Two-Dimensional Position Determination
Consider the case of a mariner at sea determining his or her vessel’s position from a
foghorn. (This introductory example was originally presented in [1] and is contained
herein because it provides an excellent overview of TOA position determination
concepts.) Assume that the vessel is equipped with an accurate clock and the
mariner has an approximate knowledge of the vessel’s position. Also, assume that
the foghorn whistle is sounded precisely on the minute mark and that the vessel’s
clock is synchronized to the foghorn clock. The mariner notes the elapsed time from
the minute mark until the foghorn whistle is heard. The foghorn whistle propagation
time is the time it took for the foghorn whistle to leave the foghorn and travel to
the mariner’s ear. This propagation time multiplied by the speed of sound (approximately
335 m/s) is the distance from the foghorn to the mariner. If the foghorn signal
took 5 seconds to reach the mariner’s ear, then the distance to the foghorn is
1,675m. Let this distance be denoted as R1. Thus, with only one measurement, the
mariner knows that the vessel is somewhere on a circle with radius R1 centered
about the foghorn, which is denoted as Foghorn 1 in Figure 2.1.

Hypothetically, if the mariner simultaneously measured the range from a second
foghorn in the same way, the vessel would be at range R1 from Foghorn 1 and range
R2 from Foghorn 2, as shown in Figure 2.2. It is assumed that the foghorn transmissions
are synchronized to a common time base and the mariner has knowledge of
both foghorn whistle transmission times. Therefore, the vessel relative to the foghorns
is at one of the intersections of the range circles. Since it was assumed that the
mariner has approximate knowledge of the vessel’s position, the unlikely fix can be
discarded. Resolving the ambiguity can also be achieved by making a range measurement
to a third foghorn, as shown in Figure 2.3.

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