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    Re: Position lines, crossing.
    From: Dave Walden
    Date: 2006 Dec 12, 17:29 -0800

    More from monte-carlo.  Using 3 LOP's 120 degree apart,  no bias and
    normally distributed errors, I find that for 4% of the observations
    (based on 1,000,000 runs) the distance from the true postion to the
    most probable is greater than 10 times the length for the cocked hat
    side.  That is, a small 'cocked hat' but a large distance from the true
    postion. 20% of observations are greater than 2 times the cocked hat
    side.  Or, 80% of the time, you're within 2 cocked hat side lengths of
    the true postion.  72% of the observations are greater than .3989
    cocked side lengths from the true postion.   .3989 is the radius of a
    circle with the same area as the cocked hat.  So the 28% inside the
    circle corresponds to the 25% inside the cocked hat.  (The difference
    being the circle includes more area closer the the center where the
    probablity is higher than the corners of the cocked hat.)
    
    Note that for a constant bias, the effect cancels out for equally
    spaced LOP's.  So, the average distance from the true position to the
    MPP is still .51 miles if the Std Dev is .5 min.  Everything scales, so
    avg dist is 1.02 miles if std dev is 1 min.  As expected, the
    monte-carlo for this case shows 100% of observations are inside the
    very large cocked hat if we use 1min std dev and a 5 min bias.  The  3
    equally spaced lines case can be thought of as a generalization of the
    2 line 180 degree difference case.
    
    
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