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    Re: Averaging
    From: Bill B
    Date: 2004 Oct 21, 17:22 -0500

    Like a political candidate, my position keeps changing ;-)
    
    My questions were, how much error from averaging is acceptable, and is there
    a rule of thumb for estimating how much error there will be by looking at
    declination, latitude, and altitude?  I think I am coming close to a crude
    working model (below).
    
    Living and sailing in the mid latitudes I think of LAN as the point the sun
    is south and at its highest, and exhibits no observable change in altitude
    for a brief period of time.  For me the altitude seldom exceeds 74d.  Note
    its azimuth is quickly changing however.  Systematic error near LAN is small
    and getting smaller as the date moves away from the summer solstice and the
    maximum altitude falls. So this is a great time to average (for a full
    reduction) to determine latitude.
    
    As an example of the extreme where the declination and latitude are
    identical (imagine yourself on the Earth's equator at equinox and ignore
    change in declination over the day) the sun would rise directly in the east.
    Its azimuth would remain at 90d until LAN when it passed directly overhead,
    the become 270 at that moment.  Trucking along approx. 1d in every 4 minutes
    all day, so a relatively large error from averaging.
    
    On the flip side of the coin, we have sunrise and sunset.  Here the change
    in altitude can approach 1d per 4 minutes, but the change in azimuth can be
    very small, so a good opportunity for determining longitude.  The problem
    being the relatively large (up to .8') shift for averaging 5 sights over 4
    minutes because of its rate of change and non-linearity.
    
    Between sunrise, LAN, and sunset the "slope" or ratio between horizontal
    movement and vertical changes from vertical to horizontal to vertical.  So
    error falls somewhere between. The above Herbert has addressed more
    mathematically than I can.
    
    To answer my own question, I came across a formula in Dutton's (article
    3004) for rate of change.
    
    Delta H per minute = 15 x cosine Lat x sine Z
    
    Where Z is the azimuth angle of the body or its supplement.
    
    They also have a nomogram for graphic calculation.
    
    So it seems with a compass bearing (corrected to true) and a hand calculator
    or the nomogram one could determine rate of change for 4 minutes by
    multiplying the 1-minute figure by 4.
    
    If we assume 1d (60') change over 4 minutes produces .8' averaging error, we
    can mentally interpolate to estimate averaging error for the current
    situation.  I am guessing I am guilty of again trying to treat a non-linear
    function as linear, and that is relationship will be explored as time
    permits.
    
    Then the navigator can determine whether that error is acceptable for
    his/her conditions.
    
    Bill
    
    
    

       
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