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    Re: The Darn Old Cocked Hat - the sequel 1
    From: Tom Sult
    Date: 2013 Mar 11, 18:22 -0500
    Looking forward to the attachment. 

    Tom Sult
    Sent from my iPhone

    On Mar 11, 2013, at 13:40, John Karl <jhkarl---.net> wrote:

    The "Cocked Hat" 9th March post of Hanni Xi was an interesting addition to our many, many, posts on the subject. It's an observation of the type made by A. S. Goudsmit (physicist at M.I.T. in 1946), that the chances are 25% that the true fix lies inside a 3-LOP cocked hat. Both are probability problems that calculate the frequency of occurrence of a large number events, given all the data necessary for the calculation. They both start with a known position of a ship and then calculate fix locations relative to that known location, using many repeated observations.

    In celnav at sea we are interested in exactly the opposite situation: Our ship is in an unknown position with just a single observation yielding the three-LOP fix. These St. Hilaire LOPs have known random standard deviation (in intercept distance) from many previous sights using a known position, the same sextant, and the same observer. (There's no azimuth error, it's calculated exactly.)

    These two cases are very different -- apparently subtly different. In the first case, the ship's position is known and we make many observations. In the second case the ship's location is unknown, and we make one observation (the three LOPs). The first is a probability problem, like calculating that a coin flip very many times produces heads and tails, each 50% of the time. It has an exact answer. The second case has no exact answer, it's a problem in estimation theory. We estimate the ship's position using simple reasonable rules: use all the best available information, make no assumptions and no contradictions. The best we can do is determine the location of the Maximum Probability Point (called the MPP).

    Attached is an expanded version of my 12/8/2010 post (14687). It discusses the results of the estimation-theory approach to the cocked hat question. The results are all very different from those of Hanno Xi and Goudsmit, after all, they're very different problems. But it's the estimation approach that's relevant for celnav at sea.

    And finally, this is all academic (while still interesting to some of us). What counts most at sea is not your best estimate of where you are, but the confident knowledge of where you're not.

    John K

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