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    Re: Systematic error and its resolution
    From: George Huxtable
    Date: 2007 Apr 6, 18:43 +0100

    Following Peter Fogg's contribution obout systematic error, let's consider
    that topic a bit more carefully.
    
    He rightly separates out the possibility of systematic error in timing, and
    its effects on longitudes and latitudes.
    
    But then, the second of his "two sources", is-
    
    | 2. A systematic error in altitude, such as using corrections with the
    | wrong sign, damage to the sextant, anomalous refraction, and the like.
    
    and he then continues-
    
    | Correction for systematic error
    |
    | If the error is systematic, each position line will be displaced equally
    by
    | the amount and in the direction of the error.
    
    Trouble is, that doesn't accord with his examples above, of what he calls a
    "systematic error in altitude".
    
    Take "using corrections with the wrong sign". Well, if you applied the
    refraction correction with the wrong sign, it wouldn't make much difference
    for high-altitude objects, but would systematically put lower bodies into
    the wrong place.
    
    Take "damage to the sextant". I wonder what damage he is thinking about
    here. If any such damage has affected the scale calibration, over different
    parts of the arc, or given rise to drum eccentricity or collimation error,
    or if a user has misinterpreted the sign of the "box corrections" on the
    certificate pasted inside his sextant box (or ignored it), these will all be
    systematic errors, but all will vary with altitude.
    
    Take "anomalous refraction". I wonder if he is thinking about "anomalous
    dip" here, which varies with the low-level refraction in the path from the
    horizon to the eye. Yes, that would indeed give rise to a systematic error,
    common to every observation. But NOT anomalous refraction, in the light path
    from the observed body. That can indeed have serious consequences, as is
    clear to anyone who has watched the distortions that often occur in the
    shape of the disc of a low Sun, and wondered how its altitude might be
    affected. And different altitudes will be affected by different amounts.
    
    Yes, indeed, there are causes that will give rise to a common, equal, error,
    in every corrected altitude, and getting the index error wrong is perhaps
    the most obvious of them. And only when it's known that the cause is such as
    to give rise to equal errors at all altitudes, will the method, that he
    described, apply. It applies, not to "systematic errors", in general, but to
    systematic common-altitude errors only.
    
    And there's another difficulty, that he doesn't go into, but it's a crucial
    one. We have recently gone at great length into the analysis of
    error-triangles in which the basic assumption was that all systematic errors
    had been corrected, and only random scatter remained. Peter is now dealing
    with a case where it's assumed that the errors are systematic ones (and he
    says so), and he must be presuming that there are no (or insignificantly
    small) random errors. If it was somehow known that the only errors in a
    round of observations were common systematic ones and that there was no
    random scatter, then I would concur that his construction method, to
    eliminate those errors, is a valid one. But how on Earth can anyone TELL
    that, from a single round of observations? In fact, both assumptions are
    unrealistic; there will always be an unknown mix of random scatter and
    systematic error, and how are they to be disentangled?. Only by considering
    the statistics of many such rounds of sights could you reach any such
    conclusions; not that I'm advocating that.
    
    Just by chance, if there was zero systematic error, one observation in four
    will show the displacements all in the same "systematic" direction, which
    would indicate to Peter that some such common error was indeed present, even
    if it wasn't, and he would "correct" the triangle accordingly. But that
    would have been done on the basis of a wrong assumption, so would he have
    "improved" anything at all? I doubt it.
    
    The technique Peter puts forward has been proposed a number of times,
    notably by George Bennett. I am going back to the example Bennett gave by
    distant memory, but as I remember he was teaching a group of students who
    were all measuring sextant altitudes of certain bodies. All had arrived at
    consistent results except one, and Bennett was later able to show, by a
    similar technique to Peter's, that there must have been a serious error in
    the index correction that had been used in that case, and to put it right.
    You can see that in that particular example, the method was indeed valid.
    The consistency of the other students' results showed clearly that random
    scatter was indeed small, by comparison with the major errors shown by the
    odd-one-out. How often does that apply in real-life?
    
    Peter's conclusions, such as that his "improved" fix will always lie inside
    the triangle if the azimuths spread by more than 180 degrees, and, indeed
    will be at its centre, are over simplistic, in that they apply only if it is
    known that there is no random scatter, or if the systematic error is
    sufficiently great as to completely overwhelm it.
    
    Indeed, if you were to find that in every case the three position lines in
    such a triangle were indeed displaced in that same "systematic" manner, in a
    series of many rounds of sights, that should excite suspicion that somewhere
    a systematic error exists, which needs to be tracked down and corrected.
    
    George.
    
    contact George Huxtable at george---.u-net.com
    or at +44 1865 820222 (from UK, 01865 820222)
    or at 1 Sandy Lane, Southmoor, Abingdon, Oxon OX13 5HX, UK.
    
    
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