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Fred Hebard wrote-

>With two LOPs, there are no degrees of freedom left to estimate the
>accuracy of the fix.  With three, if the "cocked hat" is reasonably
>small, you have some confidence; if large, you know it's not so good.
>I expect a least squares solution also could not estimate errors from
>two LOPs.

I don't disagree with what Fred wrote, but wish to point out that care is
needed in interpreting a single "cocked hat".

What is certainly true is that one single LARGE cocked hat immediately
shows that there are errors somewhere in the observations or their
processing: no question about it.

However, a single SMALL cocked hat does not, of itself, prove freedom from
such errors. A small cocked hat can happen by accident, even if there is
considerable random scatter in the observations, so that the result may
appear to be precise, even if it isn't.

Position lines from two bodies of a fix must always cross at some point
(well, two points actually, but let's presume there's no ambiguity about
which is the "right" one). Now draw in the position line from a third body.
This can be better represented as a band rather that a narrow line. If
errors are entirely random ones, that band will be centred on the true
position of the body, and its width will be related to the random scatter
in that third observation. The third position line will be somewhere within
that band.

So, by chance, that third line might happen to pass through, or close to,
the intersection of the first two lines, in which case a tiny cocked hat
would result, and an observer might deduce that he has made a particularly
accurate observation. Even the least-squares fit procedure, described at
the back of the Nautical Almanac, would deduce a small or zero error, and
the AstroNavPC package, based on that procedure, would display on its
screen a tiny error-ellipse. There's nothing else it can do, with those
three observations. But it could all be quite illusory, if the statistical
scatter is in fact high, and if that small cocked-hat came about only by
chance.

Of course, if a SERIES of three-point fixes gives rise to a series of small
cocked hats, then the observer is entitled to draw extra confidence, from
that evidence, that he is navigating precisely. Also if, in a single
observation-set, he takes more than three altitudes (a situation that the
least-squares fit method can handle without difficulty), perhaps many more,
then the resulting error-ellipse (no longer based on a simple cocked-hat
triangle) will show more robustly the true scatter in his observations.

=================

Fred ended- "BTW, where is George?  It's unusual for him not to have posted
lately."

It's nice to know one's been missed, Fred.

George.

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