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> I'll dig out my calculations again and get back  to you.

Frank

Still playing with your beach shots.  It would appear that either your
observations are approx. 4.2' minutes high or you are the victim of
refraction.  Since 4.2' minutes is a ton, especially for someone with your
experience (and the differences in observed vs. calculated elevations are so
close), I assume the Bowditch formula has problems with refraction
(especially at larger distances) and/or the heat from the mills and Chicago
are playing games with refraction.

Following are results of my calculations using calculated angles and your
observations with the Bowditch formula:

              ANGLES
      Observed   Calculated  Diff
Sears     30.8'  26.6'       4.2'
Hancock   22.1'  17.8'       4.3'
         -----   -----
Diff       8.7     8.8

NOTE:  Refraction is not considered in calculated angles.

              DISTANCE (nm)
     Observed   Calculated  GPS
Sears   21.07'  23.05       23.08
Hancock 21.02'  23.50       23.53

NOTE:  All of these calculations assume building bases 30 ft above water
level.  As base height above water level increases, differences between
calculated angles and observe may decease.

NOTE:  As Hancock is shown to be closer by observations and both fall short,
I assume observed Hancock is to large.

                  METHOD

As distance was known I flipped the Bowditch formula D = 1.17 sqr rt (H-h)
with height of eye as 11 ft. to establish how much of the buildings were
hidden.

Then I subtracted the hidden portion from the heights, and added 30ft for
distance of base above water level.

With two adjacent sides (distance converted to a chords) and an angle of the
"tilt" of the buildings off the observer's plumb (determined by height of
eye and distance) I used the law of cosines on the oblique triangle to
establish the length of the third sides.

With those established I used the law of sines to determine the angles.

With the angles for Sears and Hancock established, I plugged the building
heights (plus 30 feet) and the angles into the Bowditch formula.

           OBSERVATIONS

Won't be dropping smart bombs down smokestacks with the results, but with
refraction ignored, very close.  Both fell slightly short.

In another test I used a theoretical star at observed angles (not corrected
for refraction) of 10d, 20d, and 40d with the Earth's nominal radius
(converted to feet) as height of object.  The diagram forms a rhombus with
sides equal to the earth's radius, which seems fair game as I am not clear
Bowditch addresses the foreshortening and/or backwards lean relative to the
observers plumb line of tall objects at larger distances. As the altitude
angle increased the distance calculated using the Bowditch formula fell
further and further short of the great circle distance from observer to GP
of the body.

If nothing else, it supports your feeling that refraction is not adequately
accounted for.

Bill

   

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