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    Re: Still on LOP's
    From: George Huxtable
    Date: 2002 May 23, 21:23 +0100

    Bill Murdoch wrote-
    >In a message dated 5/20/02 1:16:27 PM Eastern Daylight Time,
    >george@HUXTABLE.U-NET.COM writes:
    >> >> Bill Noyce made a perceptive contribution a few days ago, about systematic
    >> >> errors in celestial observations that can increase the probability of the
    >> >> true position lying within the cocked hat. This happens because those
    >> >> errors expand the cocked hats to surround the true position.
    >> >>
    >This is the only part of the discussion that I have not worked my way
    >through.  Let me explain my problem with it again in a different sort of way.
    >Let's say I take three sights equally spaced around the horizon.  The three
    >sights put three circular LOPs on the earth's surface.  If there are no
    >errors, they intersect in one spot which is my true position.  If there are
    >errors, the LOPs intersect forming a triangle.  The sides of the triangle are
    >either toward the sighted body or away.  I could have several combinations of
    >sides; TTT (one kind), TTA (three kinds), TAA (three kinds), AAA (one kind).
    >The Ts happen because the circular LOP was too small in diameter, because the
    >zenith distance of the body was measured too small, because the altitude was
    >measured erroneously large.  The As happen for the opposite reason.  If my
    >systematic error is to measure angles erroneously large because I tilt my
    >sextant, my TTT triangles will be systematically erroneously large and my AAA
    >triangles systematically erroneously small.  It seems to me that my
    >systematic error has one effect on one type of cocked hat and the opposite
    >effect on another.  Systematic errors expand some cocked hats and contract
    >Bill Murdoch
    Comment from George Huxtable.
    I agree with Bill's analysis of the situation.
    Consider sights of 3 objects at azimuths of 0 deg, 120 deg, 240 deg equally
    spaced around the horizon.
    I'm not entirely certain whether Bill is considering (a) systematic errors
    combined with observational random scatter, or (b) systematic errors on
    their own. The latter case is rather simple to deal with first.
    Case b.(in which there is no  random scatter).
    If the systematic errors also were zero, then every round of sights would
    give three perfect position lines which intersect at a point, that of the
    true position. The triangle is always of zero size.
    Now consider a non-zero systematic sextant error, common to all three
    observations. If each sextant altitude shows a systematically high reading,
    due perhaps to a constant tilt-error or to an uncorrected on-the-arc index
    error, then for each minute of such error, each of the three position lines
    will be displaced by a mile from the true position, and toward the body
    being observed. However small the systematic errors are, as long as they
    are non-zero, there can be only TTT triangles; no others are possible. All
    these triangles will embrace the true position of the observer. If the
    systematic errors had been always the other way, such as an uncorrected
    off-the-arc index error, then every triangle would be an AAA triangle,
    again always embracing the true position.
    Case a (in which observational random scatter is taken into account).
    If the systematic errors were zero, we will get get, as we have considered
    before at some length, a randomly varying set of triangles, such that only
    one in four of those triangles will embrace the true position. Those will
    be the TTT or AAA triangles
    But now consider adding some systematic error, in the same sense to each
    sight. This will shift the position lines, to enhance the size of the
    triangles we label TTT (if the systematic errors are such as to always
    increase the sextant reading), and enhance the probability that such a
    triangle will embrace the true position. For other combinations, the
    probability will be correspondingly reduced. If the systematic error
    becomes big enough to overwhelm the random scatter, then the TTT triangles
    become the only ones that are possible, and will always contain the true
    The same applies if the systematic errors are such as to always decrease
    the sextant reading, when a preponderance of AAA triangles will result,
    which also embrace the true position.
    This corresponds closely with Bill Murdoch's analysis, as I see it. He has
    got it right. Why is he so worried about it?
    The situation, as outlined above, applies only when the three sights
    surround the observer, as at azimuths 0 deg, 120 deg, 240 deg. With a
    different geometry, in which the sights were to all one side, such as at 0
    deg, 60 deg, 120 deg, then that would cause a common systematic error in
    sextant readings (or, similarly, in compass bearings) to push the
    boundaries of the triangle in a different combination of directions. The
    effect would be to reduce the probability of the triangle containing the
    true position, and if the systematic error was large enough to overwhelm
    the random scatter, the triangle would then NEVER contain the true
    As I have said in an earlier mailing, I think (but haven't attempted to
    prove) these two different geometrical situations above are distinguished
    by the true position lying within, or without, the triangle connecting the
    three landmarks. In the case of a celestial fix, this (I think) would
    correspond to testing whether the true position was inside or outside the
    triangle formed by drawing great circles between the geographical positions
    of the three celestial objects. These suggestions in this paragraph are put
    forward rather tentatively.
    George Huxtable.
    George Huxtable, 1 Sandy Lane, Southmoor, Abingdon, Oxon OX13 5HX, UK.
    Tel. 01865 820222 or (int.) +44 1865 820222.

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