Geoscience Reference
In-Depth Information
where
w
denotes a set of weights for summation as expressed by Equation (2) and
R
is a
covariance matrix as expressed by Equation (4).
w
is constrained by the condition of
constant gain to waves coming from the target range, and the constraint is given by
H
ew
,
1
(13)
where
e
is a range steering vector and given by
jkr
(2
)
jkr
(2
)
jkr
(2
)
T
e
(
e
,
e
,
,
e
)
,
(14)
1
I
1
2
I
2
N
I
N
where
k
denotes the wavenumber of
m
th frequency,
r
represents the range between the
target and radar, and
m
is the initial phase of
m
th frequency.
Because
m
is determined not only by the total system delay throughout the transmitter and
receiver chains but also by
k
, the values of
1
, … ,
N
are different. Therefore the total system
delay, from which the values of
1
, … ,
N
are computed, needs to be known in order to
determine
e
correctly. Chilson et al. (2004) and Palmer et al. (2001) measured the total system
delay by leaking the transmitted signal back to the receiver through an ultrasonic delay line. In
a practical manner, only the relative phase differences among the frequencies are necessary to
correct the effects of system delay. Therefore, the correction is able to be attained by calculating
the phase term of cross correlation between the two time series of received signals measured at
different frequencies (Chen, 2004). Measurement results of phase correction using the clear-air
echoes are shown by Chen et al. (2009, 2010) and Chen & Zecha (2009).
The optimal weight
r
w
is given by
1
H1
Re
w
eRe
.
(15)
rC
Using Equations (11) and (15), scalar output of the filter
y t
is given by
()
H
y t
()
r
ws
.
()
t
(16)
By calculating Doppler spectrum of
y t
, brightness
r
B
(i.e., power density) and other
spectral parameters are able to be computed with improved range resolution.
()
B
is given by
rC
1
B
eRe
.
(17)
rC
H1
Brightness at arbitrary Doppler velocity is also able to be calculated in the same manner as
described in section 2.1.1.
Because of the limited width of transmitted pulse (i.e., wave form of transmitted pulse),
received signal power within the range gate has range dependency (i.e., the received signal
power decreases near the edge of range gate). Chen and Zecha (2009) proposed a practical
method in order to correct the range weighting, and the correction results are also presented
by Chen et al. (2009, 2010) and Chen & Zecha (2009).
