Global Positioning System Reference
In-Depth Information
The Cocked Hat
The tricorn hat, popular in the eighteenth century, gave its name to the
error triangle that results from incompatible angle readings, when con-
verted into LoPs and drawn on a chart. Let us say that a certain captain
Jean-Luc Picard (a distant relative of the famous surveyor of France) of the
French brig
Enterprise
has made three readings with his sextant and is now
staring at the three corresponding LoPs that he has drawn on his chart of
French Polynesia. Two lines intersect, indicating his ship's location, but the
good captain has decided to double-check and has made a third reading.
Unfortunately, the LoP for this third reading does not intersect at the same
point as the other two LoPs, as shown in figure 7.6a. Because of measure-
ment error, none of the positions of the LoPs is exactly right, and the three
LoPs show three intersection points that define a triangle (the cocked hat).
Which of the intersection points is the right one? Probably none of
them. It was generally assumed in the eighteenth century that the true
position of the ship was somewhere inside the cocked hat, but this is not
necessarily correct, as we will see.
10
It can be shown from elementary
statistical arguments, for example, that the probability of the ship being
inside the cocked hat is 0.25, which means that there is a 75% chance that
the ship is outside. The analysis leading to this conclusion assumes that the
error is purely random and unbiased (i.e., that there is no systematic error,
and there is an equal chance of the ship's true position being on either side
of any LoP). The probabilities of finding the ship in any of the other areas
outside the cocked hat are shown in figure 7.6a. While this argument
shows that the most likely place for the ship is indeed within the cocked
hat, this likelihood is small.
11
Suppose that Captain Picard is aware that he is not very good at using a
sextant and that, in particular, he is aware that he always reads the angle
incorrectly—say, he consistently overestimates the angle by one minute
of arc. This systematic error converts into shifted LoPs, as suggested in
figure 7.6b. He can correct for this known error, and the result is a much
10. The phrase ''knock into a cocked hat'' derives from the corrections applied to
mitigate this error of navigation, though its meaning has shifted since it was first coined.
11. Cocked hats are still a part of navigation today, and the technique is described in
many practical handbooks, such as Bartlett (2009, p. 59), Karl (2004, pp. 131-32), and
Toghill (2003, p. 65). The statistics of cocked hat positioning errors have been extensively
analyzed; you can read about them in Anderson (1997), Daniels and Wishart (1951), and
Hiraiwa (1967).
