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    Re: Averaging
    From: Alexandre Eremenko
    Date: 2004 Oct 23, 04:46 -0500

    Jim,
    Yes, indeed there were so many messages on the subject
    that following this discussion should have been hard:-)
    And especially hard was to find out what the "consensus"
    was, as the people's positions changed during the discussion.
    (Which is the purpose of any discussion, is not it?)
    
    On Sat, 23 Oct 2004, Jim Thompson wrote:
    
    > Is that not true only if the
    > navigator plots all the acceptable-looking
    > sights and then uses a plotting method
    > to find the center of the sights?
    
    I am sorry, but I am not sure what do you mean by
    "plotting the sights". Do you mean plotting the observed
    altitudes against time, or plotting the position lines
    (after sight reduction)?
    
    So let me repeat the setting of the discussion
    from the very beginning.
    We observe several altitudes over a short period.
    
    First. You may plot these altitudes against time
    to see whether they are on a straight line.
    The purpose of this procedure is to reject blunders.
    Nothing else.
    (Some people said that they never make blunders.
    So these people skip this step:-)
    
    Second. You have a choice: a) to reduce each site
    and obtain several position lines (which I think is a wrong
    thing to do, but of course if you use a computer instead
    of an almanac and for reducing, it does not matter).
    
    b) to take the average (arithmetically) and to reduce
    the average as one sight
    and to obtain and plot one position line.
    (which I think is the right thing to do, and which is also
    recommended by many authors).
    
    > Alternatively, why not average them arthmetically,
    > and then plot that
    > single average, as the single best estimate?
    
    Depends on what do you mean by "them".
    I was in favor of averaging the altitudes.
    That is the "row observations".
    
    > Seems to me that saves
    > plotting 3-5 separate LOP's.
    
    Yes. And averaging altitudes also saves you 3-5 separate
    reductions.
    
    > I thought that, strictly speaking,
    > the bodies all move in a non-linear
    > fashion,
    
    Yes. Strictly speaking. As opposite to "for practical purposes"
    For practical purposes they move in LINEAR fasion over
    short periods of time, except when they are near their
    maximal altitude and this maximal altitude is large.
    
    In these exceptional cases (mainly Sun observations near noon),
    a slightly different procedure of "averaging" can be applied
    as recommended by Chauvenet (Astronomy: suggestion to US naval
    oficers, now available on Frank Reed web site).
    See also Bowdich on line, page 324, especially the figure.
    It shows how to average in the case of strong non-linearity.
    
    > our ability to detect that non-linearity throughout most of
    > the celestial "window" with a sextant is very limited,
    > so to all intents and
    > purposes their movement can be considered linear over short time durations
    > of observation.
    
    Correct. Except the above-mentioned exception of a body
    passing near the meridian at high altitude.
    In this (and only this) case,
    the non-linearity is strong indeed, and
    a different averaging method
    is used.
    
    Alex.
    
    
    

       
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