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    Re: Still on LOP's
    From: George Huxtable
    Date: 2002 Apr 21, 02:47 +0100

    Response from George, to the message from my friend and near-neighbour
    Clive Sutherland, who said-
    
    >George�s presentation , for example, of three intersecting position lines
    >resulting in only 25% probability of the position being within the cocked
    >hat >has long been the argument in textbooks on this subject, but it
    >doesn�t gel
    >with me.
    
    I would be most interested to learn in which textbooks it's to be found
    
    >To support this contention we are supposed to agree that for each one
    >of these position lines the true position lies equally likely to the left or to
    >the right but not actually on the line! This is where I get off!
    
    I think Clive and I agree better than he thinks we do.
    
    We just need to define rather carefully exactly what we are talking about.
    
    Imagine dividing your chart up into thousands and thousands of (say)
    one-foot square cells. Along the line of a bearing, it's the "probability
    density" that is at a maximum: the (small) probability of the vessel being
    in one of those one-foot cells. To find the probability of the vessel being
    inside a particular zone, one has to integrate the probability density over
    the area of that zone. That is, sum up all the individual probabilities of
    the squares within that zone
    
    Though Clive would be right to claim that along the line of a bearing, the
    probability density is at a maximum, the probability of being exactly on
    that line is zero because the area of the line is zero, as it is with all
    lines (unless some finite line-width has been specified).
    
    >
    >Let�s imagine for a moment that the earth is still and many observations of the
    >same LOP can be taken. If these are analysed statistically ( assuming that all
    >errors are random) the most probable value will be represented by the average
    >and the further off this average  line you inspect the less likely you are to
    >find the true  position line.
    >
    > If we assume that the line is the actually the peak of a Gaussian
    >distribution, yes, it has an equal probability either side, but this is  a
    >diminishing probability value the further you move away from it and the MOST
    >PROBABLE VALUE MUST BE  THE LINE ITSELF.
    
    I agree with all the above.
    >
    >If we take a three dimensional view, our observations could be better
    >represented by a solid figure rather like a length of wood cut so that it has
    >Gaussian cross-section the same size and shape all along its length. We could
    >then lay this on the chart to represent our LOP.
    >
    >If you imagine two such position lines intersecting , merging and adding. The
    >probability curve at the point  of intersection would become a Bell shape. It
    >would have circular contours only if the error distributions of both LOPs are
    >the same , but would have elliptical contours if one LOP group is tighter  than
    >the other.
    >
    >Each contour will represent an uncertainty area at a particular confidence
    >level.(the smaller the ellipse the less confident you are of being inside it).
    >For simplicity it is usual to show only one contour for the confidence level
    >(usually 95%) the navigator prefers.
    
    I agree with that too.
    >
    >Now it is possible to extend this theory to include three or more LOPs, but my
    >maths is not up to this, however I can convince myself that the true position
    >lies inside the cocked hat if I look at this plot in a more colouful way.
    >This is my way of imagination. On maps contours are often coloured. If we
    >wereto use dark transparent colour for high probability and  a light colour for
    >low,  this would reveal the shape of the probability terrain. The darkest area
    >would represent the highest confidence in your position. For example, if we
    >plot the Gaussian distribution of a single group of LOPs  as a ridge line, the
    >center would be a dark  line and the further off either side of this line the
    >lighter the shading  would be.
    >
    >For two position lines, at the point of intersection, the shading would add and
    >this would then produce a single darker peak with  the shading getting lighter
    >in all directions away from this point. This would be consistent with the
    >circular Bell shape,  and would reveal a ellipsoid bell, if this were in fact
    >the case as above.
    >
    >If you use this trick to imagine what three or more curves intersecting as a
    >cocked hat would look like, personally I am convinced that the darkest
    >cumulative shading would be a plateau inside the triangle. This would show that
    >the further you go from the �Center of gravity� of the triangle the lighter the
    >shading would be. Therefore by inference the highest probability of the true
    >position fix would be inside  the triangle.
    >
    >PS.  In surveying it is common to find �the most probable position� in a
    >triangle of error produced by three sight lines,  by drawing lines inside this
    >plotted  triangle, parallel to each side, at a distance from the side
    >proportional to the length of the observed ray so as to produce a smaller
    >triangle inside the first. This is applied successively until the smallest
    >acceptable triangle is found and this then becomes the surveyed point.
    >
    >Clive Sutherland.
    
    I don't think I find any serious disagreement with anything Clive has said.
    I agree that the most probable position for a three-bearing fix is going to
    be somewhere within the triangle. I agree that when entering on the chart
    the result of a cocked hat plot, it is sensible to plot a point somewhere
    within the triangle as being the best estimate that can be made of the
    vessel's position, though keeping aware of the deficiencies of that process
    when taking account of nearby dangers.
    
    But that's not what we have been discussing. We have been asking "What's
    the probability of the position being within that triangle AT ALL?" And
    that's what has come out to be 25%, according to me and to JED Williams.
    That is quite compatible with the notion that the most probable position
    (the point where the probability density, or probability per square foot,
    is greatest) is within the triangle.
    
    Does Clive maintain, I wonder, that the triangle of the cocked hat MUST
    contain the true position, the conventional wisdom that so many have
    accepted in the past?
    
    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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