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I'd like to wade into this one but not to argue about stats. Seems to me
that this is a done deal anyway.

I think that we are all focusing on only one aspect of Gary's excellent and
well laid out post while perhaps overlooking some rather salient points that
he brought up. I quote (clipped):
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"So, what does this tell us. There is about a 30% chance that the
position of the observer will be more than .48 NM but less than 1.0 NM
and about a 61% chance that the position of the observer will be more
than 1.0 NM from the plotted fix. So what do we do with this
knowledge? We use the plotted fix at the intersection of the two LOPs
for our navigational purposes such as measuring our progress and
planning the next leg.  We also use this fix  to deal with the
proximity of danger keeping always in mind that the vessel may
actually be almost 4 NM from the fix in any direction. Why do we use
the intersection as the fix, because there is no better one available
since this spot marks the center of possible positions of the
observer. No other spot would be as useful for planning purposes or
avoiding danger. Also, what methodology would you use in determining
another spot to mark the fix?"

"Now moving onto the three line fix derived from three observations
resulting in a triangle. The same analysis holds with the same circles
of uncertainty since the boat doesn't know that you took three sights
this time instead of just two. The only question left is where should
we plot the fix to mark the center of these circles? For very
practical reasons we take the center of the triangle as the fix. The
size of the triangle is limited. If you wanted to plot the fix
somewhere outside the triangle how would you decide where to place it,
the choices are unlimited with no way to chose between. Again, no one
is suggesting that the position of the observer is at the center of
the triangle but this represents the center of possible positions of
the observer.  In fact, the position of the observer will be outside
of the triangle often but I don't agree with the three out of four
allegation. Counter intuitively, the smaller the triangle the more
likely that the position of the observer is outside the triangle! If
you think about it, this should be obvious. Using   reducio ad
absurdum, think about a triangle only one inch in size, it would be
impossible for the observer to be within the triangle. At the other
extreme, a very large triangle with all of the displacements of the
LOPs from the center of the triangle equal to 3.3 NM (3.3 sigma's,
linear sigma's are slightly different than circular sigma's, see
Bowditch), the only place that the position of the observer could be
is at the fix in the center of the triangle!"

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From the standpoint of a practical navigator, the above two paragraphs make
perfect sense to me. My interpretation of what Gary is trying to say is
this:

the reason why navigators place the fix at the intersection of two or more
LOPs and/or at the centre of the cocked hat formed by 3 LOPs, is because it
is the most practical and realistic thing to do. Furthermore, even
recognizing that these points may not be the actual fix, there is no
practical way to determine where the actual fix is located as the
possibilities are many and there does not appear to be a reliable way to
ascertain which one is the real McCoy. Gary, correct me if I am wrong in my
interpretation.

This gets back to the point that I was trying to make when this discussion
first came up: for purposes of practical navigation, there is no use in
getting wound around the axle about whether or not the fix only has a 25%
chance of being inside the cocked hat. It is all we've got and in the middle
of the ocean it doesn't matter.

Having said this, I would be interested in hearing from folks out there who
teach celestial navigation to find out what lessons they impart in their
students. Do you teach them to take the fix at the intersection of the LOPs
and/or centre of the cocked hat or do you hose them down with cold
statistical sea water just as they are beginning to grasp the fundamentals?
Further, I would be interested in knowing what is taught in military
navigation schools (including the air force). Does this topic even come up?
If so, how is it presented and what kind of advice is given to the students?

By the way, welcome back George. I was getting a little worried that you
appeared not to be biting at all of the delicious bait that has been thrown
out over the past two weeks!

Robert




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