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    Re: Systematic Error (LOPs revisited)
    From: Trevor Kenchington
    Date: 2003 May 28, 20:52 -0300

    Peter Fogg wrote:
    > 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.
    That would be (as George Huxtable has noted) the most probable position
    but it would not necessarily be the true position.
    In navigation, as in many other situations in life, the best policy is
    _not_ to treat the best point estimate as though it is necessarily
    correct. You have to remain aware that your true position might be in
    less probable points on your plot, particularly those that would mean
    that you were heading straight for some nasty hazard.
    > 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.
    If there is only a fixed systematic error, then there is no point in
    dealing with probabilities where the radii around the intersection are
    concerned. The bisector would be a locus on which your true position
    must lie -- an absolutely certain LOP, in effect. (Probabilities would
    still be relevant to a consideration of how far from the intersection of
    the LOPs you might be, unless you knew the magnitude of the systematic
    If there is a combination of random and systematic errors, then the
    above analysis does not hold true. To the extent that the systematic
    errors were large relative to the random ones, the probability that your
    true position lay in two of the quadrants would be higher than the
    probability that it lay in the other two, but the probabilities would
    only approach 0.5, 0, 0.5 0 as the random error dropped to being
    negligible compared to the systematic error.
    George Huxtable then contributed to this thread:
    > True for four QUADRANTS, but is it this relevant? The two lines crossing at
    > a point divide the plane they lie in into four triangles which snug
    > together at their intersection. Only if the lines are at right-angles are
    > the triangles quadrants. Otherwise they are two wide triangles and two
    > narrow ones, and the chances of the true position being in any one is the
    > angle at its apex divided by 360 degrees, if only random errors exist.
    I think that is a mistake. The probability contours around the
    intersection of two LOPs which meet at an acute angle will not be
    circular but elliptical -- assuming simple random errors in both LOPs.
    [If we had two closely aligned LOPs, we could be relatively confident
    that we lay somewhere along their combined alignment but we would have
    almost no idea where along that line we were.] Thus the probability of
    being in either of the "narrow" sectors is higher than the proportion of
    the circle that they occupy. On the other hand, I doubt that the
    probability of being in each sector remains 25% as they narrow. I think
    they must drop but not as fast as their apex angles do.
    Perhaps some mathematician on the list can provide an exact equation for
    the probability.
    Or did we already get that far last time we discussed this?
    Trevor Kenchington
    Trevor J. Kenchington PhD                         Gadus@iStar.ca
    Gadus Associates,                                 Office(902) 889-9250
    R.R.#1, Musquodoboit Harbour,                     Fax   (902) 889-9251
    Nova Scotia  B0J 2L0, CANADA                      Home  (902) 889-3555
                         Science Serving the Fisheries

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