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    Systematic Error (LOPs revisited)
    From: Peter Fogg
    Date: 2003 May 28, 15:51 +1000

    Some time ago what turned out to be the biggest controversy of my experience
    of the Nav. List erupted, with the claim by George Huxtable that the fix
    position was 3 times more likely to lie outside 3 intersecting position
    lines (Lines of Position, or LOPs in Americanese) than it was to lie within
    them. I was mightily impressed by the efforts different people put into this
    discussion; diagrams drawn, computer programs written, websites set up,
    people 'reluctantly converted'. All good stuff.
    To recap briefly (and hopefully accurately enough): The claim was that the
    fix position cannot lie along any LOP since one is a line, by definition of
    infinitely thin width, and the other a point without area. Since an
    infinitely fine point cannot occupy the same (non-existent) space as an
    infinitely thin line the fix position must lie on one side or the other,
    thus a 50% chance of either.
    When 2 LOPs intersect this becomes a 25% chance of it lying in any of the 4
    quadrants. With 3 LOPs there are 3 of these intersections, thus a 25% chance
    of the fix lying inside the triangle and a 75% chance of it lying outside.
    I was intrigued by this 3 to 1 claim and followed the discussion with great
    interest and not a little scepticism. After a while I stated that it is an
    irrelevance. My reasoning was that if you take any triangle of LOPs or
    'cocket hat' there is only one fix position that can be found by the
    doctrine of least squares. If you move these LOPs outwards to encompass any
    possibily of the fix position lying outside them they must be moved in
    proportion. You end up with a larger triangle and an identical fix position
    at its centre. Once the triangle is bigger than the earth the possibility of
    it encompassing the fix is 100% but the fix position hasn't changed.
    What would be useful is a method of calculating where, if not in the
    triangle, the fix might lie. And even better, a method of quantifying the
    error; putting a number to it; establishing a better fix position than the
    centre of the triangle.
    In what seems so far a little noticed posting by George Bennett on the 24th
    of May called 'Position Line Plots' a practical method for identifying and
    quantifying systematic error has been proposed:
    'There is a technique ..... that allows the navigator to make a simple
    analysis and assessment of his work'
    I am still working my way through this posting but the implications for the
    LOP controversy already seem profound.
    This discussion will be easier with diagrams so please draw your own as we
    go along.
    Lets start with those 2 intersecting LOPs. If there is a systematic error
    then it will be either towards the direction of the 2 azimuths or away. So
    instead of a 25% chance of the fix lying in 4 quadrants there is a 0% chance
    of it lying in 2 opposing directions and 50% each for the other 2. In this
    case the fix position lies along the line bisecting the two 50% quadrants,
    passing through the intersection. It is still just a position line. For a
    fix we need more.
    With 3 LOPs it gets a little more complicated and a little more interesting.
    This is where I came in, prompted by a half-remembered claim in a book by
    George Bennett that with an azimuth spread of less than 180 degrees the fix
    position will lie outside the triangle. So, draw 3 LOPs with an azimuth
    spread of less than 180 degrees and add a direction arrow (or more as they
    get extended) to each indicating the azimuth. Use the same technique as
    above for each of the 3 intersections of 2 lines. You now have 3 bisectors
    with a common meeting point in the area away from the direction of azimuths,
    and the distance from this point to the 3 extended LOPs is the radius of a
    circle that touches each LOP. Where the bisectors meet is the indicated fix
    With any 'cocked hat' triangle, if all the azimuth arrows point outwards or
    all inwards then the fix position lies within the triangle. But with 1 in 2
    out, or 2 in 1 out (as with an azimuth spread of less than 180 degrees) the
    fix position lies outside the triangle.
    The technique George Bennett describes, of drawing auxiliary LOPs some
    (large enough) distance from the originals, consistently either all towards
    the azimuth directions or all away, induces a large artificial systematic
    error. A circle is then drawn that touches each of the 3 auxiliary LOPs. The
    difference between the radius of this circle and the distance used for
    inducing the large systematic error is the indicated extent of the real
    systematic error.
    Apparently this technique can be used for any number of LOPs, and it seems
    that for greatest accuracy 3 is not the ideal number. But the more LOPs the
    more horribly confusing become the plots. Use different colours for
    different types of lines, and dotted lines for the bisectors, and any other
    method (thin and thick lines, etc) you can think of to avoid confusion.
    Bear in mind its only the plots which become complex, the base idea is
    brilliantly simple: a systematic error affects results systematically.
    Realizing this yesterday was like an Eureka! moment for me, I wanted to jump
    out of my bath and run naked through the streets. Fortunately enough the
    weather here is a bit too cool and wet for that at the moment.
    Speaking of confusion, I am only groping towards an understanding of this,
    and its implications, myself. I certainly don't set myself up as an expert
    on this fascinating technique.
    Good on ya, George

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