Geoscience Reference
In-Depth Information
In this case, the relationship between the measurements and the state is of the form
h
=
f
(
x
)
(where
h
is the vector of measurements,
x
is the target state and
f
(
.
)
is the function
relating the two).
Similarly, the relationship between the future state and the current state
is of the form
x
is the function
that predicts the future state). To handle these non-linearities, the EKF linearizes the two
non-linear equations using the first term of the Taylor series and then treats the problem as
the standard linear Kalman filter problem. Although conceptually simple, the filter can easily
diverge (i.e. gradually perform more and more badly) if the state estimate about which the
equations are linearized is poor. The unscented Kalman filter and particle filters are attempts
to overcome the problem of linearizing the equations.
(
t
+
1
)=
g
(
x
(
t
))
(where
x
(
t
)
is the state at time
t
and
g
(
.
)
5.3 Particle Filtering (PF)
Another example of nonlinear methods is particle filtering. This method makes no
assumptions about the distributions of the errors in the filter and neither does it require
the equations to be linear. Instead it generates a large number of random potential states
("particles") and then propagates this "cloud of particles" through the equations, resulting in a
different distribution of particles at the output. The resulting distribution of particles can then
be used to calculate a mean or variance, or whatever other statistical measure is required. The
resulting statistics are used to generate the random sample of particles for the next iteration.
However, this method also has some problems that restrict the use. This method requires
large computational operations and face severe difficulties for real-time applications. On the
other hand, this method is also not able to have suitable results in very low SNRs. In these
SNRs, PF is not able to bring us to a reasonable particle, and even using Sampling Importance
Re-sampling (SIR) method can not lead us to better results [Ristic et al., 2004]. In SIR method,
a weighted set of particles is used. These new weighted particles can face and eliminate the
noise more powerfully and present better estimation in low SNRs.
5.4 The proposed moment method
In this section, we are going to solve the problems we are faced in PF. This is done based on
the time delay and Doppler estimated in the previous section. Three sensors are used. They
are located on the vertices of an equilateral triangle. One of the sensors is a transmitter and
receiver, the other two sensors only serve as the receiver. The arrangement of the sensors and
their positions relative to the target is depicted in figure (2). The target is in the far field of the
sensors.
A signal is emitted from the first sensor to the target. When this signal comes into contact
with the target, generally speaking, it is scattered in many directions. The signal is thus
partly reflected back, hence, all three sensors receive this reflected signal. According to the
earlier discussions, the time delay and Doppler of the received signal in each sensor could be
estimated.
First, the target position is determined. Suppose the time interval between sending the signal
from the transmitter and receiving it in each sensor is shown by
T
i
for i
=
1, 2, 3, which
i
denotes the sensor number. We also use
R
i
as the distance between the target and the
i
-th
receiver. Since the transmitter is beside the first receiver, we have:
1
2
T
1
×
R
1
=
C
e
,
(25)
