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    Re: Averaging
    From: Alexandre Eremenko
    Date: 2004 Oct 8, 20:52 -0500

    Dear Bill,
    As I said before,
    the recipe is the following: take a quick sequence of sights (say, 5)
    (say, 1 minute or less time between sights). Find the average
    GMT, and the average sextant altitude.
    Then reduce the result as one, single signt.
    So the result will be ONE position line, not 5.
    This position line will refer to the AVERAGE time of your sights.
    
    > what will he use as
    > DR/EP/LHA/AP when reducing his sight(s)?
    
    Whatever you would use when reducing ONE sight.
    
    I claim this will work perfectly for a vessel moving at
    10 or even 30 knots in any direction. The precision will be
    improved by a factor of about 2 (theoretically, by a factor of
    sqrt{5}, in
    comparison with a single
    observation.
    
    > The above implies symmetry.  This is where it gets tricky for this
    > liberal-arts major.
    
    It does not imply any symmetry. The position line refers to the AVERAGE
    time, not to the MIDDLE time. Example:
    GMT 20:30:31, 20:30:50, 20:32:00.
    The average is 20:31:27. The middle of the time interval is 20:31:15.
    
    > Sight 1:    00:00
    > Sight 2:    00:21
    > Sight 3:    00:43
    > Sight 4:    02:35
    > Sight 5:    04:00
    >
    > This is where I ask the mathematicians for help.  It looks to me that if one
    > simply averages the altitudes and times, the altitude and time will be
    > skewed toward the initial three sights.  In the case of a moving vessel, how
    > does one determine where to place their position for the average used for
    > reduction?
    
    In your example, the position line will refer to the time
    01:32, the average time. (The middle of the interval is 02:00)
    
    Space and time limitations do not permit me to
    include more detail of a complete mathematical justification of the rule
    (this justification was briefly mentioned in my first message on
    the subject)
    and I do not think it is necessary.
    
    Alex.
    
    
    

       
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