Geoscience Reference
In-Depth Information
w
D
u
ı
v
;
w
i
D
u
i
v
i
;i
D
1
W
n
(9.60)
Y
2
R
n
m
For two matrices of the same order
X
;
Y
i
R
n
m
D
Tr
XY
T
D
Tr
X
T
Y
h
X
;
(9.61)
denotes the Frobenius inner product that is expressed in terms of the matrix trace
operator
. Given the vectors
u
2
R
n
,
v
2
R
m
,andthematrix
X
2
R
n
m
,
Tr
Xv
i
R
n
D
u
T
Xv
Dh
uv
T
h
u
;
;
X
i
R
n
m
(9.62)
e
W
R
n
m
!
R
2
R
n
m
, the sensitivity
Given a functional
of matrix argument
X
of
e
with respect to
X
is the matrix of the first order partial derivatives denoted as
@e
@X
i;j
@e
@
2
R
n
m
X
D
(9.63)
i
D
1;n
I
j
D
1;m
The first order variation
ıe
induced by a variation
ı
X
is expressed as
@e
@
X
@e
@
T
ıe
D
X
;ı
R
n
m
D
Tr
X
.ı
X
/
(9.64)
2
R
n
n
, the first order variation
X
1
in the inverse
For a nonsingular matrix
X
ı
matrix
X
1
induced by a variation
ı
X
is expressed as
X
1
D
X
1
ı
XX
1
ı
(9.65)
References
Baker NL, Daley R (2000) Observation and background adjoint sensitivity in the adaptive
observation-targeting problem. Q J R Meteorol Soc 126:1431-1454
Baker NL, Langland RH (2009) Diagnostics for evaluating the impact of satellite observations. In
Park SK, Xu L (eds) Data assimilation for atmospheric, oceanic and hydrologic applications.
Springer, Berlin, pp 177-196
Bannister RN (2008a) A review of forecast error covariance statistics in atmospheric variational
data assimilation. I: characteristics and measurements of forecast error covariances. Q J R
Meteorol Soc 134:1951-1970
Bannister RN (2008b) A review of forecast error covariance statistics in atmospheric variational
data assimilation. II: modelling the forecast error covariance statistics. Q J R Meteorol Soc
134:1971-1996
Bormann N, Bauer P (2010) Estimates of spatial and interchannel observation-error characteristics
for current sounder radiances for numerical weather prediction. I: methods and applications to
ATOVS data. Q J R Meteorol Soc 136:1036-1050
Search WWH ::
Custom Search