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Brad proposed making a simulation test of the "latitude around noon" method, 
and I am happy to go along with the notion.

But first, we need to fix a misconception. Brad wrote-
"George - Please post ONE example where this method doesn't work.  You have 
claimed
that Lat/Lon by Noon Sun Method doesn't work at high latitudes (if I read 
your postings correctly)."

No. I haven't ever claimed it "doesn't work". I have pointed out that its 
accuracy is degraded under certain conditions, calling for observation over 
a longer time-span, so shifting it further from noon. It will be interesting 
to see if that degredation brings it outside the claim of 5-mile accuracy, 
or not.

Brad continued-
Fine.  Suppose your DR is around the Antarctic Circle.  Make the declination 
as north as
you would like it, within reason, as nobody expects this to work if you 
cannot see the
sun.  Provide the NavList and therefore Frank with the 13 altitudes and 
supposed times.
You may even skew the data by injecting a course and a speed, up to 20 
knots, your choice.

I think we should avoid anything so extreme. I've suggested, in the past 
taking as an example a vessel approaching the Clyde in Winter, a regular 
route used by thousands of vessels, and which was particularly relevant 
during Word-war 2, when navigation aids were unavailable and there was no 
alternative to celestial nav.

I will take as an example a voyage in a standard steamer of that era, doing 
10 knots Northwards.

In midwinter, we can take the decination to be -23.4�, and  ignore any 
changes during the observation. Near midwinter, equation of time is also 
small, and for simplicity we can take that to be zero, and again ignore any 
change during the hour of an observation. Is that reasonable?

The nominal position would be, I suggest, 56�N, 10�W, and we could take the 
vessel as being somewhere  within a box centred on that position, the box 
being 1� in latitude and 2� in longitude. I will simulate an hour's worth of 
13 observations, at 5-min intervals, to be centred around 12:40 GMT, which 
would correspond to the moment of noon when seen from a vessel at the centre 
of the box. I can promise that  its actual position, at 12:40 will be 
somewhere between N55� 30' and N56� 30', and W9� to 10�. That actual 
position, at that moment, is what's being sought.

Northward component of the nominal speed is taken to be 10 knots Northwards, 
but with an added Gaussian scatter in the actual northward component of +/- 
1 knot (1 standard deviation) to account for uncertainties in the ship's 
log, course, and tidal stream. That actual speed will hold constant over the 
1 hour observation period.

Altitude corrections for index, dip, refraction, semidiameter are ignored, 
presuming that any such corrections have already been made precisely, and 
without error. The altitudes, then, are taken to be true altitudes of the 
centre of the Sun.

Each of the 13 altitudes is calculated according to the formula-
sin alt = sin lat sin dec + cos alt cos dec cos hour-angle

An observational Gaussian scatter of +/- 1 arc-minute (1 standard deviation) 
is then applied to each calculated altitude. Is that accepted as fair?

For each "data set" will be provided a set number, followed by a string of 
13 altitudes, given as decimal degrees, to 3 decimal places. For each such 
numbered set, I will record the initial latitude and longitude chosen (and 
the actual speed), but not disclose them until later.

The challenge, for anyone who wishes to accept it, is to use any method you 
choose to determine that latitude and longitude, as well as possible. It 
should be done for a number of runs, in order to compare, later, with the 
values that until then have been withheld. The scatter in the difference can 
then be compared with Frank Reed's expectation, with seems to be +/- 2 miles 
(= 2') in latitude, +/-5 miles (= 8.9') in longitude. I can provide an 
endless stream of such sets if requested; please suggest a convenient format 
(an Excel worksheet?).

Gaussian simulation will be done using the Excel function-
NORMSINV(RAND()) . I have convinced myself that it supplies a string of 
numbers averaging zero with a Gaussian distribution having a standard 
deviation of 1 unit.

None of this is set in stone and if anyone has any objections or 
suggestions, say so now. I don't regard myself as particularly adept with 
Excel; if anyone would like to check over the spreadsheet, it will be 
provided.

I have no idea whether Frank's criteria will be met; if they are, that will 
be to the credit of the method, because this is a demanding test.

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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