319
Radar Polarimetry for Rain Estimation
where σ ext
H,V is the extinction cross section at horizontal or vertical polarizations,
respectively. The specific differential attenuation A DP is defined as
A
A A
DP
H
V
1
(dB km )
=
−
− .
(13.25)
If specific attenuations are known, the attenuated Z H and Z DR at each range gate can
be calculated by
Z n Z n
A i r
i
n
H
a
H
H
( )
( )
( )
=
−
=
−
∑
2
1
1
∆
(13.26)
and
Z n Z n
A i r
a
i
n
DR
DR
DP
( )
( )
( )
=
−
=
−
∑
2
1
1
∆ ,
(13.27)
where numbers i and n denote the ith and nth range gates from the radar location,
respectively, and Δr is the range resolution.
It is expensive to directly compute the transpose of linearized operator H, which
is the matrix of the partial derivatives. In general, the adjoint method is applied to
compute H T efficiently without storing the full matrix. However, it is hard to represent the derivatives functionally in terms of DSD parameters, given that the scattering amplitude of raindrop is precalculated using the T-matrix method. Therefore,
the lookup table method is applied for the H calculation. There are a total of six
tables of the derivatives, i.e.,
H
DR
DP
H
DR
DP
∂
∂
∂
∂
∂
∂
∂
∂
∂
∂
∂
∂
Z
Z
K
Z
N
Z
N
K
Λ
Λ
Λ
,
,
,
,
,
0
0
∗
∗
N N 0
∗
. In each lookup
table, the derivative values are precalculated for parameters Λ and N 0
∗ discretized at
an interval of 0.1. Interpolation between the intervals can be performed to further
improve the accuracy. Similarly, the calculations of intrinsic (i.e., nonattenuated) Z H ,
Z DR , K DP , A H , and A DP are made efficient by the lookup table method as well, given
any two state parameters. As a result, the observational operator H is computed as
the combination of different values found in various lookup tables, avoiding integral
calculations in the forward model.
The iteration procedure for minimizing the cost function is shown in Figure 13.5.
At the beginning of the program, necessary data files such as all lookup tables,
the background, radar-measured Z H , Z DR , K DP , and SNR are loaded. With the initial
state vector (e.g., set v = 0), intrinsic variables (i.e., Z H , Z DR , K DP , A H , and A DP ) are
found for each grid point through lookup tables. Corresponding Jacobian matrices
Hs are constructed based on the lookup tables as well. After the interpolation from
grid points to the observation points, attenuated Z H and Z DR are calculated according to Equations 13.26 and 13.27. The calculated and measured Z H , Z DR , and K DP are
used in Equation 13.22 to calculate the gradient of the cost function. The initial first
guess is always assumed to be the background. During the minimization process, the
state vector is updated at each loop until the iteration converges. If the background
Radar Polarimetry for Rain Estimation
where σ ext
H,V is the extinction cross section at horizontal or vertical polarizations,
respectively. The specific differential attenuation A DP is defined as
A
A A
DP
H
V
1
(dB km )
=
−
− .
(13.25)
If specific attenuations are known, the attenuated Z H and Z DR at each range gate can
be calculated by
Z n Z n
A i r
i
n
H
a
H
H
( )
( )
( )
=
−
=
−
∑
2
1
1
∆
(13.26)
and
Z n Z n
A i r
a
i
n
DR
DR
DP
( )
( )
( )
=
−
=
−
∑
2
1
1
∆ ,
(13.27)
where numbers i and n denote the ith and nth range gates from the radar location,
respectively, and Δr is the range resolution.
It is expensive to directly compute the transpose of linearized operator H, which
is the matrix of the partial derivatives. In general, the adjoint method is applied to
compute H T efficiently without storing the full matrix. However, it is hard to represent the derivatives functionally in terms of DSD parameters, given that the scattering amplitude of raindrop is precalculated using the T-matrix method. Therefore,
the lookup table method is applied for the H calculation. There are a total of six
tables of the derivatives, i.e.,
H
DR
DP
H
DR
DP
∂
∂
∂
∂
∂
∂
∂
∂
∂
∂
∂
∂
Z
Z
K
Z
N
Z
N
K
Λ
Λ
Λ
,
,
,
,
,
0
0
∗
∗
N N 0
∗
. In each lookup
table, the derivative values are precalculated for parameters Λ and N 0
∗ discretized at
an interval of 0.1. Interpolation between the intervals can be performed to further
improve the accuracy. Similarly, the calculations of intrinsic (i.e., nonattenuated) Z H ,
Z DR , K DP , A H , and A DP are made efficient by the lookup table method as well, given
any two state parameters. As a result, the observational operator H is computed as
the combination of different values found in various lookup tables, avoiding integral
calculations in the forward model.
The iteration procedure for minimizing the cost function is shown in Figure 13.5.
At the beginning of the program, necessary data files such as all lookup tables,
the background, radar-measured Z H , Z DR , K DP , and SNR are loaded. With the initial
state vector (e.g., set v = 0), intrinsic variables (i.e., Z H , Z DR , K DP , A H , and A DP ) are
found for each grid point through lookup tables. Corresponding Jacobian matrices
Hs are constructed based on the lookup tables as well. After the interpolation from
grid points to the observation points, attenuated Z H and Z DR are calculated according to Equations 13.26 and 13.27. The calculated and measured Z H , Z DR , and K DP are
used in Equation 13.22 to calculate the gradient of the cost function. The initial first
guess is always assumed to be the background. During the minimization process, the
state vector is updated at each loop until the iteration converges. If the background