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I had referred to the section in AstroNavPC, the UK Nautical Amanac office 
publication, page 68 in the 2000-2005 edition, which took the least-squares 
procedure a bit further and showed how the error ellipse was deduced for 
their software.

In passing, I referred to "...a minor error in its text..."

To which Herbert Prinz has responded-

"This rings a bell. Checking my copy of AstroNavPC, from 2001, I find that I 
had penciled "A = C and B = 0" into the margin of p. 68. Is that it? "

That's it, exactly. The same is scribbled in mine. We agree. These are the 
conditions that produce a confidence-circle rather than a 
confidence-ellipse, and not A=B and C=0, as was stated.

There's another error, actually. The text tells us that two quantities, 
sigmaL and sigmaB,  are  the standard deviations of the estimated position 
in longitude and latitude, respectively. Indeed, one is the standard 
deviation of the latitude, in arc-minutes or miles. But the other is the 
standard deviation of the East-West position, in miles, which isn't the same 
(except at the equator) as the SD of longitude. It's what old-salts termed 
the "departure".

Herbert continues-

"But why do you think "the whole concept of deducing an error-ellipse from a 
single cocked-hat is fundamentally flawed" ? The ellipse serves to measure 
the quality of the data set which is at the basis of that single cocked hat. 
Specifically, it does a better job of it than the cocked-hat itself."

Well, it does a very similar job to the cocked hat. The problem is that that 
"measure of the quality of the data", cocked-hat or confidence-ellipse, is 
based on only a single sample. It's relying on the statistics of a single 
event.

This is a problem that affects not just the computer predictions of the 
program supplied with the AstronavPC booklet, but presumably applies to 
other software that uses similar algorithms to reduce multple observations.

To simplify the picture, assume that just three stars are observed, equally 
spaced around the horizon 120º apart, in which case any cocked-hat will be 
an equilateral triangle, and any confidence-ellipse will become a simple 
circle. Assume, also, that any systematic errors have been corrected for and 
nulled out, and only random scatter remains.

When you plot the position lines that result from a round of three star 
sights, any pair of such lines will cross at a point that's somewhere in the 
general vicinity of the true position. When you draw in the third position 
line, it will, in general, miss that point, but will go past somewhere 
close, to one side or the other. How close? It depends on chance, to a large 
extent. Sometimes, just by chance, it will happen to pass indistinguishably 
close to it. In which case, the cocked-hat, and the deduced error-circle, 
will be minutely small. But that doesn't allow you to draw any firm 
conclusions about that particular round of sights, that it was somehow 
"better" than the round of sights you took a few hours before, which 
provided a more reasonable-looking cocked-hat, a few miles across, or a 
confidence-circle a few miles in diameter. It was just a matter of chance.

So, let's say you were attempting a narrow unmarked passage through a reef, 
which relied on that round of star-sights giving a precise position. Would 
you feel emboldened to try it, by that tiny cocked-hat, and 
confidence-circle? Only a gambler would do so. The mariner should be guided, 
not by the confidence-circle resulting from that one round, but by the 
general run of such confidence-circles that his rounds of star-sight have 
been producing recently.

And that leads to what I regard as another flaw in the text on that page of 
the AstronavPC manual. The confidence-ellipse is suggested to be drawn with 
a scale-factor k, of 2.4, to give a confidence level P of 95%.  The text 
then says- "statistical theory shows that the estimated position has a 
probability P of lying within" such a confidence ellipse. Whatever that 
means. As I see it, the confidence ellipse is intended to be plotted around 
that estimated position, so the estimated position will therefore ALWAYS be 
placed at the exact centre of such a confidence ellipse! That is the only 
place in the book I have been able to find any explanation of the meaning of 
the confidence ellipse. The associated computer program simply displays the 
ellipse on the screen, gives its dimensions and states the 95% confidence 
level, without giving any further guidance about what it's supposed to mean.

What would be nice to have, if it was possible, is the probability that such 
a confidence-ellipse will contain the TRUE position, not the estimated 
position. But say a star-round has, just by chance, produced a tiny 
cocked-hat and ellipse, and we have nothing else to go on? Can we then claim 
that this tiny ellipse has a 95% probability of containing the true 
position? We can not! However, we have nothing better, to tell us otherwise, 
so it's the best that can be done with that data, a single star-round. But 
the apparently "scientific" nature of the confidence ellipse lends an 
entirely spurious respectability to its predictions, to which it is 
unentitled.

I am not suggesting that such a confidence-ellipse is valueless - far from 
it! All I'm saying is that it has to be grouped with others before a savvy 
navigator is able to assess its value.

Those warnings apply particularly to a cocked-hat triangle, when only three 
bodies are involved. The more star-sights that are included in the 
assessment, the less likely it will be that they will accidentally combine 
to produce a spuriously-small confidence ellipse.

Several years ago, before the next edition became due, I drew some of these 
defects in the 2000-2005 edition to the attention of the UK Nautical Almanac 
Office, so it's possible that changes have been made in later editions. I 
would be interested to learn if they have.

George.

contact George Huxtable, at  george@hux.me.uk
or at +44 1865 820222 (from UK, 01865 820222)
or at 1 Sandy Lane, Southmoor, Abingdon, Oxon OX13 5HX, UK. 

   

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