorol Soc
135:239-250
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