Geoscience Reference
In-Depth Information
So,
x
and
y
coordinates of the intersection point is:
R
2
−
R
1
x
=
4
x
0
R
2
−
R
1
4
x
0
2
y
0
+
R
cir
+
=
y
.
(33)
2
y
0
Using this two values and the equation (31), the third coordinates is also calculated:
⎛
⎝
⎞
⎠
R
2
−
R
1
4
x
0
2
2
R
2
−
2
y
0
+
R
cir
+
R
1
R
3
−
=
±
−
−
z
y
0
.
(34)
4
x
0
2
y
0
As mentioned, this intersection contains only two points which are located in the two sides of
the plane
xy
and in front of each other. But in reality, only one of these points has a positive
height and coincides with the coordinate of a target in sky.
After this proof, we continue our discussion about the tracking. On the one hand, the target
position is achievable by using
R
i
s, and on the other hand, the equations (25) and (26) inform
about the relation between
R
i
s and
T
i
s. Therefore, the target position can be determined if
T
i
is known. For calculating this parameter, it should be considered as the signal's time delay
to reach to the
i
-th receiver. Let's assume the first sensor in the section (IV), is the transmitter
now, and the second sensor in there is one of the three receivers in here. By using the proposed
moment method three times, the time delay can be estimated for all the three receivers.
T
i
is
denoted as the estimated time delay for
i
-th receiver. Now, all unknowns are obtained, so the
position is easily predicted.
Finally, the target velocity should be obtained.
The receivers compute three values for
Doppler, ˆ
ε
i
, by the proposed moment technique. Since the transmitter and the first receiver
are at the same sensor, the velocity component along the connecting line between the target
and the first sensor is:
d
dt
R
1
=
C
2
f
t
υ
1
=
ˆ
ε
1
,
(35)
where
R
1
is the vector connecting the first sensor to the
target.
C
is the speed of light and
f
t
is the frequency of the emitted signal.
.
represents Euclidean norm, and
υ
1
,we
determine the velocity components along the connecting line between the target and two other
sensors (receivers):
Using
C
f
t
υ
i
=
ε
i
ˆ
−
υ
1
,
i
=
2, 3.
(36)
In the next section, there are results that compare the different methods available for
estimating the time delay and Doppler. There are also some results about tracking a target
which has a nonlinear motion. In the parameter estimation results, the proposed moment
method is compared with the methods Wigner-Ville (WV), fractional lower order ambiguity
function (FLOAF) and wavelet, and in the tracking part, there is a comparison between the
proposed method and EKF and PF ones.
