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On 2016-06-22 22:03, Paul Hirose wrote:
> alt   alt   alt   az    az     az
> RMS   >.5'  max   RMS   >.5°   max     table type
>
> 0.69' 4.24% 23.7' 1.71' 0.054% 1.457°  0.2' 84° no interp
> 0.55' 3.74% 21.1' 1.65' 0.05%  1.445°  0.2' 84° interp t<5
> 0.55' 3.74% 23.8' 1.62' 0.047% 1.102°  0.2' 84° interp K<8
>
> 0.62' 2.96% 19.3' 1.41' 0.034% 0.852°  0.2' 54° no interp
> 0.28' 1.95% 12.3' 1.30' 0.027% 1.045°  0.2' 54° interp t<5
> 0.27' 1.52% 12.5' 1.32' 0.027% 1.015°  0.2' 54° interp t<8
> 0.27' 1.65% 12.0' 1.26' 0.021% 0.787°  0.2' 54° interp K<5
> 0.28' 1.42% 12.1' 1.30' 0.023% 0.987°  0.2' 54° interp K<8
>
> 0.42' 11.5% 14.2' 2.53' 0.16%  1.650°  0.5' 54° interp t<5

I forgot to examine a couple cases in my previous message on a 0.2' table.

If such a table is used with no interpolation and you exclude sights
where t is within 5° of 90, and also exclude stars with declination
greater than 75°, root mean square altitude error is 0.17', 1.25% of the
solutions have more than half a minute error, and the worst is 3.1'.

In addition to the above, suppose the precision of function A changes
from whole numbers to tenths at 54° instead of 84°, and there's a
corresponding change to B. Then RMS altitude error decreases to 0.14',
0.648% exceed 0.5' error, and the worst altitude is 1.7'.

The significance of 54° is that it's the point where the A function no
longer changes one count per tenth minute difference in angle.

Now consider a hypothetical HO 211 tabulated every tenth minute. With
600 entries per degree, I don't think it would be practical, but it may
be interesting to see how such a table performs in a Monte Carlo simulation.

With no interpolation, altitude RMS error is 0.60', 3.24% exceed 0.5'
and worst altitude is 21.2' off. Moving the change of precision to 54°
makes a lot of sense at this tabulation interval, since otherwise you
get many adjacent table entries with the same values.

That improves altitude RMS from 0.60' to 0.36', but there's little
difference in the worst error. Increasing precision to two decimal
places at 85° doesn't help much either.

Interpolation of B(R) from A(R) when t is near 90° gets RMS altitude
error down to a quarter minute. However, the biggest gain occurs if we
simply exclude t near 90 and high declinations. Then RMS altitude error
is 0.08' and the worst sight is only 1.1'.

Of course it's not always practical to exclude the 8% of sights that
meet those criteria. And as I've shown, interpolation when a sight is in
the danger zone doesn't guarantee accurate results. We need a different
solution for such sights. One is the Sadler technique, which will be
evaluated in my next message.

alt   alt   alt   az    az     az
RMS   >.5'  max   RMS   >.5°   max     table type

0.17' 1.25%  3.1' 1.65' 0.05%  1.362°  .2' 84° no interp t<5 d<75
0.14' 0.648% 1.7' 1.32' 0.022% 0.882°  .2' 54° no interp t<5 d<75

0.60' 3.24% 21.2' 1.43' 0.037% 1.153°  .1' 84° no interp
0.36' 1.70% 18.2' 0.85' 0.006% 0.652°  .1' 54° no interp
0.25' 1.30% 14.4' 0.79' 0.006% 0.773°  .1' 54° interp t<5
0.38' 1.71% 14.6' 0.86' 0.002% 0.733°  .1' 54°/85 no interp
0.26' 1.18% 12.4' 0.82' 0.004% 0.808°  .1' 54°/85 interp t>5
0.08' 0.14%  1.1' 0.85' 0.002% 0.672°  .1' 54° no interp t<5 d<75

   

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