309
Radar Polarimetry for Rain Estimation
where D min/max is the minimum/maximum raindrop diameter, and v(D) is the falling velocity of raindrop. Empirical relations can be found between radar variables and rain variables. These relations are not directly associated with the DSD.
Each relation works only with given radar–rain variables and, more specifically,
for one radar frequency. Therefore, the applications of empirical relations are not
flexible. However, using DSD retrievals, all the rain variables of interest can be
calculated.
13.3.1 eMPiRical RadaR–Rain eStiMation
The traditional radar–rain estimation applies power-law R–Z h relations. It has been
realized that the coefficient and the exponent in the power-law relation have large
variability (Doviak and Zrnić 1993). Many R–Z h relations have been reported for
different rain types, seasons, and locations. Rosenfeld and Ulbrich (2003) gave a
complete review of those R–Z h relations and summarized microphysical processes
that might cause R–Z h variability. The essential reason is that Z h alone cannot provide a unique quantification of R, given the DSD variability.
Dual-polarization measurements help better represent DSD variability and therefore improve empirical estimation. Polarimetric radar–rain estimators generally
have the following forms:
R Z Z
aZ Z
b c
( , )
h
dr
h dr
=
(13.9)
R K
aK
b
(
)
DP
DP
=
(13.10)
R K Z
aK Z
b
c
(
, )
DP
dr
DP dr
=
,
(13.11)
where a, b, and c are constant parameters. The derivation of those relations requires
a key assumption of raindrop shape, which is important for the quantification of
polarimetric measurements. Generally, there are three kinds of raindrop axis ratio
relations. The empirical relations introduced by Pruppacher and Beard (1970), Green
(1975), and Chuang and Beard (1990) focus on the raindrop shape under an equilibrium condition. Other studies, such as Pruppacher and Pitter (1971), Beard et al.
(1983), Beard and Jameson (1983), and Beard and Tokay (1991), found that collision,
wind shear, and turbulence could lead to the oscillation of raindrops, whose shapes
would be more spherical than shapes under an equilibrium condition. Keenan et al.
(2001) and Brandes et al. (2002; BZV model) derived raindrop axis ratio relations
from previous observations or relations with an experimental regression procedure.
Different raindrop shape assumptions could result in different calculations of
polarimetric variables. Brandes et al. (2002) illustrated with a specific example that
the simulated Z DR using the equilibrium shape model is 0.2 dB larger than the corresponding value calculated using the experimental shape model. As a result, constant parameters for polarimetric rain estimators (e.g., Equations 13.9 through 13.11)
also depend on the assumption of raindrop shape. Table 13.1 lists some polarimetric
Radar Polarimetry for Rain Estimation
where D min/max is the minimum/maximum raindrop diameter, and v(D) is the falling velocity of raindrop. Empirical relations can be found between radar variables and rain variables. These relations are not directly associated with the DSD.
Each relation works only with given radar–rain variables and, more specifically,
for one radar frequency. Therefore, the applications of empirical relations are not
flexible. However, using DSD retrievals, all the rain variables of interest can be
calculated.
13.3.1 eMPiRical RadaR–Rain eStiMation
The traditional radar–rain estimation applies power-law R–Z h relations. It has been
realized that the coefficient and the exponent in the power-law relation have large
variability (Doviak and Zrnić 1993). Many R–Z h relations have been reported for
different rain types, seasons, and locations. Rosenfeld and Ulbrich (2003) gave a
complete review of those R–Z h relations and summarized microphysical processes
that might cause R–Z h variability. The essential reason is that Z h alone cannot provide a unique quantification of R, given the DSD variability.
Dual-polarization measurements help better represent DSD variability and therefore improve empirical estimation. Polarimetric radar–rain estimators generally
have the following forms:
R Z Z
aZ Z
b c
( , )
h
dr
h dr
=
(13.9)
R K
aK
b
(
)
DP
DP
=
(13.10)
R K Z
aK Z
b
c
(
, )
DP
dr
DP dr
=
,
(13.11)
where a, b, and c are constant parameters. The derivation of those relations requires
a key assumption of raindrop shape, which is important for the quantification of
polarimetric measurements. Generally, there are three kinds of raindrop axis ratio
relations. The empirical relations introduced by Pruppacher and Beard (1970), Green
(1975), and Chuang and Beard (1990) focus on the raindrop shape under an equilibrium condition. Other studies, such as Pruppacher and Pitter (1971), Beard et al.
(1983), Beard and Jameson (1983), and Beard and Tokay (1991), found that collision,
wind shear, and turbulence could lead to the oscillation of raindrops, whose shapes
would be more spherical than shapes under an equilibrium condition. Keenan et al.
(2001) and Brandes et al. (2002; BZV model) derived raindrop axis ratio relations
from previous observations or relations with an experimental regression procedure.
Different raindrop shape assumptions could result in different calculations of
polarimetric variables. Brandes et al. (2002) illustrated with a specific example that
the simulated Z DR using the equilibrium shape model is 0.2 dB larger than the corresponding value calculated using the experimental shape model. As a result, constant parameters for polarimetric rain estimators (e.g., Equations 13.9 through 13.11)
also depend on the assumption of raindrop shape. Table 13.1 lists some polarimetric