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    Re: Timing Noon
    From: George Huxtable
    Date: 2002 Apr 17, 12:49 +0100

    Trevor Kenchington wrote-
    
    
    >I hesitate to question George's comments.
    
    Please don't hesitate. It's only by arguing such matters out that we can
    hope to converge on an answer, which with luck may even be the right answer
    
    >However, in response to his:
    >
    >
    >>the
    >>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.
    
    I agree with Trevor's logical analysis (above) of the situation. For
    example, assume a navigator has allowed a long interval between noon and
    his other observation, so as to achieve a large angle of cut between the
    two azimuths. Once that angle has reached a large enough value (70 or 80
    degrees, perhaps), there isn't much to be gained by increasing it further.
    The navigator has reached the realm of diminishing returns. However, any
    such increase in that time interval must add somewhat to the uncertainty in
    the DR. There is a compromise to be struck concerning the optimum length of
    this time window, which depends on the circumstances. However, I suggest
    that the length of this time window should always be measured in hours, not
    minutes.
    
    I'm not entirely happy with the following statement from Trevor-
    
    >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.
    
    Just a comment. I'm not sure why Trevor makes such a distinction between
    the two types of vessel. The faster vessel will travel much further in the
    period between morning and noon. However, the unknown perturbation to its
    motion during this period, due to unexpected currents, would not be less
    than the similar perturbation to a small sailing vessel over that same
    period. Or have I misunderstood?
    >
    >
    >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
    >>right.
    >>
    >
    >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.
    
    Well, let's say we are determining our position by bearings on three
    distant landmarks, 1, 2, and 3.
    
    There is an equal chance that, due to errors in taking the bearing from
    landmark 1, that bearing will lie to the left of the true position as to
    the right of it. If we take the possibility that the bearing can be exactly
    on the line of the true position to be zero, the probability of it being on
    the left (L) is 0.5, the same as it being on the right (R). We can say the
    same about landmarks 2 and 3.
    
    There are eight possible combinations, if we list the three bearings, taken
    in the order 1, 2, 3, as follows-
    
    LLL, LLR, LRL, LRR, RLL, RLR, RRL, RRR.
    
    For each such combination, because it combines 3 terms each with a
    probability of 0.5, its probability is (0.5) cubed, or 0.125. There are 8
    such combinations, each with a probability if 0.125, so that looks right,
    doesn't it?
    
    However, of those 8 combinations, there are only two which place the true
    position inside the cocked hat. These are LLL and RRR. This can be seen if
    a drawing is made showing all the 8 options. The other combinations put the
    true position outside a side or outside a corner. So the probability of the
    true position being inside the cocked hat is exactly 0.125 x 2, or 0.25,
    which is what we set out to show.
    
    What seems at first so surprising is that this result is quite independent
    of the skill of the navigator. The reason for this is that the better
    navigator will produce, on average, a smaller cocked hat, But the
    probability of it embracing the true position will remain at 1 in 4.
    
    George Huxtable.
    
    ------------------------------
    
    george---.u-net.com
    George Huxtable, 1 Sandy Lane, Southmoor, Abingdon, Oxon OX13 5HX, UK.
    Tel. 01865 820222 or (int.) +44 1865 820222.
    ------------------------------
    
    
    

       
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