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    Again on LOP's
    From: Trevor Kenchington
    Date: 2002 May 19, 14:26 -0300

    Following up on all of the discussion of cocked hats, I read through the
    exchange in the Journal of Navigation (the UK one) that was cited here
    some weeks back and I also attempted to work my way through Daniels'
    1951 paper in the Journal of the Royal Statistical Society (vol.13B, pp.
    186-207).
    
    For those looking for practical guidance who do not have access to the
    journals, I can't say that I found a whole lot to pass on. As has
    already been stated in this thread, the Journal of Navigation discussion
    covered much the same ground as we have done. If it is any comfort to
    the doubters, all contributors to the earlier debate seem to have taken
    it as proven that there is only a 25% probability of the true position
    falling within a cocked hat. Williams' argument that the probability of
    being off any one side of the cocked hat is always one sixth and that of
    being in the sector beyond any one corner is always 1/12 (an argument
    repeated here by, I think, George) was disputed by one Ian Cook of the
    University of Essex and Williams' attempt to rebut the rebuttal seem
    unconvincing to me. But neither of them, nor anyone else, questioned the
    25% inside the cocked hat conclusion. (They did argue over whether or
    not the 25% was already common knowledge. Clearly, for far too many,
    myself included, it was and is not.)
    
    On that point, in a discussion of Daniels' earlier paper, one of his
    colleagues who had been involved in operations research during WW2
    suggested that the average RAF Squadron Leader was convinced that his
    position necessarily lay within any cocked hat he might plot. If so, an
    understanding of the 25% probability has only filtered through to
    practical navigators in the past half century. Or maybe the RAF's
    navigational instruction was way behind that of the RN and the services
    of other nations!
    
    Otherwise, Daniels provided a lot of math, most of which is beyond me. I
    dare say that it is important to people designing the software for GPS
    receivers but I'm not sure that it has any more direct practical
    application.
    
    He did provide a graphical method for finding the most probable position
    within a cocked hat:
    
    1: Place a ruler across LOPs 2 & 3, aligned so that it make the same
    angle  with LOP 3 as LOP 1 does with LOP 2.
    
    2: Divide the length of the ruler between LOPs 2 & 3 in proportion to
    the variances of those two LOPs.
    
    3: Join the point so identified to the intersection of LOPs 2 & 3.
    
    4: Repeat with either LOPs 1 & 2 or LOPs 1 & 3.
    
    5: Where the two plotted lines cross is the MPP.
    
    He gave an even more complex way of doing the same thing with four LOPs
    to the fix. Personally, however, I'd be more concerned about the ellipse
    around the cocked hat that I was likely to be inside than I was about
    just where exactly in the middle of that the MPP lies. So carefully
    plotting the MPP seems a bit of a waste of time. Daniels gave equations
    for the ellipse but I'd not fancy trying to use them to plot it onto a
    chart.
    
    
    The one remaining point that I got out of the paper concerned the
    formula already quoted on this thread for calculating the probability of
    being within the "cocked hat" when there are more than three LOPs. That
    probability is 1-n/x where x equals 2 raised to the power (n-1) and n is
    the number of LOPs. Perhaps everyone but me already realized this,
    however that formula gives the probability of being within, in Daniels'
    words, the "largest closed polygon". If there were four LOPs, they will
    typically plot out as enclosing an irregular quadrilateral with
    triangles projecting from two of its sides and the rest of the surface
    of the globe divided into 8 segments by the projection of the LOPs.
    Daniels' formula gives the probability of being somewhere in the
    combination of the quadrilateral and the two triangles (all of those
    being "closed" polygons, as is their combination) versus in the "open"
    segments extending away.
    
    One could, in theory, take 10 LOPs, in which case the formula would lead
    to a probability of 0.98. That would not mean a 98% probability of being
    in the small decagon in the centre of your fix but somewhere in the
    complex (likely 20-sided) polygon that includes all areas closed off on
    your plot by the intersection of the many LOPs. I guess that is one way
    of figuring where you are not, with a high degree of probability, but it
    would be a lot of work for limited benefit.
    
    
    Trevor Kenchington
    
    
    
    
    --
    Trevor J. Kenchington PhD                         Gadus{at}iStar.ca
    Gadus Associates,                                 Office(902) 889-9250
    R.R.#1, Musquodoboit Harbour,                     Fax   (902) 889-9251
    Nova Scotia  B0J 2L0, CANADA                      Home  (902) 889-3555
    
                         Science Serving the Fisheries
                          http://home.istar.ca/~gadus
    
    
    

       
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