Environmental Engineering Reference
In-Depth Information
15.2.2 Statistical Rain Field Model
The rainfall model used is a mixed distribution model, consisting of a no rain
probability of (1
p
) at zero rain rate and a lognormal distribution for the rainy part
(rain rate
>
0 mm/day) as follows (Kedem et al.
1990
):
GðxÞ¼ð
1
pÞHðxÞþpFðxÞ
(15.2)
where
p
is the rain probability (or frequency),
x
is rain rate, and
H
(
x
) is the
Heaviside step function
0
;
if
x<
0
HðxÞ¼
1
;
if
x
0
And
F
(
x
) is a Lognormal Distribution with parameters of
μ
and
σ
,
"
#
d
x
x
2
1
exp
ð
ln
x μÞ
FðxÞ
d
x ¼
p
2
(15.3)
2
σ
2
σ
π
The expected value of the mean of the mixed distribution is
2
EðxÞ¼p
exp
ðμ þ σ
=
2
Þ
(15.4)
Other statistical models have also been used to describe the rainy portion of the
distribution (Kedem and Chiu
1987a
; Kedem et al.
1990
). The lognormal
distributions have often been used to describe geophysical parameters which
show skew distributions. Based on a simple model, Kedem and Chiu (
1987a
)
argued that the lognormal distribution is not unreasonable for rain rate distributions.
15.2.3 Beamfilling Correction
One of the disadvantages of the use of passive remote sensors is the coarse
resolution of the sensor field of view (FOV) compared to the spatial scale of rain
clouds. The beamfilling error refers to a bias associated with the nonuniformly filled
FOV coupled with a nonlinear relation between the observed and the estimated
parameter, i.e.,
T-R
relation (Eq.
15.1
) (Short and North
1990
). Chiu et al. (
1990
)
examined radar rainfall observed at the GATE and derived an approximate formula
for the beamfilling correction (BFC). The beamfilling bias depends on nonlinearity
of
T-R
relation and rain rate variance within field of view
h
i
T
00
2
T
0
ðR ½RÞ
2
R
E
½R¼
(15.5)
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