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    Re: Timing Noon
    From: Trevor Kenchington
    Date: 2002 Apr 16, 21:52 -0300

    George Huxtable wrote:
    
    >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?
    >
    
    On further thought, I'll withdraw one part of my suggestion but maintain
    the rest.
    
    It does not matter what the speed of the ship is relative to any
    current. The effect of the current is, of course, additive (not
    multiplicative) and hence a one-knot current will cause a six mile error
    in DR over six hours whether the vessel is struggling to make half a
    knot in the doldrums under sail or is a fast catamaran ferry doing a
    steady 50.
    
    However, my postulated large steamer can be relied upon to maintain a
    heading close to her course and a speed close to her master's intent.
    Indeed, she probably carries some form of log that reads out the miles
    she has move dthrough the water. Hence, her officer's DR can be very
    close to correct, saving only the effects of any unknown components of
    the navigational current.
    
    In contrast, the small yacht beating in light weather will have a
    constantly-varying speed and a constantly-varying heading. Her speed may
    well frequently drop too low for her log to give a reliable reading.
    Under those circumstances, I would doubt that her skipper's DR would
    even begin to approach the accuracy achievable on the steamer.
    
    George also provided his explanation of his reasoning why only 25% of
    true positions lie within observed cocked hats:
    
    >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.
    >
    
    OK, that makes sense. (I'm still trying to get my head around how my
    version, in my reply to Herbert of earlier this evening, came out at
    12.5% when Georges comes out at 25% but that is my problem to ponder over!)
    
    But if the cocked hat is most likely where you are NOT, how come so many
    navigators have got along for so many years assuming that it is where
    they were?
    
    
    Trevor Kenchington
    
    
    

       
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