Geoscience Reference
In-Depth Information
where
P
a
is the power radiated by the antenna,
r
is the distance,
f
is the antenna
pattern function (one way voltage),
η
is the free space impedance (120
π
Ω),
c
speed of light,
f
radar frequency, and
ψ
t
arbitrary phase at the antenna.
U
(
t-r/c
) designates the pulse function
such that it is 1 if its argument is between 0 and
τ
(the pulse width).
(,)
2.2.1 Radar signal and Doppler shift
The effective beam cross section and pulse width define the intrinsic radar resolution
volume but processing by the receiver increases it in range. Scatterers (hydrometeors such
as rain, hail, snow and also insects, birds etc.,) within the resolution volume contribute to
the backscattered electric field which upon reception by the antenna is transformed into a
microwave signal. The signal is converted to an intermediate frequency
f
if
then passed
through anti-alias filter (nominal passband ~ 14 MHz), digitized (as per Table 2), and down
converted to audio frequencies (base band) for further processing.
At intermediate frequency the signal coming from a continuum of scatterers can be
represented as
where the amplitude
A
(
t
) fluctuates due to contribution by
scatterers and
ω
d
is the instantaneous Doppler shift caused by their motions toward (positive
shift) and/or away (negative shift) from the radar. To determine the mean sense of motion
(sign of Doppler shift) the intermediate frequency is removed and the signal is decomposed
into its sinusoidal and cosinusoidal components, the inphase I and quadrature phase Q
parts. These carry information about the number and sizes of scatterers as well as their
motion. Samples of I and Q components are taken at consecutive delays with respect to the
transmitted pulse. The delays are proportional to the range within the cloud from which the
transmitted pulse is reflected. Samples from the same range locations (delays) are combined
to obtain estimates of the spectral moments: reflectivity factor
Z
, mean Doppler velocity
v,
and spectrum width
σ
v
(Doviak & Zrnic, 2006). The Doppler velocity
v
is related to the
frequency shift
f
d
and wavelength λ via the Doppler equation
At
()cos(
t
t
)
if
d
f
d
=2
v/λ
,
(2)
and so is the spectrum width.
Radars display (and store) equivalent reflectivity factor (often denoted with
Z
e
)
which is
computed from the power and other parameters in the radar equation (Doviak & Zrnic
2006) assuming the scatterers have refractive index of liquid water. For small (compared to
wavelength) spherical scatterers,
Z
e
expressed as function of the distribution of sizes
N
(
D
),
equals
D
max
6
Z
()
dD
.
(3)
e
0
2.2.2 Processing path from signals to algorithms
Top left part in Fig 2 illustrates the continuum of returns (either I or Q), after each
transmitted pulse from 1,…to
M
. Thus
M
samples at a fixed range delay (double vertical
line) are operated on in various ways to produce estimates. There are as many estimates
