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Here is some more food for thought:

We take 3 bearings on fixed objects.

Assume that the error in taking a bearing follows a Gaussian
(Normal) distribution with mean 0 and standard deviation 1
degree.  These numbers seem to me to be not unreasonable.

The probability that any one bearing is within +/- 1 degree of
the true bearing is 0.6826.

The probability that all 3 bearings are within +/- 1 degree of
the true bearing is (0.6826)^3 = 0.32.

The probability that any one bearing is within +/- 2 degrees of
the true bearing is 0.9544.

The probability that all 3 bearings are within +/- 2 degrees of
the true bearing is (0.9544)^3 = 0.87.

The probability that any one bearing is within +/- 3 degrees of
the true bearing is 0.9974.

The probability that all 3 bearings are within +/- 3 degrees of
the true bearing is (0.9974)^3 = 0.99.


Using this information, we can define confidence regions of a sort.

Converting 1 degree of angle to distance depends on distance from
the landmark.  Here is a rough table, computed by multiplying the
tangent of 1 degree by the distance to the landmark:

   Distance         Width of
   to landmark      1 degree

   1 nm             0.017 nm
   2                0.035
   3                0.052
   4                0.070
   5                0.087
   6                0.105
   7                0.122
   8                0.140
   9                0.157
  10                0.175

Interpreting these numbers for a particular instance, if all 3
landmarks are at a distance of just under 6 miles, then we can
say that each line of position is within +/- 0.1 nm (about 200
yards) of the true line of position with probability 0.32.
Similarly, we can say that each line of position is within +/-
0.2 nm (about 400 yards) of true position with probability 0.87,
and within +/- 0.3 nm (about 600 yards) with probability 0.99.

Note that I have said nothing about whether a particular cocked
hat covers the true position.  Still, a feel for how far off a
particular bearing line might be can be useful information to a
navigator.

Chuck Taylor
Everett, WA, USA

   

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