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    Re: Timing Noon
    From: Trevor Kenchington
    Date: 2002 Apr 14, 18:54 -0300

    I hesitate to question George's comments. However, in response to his:
    >length of the time-scale, before and after LAN, that he thinks the plot
    >should cover, which is an important matter. I have come many navigators who
    >think the job can be done over a few minutes around noon, Where I differ is
    >in maintaining that to get any reasonable precision in determining the
    >moment of noon, the plot should cover several hours, not several minutes. [snip]
    >I have no objection at all to using this method: it's fine. The point I
    >wished to make was that it should measure time of noon, not AT or near
    >noon, but on either side of noon. Within limits, the further from noon the
    >measurement is made, the more accurate it will be.
    Measuring the time of local noon at any time other than local noon seems
    to have limited utility. It serves to check the chronometer but cannot
    directly contribute to fixing your position. (Even the chronometer check
    doesn't help much with LAN if you don't know how much your longitude has
    changed between the check and the following noon.)
    Assuming that the final aim of celestial navigation is to develop a fix,
    we have a more complex problem that George appears to suggest. At noon,
    and using Sun sights exclusively, we can get a precise estimate of
    latitude but only a very poor one of longitude. Some hours earlier or
    later, we can get precise LOPs that approximate to meridians but we
    cannot determine latitude with high precision. To get a fix, therefore,
    we must either advance or retard one or another LOP based on dead
    reckoning. And the longer the delay between one sight and the next, the
    less precise the DR will be. Increasing that delay will improve the
    angle of cut of the two LOPs (ultimately making it 90� if the morning or
    evening sight has an azimuth of 090 or 270) but at the cost of greater
    uncertainty in the DR.
    It would therefore seem to me that the greatest accuracy in the fix will
    NOT result from moving the second sight as far away from noon as
    possible (within limits, as George noted) but requires a balanced
    response to both the improved precision in estimating longitude and the
    worsened errors in DR. If so, the optimum timing of sights should depend
    on the kind of vessel: The officers of a large steamer, maintaining a
    steady course at a speed far in excess of likely ocean currents, might
    be best advised to take dawn and noon sights, while a sailor on a small
    yacht, working against light and variable headwinds, might do better to
    keep the sights much closer in time.
    George also wrote:
    >Well, the meaningfulness of a "cocked hat", whether from land bearings or
    >astro sights, is frequently misunderstood. It's a surprising fact that no
    >matter how good the navigator, only one time in four will his cocked hat
    >embrace his actual position, which is three times more likely to lie
    >outside it. This is a universal truth, relying in no more than this
    >proposition: that each position line, being the best estimate that can be
    >made, is just as likely to lie to the left of the true position as to the
    I understand the proposition but not how it leads to the conclusion that
    the true position has a probability of only 0.25 of falling within the
    cocked hat.
    Given a single LOP, the true position could lie to either side of it,
    with the probability of it being some particular distance away taking
    the form of some probability density function -- hopefully a Gaussian
    one but most likely not. 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. 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�).
    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). The question must then be: How much of
    the overlap between the rhombus around the intersection of two LOPs and
    the zone of uncertainty (with width x) around the third LOP falls within
    the cocked hat formed by the three LOPs?
     From sketching out a few examples, I would question whether there is
    any fixed proportion (0.25 or anything else). If the cocked hat chances
    to be large, relative to the precision of the LOPs, then the true
    position is almost certain to lie within the cocked hat. On the other
    hand, when we get lucky and have three LOPs passing through the same
    pencil point, the true position is almost certain to lie outside that
    (very, very small) cocked hat -- or so it seems to me.
    The problem for most navigators, of course, is that we attempt to judge
    the precision of our LOPs from the size of the cocked hat, whereas that
    actually arises from a combination of precision and chance. Hence,
    unless an individual studies his/her performance over an extended period
    and gets a clear idea of how precise his/her LOPs are with the effects
    of chance averaged out (but with the effects of difficult conditions and
    so forth factored in), it is impossible to say whether a particular
    cocked hat is large or small compared to the precision of the LOPs.
    Trevor Kenchington

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