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Re: Exercise #14 Multi-Moon LOP's
From: George Huxtable
Date: 2009 May 3, 15:31 +0100

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Frank Reed wrote-

"most of these software packages do exactly the same thing: they use the
least squares solution for multiple sights provided in many sources,
including every copy of the Nautical Almanac for some twenty-odd years. "

Indeed, that's true. The Nautical Almanac provides a routine by which an
increasingly good fit can be made to a series of observations by a
least-squares process, under the heading "Position from intercept and
altitude by calculation". What the almanac omits is any estimate of the
errors involved in that process. That omission is made up for, in
"AstroNavPC and compact data", from her majesty's nautical almanac office
(in my now-outdated edition 2001-5, the error ellipse estimation procedure
is described in para 7.5).

"In a (typical) rapid-fire fix, the error ellipse will be aligned with its
long axis perpendicular to the mean azimuth of the sights (quite comparable
to lat/lon at noon, by the way). The error in this direction is given
approximately by (3/2)*S/(sqrt(N)*A) where S is the standard deviation of
the observations themselves, N is the number of sights, and A is the range
of azimuth in the sights (azimuth expressed as a pure number, "in

That may perhaps be so, but Frank gives no reference or reasoning to back
it, for us to check for ourselves. It's somewhat disconcerting to find that
the multiplier constant is uncertain by a factor of nearly 2 (0.8 to 1.5).
Does that expression take account of the uniform spacing of the observations
over the time-span, as Frank proposes? How would it differ, then, if
instead, half were closely grouped at the start of that time-span, and the
rest at its end? That was a question that's been asked but not yet

and continues-

"So take Jeremy's moon sights in his "Exercise #14". First, we need the
standard deviation of observations. Jeremy has posted quite a few cases, and
he gets pretty good sights, with s.d. around 0.5 minutes of arc. In this
particular set, N is 11, A is 0.1 (5.5 degrees is a tenth of a radian). So
the standard deviation "cross-range" would be 2.3 nautical miles. Pretty
good."

Not quite that good. Indeed, Jeremy's observations show little scatter; my
own estimate of the standard deviation, in [8062]  was somewhat less than
Frank's. It shows what can be done from a big-ship, in good conditions.
Taking Frank's error formula at face value, if you refer to the data-sets in
http://www.fer3.com/arc/imgx/f1-Rapid-Moon.pdf
the range of azimuths is 4.4 degrees, not 5.5, which makes the factor A
0.077, not 0.1, making the calculated scatter 30% greater, or now, 3
nautical miles. So our position ends up as somewhere within a bracket 6
miles or so either side of the estimated position. Not a marvellous result,
when you take the high  precision of the original observations, and the fact
that it called for 11 such observations to be made. Alternatively, if, after
just a single such observation, Jeremy had simply waited a few hours, then
taken one additional Moon altitude, at a moment to suit himself, the
uncertainty should have been within a mile. Or instead, he could have
immediately taken a sight of another object in a different direction. So,
hardly a saving of effort, this "rapid-fire fix". Instead, a
labour-intensive method for discarding most of the inherent precision
achievable from celestial observation, and ending up with a second-rate
result.

Frank has presented this particular observation as an example of his
proposed method; indeed, the only example he has offered so far. Was this a
typical situation, then? Far from it. Jeremy's own words, in [6066] explain
"that is why I shot the moon, as the azimuth is changing rapidly, even away
from the time or meridian transit" It was selected, deliberately, for
enhanced effect.

And indeed, that was the case. In the 7min 48 sec of the observation set,
the Moon's azimuth changed by 4.4�, or at a rate of 34� per hour, mainly
because it was rather high in the sky. In most circumstances, a typical rate
of change is more like 15� per hour, corresponding to the spin rate of the
Earth. In which case, the errors that we've seen in this example would be
more than doubled.

Frank had a dig at me, in [8145], when he wrote- "I reminded the group of an
interesting case where someone on NavList actually tried this out for
himself, rather than just pontificating and declaring it impossible from his
armchair, as you have done, George. It was an EXAMPLE of the sort of results
you can expect in the real world." It's the responsibility of the proponent
of such a notion to rise from his own armchair and try it out to show
results that prove its  worth, but Frank has not bothered to do so. It's
left to others to point out the weaknesses in the optimistic gloss, which is
all we've been offered so far.

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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