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    Re: HO 211 with Sadler method
    From: Paul Hirose
    Date: 2016 Jul 22, 21:34 -0700

    On 2016-07-19 21:18, Robert VanderPol II wrote:
    > So It sounds like you get a moderate improvement in RMS accuracy and worst 
    error by using Sadler very near K.  Is that correct.
    I'd call the improvement more than moderate. Without the Sadler
    technique, but interpolating B(R) from A(R) when K is within 8° of 90,
    gives these statistics:
    > alt   alt   alt   az    az     az
    > RMS   >.5'  max   RMS   >.5°   max   table type
    > 0.57' 12.8% 25.6' 2.65  0.2% 1.9°    interp K<8
    But instead of interpolating in the danger zone, use Sadler without
    interpolation and get this:
    > 0.54' 13.0% 32.9' 2.67' 0.2% 1.782°  Sadler K<8
    > 0.39' 12.3% 12.5' 2.67' 0.2% 1.720°  Sadler K<5
    > 0.38' 12.6%  8.5' 2.57' 0.2% 1.858°  Sadler K<3
    > 0.42' 13.2%  7.4' 2.78' 0.2% 1.447°  Sadler K<2
    The percentage of sights with error greater than half a minute is the
    same, but the max error is far less. It drops even lower on the last
    line, where the Sadler zone is narrowed to plus or minus 2°. However,
    the overall altitude statistic (RMS error in the left column) begins to
    rise. The optimum K where you change methods is a compromise between
    poor Sadler performance away from 90, and poor Ageton performance near 90.
    If the above table is extended one more line (Sadler employed when K
    within one degree of 90), both RMS and peak altitude error statistics
    are clearly degraded. So I think K within 2 or 3 degrees of 90 is the
    optimum point to switch to Sadler.
    > For T<75 &  Dec<75  how do Ageton 0.5 and 0.2 tables stack up against the 
    same tables with Sadler for 87 75. (These amount to 4.3% of sights.) I also
    show what happens when the changeover point for tenths is moved from 84°
    to 54°. That helps only a little in the .5' table, but helps a lot in
    the .2 table. Worst altitude error in a million random problems was only
    alt     alt    alt    az    az     az
    RMS     >.5'   max    RMS   >.5°   max   table type
    0.33'  10.33%  4.7'  2.68'  0.18%  1.8°  .5' 84° no t>5 no d>75
    0.31'   9.31%  3.3'  2.55'  0.16%  1.6°  .5' 54° no t>5 no d>75
    0.16'   1.18%  3.0'  1.64'  0.05%  1.6°  .2' 84° no t>5 no d>75
    0.13'   0.54%  1.6'  1.31'  0.03%  1.1°  .2' 54° no t>5 no d>75
    Instead of excluding sights in the danger zone, you can interpolate
    B(R), but the results are not nearly as good.
    Below is what happens if you use all sights, with the Sadler method when
    K is within 3° of 90. I tested tables at .5 and .2 minute intervals, and
    the switch to tenths at 84° (standard Ageton) and 54°.
    0.38'  12.72%  10.5'  2.63'  0.17%  2.0°  .5' 84° Sadler if K<3°
    0.37'  11.47%   8.8'  2.54'  0.16%  1.8°  .5' 54° Sadler if K<3°
    0.21'   2.52%  10.4'  1.63'  0.05%  1.6°  .2' 84° Sadler if K<3°
    0.18'   1.53%   9.9'  1.30'  0.03%  1.0°  .2' 54° Sadler if K<3°
    It's still more accurate to exclude (or avoid by careful planning)
    sights in the danger zone.
    The significance of 54° is that it preserves the ability to resolve
    tenths of a minute. That is, the standard Ageton table where the A
    values change to tenths at 84° doesn't guarantee the last digit of A
    value will flip when angle changes by a tenth minute.
    It follows that as you get near 90 there's a point where precision ought
    to increase further to .01. I place it at 86°. However, my simulation
    statistics show no significant benefit for that refinement.
    It's interesting that a 0.2 table, with precision step-up at 54° and
    interpolation of B(R) near 90, has better RMS than the best I got with
    Sadler and the standard table. And the percentage of sights with error >
    0.5' was almost ten times less. About 98% of reductions are within half
    a minute altitude error. Yet the worst result is better when Sadler is
    However, the circumstances that create the worst sight reductions are so
    uncommon that I normally get the statistic from a Monte Carlo run of a
    million random test problems, despite an annoying pause of a few seconds
    while the computer works. If I use "only" 100,000 problems the result is
    immediate but there's significant variation from run to run.

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