A Correlation Algorithm for
Geo-Location Position Measurements
Mike Grabbe
The Johns Hopkins University Applied Physics Laboratory
Laurel, MD
Memo Number FPD-M-11-0073
November 10, 2011
1
Introduction
Most geo-location scenarios faced by surveillance systems involve the existence of multiple targets
within the area of operation. As a result, each new sensor measurement must be correlated with
a speci…c target before any target position estimate can be updated by the geo-location algorithm.
Geo-location performance is typically degraded in a dense target environment due to the di¢ culty of
correlating each measurement with the correct target, and of preventing the generation of numerous
false or "ghost" targets. See reference [1] for more details.
A batch processing algorithm that solves the correlation problem for stationary ground targets
when the measurements are of Direction Finding (DF) angles was given in [2]. This paper presents
a modi…cation of the algorithm that can be used when the measurements are of target position
and uses a geo-location simulation example to illustrate each step. No assumption is made as
to how the position measurements are generated. They could be the output of an active radar,
an imaging system, or a passive geo-location system that processes measurements of DF angles,
Time Di¤erence of Arrival, or Frequency Di¤erence of Arrival. The algorithm determines the
cluster of measurements that maximizes the target position log-likelihood function when compared
to all candidate clusters. The candidate clusters are those that pass a Mahalanobis distance
[1] association criterion - an exhaustive search over all possible measurement combinations is not
performed. Once the optimal cluster of measurements has been determined, a target position
estimate is computed using the cluster, the cluster is removed from the set of all measurements,
and the process is repeated. This continues until no additional clusters can be formed.
1
Distribution Statement A: Approved for Public Release; Distribution is Unlimited
2
Simulation Example
R
Matlab was used to simulate the correlation algorithm developed in this paper for the following
geo-location scenario. An aircraft is receiving RF transmissions from multiple stationary ground
target emitters within its area of operation. The number and positions of the targets are unknown. The received RF transmissions are used to compute target position measurements which
are processed by the correlation algorithm to determine the …nal target position estimates. The
values of the scenario parameters are given in the table below.
Scenario Parameter
number of targets
number of target position measurements
3
3.1
Value
7
100
Sensor Measurements
De…nitions
Let xi be the r 1 true target position vector for i = 1; 2; : : : ; N , where N is the number of position
measurements. Note that the number of unique values of xi will typically be much less than N .
The target position x could be a 2 1 vector giving position in a plane tangent to the earth’s
surface as described in [2], a 2 1 vector containing WGS84 longitude and geodetic latitude [3],
or a 3 1 vector giving Cartesian position relative to the Earth Centered Earth Fixed coordinate
frame. Let z i be a measurement of xi . We will assume that these measurements are independent
and that the measurement errors are unbiased and Gaussian with known covariances. As a result,
we have
z i = xi + i
(1)
i
N (0; Ri )
(2)
where N represents a Gaussian distribution, i is the measurement error, and Ri is the r
measurement error covariance matrix. From the above we have
zi
N (xi ; Ri )
r
(3)
and therefore that
(z i
xi )T Ri 1 (z i
xi )
2
(r)
(4)
where 2 (r) represents a chi-square distribution with r degrees of freedom. The sensor measurements processed by the correlation algorithm are the pairs (z i ; Ri ) for i = 1; 2; : : : ; N .
2
3.2
Simulation Example
For each of the N = 100 target position measurements, a random draw was made to determine
which of the 7 targets was transmitting a signal. For that target, a WGS84 position measurement
was generated by processing simulated azimuth and elevation measurements using the algorithm
described in [4]. The resulting position measurements and 95% con…dence error ellipses are shown
in Figure 1 below.
true position
measured position
36
35.9
latitude (deg)
35.8
35.7
35.6
35.5
35.4
35.3
35.2
-117
-116.8
-116.6
-116.4
longitude (deg)
-116.2
-116
Figure 1: True and Measured Target Positions
4
4.1
Step 1
Algorithm
The clustering algorithm requires that each z be treated both as a position measurement with
associated error covariance matrix R, and as a candidate cluster point. We will let z i represent a
position measurement and z j represent a candidate cluster point, for i; j = 1; 2; : : : ; N . The …rst
step of the algorithm is to determine the set of position measurements that can be associated with
each candidate cluster point based on Mahalanobis distance. The Mahalanobis distance between
measurement z i and candidate cluster point z j is
mij = (z i
z j )T Ri 1 (z i
zj )
(5)
Note from (4) that if z j is the true target position associated with z i , then mij has a 2 (r)
distribution. This fact is the basis for the measurement association criterion. Our null hypothesis
is that z j is the target position associated with z i . Let be the probability of a Type 1 error, i.e.,
the probability of rejecting the null hypothesis when it’s true. Let k be the critical value from a
2 (r) distribution such that
Pr 2 (r) k = 1
(6)
3
The association criterion is that position measurement z i can be associated with candidate cluster
point z j only if mij k. The set of indices of measurements that can be associated with z j is
Sj = fi : mij
kg
(7)
We will require that the number of elements in Sj be at least 3 in order for Sj to be a candidate
cluster for measurement correlation. Two facts about (7) are worth noting. One is that each
position measurement can typically be associated with numerous candidate cluster points. The
other is that the elements in each set Sj are not necessarily unique: Sj1 and Sj2 may contain the
same elements for j1 6= j2 .
4.2
Simulation Example
The value of used was 0:05, which means that k used in (6) and (7) is the 95% critical value
from a 2 (2) distribution. Of the 100 candidate cluster points, 96 produced a candidate cluster
for measurement correlation.
5
5.1
Step 2
Algorithm
The second step of the algorithm is to determine which of the candidate clusters gives the largest
log-likelihood function value when the measurements in that cluster are combined to compute a
target position estimate. Let Sj be a candidate cluster and let y be the stacked vector containing
all z i such that i 2 Sj . We will assume that all such z i are associated with the same true target
position x. Using (3) gives that the conditional probability density function for the measurements
given the target position x is
f (y j x) =
Y
i2Sj
f (z i j x) =
Y
i2Sj
1
r=2
(2 )
jRi j
1=2
1
(z i
2
exp
x)T Ri 1 (z i
(8)
x)
The Maximum Likelihood Estimate of x is de…ned as
x
^ = arg max f (y j x) = arg max ln (f (y j x)) = arg min
x
x
x
X
(z i
x)T Ri 1 (z i
x)
(9)
i2Sj
The value of the estimate is
2
x
^=4
X
i2Sj
3
10
Ri 1 5
@
X
i2Sj
The associated estimation error covariance matrix is
h
P = E (x
x
^) (x
T
x
^)
i
1
Ri 1 z i A
2
=4
X
i2Sj
Ri
(10)
3
15
1
(11)
Because the measurement errors are assumed to be Gaussian, we have that
x
^
N (x; P )
4
(12)
The log-likelihood function value associated with candidate cluster Sj is
lj = ln (f (y j x
^))
(13)
The cluster giving the largest log-likelihood function value is Sjmax , where
jmax = arg max lj
(14)
j
5.2
Simulation Example
Of the 96 candidate clusters, only 1 achieved the maximum log-likelihood function value and contained 22 position measurements. The optimal cluster is shown in Figure 2 below.
optimal cluster
36
35.9
latitude (deg)
35.8
35.7
35.6
35.5
35.4
35.3
35.2
-117
-116.8
-116.6
-116.4
longitude (deg)
-116.2
-116
Figure 2: Optimal Cluster of Position Measurements
6
6.1
Step 3
Algorithm
The third step of the algorithm is to compute the target position estimate (10) and estimation
error covariance matrix (11) using the optimal cluster, then remove the cluster from the set of all
measurements. Note that the position estimate has already been computed as an intermediate
step in determining the log-likelihood function value (13).
5
6.2
Simulation Example
The target position estimate for the optimal cluster and the 95% con…dence error ellipse are shown
in Figure 3. The 22 measurements in the optimal cluster have been removed.
estimated position
36
35.9
latitude (deg)
35.8
35.7
35.6
35.5
35.4
35.3
35.2
-117
-116.8
-116.6
-116.4
longitude (deg)
-116.2
-116
Figure 3: Target Position Estimate and Error Ellipse for Optimal Cluster
7
7.1
Remaining Steps
Algorithm
The remaining steps of the algorithm involve repeating steps 1-3 above until no additional measurement clusters are formed.
7.2
Simulation Example
The …nal target position estimates and 95% con…dence error ellipses are shown in Figures 4 and 5.
Note that 3 position measurements were not assigned to a cluster and that the center error ellipse
did not contain the true target position.
6
estimated position
36
35.9
latitude (deg)
35.8
35.7
35.6
35.5
35.4
35.3
35.2
-117
-116.8
-116.6
-116.4
longitude (deg)
-116.2
-116
Figure 4: Final Measurement Correlation and Geo-Location Results
true position
estimated position
35.9
35.85
35.8
latitude (deg)
35.75
35.7
35.65
35.6
35.55
35.5
35.45
-116.8
-116.7
-116.6
-116.5
longitude (deg)
-116.4
-116.3
-116.2
Figure 5: Final Target Position Estimates and Error Ellipses
7
8
References
[1] Bar-Shalom, Y., P.K. Willett, and X. Tian, Tracking and Data Fusion, A Handbook of Algorithms, YBS Publishing, 2011.
[2] Grabbe, M.T., “A Measurement Correlation Algorithm for Line-of-Bearing Geo-Location”,
Johns Hopkins APL Memo GVW-0-11U-002, July 2011.
[3] NIMA Technical Report TR8350.2, Department of Defense World Geodetic System 1984, Its
De…nition and Relationships With Local Geodetic Systems, 3 rd Ed., January 2000.
[4] Grabbe, M.T. and B.M. Hamschin, “Geo-Location Using Direction Finding Angles”, Johns
Hopkins APL Technical Digest, (in press).
8