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    Re: On potential error introduced by rounded values
    From: George Huxtable
    Date: 2005 Jan 9, 16:45 +0000

    In the past, when Peter Fogg and I have discussed Bennett's "The Complete
    On-Board Celestial Navigator", we have disagreed strongly: but not this
    time. I agree with the general thrust of his argument, with his numerical
    model, and his conclusions, though there might be a bit of quibbling about
    Because, in Bennett's book, all the tabulations are only to the nearest
    whole minute, many "rounding" operations, to the nearest minute, are called
    for. Each such rounding introduces an error, at the most ? 0.5', but any
    value within that range is equally probable. Statistically, this is a
    "square" or "flat" error distribution, just 1' wide, differing markedly in
    shape from the familiar Gaussian, or "bell curve" error distribution.
    If there are two such roundings, they need to be combined by a "folding"
    operation, which results in a triangular error distribution, with a
    base-width of 2', and at the 50% level being 1' wide. Already, this is
    looking a bit more like the shape of a bell curve.
    Each successive rounding operation that's folded in brings the resulting
    error distribution closer to the shape of a bell curve, and widens it
    slightly. But the resulting width doesn't increase in proportion to the
    number of roundings, but to the square root of that number. Even so, the
    maximum conceivable error, if everything conspires in the same direction,
    is still ? 0.5 x the number of roundings. However, the more such roundings
    there are, the less likely it will be that they will all work, in the same
    direction, to their maximum extent: this is Peter's argument, and a valid
    Peter suggests- "There are 5 or 6 values (6 for stars, 5 for other bodies)
    entered during the sight reduction process used in George Bennett's book",
    and I won't argue with that assessment. If we take 6 such roundings, then
    in theory the maximum amount they could possibly (but most unlikely)
    contribute to an error in position is ?3 miles, though in practice the
    error is unlikely to exceed about ?1.5 miles, as Peter's numerical model
    The question then arises: what does a prudent navigator take as the error
    introduced by the roundings in this calculation process, to combine with
    his estimated errors of observation? How much skirting distance should he
    allow around a charted but unmarked rock in mid-ocean, relying only on
    astro observations to avoid it? How narrow a passage between two such
    hazards would he tackle? A careful family-man might take the view that if
    there's a possibility (even a tiny one) of the calculation being 3 miles
    out either way, that's what he should allow for. A gambler might assess
    that if there's no more than (say) a 1 in 50 chance of the calculation
    error exceeding 1.5 miles, he will accept the risk and plot his course in
    that basis. After all, we're all risk-takers to some extent, or we wouldn't
    be out there on small boats in big waters.
    Of course, no navigator makes such a numerical hazard-assessment, or has
    the detailed information on which to base it: but still, those are the
    estimations that should be going through his mind, perhaps unconsciously.
    Peter offers two numerical models. One draws a parallel with the numbers of
    boys and girls in a family. I suggest that this is a very over-simplified
    approach to the problem, presuming that errors would always be the same in
    amount with just the sign being random, and his numerical results are
    therefore not very relevant. However, Peter's second model, using random
    numbers, is to my mind a good analogy with the real-life error situation.
    The results are meaningful, and the simulation could usefully be taken
    Peter's conclusion, that " rounding to whole numbers in a series does not
    lead to a great chance of the rounded amounts adding up to significant
    error." is somewhat woolly, but I can see what he's getting at.
    He states- "The result of this sight reduction is an intercept that is
    expressed to a whole minute of arc, leading to a fix consisting of latitude
    and longitude expressed as whole minutes of arc." Which is fair enough, as
    long as the user appreciates that a resulting intercept, given to an
    arc-minute, may be in error; that the true answer could very likely be the
    adjacent minute either way, and much less likely, yet another minute away
    than that.
    He ends- "I suspect there is also some wooly thinking involved in the
    assumption that a fix expressed to the nearest tenth of a minute of arc is
    better for the purpose of position finding from a small boat ..."
    Agreed. The Nav-l list frequently concerns itself with a quest for
    perfection, in observing to the nearest 0.1'. However, such ambitions
    crumble when faced with the reality of a small boat on an unquiet sea with
    an indefinite horizon. I accept that the standard of precision for
    intercepts, achieved in the Bennett tables (generally speaking, to an
    arc-minute or two) is perfectly appropriate for small craft, when viewed in
    the light of the limited precision achievable in such observations. The
    important proviso is that these limitations should be understood, and
    accepted, by the user.
    Finally, after all that accord, and if only to tweak Peter Fogg's tail a
    little, I can't resist pointing out that earlier criticisms of the Bennett
    tables revolved around the major errors in azimuth, of many degrees, that
    can occur when using Bennett's azimuth tables for directions anywhere near
    to due East or West: not the errors of a minute or two in intercept that
    are being considered at present.
    contact George Huxtable by email at george@huxtable.u-net.com, by phone at
    01865 820222 (from outside UK, +44 1865 820222), or by mail at 1 Sandy
    Lane, Southmoor, Abingdon, Oxon OX13 5HX, UK.

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