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To follow up Brad's suggestion, here's a simulation of longitude around 
noon, taken from an unknown position that, on 21 Dec 2008, at 12.40 GMT, is 
somewhere between 55º 30'N and 56º 30'N in latitude, and between 9º W and 
11ºW in longitude.

13 supposed observations of Sun altitude were taken at exact intervals of 5 
minutes, for an hour, centred around 12.40 GMT. It's assumed that these have 
been corrected without error to provide true Sun altitude.

The Sun dec. has been taken as -23.442º and the equation of time 1min 44 sec 
(the Sun transiting at Greenwich that much before noon GMT). No allowance 
has been made for changes in these quantities over the hour of observation; 
that might become necessary if other dates, nearer the equinoxes, were being 
considered..

A nominal ship's speed of 10 knots (Northerly) has been assumed, but to 
allow for uncertainties in speed measurement and course, and tidal currents, 
a possible scatter of +/- 1knot rms has been added. That adjusted speed is 
assumed to be constant over the hour of observation. That leads to 13 
latitudes, over the hour, and thus to 13 calculated altitudes, which should 
reach a shallow maximum somewhere within the period of the scan.

To each of those 13 calculated altitudes, to represent observational 
uncertainty, a scatter has been added, of +/- 1 arc-minute rms, so 13 
differing perturbations, to provide 13 perturbed altitudes. On the basis of 
those 13 numbers, the challenge is to determine the observer's lat and long, 
at 12:40 GMT, as precisely as possible. The interest will be in observing 
the scatter between that number and the original assumed lat and long, from 
which the altitudes were calculated.

I attach a table, noon1a.rtf , which covers 20 "runs" of the simulation. 
Each run has 13 values of Sun altitude, which should provide sufficient 
information to deduce lat and long. The "identity code" is just an 
increasing number to relate each data-set unambiguously with the excluded 
data, which would show lat, long, and speed. For the first two data-sets 
only, these answers are shown, so you can get an idea of what's expected. 
Don't try to identify the vessel's speed, just its position. The remaining 
18 will provide something to get your teeth into, but you aren't expected to 
tackle the lot.

If anyone would like to see the spreadsheet that calculates one line of 
those numbers, it's attached. The 13 times correspond to columns D to P. 
Rows 20/21, copied in 26, are the GMT. Row 22 shows the latitude, varying 
according to the presumed speed of around 10 knots around the presumed 
latitude at 12:40. Row 23 shows calculated Sun altitude, corresponding to 
that latitude and Sun position.
Row 24 shows a random Gaussian fluctiation, of +/- 1 arc-min.
And Row 25 shows this summed with Sun altitude, to provide one row of the 
output printed in the table.

======================
If anyone can see snags or any unfairness, say so, quickly.

You may note that this simulation differs from Brad's condition 4, which 
read-

"4) 13 values scattered in time.  About 1/2 before and about 1/2 after LAN, 
with no periodicity (not every 5 minutes, some variation can be expected."

Instead, I've shosen the observations to be made "on the dot", at 5 min 
intervals, to give the method an easier ride.

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: "Brad Morris" 
To: 
Sent: Tuesday, April 14, 2009 3:13 PM
Subject: [NavList 7940] Re: Lat/Lon by "Noon Sun" & The Noon Fix PROVE IT



Excellent, George.  I make no apologies about the one example not working. 
It
was meant in the spirit of degraded accuracy vs stated criteria.

In the sense of fairness, George, would you mind posting the spreadsheet? 
It shouldn't
contain the final set of values that will be applied to the challenge set, 
but
it will permit others to review the calculation.

I believe at this point that we just need Frank's buy-in.  Mr. Reed?

Jim - do you accept the conditions that are posted by George.  They are
        1) 10% (1 knot in 10) velocity scatter
        2) 1 arc minute in true altitude scatter
        3) Centered around Noon (but not precisely so. We don't want the 
apex of the parabola to contain an altitude)
        4) 13 values scattered in time.  About 1/2 before and about 1/2 
after LAN, with
            no periodicity (not every 5 minutes, some variation can be 
expected.

In review of other posts by Henry and Geoffrey, it seems that there are 
published sets of limitations.  Would
one of you gentlemen post either the reference or scan in the appropriate 
section, since it seems to be at your
fingertips?  Thanks!

Best Regards
Brad

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