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And here is my message from last night that got bounced by system problems...

Dear Antoine,

Thanks for you question about my post concerning the "quality of fit" for multiple LOP fixes. I'm afraid I didn't make the point about similar LOP triangles of different sizes clear enough. For all such similar triangles regardless of their size, the one standard deviation error ellipse that is centered on the MPP has precisely the same shape and size.

This means that a probability contour plot, such as the ones that John Karl has produced, will also be identical regardless of the triangle size. This being the case then a larger triangle will enclose more integrated probability, meaning it is more probable that the actual location is enclosed within its sides.

In John Karl's recent posting you can see this clearly when he shows the contours for two similar LOP triangles where one of them is shrunk to a point - both cases the contours are the same (this is a little hard to see since the contour scales are different, but if you focus on the 20% contour it is pretty clear).

Finally let's consider the second case that JK has shown, namely a triangle that is shrunk to a point with zero area (with the MPP obviously at the intersection). At the intersection the probability density is finite (there is no singularity at the MPP) and the area of the triangle is zero. This means that the probability of the actual location is enclosed is zero just as you would expect.

I hope this is helpful.
George B

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From: antoine.m.couette---fr
Date: 13 Dec 2010 21:23

Dear George,
in your post you wrote the following :

QUOTE

Now if we scale the triangle up in size keeping its shape the same we will get the same MPP AND the same error ellipse. In other words the error ellipse and the associated probability contours do not depend on the size of this triangle.

UNQUOTE

Should I interpret your statement as meaning that "for all similar cocked hat shapes, and whatever their actual dimensions/sizes, the probability for the fix to be inside the cocked hat remains constant" ?

If my interpretation of your statement is correct, I should conclude that in the limiting case when the size of the cocked hat shrinks down to be only one single point (which becomes a similar case as the one addressed in the "REMARK" to John hereabove, i.e. 3 LOP's crossing in one single point) a "zero surface" cocked hat a NON zero probability to contain/to be the actual fix, such probability being equal to the constant probability of the (earlier shrinking) similar shape cocked hats. It also brings an interesting conclusion about different Crossing LOP's having different probabilities according to the various body azimuts.

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