V. describe the simulations that were performed as basis


In this chapter we describe the simulations that were performed as
basis for the geolocation accuracy analyses. The results are presented in
Chapter VII. Software for the simulations is written in Matlab and is given in Appendix E.

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Before performing the simulations we should know the approximate
range of the sensors. Radar waves do not travel along perfectly straight lines,
but curve somewhat along the Earth surface. As a result, it is possible to see
radars at some distance behind the visual horizon. For a sensor at

50 m height above the sea surface, the radar horizon is at
approximately 30 km distance. If the emitter is at the same height as the
sensor, this increases to 60 km distance, see Figure 3.1. If conditions are favorable,
the radar signal may be subject to ‘ducting’, in which case the signal is
trapped close to the surface and may travel long distances following the
Earth’s curvature. When this happens, the sensor range increases compared to
what is shown here.

Figure 3.1   Sensor coverage.
The red and green circles mark the radar horizon for each of the sensors.


We study TDOA localization method. We therefore define a TDOA (or
angle of arrival) measurement to be the average value for the TDOA (or angle of
arrival) measured over the measurement period. The measurements are associated
with corresponding variances. We have assumed that the TDOA variance is (50
ns)2, the aperture angle variance is (1º)2, and the angle of arrival variance
is (2º)2, see Table 3.1. An error in the TDOA of 50 ns corresponds to an error
in the measured pulse time of arrival (TOA) of about 35 ns (on each of the two
sensors), which can be obtained with a very precise clock such as for instance
a rubidium clock.


The method (CRLB) for determining the geolocation accuracy does not
distinguish between a scenario where all measurements are made simultaneously
by many sensors or a scenario where the measurements are made one after the
other by one (or a pair of) sensor(s) over a finite time period. Hence, neither
observation time nor number of sensors (only number of sensor positions) are
directly relevant for the calculations. However, in order to keep the
simulation results realistic for a maritime scenario, we will make some
assumptions both about the number of sensors used, the observation time and
other relevant parameters.


We have set the maximum acceptable observation time to be 15 min,
and the maximum vessel (sensor) speed to be 8 m/s (16 knots). The maximum
distance a sensor can move during the observation time is then 7.2 km. The maximum
distance per measurement period is 20 m and the maximum number of measurement
periods is 360. Values for all key parameters are listed in

Table 3.1.


Radar rotation period   2.5
s  (24 rpm)

Measurement period    2.5 s

Maximum observation time     15

Maximum number of measuremement periods            360

Maximum vessel (sensor) speed          8
m/s (16 knots)

Maximum sensor movement during the measurement period  20 m

Maximum sensor movement during the observation time        7.2 km

Variance, TDOA          (50

Variance, aperture angle          (1º)2

Variance, angle of arrival        (2º)2



Table 3.1     Key parameters
for the simulations.


We have investigated the following methods or combination of


1.   TDOA


The TDOA method require the use of two sensors.


The sensors can be placed at different distances from each other and
may be stationary or move with respect to the emitter. For AOA (single sensor),
scanphase or TDOA the sensors must always move some distance in order to obtain
a geolocation for the emitter. A single measurement period is then sufficient
to make a geolocation for the emitter, i.e., momentaneous geolocation can be
performed. We assume that the emitter does not move during the observation


Each calculation is done for 1 emitter position (stationary
emitter), m sensor (pair) positions, and

N measurements with corresponding variances. The sensor positions
can be the same for all m (stationary sensors) or vary for different m (moving
sensors). The number of measurement periods equals the number of sensor
positions m , while the number of measurements is equal to or larger than the
number of sensor positions, i.e., N ? m , depending on which method or combination
of methods are used.


Input and output from the simulations are:



•   Emitter position, ( xe , ye )

•   Sensor 1 position(s), ( x1
, y1 )

•   Sensor 2 position(s), ( x2
, y2 )

•   Variance ? 2  for each measured parameter (TDOA, aperture
angle or angle of arrival)




•   CEP-radius at the emitter

•   Major semi-axis of the
CRLB error ellipse at the emitter position

•   Minor semi-axis of the
CRLB error ellipse at the emitter position