318
Multiscale Hydrologic Remote Sensing: Perspectives and Applications
The size of matrix B is n 2 , where n is the size of state vector x. The full matrix is
usually large (e.g., n might be an order of magnitude of 6). Matrix computation and
storage, especially for the inversion of B, can be a major problem during the iterative
minimization of the cost function. To solve this problem, a new state variable v is
introduced, written as
v = D –1 δx
(13.21)
with δx = x – x b and DD T = B (Parrish and Derber 1992). Notation δ indicates the
increment. D is the square root of the background error covariance matrix B. This
way, the inversion of B is avoided. The minimization of cost function J can be
achieved by searching the minimum gradient of cost function ∇ v J, which is given by
= +
−
(
) +
−
v
v w D H R H Dv d
w D H R
J
Z
Z
Z
Z
Z
Z
Z
Z
H
H
H
H
H
DR
DR
D
T T
T
T T
1
R R
DR
DR
DP
DP
DP
DP
D
T
T T
T
−
−
−
(
)
+
−
1
1
H Dv d
w D H R
H Dv d
Z
Z
K
K
K
K
K P P
(
) ,
(13.22)
where H represents the Jacobian operator, a matrix containing the partial derivative
of observation operator H with respective to each element of the state vector, and d
is the innovation vector of the observation, that is, d = y – H(x b ).
The spatial influence of the observation is determined by the background error
covariance matrix B. Huang (2000) showed that the element b ij of matrix B could be
modeled as a spatial filter:
b
r
r
ij
b
ij
L
=
−
σ
2
2
2
2
exp
,
(13.23)
where subscripts i, j denote two grid points in the analysis space, σ b
2 is the background error covariance, r ij indicates the distance between the ith and jth grid points,
and r L is the decorrelation length of observed storm. In this study, r L is assumed to
be constant in the two-dimensional analysis space, that is, the error covariance is
spatially homogeneous in the horizontal plane, as is the isotropic covariance in the
work of Liu and Xue (2006). The square root of B, D can be computed by applying
a recursive filter described by Gao et al. (2004) and Liu and Xue (2006). This way,
the cost of computation and storage can be reduced significantly (by a factor of B’s
dimension), compared to the computation of inversion of B.
The parameters in state vector x are DSD parameters N 0 and Λ at every grid point
of the analysis region. The C-G model (Equation 13.14) is applied, reducing the
gamma model freedom to 2. Forward operators of intrinsic Z H , Z DR , and K DP follow
Equations 13.1, 13.2, and 13.4. Specific attenuations at the horizontal (A H ) and vertical (A V ) polarizations are calculated by
A
D N D dD
H,V
ext
H,V
1
( dB km )
=
×
−
∞
∫
4 343 10
3
0
.
( ) ( )
σ
,
(13.24)
Multiscale Hydrologic Remote Sensing: Perspectives and Applications
The size of matrix B is n 2 , where n is the size of state vector x. The full matrix is
usually large (e.g., n might be an order of magnitude of 6). Matrix computation and
storage, especially for the inversion of B, can be a major problem during the iterative
minimization of the cost function. To solve this problem, a new state variable v is
introduced, written as
v = D –1 δx
(13.21)
with δx = x – x b and DD T = B (Parrish and Derber 1992). Notation δ indicates the
increment. D is the square root of the background error covariance matrix B. This
way, the inversion of B is avoided. The minimization of cost function J can be
achieved by searching the minimum gradient of cost function ∇ v J, which is given by
= +
−
(
) +
−
v
v w D H R H Dv d
w D H R
J
Z
Z
Z
Z
Z
Z
Z
Z
H
H
H
H
H
DR
DR
D
T T
T
T T
1
R R
DR
DR
DP
DP
DP
DP
D
T
T T
T
−
−
−
(
)
+
−
1
1
H Dv d
w D H R
H Dv d
Z
Z
K
K
K
K
K P P
(
) ,
(13.22)
where H represents the Jacobian operator, a matrix containing the partial derivative
of observation operator H with respective to each element of the state vector, and d
is the innovation vector of the observation, that is, d = y – H(x b ).
The spatial influence of the observation is determined by the background error
covariance matrix B. Huang (2000) showed that the element b ij of matrix B could be
modeled as a spatial filter:
b
r
r
ij
b
ij
L
=
−
σ
2
2
2
2
exp
,
(13.23)
where subscripts i, j denote two grid points in the analysis space, σ b
2 is the background error covariance, r ij indicates the distance between the ith and jth grid points,
and r L is the decorrelation length of observed storm. In this study, r L is assumed to
be constant in the two-dimensional analysis space, that is, the error covariance is
spatially homogeneous in the horizontal plane, as is the isotropic covariance in the
work of Liu and Xue (2006). The square root of B, D can be computed by applying
a recursive filter described by Gao et al. (2004) and Liu and Xue (2006). This way,
the cost of computation and storage can be reduced significantly (by a factor of B’s
dimension), compared to the computation of inversion of B.
The parameters in state vector x are DSD parameters N 0 and Λ at every grid point
of the analysis region. The C-G model (Equation 13.14) is applied, reducing the
gamma model freedom to 2. Forward operators of intrinsic Z H , Z DR , and K DP follow
Equations 13.1, 13.2, and 13.4. Specific attenuations at the horizontal (A H ) and vertical (A V ) polarizations are calculated by
A
D N D dD
H,V
ext
H,V
1
( dB km )
=
×
−
∞
∫
4 343 10
3
0
.
( ) ( )
σ
,
(13.24)