Environmental Engineering Reference
In-Depth Information
and Anthes 1989). The different sizes of raindrops
define a drop size distribution (DSD) given by
N(D). The DSD describes the probability density
function of the raindrop sizes and it is one of the
most important functions in rainfall rate estima-
tion algorithms. All the microphysical processes
involved in the interaction between the raindrops
are reflected in the DSD, and for a given DSD the
rain rate can be obtained from:
mean particle shape and large raindrops produce
large values of Z
dr
. The sensitivity of Z
dr
to particle
shape is less for ice than for water (Herzegh and
Jameson 1992), and ice particles tend to wobble
and spin in their descent resulting in Z
dr
values
being closer to zero.
Specific differential phase
ð
v
Electromagnetic waves experience phase shifts as
they propagate through regions of precipitation.
The horizontally polarized wave suffers larger
phase shifts than the vertically polarized wave
because raindrops are horizontally orientated as
they fall. The differential phase
W
dp
is the differ-
ence between the received phases of horizontally
and vertically polarized electromagnetic waves
(
W
dp
¼W
hh
W
vv
), and the specific differential
phase (K
dp
) is the rate of change of
W
dp
along the
range and it is then given by:
dDmm h
1
D
3
N
R
¼
0
:
0006
p
ð
D
Þ
ð
D
Þ
ð
7
:
6
Þ
where D is the raindrop diameter and v(D) is the
terminal velocity of the raindrops. Atlas and
Ulbrich (1977) expressed the terminal velocity as
a function of the particle diameter, given by
v
ð
D
Þ¼
3
:
78D
0
:
67
ms
1
, assuming the absence of
vertical air motions. Thus, R represents the 3.67th
moment of the DSD whereas Z represents the 6th
moment (see Equation 7.2), with Z being more
sensitive to large drops than R. Knowledge of the
DSD is important because it establishes the inter-
action between the radar reflectivity and the rain-
fall rate. Marshall and Palmer (1948) observed an
exponential DSD, and the exponential form of the
DSDcan always be appliedwhen a large number of
DSD are averaged in space or time (Bringi and
Chandrasekar 2001). The Marshall and Palmer
DSD can be expressed as:
1
2
d
W
dp
dr
km
1
K
dp
¼
ð
7
:
4
Þ
It can also be expressed as (Bringi and Chandra-
sekar 2001):
10
3
CW
180
l
K
dp
¼
ð
0
:
062D
m
Þ
with
Ð
D
4
N
ð
7
:
5
Þ
ð
D
Þ
dD
Ð
D
3
N
D
m
¼
N
ð
D
Þ¼
N
0
exp
ð
3
:
67D
=
D
0
Þ
ð
7
:
7
Þ
ð
D
Þ
dD
where N
0
is 8000m
3
mm
1
and D
0
is the median
drop diameter, which is defined as the drop
diameter such that 50% of the water content
comprises drops with diameters less than D
0
(Doviak and Zrnic 1993). According to Marshall
and Palmer (1948) the exponential DSD slightly
overestimates for raindrop diameters less than
1.5mm in diameter. However, in reality there are
larger variations in the shape of the DSD not
represented by the Marshall and Palmer DSD, and
Ulbrich (1983) proposed a more general three-
parameter gamma DSD given by:
3.75,Wis
the water content in grams per cubic metre, and
D
m
is the mass-weighted mean diameter in milli-
metres. Equation 7.5 is important because K
dp
is
almost linearly related to the liquid water content
(LWC) multiplied by the mean raindrop shape and
therefore it provides the possibility of better esti-
mates of the actual rainfall rate.
where l is thewavelength inmetres, C
Estimation of Precipitation Using Weather
Radar
N
0
D
m
exp
ð
3
:
67
þ
m
Þ
D
Raindrops grow to a critical size and then suffer
break-up due to hydrodynamic instability (Cotton
N
ð
D
Þ¼
ð
7
:
8
Þ
D
0
