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On 2016-06-11 22:08, Paul Hirose wrote:
> alt     alt   alt    az      az    az
> RMS     >.5'  max    RMS     >.5°  max
>
> 1.23'  14.6%  31.3   2.76'  .184%  2.0°  Ageton no interp
> 1.26'  14.9%  31.5   2.87'  .184%  2.2°  no interp, whole numbers
>  .55'   8.8%  25.0'  2.25'  .111%  2.1°  interp all
>  .57'  12.8%  25.6'  2.65   .170%  1.9°  interp K<8 8.5%
>  .59'  13.6%  25.4'  2.65   .168%  2.1°  interp K<5 5.3%
>  .58'  12.8%  24.6'  2.68'  .174%  2.1°  interp t<8 7.3%
>  .59'  13.5%  24.4'  2.65'  .171%  2.0°  interp t<5 4.4%
>  .68'  14.0%  24.9'  2.82'  .195%  2.3°  interp t<5, whole numbers


Standard Ageton tabulates the A and B functions every 0.5'. If that is
changed to 0.2', there's a noticeable accuracy increase. If you don't
interpolate, root mean square altitude error improves from 1.23' to
0.69', and the sights with altitude errors greater than 0.5' decrease
from 14.6% to 4.2%. Interpolation of B(R) when K or t is near 90 makes
only a marginal improvement to those numbers.

In the standard table, the precision of function A increases from whole
numbers to tenths at 84°. If we change that to 54° (with a corresponding
change to B), plus 0.2' tabulation interval, and interpolate B(R) when t
is within 5 degrees of 90, RMS altitude error is only 0.28', with 98% of
altitudes within 0.5' of the truth.

Increasing the size of the danger zone from 5° to 8° doubles the number
of sights to interpolate (from 4.3% to 8.6%), with almost no change in
the statistics. Likewise using K instead of t as the basis of the danger
zone. As I explained in an earlier message, t is less work for the
navigator than K.

Below are my statistics. As before, the randomly generated test problems
are limited to altitudes from 5° to 80° and latitudes from 0 to 70°.
Columns are altitude root mean square error, percentage of sights with
more than 0.5' altitude error, max altitude error detected during a
Monte Carlo simulation run of 100,000 test points, azimuth RMS error,
percentage with more than 0.5 degree azimuth error, max azimuth error.

The last column shows the characteristics of the simulated table, and
the interpolation criterion. For example, "interp t<5" means interpolate
B(R) when t is within 5° of 90°. Precision of the tabulated A values
changes to tenth at either 84° or 54° Table interval is 0.2' throughout,
except the last line.

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 think the optimum "HO 211+" method combines a 0.2' tabulation interval
with interpolation of B(R) when t is within 5° of 90°. Of course the
table would be larger: 90 pages vs. 36. That's still not very big, and
could be cut in half by "turning around" at 45°, as in the Bayless
table. The price you pay is that's it's easier to make a blunder, as the
A and B columns exchange places depending on the angle.

What if you tabulate A and B values every tenth minute? Stay tuned.

   

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