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Subject: Again on LOP's
From: Trevor J. Kenchington (Gadus@XXX.XXX)
Date: Sun May 19 2002 - 13:26:20 EDT
Following up on all of the discussion of cocked hats, I read through the
exchange in the Journal of Navigation (the UK one) that was cited here
some weeks back and I also attempted to work my way through Daniels'
1951 paper in the Journal of the Royal Statistical Society (vol.13B, pp.
186-207).
For those looking for practical guidance who do not have access to the
journals, I can't say that I found a whole lot to pass on. As has
already been stated in this thread, the Journal of Navigation discussion
covered much the same ground as we have done. If it is any comfort to
the doubters, all contributors to the earlier debate seem to have taken
it as proven that there is only a 25% probability of the true position
falling within a cocked hat. Williams' argument that the probability of
being off any one side of the cocked hat is always one sixth and that of
being in the sector beyond any one corner is always 1/12 (an argument
repeated here by, I think, George) was disputed by one Ian Cook of the
University of Essex and Williams' attempt to rebut the rebuttal seem
unconvincing to me. But neither of them, nor anyone else, questioned the
25% inside the cocked hat conclusion. (They did argue over whether or
not the 25% was already common knowledge. Clearly, for far too many,
myself included, it was and is not.)
On that point, in a discussion of Daniels' earlier paper, one of his
colleagues who had been involved in operations research during WW2
suggested that the average RAF Squadron Leader was convinced that his
position necessarily lay within any cocked hat he might plot. If so, an
understanding of the 25% probability has only filtered through to
practical navigators in the past half century. Or maybe the RAF's
navigational instruction was way behind that of the RN and the services
of other nations!
Otherwise, Daniels provided a lot of math, most of which is beyond me. I
dare say that it is important to people designing the software for GPS
receivers but I'm not sure that it has any more direct practical
application.
He did provide a graphical method for finding the most probable position
within a cocked hat:
1: Place a ruler across LOPs 2 & 3, aligned so that it make the same
angle with LOP 3 as LOP 1 does with LOP 2.
2: Divide the length of the ruler between LOPs 2 & 3 in proportion to
the variances of those two LOPs.
3: Join the point so identified to the intersection of LOPs 2 & 3.
4: Repeat with either LOPs 1 & 2 or LOPs 1 & 3.
5: Where the two plotted lines cross is the MPP.
He gave an even more complex way of doing the same thing with four LOPs
to the fix. Personally, however, I'd be more concerned about the ellipse
around the cocked hat that I was likely to be inside than I was about
just where exactly in the middle of that the MPP lies. So carefully
plotting the MPP seems a bit of a waste of time. Daniels gave equations
for the ellipse but I'd not fancy trying to use them to plot it onto a
chart.
The one remaining point that I got out of the paper concerned the
formula already quoted on this thread for calculating the probability of
being within the "cocked hat" when there are more than three LOPs. That
probability is 1-n/x where x equals 2 raised to the power (n-1) and n is
the number of LOPs. Perhaps everyone but me already realized this,
however that formula gives the probability of being within, in Daniels'
words, the "largest closed polygon". If there were four LOPs, they will
typically plot out as enclosing an irregular quadrilateral with
triangles projecting from two of its sides and the rest of the surface
of the globe divided into 8 segments by the projection of the LOPs.
Daniels' formula gives the probability of being somewhere in the
combination of the quadrilateral and the two triangles (all of those
being "closed" polygons, as is their combination) versus in the "open"
segments extending away.
One could, in theory, take 10 LOPs, in which case the formula would lead
to a probability of 0.98. That would not mean a 98% probability of being
in the small decagon in the centre of your fix but somewhere in the
complex (likely 20-sided) polygon that includes all areas closed off on
your plot by the intersection of the many LOPs. I guess that is one way
of figuring where you are not, with a high degree of probability, but it
would be a lot of work for limited benefit.
Trevor Kenchington
-- Trevor J. Kenchington PhD Gadus@XXX.XXX Gadus Associates, Office(902) 889-9250 R.R.#1, Musquodoboit Harbour, Fax (902) 889-9251 Nova Scotia B0J 2L0, CANADA Home (902) 889-3555Science Serving the Fisheries http://home.istar.ca/~gadus
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