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    Re: Timing Noon
    From: Trevor Kenchington
    Date: 2002 Apr 16, 21:07 -0300

    You wrote:
    >>To simplify for the purposes of illustration,
    >>we could pretend that the true position is equally likely to lie
    >>anywhere within a distance "x" on either side of the LOP.
    >Unfortunately, you can't make this simplification. Doing this, you are implying
    >rectangular error distribution (such as you get from limited resolution of your
    >sextant) and this is an entirely different problem.
    Obviously -- in practical applications. However, in an attempt to
    explain the point through the limited medium of e-mail, I still think
    that the assumption is useful and warranted.
    >>Given a second LOP (assumed, for present purposes, to have a precision equal to
    >>that of
    >>the first LOP), the true position must lie within x of each LOP (still one either
    >>side). Thus, it must lie within a rhombus symmetrically arranged around the
    >>intersection of the two LOPs, the sides of the
    >>rhombus being longer than x (unless the two LOPs cut at 90�).
    >For rectangular error, this is correct. But then the true position cannot possibly
    >lie outside. For the consequences, see below. But not so for normal error. In this
    >case, the contours of equal probability density are ellipses.
    Again: Obviously. My rhombus is only an approximation to an ellipse,
    just as my assumption of uniform probability is only an approximation to
    the far more complex probability densities within (and outside) that
    I would, however, doubt that the uncertainties around a real LOP have a
    normal/Gaussian distribution or even that they are symmetrical on either
    side of the line. If nothing else, the limited precision of our
    calculations (which iself stems from the limited precision with which
    our instruments can be read) will give the uncertainties a discrete,
    rather than a continuous, distribution.
    >>If a third LOP is added (still with the same precision), the true
    >>position must lie within the previous rhombus _and_ within x of the
    >>third LOP (on either side of it).
    >For rectangular error, yes. The band of the third LOP _must_ intersect the rhombus.
    >If it didn't, you would not just have a case where the true position lies outside the
    >cocked hat. You would, in fact, have a logical contradiction, and that is to say,
    >invalid data.
    >For normal error, however, the true position may be anywhere, with varying
    >probability. In particular, it may be outside any given confidence ellipse, and
    >certainly outside the previously obtained rhombus.
    It certainly _may_ be outside that rhombus but it is not certainly
    outside of that area.
    What you say is true but can only be shown by invoking the greater
    complexity of statistical distributions which, I suspect, is taking this
    exchange beyond the limits of tolerance of most members of Navigation-L.
    >To evaluate the probability of the true position being in a given area, you have to
    >find the integral over the probability density over this area. In case of rectangular
    >error, the probability density is constant. Therefore you get away with comparing the
    >sizes of areas. But in case of Gaussian error, the probability density is not uniform
    >within the cocked hat. Things are becoming more tedious. It would actually find it
    >very interesting to see whether, by evaluating this integral  for the cocked hat, we
    >can confirm George's assertion that it comes out to 25% probability.
    Indeed it would, though I would settle for George explaining how he came
    to that conclusion.
    >Until we find a volunteer to do this, the following thought experiment is much
    >easier: From your true position "shoot" 10000 rounds of 3 sights in three directions.
    >Assume _nothing_ about error distribution other than symmetry around 0. For each
    >azimuth we expect 5000 LOPs towards and 5000 away. Draw, therefore, any 10000 LOPs
    >satisfying this criterion. Examining all the 10000 cocked hats resulting from these
    >"observations", you will find that they enclose the true position in exactly those
    >2500 cases where all 3 LOPs in the series are on the same side (i.e. either all
    >towards or all away). (There are really only 8 cases to distinguish. I say 10000 to
    >make it statistically valid.)
    I'm afraid that I don't see the relevance of whether the intercepts are
    towards or away, even if you mean error in the intercepts. (The same LOP
    could be generated by a meridian sight on some suitable star or by a
    sight on Polaris. What would be an intercept towards with one would be
    an intercept away with the other.]
    The simplest class of cases might be with all errors being away from the
    centre of the cocked hat, meaning that the true position lay on the
    cocked-hat side of each LOP. In all of those, the true position must lie
    within the cocked hat, as one can readily demonstrate by sketching all
    possible cases within that class. Given symmetrical errors (which I
    would doubt in reality), the class of cases in which all errors were
    towards the centre of the cocked hat would be equally probable. In that
    class, the true position must lie outside of the cocked hat. Still
    maintaining the assumption of symmetry, each of those classes will occur
    in 1/8 of three-LOP fixes on average (i.e. in 12.5% of fixes the true
    position should lie inside the cocked hat and in 12.5% it should lie
    outside). The remaining 75% of cases would see either one or two LOPs
    drawn on the cocked-hat side of the true position, meaning that that
    position must be outside the cocked hat. That logic therefore leads (or
    leads me) to the conclusion that only 12.5% of of fixes will produce a
    cocked hat that contains the true position. In the other 87.5%, the
    navigator would be outside his carefully-plotted triangle.
    But that assumes symmetrical errors. Besides all other forms of
    asymmetry (due to instrument errors, observational errors and whatever
    else), I would suggest that the intersection of any two LOPs gives us an
    estimate of position. If a third LOP fails to pass through that
    intersection and passes, for example, further east, then it is already
    likely (though not, of course, certain) that the third LOP lies east of
    the true position. I am nowhere near statistician enough to determine
    what the real probabilities are but I suspect that the class of cases in
    which the true position lies on the cocked-hat side of all three LOPs is
    a lot more frequent than 12.5%.
    If I am wrong, I would welcome an explanation of where I have gone astray.
    Trevor Kenchington

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