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Frank Reed, in trying to make the best of the disastrous performance of his 
"rapid-fire fix", proposed measuring a set of Sun
altitudes every two minutes for an hour, and then-

"For a small-boat navigator, if you shoot four Sun sights at the beginning of 
the hour, average for one LOP, and then shoot four
sights at the end of the hour, average for another LOP, and cross, you would 
only have about twice the uncertainty of the procedure
above."

Yes, and if the navigator waited three or four hours, as in the traditional 
procedure for obtaining a fix from a single object, to
obtain a decent angle of cut, a single observation, before and after, would do 
even better. He would have made much better use of
his time than standing in the cockpit with his sextant for an hour. Most 
small-boat sailors have other demands on their attention.

So where is the virtue of this "rapid-fire fix", then? I can see one possible 
scenario, when, shortly after the Sun becomes visible,
we know it's going to disappear, in cloud or in sunset, and a navigator must 
make the best of the limited time available, having no
alternative. Otherwise, a mid-morning Sun altitude, another around noon, and 
another mid-afternoon, the traditional recipe for good
Sun navigation, does exactly what's needed, with minimum effort.

When a novel procedure is dreamed up, given a catchy name-tag, and proposed 
for us all to use, we need to ask why no navigator has
used it in the past. There's usually a good reason.

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.
----- Original Message -----
From: "Frank Reed" 
To: 
Sent: Thursday, March 18, 2010 6:16 PM
Subject: [NavList] Re: Today's octant sights


Antoine you wrote:
"IN OTHER WORDS, OUR POSITIONS ARE WRONG BY ALMOST 20 MILES, WITH LOP'S ALMOST 
PARALLEL AND WITH INDIVIDUAL KNOWN OBSERVATION
ACCURACIES OF SOME 1 NM. SO, WHAT IS THE REAL OPERATIONAL VALUE OF ANY FIX 
DERIVED UNDER SUCH CONDITIONS ???

If I were a Skipper - Aircraft Carrier, Big Container Ship or small sailboat 
whatever - and as indicated yesterday, from this set of
observations, I would MOSTLY and UNIQUELY keep the main and almost ONLY 
information they can confidently give : i.e. just one LOP as
depicted yesterday. Anything else seems to me HIGHLY SUBJECT TO CAUTION.

Only in high open oceans, and far away from everything, would I give such Fix 
precedence over to my DR position. Unless I am totally
lost without any DR whatsoever, in which case ... better "a little something" 
than a "big nothing" ... But in such a case I would
have to be be prepared for a big position update sooner or later."

Sure. This is ten sights in a row over twenty minutes with typical standard 
deviation of the sights of somewhat more than a minute
of arc. You said you would have to be prepared for a big position update. Ok. 
Just keep shooting. Suppose I had continued what I was
doing yesterday morning, taking an altitude every two minutes. We're not 
particularly interested in the changing error of the fix in
the direction towards the Sun's azimuth. That's good enough after a single 
sight. But what happens to the uncertainty in position
perpendicular to that? The output from any position calculation is not a 
point. It's an error ellipse. With ten sights over twenty
minutes, we end up with a long thin ellipse oriented with its long axis 
perpendicular to the mean azimuth of the Sun. It's that long
axis that we want to improve. If we continue shooting every two minutes, 
first, we get an improvement proportional to sqrt(N) where
N is the total number of sights just due to statistical averaging. We also get 
an improvement that will usually be at least roughly
proportional to N since the spread in azimuths of the individual sights is 
proportional to the number of sights (for sights at a
constant time interval, like two minutes). So that means that the long axis of 
the ellipse is shrinking at a rate of approximately
N^(3/2). If the long axis of the ellipse is 49 n.m. after ten sights, then 
after twenty sights, it will be 17 n.m. After thirty
sights (an hour's worth), it would be just 9 n.m. And that's the 95% certainty 
ellipse. The 1 s.d. ellipse is half that size.

I don't see this as particularly practical since it requires a computing 
device of some sort. And if we have that, then we miaght as
well just turn on the GPS, unless it's just for fun (which is fine, too). For 
a small-boat navigator, if you shoot four Sun sights
at the beginning of the hour, average for one LOP, and then shoot four sights 
at the end of the hour, average for another LOP, and
cross, you would only have about twice the uncertainty of the procedure above. 
Of course the traditional LOP plotting procedure
provides no error ellipse so the comparison is obscured. The only 
semi-practical case worth considering is some automated system.
Imagine an automated sextant shooting altitudes a hundred times faster (every 
1.2 seconds). While the azimuth range obviously won't
be affected, the statistical sqrt(N) factor would be ten times bigger and the 
position would converge ten times faster. You could
expect a position accurate to 9 n.m. at 95% confidence level in just six 
minutes. Now suppose that this auto-sextant can measure
altitudes with four times better resolution than my "octant" data example. 
That wouldn't be an unreasonable level of accuracy to
expect. That improvement applies in direct proportion to the error ellipse so 
now we're down to a little over 2 n.m. for the length
of the error ellipse perpendicular to the Sun's azimuth in just six minutes. 
That's a pretty good fix in a short period of time. And
yes, I realize that an automated system like that is a long, long way from 
traditional navigation, but at least it would still be
celestial! :-)

-FER



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