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    Distance off with Chicago buildings
    From: Frank Reed CT
    Date: 2005 Oct 6, 18:51 EDT

    Bill wrote:
    "As to distance by height, I have  done many trials with Chicago buildings,
    and the Michigan City (Indiana)  light house, cooling tower and smokestack
    with an Astra IIIB. I was amazed  at my very poor results, using Bowditch
    tables or trig. I won't discount my  technique, but results were often 30%
    off our DR/GPS position. "
    I  have a hunch I know what the problem is here. The bases of those buildings
    are  usually well below the horizon unless you're within just a few miles of
    shore. I  did some beach sights last week as follows (corrected for IC):
    Sears Tower  Altitude: 30.8'
    Hancock Tower Altitude: 22.1'
    Hancock-Sears Separation: 3d  7' (antennas aligned)
    Michigan City Cooling Tower: 10.7'
    The  approximate heights of these in feet are: Sears 1450, Hancock 1127,
    Cooling  Tower 361. Since my horizon was 4 or 5 miles away, the lower parts of all
    of  these buildings were below the horizon, something like 8 minutes of arc
    for the  two Chicago towers were hidden below the horizon. So instead of using
    the  altitudes directly, use their difference: 8.7 minutes of arc.
    Do we need  trigonometry or tables now? No. Just memorize one number: 3438.
    An angle of 8.7  minutes is a ratio of 8.7/3438 or just about 1/395. That means
    that my distance  from the two towers (assumed to be the same distance away,
    which was roughly  true) is 395 times larger than the difference in their
    heights in feet. The  difference in height is 323 feet so I must be about 127,000
    feet or about 24  statute miles away. Notice that if I had done the
    calculation for the Sears  Tower without realizing that a big piece of it is hidden
    beyond the curve of the  Earth, I would have calculated the distance at around 31
    statute miles, so we're  dealing with a substantial difference here. Also note
    that there are ways of  making this calculation more accurate but they're
    probably not worth the  trouble.
    42.0N 87.7W, or 41.4N  72.1W.

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