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    Re: Lat/Lon by "Noon Sun" & The Noon Fix PROVE IT
    From: George Huxtable
    Date: 2009 Apr 16, 20:20 +0100

    Thanks to Dave Walden for making an analysis of the longitude-around-noon 
    problem, and in record-time (wish I was so fluent in Excel!).
    His result, a scatter in deduced longitude of +/- 5.68 miles rms , bears out 
    Frank Reed's claim of +/- 5 miles rms, pretty well. I have no wish to 
    quibble about that difference of 0.68 miles, in this demanding test. So I 
    must concede that, when the resut is deduced by a full least-squares fit to 
    the data, the method meets the claims that have been made for it by Frank. I 
    expected it to be significantly worse than that. I was wrong.
    That full statistical analysis, that Excel makes as a fit to the parabola, 
    is, no doubt, as good as its possible to get, and extracts the best value 
    from the input information. In practice, we are not expecting to have such a 
    tool at hand (though many navigators will) and at sea, will fall back on 
    some graphical method. I expect that it won't do quite as well as Excel 
    does, but I agree with Marcel, that the human eye is remarkably capable in 
    that regard, so I don't expect a great worsening from graphical methods, 
    such as paper-folding. We shall see, I hope.
    Dave managed to turn-round my spreadsheet into a result within 2 hours, 
    which is remarkably good going. He has even taken-in my provision of 4 
    switches, locations that by setting to 0 or 1 can turn off or on the 
    individual components of the scatter, which I hadn't even got round to 
    explaining. Only one thing he got wrong; The second switch doesn't affect 
    the equation of time, as he wrote. That always stays constant. The second 
    switch, when set to 0, turns off any scatter in latitude, so it's then held 
    constant at 56�.
    Dave added, in [7973], "Here are results for George's 20 cases.", but his 
    attachment just showed my original spreadsheet. If anyone is interested in 
    that spreadsheet, they can get it to "work", by simply pressing F9, when a 
    new set of random numbers is recalculated to generate a new scan of data. I 
    would like to see his detailed results, but perhaps he might prefer to send 
    them to me off-list, at
    george-AT-hux.me.uk, so as not to pre-empt others working on that data in 
    more traditional ways.
    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 ----- 
    Sent: Thursday, April 16, 2009 4:23 PM
    Subject: [NavList 7970] Re: Lat/Lon by "Noon Sun" & The Noon Fix PROVE IT
    The .doc file works great. Thanks.
    I took George's spreadsheet and automated it with a macro.
    First, I used it to generate 1000 sets of 13 observations.  For each set, I 
    let excel fit the parabola, and find the time and altitude of its peak.  I 
    then find the difference between the resulting latitude and longitude thus 
    calculated and the true values.  Thus for each set of 1000, I get a mean and 
    standard deviation of the error.
    Using George's convenient "scatter switches", I tried various combinations. 
    He has four switches. In order, the first turns on/off the scatter in ship 
    speed (if one always uses the nominal 10 knots in the calculation), the 
    second changes the equation of time (as I understand it, changing the value 
    to be calculated as opposed to introducing scatter into the result), the 
    third changes the longitude (same comment), the fourth adds scatter to the 
    observed altitude. In the table below, the switches are listed in order, 
    1=on, 0=off.
    As can be seen for everything on, the standard deviation of longitude error 
    in nm is 5.68.  It can be seen the ship speed scatter seems to have a 
    slightly greater effect than the altitude reading scatter.
    The 1 mile lat and 5 mile long estimates look pretty good to me. YMMV.
    1,1,1,1 min lat min long nm long
    mean 0.00 0.19 0.11
    stdev 0.40 10.16 5.68
    max 1.29 48.02 26.85
    0 1 1 1
    mean 0.01 0.19 0.10
    stdev 0.44 6.57 3.67
    max 1.55 22.64 12.66
    0 0 1 1
    mean 0.00 -1.00 -0.56
    stdev 0.42 6.73 3.77
    max 1.41 20.77 11.61
    0 0 0 1
    mean -0.01 -0.38 -0.21
    stdev 0.44 6.72 3.76
    max 1.14 20.50 11.46
    1 1 1 0
    mean 0.00 -0.22 -0.12
    stdev 0.05 7.48 4.18
    max 0.25 24.18 13.52
    0 1 1 0
    mean -0.01 -0.38 -0.21
    stdev 0.00 0.01 0.01
    max 0.00 -0.34 -0.19
    1 0 0 1
    mean 0.01 0.17 0.09
    stdev 0.42 9.97 5.58
    max 1.33 32.45 18.14
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