A Measurement Correlation Algorithm for
Line-of-Bearing Geo-Location
Mike Grabbe
The Johns Hopkins University Applied Physics Laboratory
Laurel, MD
Memo Number GVW-0-11U-002
July 26, 2011
1
Introduction
Passive geo-location of ground targets is commonly performed by surveillance aircraft using Direction Finding (DF) angles. These angles de…ne the line-of-sight from the aircraft to the target and
are computed using the response of an antenna array on the aircraft to the target’s RF emissions.
Aircraft that depend entirely upon DF angles for geo-location will often convert each DF angle
measurement to a Direction of Arrival (DOA) angle measurement and use these values for geolocation. DOA is the angle equivalent to azimuth or Angle of Arrival (AOA) when de…ned relative
to a local-level coordinate frame at the current aircraft position. DOA is computed using azimuth
or AOA, an estimate of the elevation angle to the target, the antenna array mounting angles on
the aircraft, and aircraft navigation system output. Associated with each angle measurement is
a Line-of-Bearing (LOB) that originates at the aircraft and, if perfect, passes through the target’s
position. See reference [1] for additional DF angle details.
Most geo-location scenarios faced by surveillance aircraft involve the existence of multiple target
signal sources within the area of operation. As a result, each new LOB/DOA measurement must
be correlated with a speci…c target emitter before that target’s position estimate can be updated
by the geo-location algorithm. Geo-location performance is typically degraded in a dense emitter
environment due to the di¢ culty of correlating each LOB with the correct target, and of preventing
the generation of numerous false or "ghost" targets.
LOB correlation for stationary targets is addressed in references [2]-[6]. In [5], clusters are formed
by solving a combinatorial assignment problem with the number of sensor positions limited to 3
due to computational complexity. Once clusters are formed, the optimal cluster is de…ned to be
the one that maximizes a likelihood function. In [6], clusters are formed using Euclidean distance
between LOB intersection points as the metric for measurement association. The optimal cluster
is then de…ned simply as the one having the most measurements. References [7]-[9] address LOB
correlation for moving targets.
1
Distribution Statement A: Approved for Public Release; Distribution is Unlimited
This paper presents an algorithm that can be used to solve the LOB correlation problem for stationary targets and uses a geo-location simulation example to illustrate each step of the algorithm.
The algorithm is based on statistical clustering of measurements but does not require that the
number of clusters be speci…ed in advance, as with methods such as k-means clustering [10]. This
algorithm determines the cluster of LOBs that maximizes the target position log-likelihood function
when compared to all candidate clusters. The candidate clusters are those that pass a Mahalanobis
distance association criterion - an exhaustive search over all possible measurement combinations is
not performed. Once the optimal cluster of LOBs 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.
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 will be processed to generate measurements of DOA, which in turn
will be used to locate the targets. The values of the scenario parameters are given in the table
below.
Scenario Parameter
aircraft speed
collection time
approximate range to targets
number of targets
number of DOA measurements
DOA measurement error sigma
3
3.1
Units
knots
minutes
nautical miles
n/a
n/a
degrees
Value
300
10
20
7
100
1:5
Geometry
De…nitions
We will construct a …xed plane tangent to the earth’s surface within the area of operation. The
coordinate frame within this plane will have its origin at the point where the plane intersects the
earth’s surface and will be oriented such that the x axis points east and the y axis points north.
The aircraft positions will be represented in this coordinate frame by a gnomonic projection into
the tangent plane. The DF angle measurements will be converted to DOA measurements in the
local-level coordinate frame at each aircraft position, then converted to DOA values in the …xed
tangent plane. Let ai be the aircraft’s 2 1 position vector in the plane for i = 1; 2; : : : ; N ,
where N is the number of DOA measurements. We will assume that there is exactly one DOA
measurement at each aircraft position. Let i be the true DOA value at aircraft position ai for
a signal transmitted from the true target position xi . The components of the aircraft and target
position vectors are given by
a1i
ai =
(1)
a2i
and
x1i
x2i
xi =
2
(2)
DOA is de…ned as shown in Figure 1 below.
Figure 1: Direction of Arrival in Tangent Plane
3.2
Simulation Example
The N = 100 aircraft positions and 7 true but unknown target positions are shown in Figure 2
below.
aircraft
targets
30
25
↑
20
north (nm)
15
10
5
0
-5
-10
-20
-10
0
10
20
east (nm)
Figure 2: Aircraft and Targets in Tangent Plane
3
30
4
Sensor Measurements
4.1
De…nitions
Let hi be the function that gives the true DOA value in terms of the true target position. Then
from (1), (2), and Figure 1 we see that
i
= hi (xi ) = tan
1
x2i
x1i
a2i
a1i
(3)
Let zi be a measurement of i . We will assume that the measurement errors are unbiased and
Gaussian with known variances. As a result, we have
zi =
i
+
i
= hi (xi ) +
i
N 0;
2
i
(5)
where N represents a Gaussian distribution, i is the measurement error, and
measurement error sigma value. From the above we have
zi
N hi (xi ) ;
and therefore that
zi
hi (xi )
(4)
i
2
i
i
is the known
(6)
2
2
(1)
(7)
i
where
4.2
2 (1)
represents a chi-square distribution with 1 degree of freedom.
Simulation Example
At each of the 100 aircraft positions, a random draw was made to determine which of the 7 targets
was transmitting a signal. For that aircraft position and target, the true DOA value was computed
using (3). The measured DOA value was computed as in (4) and (5) using a random draw with
= 1:5 degrees.
4
The resulting DOA measurements are shown by the 100 LOBs in Figure 3 below.
aircraft
targets
30
25
↑
20
north (nm)
15
10
5
0
-5
-10
-20
-10
0
10
20
30
east (nm)
Figure 3: Lines-of-Bearing
5
5.1
Step 1
Algorithm
The …rst step of the LOB correlation algorithm is to compute all 2-LOB intersection points within
speci…ed distances from the aircraft. Each 2-LOB intersection is computed as follows. From
Figure 1 we see that the unit vector along the measured LOB from aircraft position ai is
ui =
cos (zi )
sin (zi )
(8)
The LOBs from aircraft positions ai and aj will intersect at the point p such that
p = ai + ri ui = aj + rj uj
(9)
where ri and rj are the ranges from the aircraft positions to the intersection point. From the above
we have
ri
ui
uj
= (aj ai )
(10)
rj
and therefore that
ri
rj
=
ui
uj
1
(aj
ai )
(11)
If either of the computed ranges falls outside of the speci…ed minimum and maximum allowed
values, then this intersection point is discarded. The intersection point is computed using either
of the computed ranges and (9). Note that a negative range indicates that the 2 LOBs intersect
on the wrong side of the aircraft. We will let M be the number of computed intersection points.
5
5.2
Simulation Example
The 2-LOB intersection points were required to lie between 10 and 50 nautical miles from the
aircraft positions. The resulting M = 2651 2-LOB intersection points are shown in Figure 4
below.
↑
20
15
10
north (nm)
5
0
-5
-10
-15
-20
-25
aircraft
targets
2-LOB intersections
-30
-30
-20
-10
0
10
20
30
40
east (nm)
Figure 4: 2-LOB Intersection Points
6
6.1
Step 2
Algorithm
The second step of the algorithm is to determine the set of LOBs that can be associated with each
2-LOB intersection point based on Mahalanobis distance [11]. The Mahalanobis distance between
DOA measurement zi and 2-LOB intersection point pj is
mij =
zi
hi pj
i
!2
(12)
for i = 1; 2; : : : ; N and j = 1; 2; : : : ; M . Note from (7) that if pj is the true target position associated
with zi , then mij has a 2 (1) distribution. This fact is the basis for the LOB association criterion.
Our null hypothesis is that pj is the target position associated with zi . 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 (1) distribution such that
Pr
2
(1)
k =1
6
(13)
The association criterion is that LOB/DOA measurement zi can be associated with pj if mij
The set of indices of LOBs that can be associated with pj is
Sj = fi : mij
kg
k.
(14)
Several facts about (14) are worth noting. One is that each LOB can typically be associated
with numerous 2-LOB intersection points. Another is that the elements in each set Sj are not
necessarily unique: Sj1 and Sj2 may contain the same elements for j1 6= j2 . Finally, the number
of elements in Sj , jSj j, is at least 2 for each j since 2 LOBs were used to compute pj . We will
require that jSj j 3 in order for Sj to be a candidate cluster for LOB correlation.
6.2
Simulation Example
The value of used was 0:05, which means that k used in (13) and (14) is the 95% critical value
from a 2 (1) distribution. Of the 2651 2-LOB intersection points, 2644 produced a candidate
cluster for LOB correlation.
7
7.1
Step 3
Algorithm
The third step of the algorithm is to determine which of the candidate clusters gives the largest
log-likelihood function value when the LOBs in that cluster are combined to compute a target
position estimate. Let Sj be a candidate cluster and let z be the vector containing all zi such that
i 2 Sj . We will assume that all such zi are associated with the same true target position x. Using
(6) gives that the conditional probability density function for the measurements given the target
position x is
!
Y
Y
1
1
zi hi (x) 2
p exp
f (zi j x) =
f (z j x) =
(15)
2
i
i 2
i2Sj
i2Sj
The Maximum Likelihood Estimate of x is
x
^ = arg max f (z j x) = arg max ln (f (z j x)) = arg min
x
x
x
X
i2Sj
zi
hi (x)
2
(16)
i
This value can be found numerically using Iterated Least-Squares [1]. The log-likelihood function
value associated with candidate cluster Sj is
lj = ln (f (z j x
^))
(17)
The cluster giving the largest log-likelihood function value is Sjmax , where
jmax = arg max lj
j
7
(18)
7.2
Simulation Example
Of the 2644 candidate clusters, 15 achieved the maximum log-likelihood function value. These 15
clusters all contained the same 20 LOBs. The optimal LOB cluster is shown in Figure 5 below.
30
optimal cluster
25
↑
20
north (nm)
15
10
5
0
-5
-10
-20
-10
0
10
20
30
east (nm)
Figure 5: Optimal LOB Cluster
8
8.1
Step 4
Algorithm
The fourth step of the algorithm is to compute a target position estimate (16) using the optimal
LOB cluster, then remove the LOBs 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 (17).
8
8.2
Simulation Example
The target position estimate for the optimal cluster and the 95% con…dence error ellipse are shown
in Figure 6. The 20 LOBs in the optimal cluster have been removed.
30
target position estimate
25
↑
20
north (nm)
15
10
5
0
-5
-10
-20
-10
0
10
20
30
east (nm)
Figure 6: Target Position Estimate and Error Ellipse for Optimal LOB Cluster
9
9.1
Remaining Steps
Algorithm
The remaining steps of the algorithm involve repeating steps 1-4 above until no additional LOB
clusters are formed.
9.2
Simulation Example
The …nal target position estimates and 95% con…dence error ellipses are shown in Figures 7 and 8.
Note that 1 LOB was not assigned to a cluster and 1 error ellipse did not contain the true target
position.
9
30
target position estimate
25
↑
20
north (nm)
15
10
5
0
-5
-10
-20
-10
0
10
20
30
east (nm)
Figure 7: Final LOB Correlation and Geo-Location Results
targets
estimates
10
north (nm)
5
0
-5
-10
-10
-5
0
5
east (nm)
10
15
20
Figure 8: Final Target Position Estimates and Error Ellipses
10
10
Summary
The purpose of this paper was to present an algorithm that can be used to solve the LOB correlation problem for stationary targets. LOB clusters were formed using a Mahalanobis distance
association criterion. This approach accounts for angle measurement error statistics and avoids the
computational complexity of an exhaustive combinatorial assignment. Once clusters were formed,
the optimal cluster was de…ned to be the one that maximized the target position log-likelihood
function. This cluster was used to compute a target position estimate then removed from the set
of measurements. The process was repeated until no additional clusters could be formed. The
algorithm was shown to provide good correlation and geo-location performance in a scenario where
100 LOBs were distributed randomly across 7 targets.
11
References
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[3] Hamschin, B.M., "A Comparison of Two Data Association Algorithms", L-3 Communications
Integrated Systems Report Number G3074.1205.01E, October 2007.
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Communications Integrated Systems, January 2008.
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[7] Beard, M. and S. Arulampalam, "Comparison of Data Association Algorithms for Bearingsonly Multi-sensor Multi-target Tracking", 2007 10th International Conference on Information
Fusion, pp. 1-7, July 2007.
[8] Paradowski, L.R., "Qualitative and Quantitative Characteristics of Deghosting in Advanced
Radar Systems", 2002 6th IEEE AFRICON, Vol. 1, pp. 19-24, October 2002.
[9] Schwartz, D., “Progress on Angle-Only Tracking and Deghosting”, Johns Hopkins APL Memo
GVE-11-0028, April 2011.
[10] Duda, R.O., P.E. Hart, and D.G. Stork, Pattern Classi…cation 2 nd Ed., John Wiley & Sons,
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[11] Bar-Shalom, Y., P.K. Willett, and X. Tian, Tracking and Data Fusion, A Handbook of Algorithms, YBS Publishing, 2011.
11