Environmental Engineering Reference
In-Depth Information
¼
190R
1
:
72
), which was
where the parameter m takes values between
3 and 8. For m
¼
0, Equation 7.8 takes the form
of the Marshall and Palmer DSD. The shape of the
gamma DSD is determined by the exponent m, and
for positive values of m the gamma DSD is concave
down whereas for negative values it is concave
upward. A gamma DSD can describe many of
the natural variations in the shape of the DSD.
When there is a substantial depth between the
melting level and the ground surface, the param-
eterization of a gamma DSD appears to be suitable
for stratiform and convective rainfall events
(Bringi and Chandrasekar 2001). In addition,
Testud et al. (2001) proposed the normalization
of the DSD to avoid any assumption about the
shape of the DSD.
relationships
(Z
later
slightly modified to Z
¼
220R
1
:
6
(Marshall and
Palmer 1948). Some years later, Marshall
et al. (1955) slightly revised the 1948 relationship,
obtaining the well-known Marshall-Palmer for-
mula Z
¼
200R
1
:
6
. The UKMeteorological Office's
Nimrod system estimates the precipitation rate
using this equation. Unfortunately, there is no
single Z-R relationship that can be applied in every
part of the world. Battan (1973) listed 69 different
Z-R relationships derived from different climato-
logical regions bymany researchers. This variabil-
ity was due to the fact that the coefficient and
exponent of the Z-R relationship depend on the
shape of the DSD. Therefore, it is necessary to
estimate in real time the parameters of the DSD to
allow flexibility in the variation of the parameters
a and b of the Z-R relationship.
One of the main goals of dual-polarization
radars is the improvement in quantitative precip-
itation estimates, and Seliga and Bringi (1976)
proposed the use of differential reflectivities at
orthogonal polarizations to estimate the para-
meters of an exponentialDSD (Equation 7.7). They
suggested that the parameter D
0
is obtained with
Z
dr
whereas N
0
is obtained with Z
h
and D
0
. The
main advantage of using the differential reflectiv-
ity Z
dr
is that the median raindrop diameter D
0
is
related to the value of Z
dr
. Using the differential
reflectivity measurements has been exploited by
several researchers to obtain amore representative
shape of the DSD.
In order to derive relationships between the
rainfall rate and the polarimetric radar measure-
ments a common method is based on varying the
parameters N
0
, D
0
and m of a theoretical gamma
DSD and then calculating R, Z
h
, Z
dr
and K
dp
assuming a scattering model. In addition, the
mean raindrop axis ratio r(D), or degree of defor-
mation as a function of the diameter D, must be
specified. The coefficients and exponents of the
different rainfall rate algorithms are usually
obtained by performing a non-linear regression
between the rainfall rate and the polarimetric
variables (Bringi and Chandrasekar 2001).
The deformation of raindrops is an important
relationship that
Algorithms to Estimate Rain from Radar
Measurements
Themost commonly used polarimetric radar mea-
surements for rainfall estimation are the reflectiv-
ity factor (Z
h
), the differential reflectivity (Z
dr
) and
the specific differential phase (K
dp
), and for many
years radar meteorologists have tried to find a
useful equation relating the reflectivity factor to
the rainfall rate. The rainfall rate given by Equa-
tion 7.6 can be obtained by assuming a drop size
distribution and terminal velocity of the rain-
drops. By comparing the rainfall rate with the
actual reflectivity measured by the radar, it is
possible to derive Z-R relationships of the form:
aR
b
a
1
=
b
Z
1
b
=
Z
¼
or R
¼
ð
7
:
9
Þ
where Z is the reflectivity factor in mm
6
m
3
, R is
the rainfall rate in mmh
1
, and a and b are the
parameters obtained from a regression analysis.
Atlas and Ulbrich (1990) showed that the first Z-R
relationship could be traced back to the research
work carried out by Ryde (1946). They also
showed that this relationship is approximately
Z
¼
320R
1
:
44
. This is very similar to that employed
to estimate the rainfall rate from reflectivity
measurements in the WSR-88D radar network,
which is Z
¼
300R
1
:
4
(Serafin and Wilson 2000).
Marshall et al. (1947) reported one of the first Z-R
leads
to different
rainfall
