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    Re: On LOPs
    From: Trevor Kenchington
    Date: 2002 Apr 16, 09:33 -0300

    You wrote:
    >The ellipse is based on formulas from the Nautical Almanac.  I wrote the
    >program more than twelve years ago based upon those formulas, but my oldest
    >Almanac (1992) doesn't seem to contain them anymore.
    >The confidence ellipse is just a two-dimensional version of the standard
    >one-dimensional confidence interval used in statistics.
    With respect, you have not advanced any logical reason why these
    confidence intervals should be elliptical and (as I implied yesterday) a
    brief graphical examination of the problem suggests that they are not
    once there are more than two LOPs involved. What you appear to have is a
    programmed version of some (unknown and thus unverifiable) equations
    that calculate an elliptical approximation to the true shape of the
    confidence interval. That makes yours one step more realistic than an
    assumption that the area of uncertainty is circular but it does not
    demonstrate that the true confidence intervals are elliptical.
    >The MPP seems to be "pulled" towards the LOP intersection which is most near
    >90 degrees.
    That is the point that George made a few days ago. Two near-parallel
    LOPs do not provide much more information than a single one on the same
    bearing. They confirm that the MPP must be somewhere along their
    combined length and generally near their intersection. A third LOP
    cutting them will then indicate where along the length of the first two
    the MPP lies -- unless it lies very far from the intersection of the two
    near-parallel lines.
    >When the intersecting angles of the LOPs are quite different the ellipse
    >will be elongated.  The MPP will be at the CENTER of the ellipse, not at a
    >focus.  The center will NOT be the center of the cocked hat.
    >Note that part of the cocked hat may be outside the confidence ellipse.
    Again: Those are properties of the particular equations that you have
    programmed. Whether they are properties of the true confidence intervals
    is a whole other question.
    >The number of LOPs does not matter, except that with more lines the size of
    >the ellipse will be less, indicating greater confidence.  This is a standard
    >statistical effect.  A larger (random) sample reduces the probability of
    >error.  Each LOP you observe is a random sample from the infinity of
    >(randomly different) lines which could have been observed at that moment.
    With more data, your ability to place the MPP near the true position
    should increase (though chance effects can reverse that in particular
    cases). However, just as the MPP is an estimate of the true position
    based on available data, so the confidence interval (elliptical or
    otherwise) is an estimate based on those same data. Your first three
    LOPs might come close to passing through a single point, leading to an
    estimate of high confidence in the MPP. Adding further LOPs which
    intersected away from that point would widen the confidence interval --
    or it would unless your equations use input values of your assumed
    precisions of the various LOPs.
    In your second message, you wrote:
    >After playing with my program a little I see that the size can actually get
    >smaller with acute angles, provided that the intercepts (in the navigation
    >sense) are similar.  It can also get larger if the intercepts are (somewhat)
    Can you explain why the size of the intercept is relevant? Assuming that
    your DR is reasonably accurate, the intercept depends only on the
    difference between your true latitude and a latitude in whole degrees,
    the difference between your true longitude and a longitude that gives an
    LHA in whole degrees, and the azimuth of the particular body that you
    observed. (At least, that is the case if you reduce your sights using
    HO229.) It is not related to errors or uncertainties in the estimate of
    the LOP and thus should not influence the size of the confidence interval.
    Trevor Kenchington

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