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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

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
1.6'.

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
employed.

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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