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    Re: Exercise #14 Multi-Moon LOP's
    From: Frank Reed
    Date: 2009 May 2, 23:14 -0700

    George H wrote previously:
    "Now, I don't know how that program operates, internally, and I doubt whether
    Jeremy or Frank know either. It seems worthwhile, then, to ask some probing
    questions about the value of such an exercise, before building on to such an
    insecure foundation, as Frank has done."
    Insecure foundation? You're way off, George. 
    As for the supposed "black box", most of these software packages do exactly 
    the same thing: they use the least squares solution for multiple sights 
    provided in many sources, including every copy of the Nautical Almanac for 
    some twenty-odd years. 
    The calculational details are not the point of all this. The POINT is that you 
    can get a good fix with a series of sights over a fairly brief period of 
    time. This is not widely known among celestial navigation 
    In a (typical) rapid-fire fix, the error ellipse will be aligned with its long 
    axis perpendicular to the mean azimuth of the sights (quite comparable to 
    lat/lon at noon, by the way). The error in this direction is given 
    approximately by (3/2)*S/(sqrt(N)*A) where S is the standard deviation of the 
    observations themselves, N is the number of sights, and A is the range of 
    azimuth in the sights (azimuth expressed as a pure number, "in radians"). So 
    take Jeremy's moon sights in his "Exercise #14". First, we need the standard 
    deviation of observations. Jeremy has posted quite a few cases, and he gets 
    pretty good sights, with s.d. around 0.5 minutes of arc. In this particular 
    set, N is 11, A is 0.1 (5.5 degrees is a tenth of a radian). So the standard 
    deviation "cross-range" would be 2.3 nautical miles. Pretty good. 
    In the direction towards the object, the error ellipse "down-range" has a 
    width of about S/sqrt(N). We're not generally worried about the error in this 
    direction, but it's worth mentioning that you would bump up againt systematic 
    error fairly quickly. So if the equation says your error is 0.05 nautical 
    miles in that direction, don't take it seriously.
    PS: The factor (3/2) above may be wrong. I got that twice, but a third time I 
    found (4/5). Whatever that constant is, it is "of order unity"
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