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Alexandre Eremenko and Herbert Prinz have been
discussing the averaging of celestial sights.  The
following is an attempt to summarize:

Consider the following hypothetical situation (the
actual numbers here are made up):

Body       Time        Altitude

Spica      20-00-00    25d 10.5'
Spica      20-01-00    25d 05.1'

Arcturus   20-03-00    35d 30.0'
Arcturus   20-04-00    35d 25.0'
Arcturus   20-05-00    35d 20.0'
Arcturus   20-06-00    35d 15.0'
Arcturus   20-07-00    35d 10.0'

Dubhe      20-09-00    42d 00.0'

We have 2 observations of Spica, 5 of Arcturus, and 1
of Dubhe, for a total of 8 observations.  How should
we proceed in obtaining a fix?

One obvious solution is to simply choose one
observation of each body, then reduce and plot.  The
fix would be plotted somewhere within the "cocked
hat".

Alex would argue that we could do better than that by
averaging the two sights of Spica (treating the
average time and the average altitude as a one sight),
averaging the 5 sights of Arcturus, and taking the
single sight of Dubhe. (The motivation for averaging
is the expection that random errors will cancel out in
the process of averaging.)

If I understand Herbert's objection correctly, he
would claim, "Now wait a minute.  You have 5 sights of
Arcturus, but only 2 of Spica and 1 of Dubhe.  How can
you give the same amount of weight to the single sight
of Dubhe as you do to the 5 sights of Arcturus?  There
is more information in the 5 sights of Arcturus than
in the 1 sight of Dubhe, yet you are giving them the
same weights."  (I hope Herbert will forgive me for
putting words in his mouth.  If I have misunderstood
him, I apologize in advance.)

In theory you can only justify giving equal weights to
each set of observations if each set contains an equal
number of sights.  In practice, you will probably not
go far wrong anyway.  Nor will you go far wrong in
practice by selecting one sight from each set ("run")
of sights.

I happen to have a copy of the paper, "On the
Overdetermined Celestial Fix", by Thomas and Frederic
Metcalf.  (Tom was once a contributor to this list.
The last I heard, he was associated with the Institute
of Astronomy, University of Hawaii, Honolulu.) I also
have a copy of a follow-on paper, "An Extension to the
Overdetermined Celestial Fix" by Tom Metcalf (alone).
I obtained my copies of these papers some years ago
directly from Tom.

At the risk of gross oversimplification, I present the
following analogy:

Suppose you have two equations in two unknowns, each
representing a line in a plane.  Assuming the two
lines are not parallel, the two lines intersect at a
single point, and that point can be found by solving
those two equations simultaneously. The solution can
be by one of several methods, including Gaussian
elimination, Gauss-Jordon elimination, or by the use
of matrix algebra.

Now suppose that we have 3 equations in two unknowns.
We observed above that a unique solution is determined
by two equations in two unknowns.  Now we have an
"overdetermined" set of equations with no unique
solution.  This is where the method of least squares
enters in.  We find the point such that the squared
distance between that point and each of the lines in
minimized.  As Herbert pointed out, we can do this
using matrix algebra.

Now, let us return to the example of our sights of
Spica, Arcturus, and Dubhe.  Herbert's argument is
that we ought to compute 8 lines of position (2 for
Spica, 5 for Arcturus, and 1 for Dubhe) and find that
point which minimizes the sum of the squared distances
between it and each of the 8 lines of position.  That
would be solving an overdetermined celestial fix by
the method of least squares.  With this method, each
observation from each body is given equal weight.

Herbert, have I stated your argument fairly?

The Metcalf papers actually address a slightly
different form of overdetermined celestial fix.
Quoting from the earlier of the two papers, "... using
these methods, an accurate fix can quite often be
determined from 10 to 20 observations of a single
celestial body spanning only 10 to 20 min of time."
The algorithm uses a Lagrange multiplier and the
simultaneous solution of a nonlinear equation for the
Lagrange multiplier and a number of linear equations.
The solution is iterative.  It's an interesting
concept.  Tom tried it out with "real observations of
the Moon taken on land with an inexpensive plastic
sextant, and with a pan of vegetable oil serving as an
artifical horizon."  He took 18 sights over
approximately 35 minutes, and was able to confirm his
known position within about 2.5 nautical miles. He
programmed his algorithm on an HP-48SX calculator.
The paper offers to make the code available to
interested parties who send a stamped, self-addressed
envelope.  I'm not sure whether there was any
expiration date on the offer, as it appeared in
"Navigation: Journal of The Institute of Navigation",
Vol 38, No. 1, Spring 1991. The follow-on paper
appeared in Vol. 39, No. 4 (Winter 1992-1993).

Best regards to all,

Chuck Taylor
Everett, WA, USA




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