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I hardly ever find anything to disagree with in Gary LaPook's
contributions. They are usually full of sense.

That applies, too to most of his 6th March posting, Navlist 2236,
labelled "resolution of systematic error". Though actually, what he
discusses there is the situation of entirely random error, where all
systematic error has been corrected out.

But Gary concludes with this-

"Again, no one
is suggesting that the position of the observer is at the center of
the triangle but this represents the center of possible positions of
the observer.  In fact, the position of the observer will be outside
of the triangle often but I don't agree with the three out of four
allegation. Counter intuitively, the smaller the triangle the more
likely that the position of the observer is outside the triangle! If
you think about it, this should be obvious. Using   reducio ad
absurdum, think about a triangle only one inch in size, it would be
impossible for the observer to be within the triangle. At the other
extreme, a very large triangle with all of the displacements of the
LOPs from the center of the triangle equal to 3.3 NM (3.3 sigma's,
linear sigma's are slightly different than circular sigma's, see
Bowditch), the only place that the position of the observer could be
is at the fix in the center of the triangle!"

Here, Gary is wrong. The statement, that "with entirely random errors,
three times out of four the triangle will not embrace the position of
the observer", is precisely correct, and based on irrefutable
statistical arguments. If Gary doesn't accept this, unlikely though it
may seem to him at first sight, he should scan through previous
discussion of cocked hats on this list, as Geoffrey Kolbe suggested.
If he remains unconvinced, we can go through the arguments once again,
until one way or another, either he accepts it, or else he convinces
us otherwise.

The best way to consider the matter is not to take a particular
triangle, and then consider "where can the true position be?", but to
take a true, known position of an observer, and three celestial
bodies. Now plot in his vicinity a set of three position lines that
have been displaced by Gaussian amounts, toward or away from the GPs
of the bodies in question. (Indeed, the argument doesn't rely on a
Gaussian distribution, just equal numbers toward and away). And then,
with the same observer position, and the same three bodies, plot
another triangle, with different displacements, and another, until you
are tired of it. The resulting triangles will be different each time;
some will be large, some will be tiny, some long and thin, others
nearly equilateral. And in the end, if you check enough triangles, you
find that 25% of those triangles will embrace the true position that
you started with.

It doesn't matter how skilled or unskilled the observer is; it remains
true. For a skilled navigator, of course, the triangles will indeed be
smaller on average, which is where his skill shows itself, but still,
only 25% of those smaller triangles will include the true position.

George.

contact George Huxtable at george@huxtable.u-net.com
or at +44 1865 820222 (from UK, 01865 820222)
or at 1 Sandy Lane, Southmoor, Abingdon, Oxon OX13 5HX, UK.


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