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Federico Rossi asked-

>George,
>I don't have this book (though I've often read about it on the internet)
>but I'm very interested in the matter of error ellipse and even if I
>found the equation to plot it, I've never been able to find the formula
>giving its orientation with respect to xy axis.
>I would greatly appreciate if you wrote it down for me.
>Thanks,
>Federico

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

With pleasure. Below is what the booklet AstronavPC 2001-2005 (which is
accompanied by a cdrom) has to say.

In emailese, I don't have the Greek letters or subscripts, but I will do my
best to substitute so that you can make sense of it. It follows directly
after 7.4, which is equivalent to section 11 in the Nautical Almanac,
"Position from intercept and azimuth by calculation", and uses the same
symbols. I presume you have a copy of that. My own comments will be in
[square brackets].

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

"7.5. Estimated position error.

If three or more position lines are obtained an estimate of the error may
be calculated. The standard deviation of the estimated position sigma in
nautical miles is given by

sigma = 60 * square root of (S/ (n-2))

where S = F - DdB - EdL cos B(F)  [the F is a subscript, in the original]

The standard deviations sigma(L) and sigma(B) in longitude and latitude are
given by

sigma(L) = sigma * square root of (A/G)  [this is in error by a factor of
cos lat, I suggest, if it's to be expressed in units of longitude rather
than units of distance)]

sigma(B) = sigma * square root of (C/G)

In general as the number of observations increases the error in the
estimated position decreases. Statistical theory shows that the estimated
position has a probability P of lying within a confidence ellipse which is
specified by the lengths of its axes a and b and the azimuth theta of the
a-axis, where

Tan 2*theta = 2B/(A - C)
a = sigma*k/(square root of (n/2 + B/sin 2*theta))
b = sigma*k/(square root of (n/2 - B/sin 2*theta))
and k = square root of(-2 * (log to the base e of(1-P))) is a scale factor.

Values of the scale factor for selected values of P are given in the table below

Probability  P   0.39  0.50  0.75  0.90  0.95
Scale factor k   1.0   1.2   1.7   2.1   2.4

Normally a confidence level of 95% is chosen, that is P = 0.95

The shape of the confidence ellipse depends only upon n and the
distribution of the observations in azimuth; while the size of the ellipse,
apart from the scale factor, depends on the errors of observation. The
method assumes that the observations have equal weight. The ideal solution
is to produce a circular distribution of errors, with A = B and C = 0, so
that the errors are the same in all directions. [This is another passage
that I question; I suggest it should be "A = C and B = 0"] This situation
is approached when the bodies are equally spaced in azimuth. In addition,
if the bodies are observed at similar altitudes, this has the effect of
minimising the systematic errors of the final calculated position."

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

A comment from George.

What makes me a bit uneasy about the resulting error ellipse is this-

Take a simple measurement of three stars at 120 deg apart, with a random
distribution of errors (i.e. no systematic errors). You will get a cocked
hat with the sides all angled 120 deg apart, but the size of the cocked hat
is unpredictable. Sometimes it will be a big triangle, sometimes a small
one, occasionally (rarely) it will be tiny. For each observation of a set
of 3 stars, the program has to base its assessment of the error ellipse on
the size of that cocked hat: it has nothing else to go on. So the error
ellipses will vary wildly from one observed set of three, to the next, just
because of these statistical fluctuations.

It's a bit like asking someone to throw a single dart at the bull's eye on
a dartboard, and then assessing his skill on the basis of how close that
one dart got. Sometimes, just by chance, he will hit the target even if he
is quite unskilled. If you were assessing his skill on the basis of that
one lucky dart, your judgment would be seriously at fault.

For that reason, I suggest that an error ellipse from a single set of 3
star altitudes is not taken too seriously. It's not due to any defect in
the program, which is doing the best it can with the limited information
available. Because a nice error-ellipse has been plotted out by the
computer, it's tempting to give it undue weight. Resist that temptation.

George.


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01865 820222 (from outside UK, +44 1865 820222), or by mail at 1 Sandy
Lane, Southmoor, Abingdon, Oxon OX13 5HX, UK.
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