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    HO 211 with Sadler method
    From: Paul Hirose
    Date: 2016 Jul 18, 22:17 -0700

    The "Compact Sight Reduction Table" by Allan E. Bayless (2nd edition,
    Cornell Maritime Press, 1989) is an abbreviated version of the HO 211
    (Ageton) table. Bayless offers an alternate solution for cases when
    angle K is near 90°, when the standard Ageton solution is inaccurate.
    "This edition includes an important addition devised by the late D.H.
    Sadler, former Superintendent of H.M. Nautical Almanac Office and
    renowned British authority on navigation and navigational tables, an
    alternative sight reduction technique which makes it possible to salvage
    otherwise entirely satisfactory sights formerly discarded because they
    fell in the forbidden zone where K is between 82° and 98°, a little less
    than 9 percent of sights."
    I've never seen anything on the theory or formulas behind this technique.
    1. A(dec) + A(lat) = A(h1).
    2. (h1 + Ho) / 2 = hm. In the book this quantity has no name, but I call
    it hm ("h mean").
    3. B(t) + B(dec) + B(lat) - B(hm) = A(h2)
    4. Hc = h1 + h2
    5. A(t) + B(dec) - B(Hc) = A(z)
    That yields Hc and Z. The rules are:
    1. h1 is negative if dec and lat are in opposite hemispheres.
    2. h2 is zero if t = 90 and negative if t is greater than 90.
    3. Z is greater than 90 if dec and lat are in opposite hemispheres.
    The last rule is not in the book, but is necessary to get the correct Z
    in all cases.
    Note that Ho is required to obtain Hc, and then of course again to
    obtain the intercept. In a Monte Carlo test of the Sadler method you
    must therefore decide how well Ho should agree with the true value. The
    amount of error you inject affects the quality of the solution.
    I use a rather pessimistic standard error of one degree. I.e., about 2/3
    of the time, Ho is within 1° of the correct value that would be observed
    at the DR position. I think it's not unreasonable that your dead
    reckoning could be off by 60 miles in an emergency.
    Unfortunately, Sadler's method is not a good general purpose reduction
    method. It's wildly inaccurate in problems where the Ageton method
    performs well. For example, if LHA = 338°16.6', dec = 22°15.7', lat =
    1°43.3', then correct Hc = 60°32.6'. The Sadler solution is 30 *degrees*
    On the other hand, if Sadler is used only when K is within 8° of 90°,
    the root of the mean squared altitude error is 0.54', and worst error
    18.5'. Compare to 1.23' and 31.3' with Ageton only.
    So rapid is the degradation of Sadler accuracy away from K=90, that you
    get best overall accuracy by limiting it to plus or minus three degrees
    from 90. In that case the RMS altitude error is .38', 88% of reductions
    are within 0.5', and the worst result in one million random problems is
    I tried to gain more accuracy by limiting declination to 75° (which
    would still allow the 57 navigational stars), but there was no
    significant improvement. And apparently the 1° standard error I applied
    to Ho had no serious effect. Cutting that in half made little difference.
    Only about 1/500 of the reductions have azimuth error greater than half
    a degree. However, the maximum error can reach nearly 2 degrees. The
    worst azimuth errors always occur at high altitudes.
    alt   alt   alt   az    az     az
    RMS   >.5'  max   RMS   >.5°   max   table type
    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
    0.37' 12.5%  8.4' 2.57' 0.2% 1.818°  Sadler K<3 d<75
    0.42' 13.2%  7.4' 2.78' 0.2% 1.447°  Sadler K<2
    The statistics below show t is not as good a criterion as K for applying
    the Sadler technique.
    1.45' 16.0% 54.6' 2.82' 0.2% 1.897°  Sadler t<8
    1.46' 16.1% 37.6' 2.74' 0.2% 1.673°  Sadler t<5
    0.41' 12.6% 22.1' 2.73' 0.2% 1.897°  Sadler t<5 d<75
    My statistics assume the standard Ageton table, tabulated at 0.5'
    intervals. The Bayless table pays a price for its compactness.
    (Equivalent Ageton statistics on lower line.)
    0.71' 41.5%  9.8' 4.31' 0.5% 2.597°  Sadler K<3 (Bayless)
    0.38' 12.6%  8.5' 2.57' 0.2% 1.858°  Sadler K<3 (Ageton)
    Just for fun, I simulated the original Ageton table (Sadler method if K
    within 3° of 90) and interpolating every table reading, not just B(R).
    The max error is no better, but the RMS altitude error decreases from
    0.38' to 0.20', and only 2.5% of sight reductions have more than half a
    minute of error.
    The result is practically identical if the same plan is followed, except
    no interpolation and the table interval is decreased to 0.2'. By
    increasing the precision of the A values to tenths at 54° instead of 84°
    (and a corresponding change to B), still more accuracy is achieved.
    Finally, if we avoid declinations (north or south) greater than 75, the
    worst altitude error comes down to 3.8'.
    0.21' 2.6% 11.5' 1.65' 0.0% 1.503°  .2, Sadler K<3
    0.18' 1.5%  7.5' 1.32' 0.0% 1.060°  .2, 54, Sadler K<3
    0.14' 0.7%  3.8' 1.31' 0.0% 1.037°  .2, 54, Sadler K<3, no d>75

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