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    Re: Still on LOP's
    From: Bill Noyce
    Date: 2002 Apr 19, 16:39 -0400

    I've also been thinking about George's assertion that the
    cocked hat contains the tru position just 25% of the time,
    and, much to my surprise, I've convinced myself it's true.
    The key question, in my mind, was whether the observations
    could properly be considered as independent.
    Let's start with the case of two LOP's. I think if they cross
    at right angles, then we can all agree the observations are
    independent, and the true position is equally likely to lie
    in any of the four quadrants so defined.
    Now suppose, as Trevor has suggested, that the azimuths are
    nearly equal, so that the two LOP's define two skinny regions
    and two very wide regions.  Intuitively, it seems that the
    true position ought to be more likely to be in a wide region
    than in a skinny one.  But I don't think that's true.  For
    concreteness, let's say both LOP's run nearly north-south.
    The first one is equally likely to be east of our true position,
    or west of it.  Let's say it is east of our true position.
    Now we lay down the second LOP.  It, too, is equally likely
    to be east of our position or west of it, since it has no
    "knowledge" of the previous observation.  (However, it is
    likely to be west of *the other LOP* near out latitude, since
    we are assuming the other LOP is too far east.  In this sense
    the relative positions of one LOP to the other is not
    independent of the error in one LOP.  But the direction from
    one LOP to our true position *is* necessarily independent of
    the direction from the other LOP to our true position.)
    Therefore, the surprising fact is that our true position is
    just as likely to be in a skinny region as in a wide one.
    I can rationalize this by noting that the intersection
    skitters away off to the north or south as the two LOP's move
    east and west, so in a sense the area in the skinny region
    grows lengthwise to make up for its lack of width.  It's still
    Let's go back to the case that's easier to accept, with two
    LOP's crossing at right angles.  Suppose one runs north-south
    and one runs east-west, so they define four quadrants: NE, NW,
    SW, and SE, and our true position is equally likely to appear
    in any quadrant.  Now let's add a third LOP, running SE-to-NW.
    Like the others, it is equally likely to be on either side of
    our true position.  Depending on where it falls with respect
    to the intersection of the other LOP's, it will define a
    cocked hat of some size in either the NE quadrant or the SW
    quadrant.  If our position was in the SE quadrant or the NW
    quadrant, then the true position is clearly outside the cocked
    hat -- this accounts for 50% of the cases.  If our position
    was in the NE quadrant, and the new LOP is equally likely to
    fall NE of us or SW of us, then half the time we'll be inside
    a cocked hat in the NE quadrant, and half the time we'll be
    outside the cocked hat (which may be in the NE quadrant or
    in the SW quadrant).  This contributes 12.5% of cases inside
    the cocked hat, and 12.5% more outside.  The same argument
    holds if we are in the SW quadrant, giving a total of 25% of
    cases where our true position was inside the cocked hat, and
    75% where it was outside.  This argument depends on the fact
    that whether the new LOP is NE or SW of *our true position*
    is independent of which quadrant we are in.  There are some
    other things we might see that are *not* independent.  For
    example, if we are in the NE quadrant, the new LOP is likely
    to fall NE of the intersection of the other LOP's.  Equivalently,
    if we are in the NE quadrant, it is more likely that the cocked
    hat falls in the NE quadrant than that it falls in the SW
    quadrant.  Renaming George's 8 possible outcomes, our true
    position could be
      N,E,NE; N,E,SW; N,W,NE; N,W,SW; S,E,NE; S,E,SW; S,W,NE; S,W,SW
    of the three LOP's.  If it's N,E,SW or S,W,NE then our true
    position falls inside the cocked hat -- but one of these doesn't
    exist!  Does that reduce the probability to 1/7 (or even 1/8)?
    No, because which one exists is not independent of our true
    position.  The only thing that *is* independent is whether the
    new LOP is NE of us or SW of us.

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