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I think the best way to explain my lunar distance algorithm is to work
one iteration by hand. These are the input conditions:

Estimated time 1999 Jan 26 22:21:04 Terrestrial Time (22:20:00 UTC)
Estimated position N20°00' W70°00'.

Observed angles:
68.323° Moon near limb to Jupiter center, refracted
66.027° Moon refracted upper limb altitude
45.243° Jupiter refracted center altitude

Observed refracted angles to centers of bodies:
68.597° Moon to Jupiter, refracted
65.753° Moon refracted center altitude
45.243° Jupiter refracted center altitude


With the USNO MICA program, predict the lunar distance and altitudes at
the estimated time and position. Compare the prediction to the observed
angles.

Refracted lunar distance:
68.685° predicted
68.597° observed
------
   .088° error in prediction

Moon refracted altitude:
58.228° predicted
65.753° observed
------
-7.525° error in prediction

Jupiter refracted altitude
49.914° predicted
45.243° observed
------
  4.671° error in prediction


Change the time by 10 minutes and note the effect on the predicted
observables.

68.750 new predicted separation angle
68.685 old predicted separation angle
------
   .065 difference (.006 per minute of time)

60.524 new predicted Moon altitude
58.228 old predicted Moon altitude
------
  2.296 difference (.230 per minute of time)

47.859 new predicted Jupiter altitude
49.914 old predicted Jupiter altitude
------
-2.055 difference (-.205 per minute of time)


Move latitude 5 degrees north and note the effect on the predicted
observables (time is restored to its original value).

68.653 new predicted separation angle
68.685 old predicted separation angle
------
  -.032 difference (-.006 per degree lat.)

57.697 new predicted Moon altitude
58.228 old predicted Moon altitude
------
  -.531 difference (-.106 per degree lat.)

47.215 new predicted Jupiter altitude
49.914 old predicted Jupiter altitude
------
-2.699 difference (-.540 per degree lat.)


Move longitude 5 degrees east and note the effect on the predicted
observables (latitude is restored to its original value).

68.616 new predicted separation angle
68.685 old predicted separation angle
------
  -.068 difference (-.014 per degree lon.)

62.990 new predicted Moon altitude
58.228 old predicted Moon altitude
------
  4.762 difference (.952 per degree lon.)

45.773 new predicted Jupiter altitude
49.914 old predicted Jupiter altitude
------
-4.141 difference (-.828 per degree lon.)


Write linear equations for the increments in predicted lunar distance,
Moon altitude, and Jupiter altitude in terms of increments to time,
latitude, and longitude. The nine coefficients were obtained above. The
constants on the right are the increments we desire, i.e., the changes
to the predicted angles that will make them equal to the observed angles.

  .006 ∆time - .006 ∆lat - .014 ∆lon =  -.088
  .230 ∆time - .106 ∆lat + .952 ∆lon =  7.525
-.205 ∆time - .540 ∆lat - .828 ∆lon = -4.671

The approximate solution is:

∆time = +.027 minutes
∆lat = -3.0°
∆lon = +7.6°

After applying the adjustments, time is 1999 Jan 26 22:21:06 Terrestrial
Time (22:20:02 UTC) and position is N17°00' W62°24'. Predict the Moon
and Jupiter positions for that time and place, and compare to the
observations.

Refracted lunar distance:
68.594 predicted
68.597 observed
------
  -.002° error in prediction (was .088)

Moon refracted altitude:
65.579 predicted
65.753 observed
------
  -.174° error in prediction (was -7.525)

Jupiter refracted altitude
44.720 predicted
45.243 observed
------
  -.524° error in prediction (was 4.671)

That's a good result, considering my rounded-off numbers. The program's
algorithm is basically the same. It repeats this process until
predictions of the three angles (lunar distance and both altitudes) all
differ by less than the requested accuracy in two successive iterations.
At that point the solution is complete.

To automatically apply appropriate increments to the unknowns, I make
the increments similar to the errors in the prediction. In this case the
initial predicted lunar distance missed the observed angle by .088°
(5.8'), so 5.8' would be converted to the equivalent time increment,
allowing for the actual angular rate. Position is easier: the average
altitude intercept of the bodies is the increment for latitude and
longitude.

A disadvantage of this method is that the three observable angles must
behave in a fairly linear way when increments are applied to the
unknowns. For example, if the Moon or target body is at a high altitude,
that altitude can vary in a seriously nonlinear way with respect to
time. The iteration might diverge in such a case. For that reason,
there's a counter to abort the loop if it runs too many times.

Also, I include a second "solver", also iterative, but with a more
traditional algorithm which "clears" the lunar of refraction and
parallax. I actually wrote that one first, being sceptical of the linear
equation method. It's not as interesting, and this message is already
too long, so I won't describe the algorithm. Anyway, the source code for
everything is at my site.

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