A Community Devoted to the Preservation and Practice of Celestial Navigation and Other Methods of Traditional Wayfinding
From: Frank Reed
Date: 2010 Mar 18, 11:16 -0700
Antoine you wrote:
"IN OTHER WORDS, OUR POSITIONS ARE WRONG BY ALMOST 20 MILES, WITH LOP'S ALMOST PARALLEL AND WITH INDIVIDUAL KNOWN OBSERVATION ACCURACIES OF SOME 1 NM. SO, WHAT IS THE REAL OPERATIONAL VALUE OF ANY FIX DERIVED UNDER SUCH CONDITIONS ???
If I were a Skipper - Aircraft Carrier, Big Container Ship or small sailboat whatever - and as indicated yesterday, from this set of observations, I would MOSTLY and UNIQUELY keep the main and almost ONLY information they can confidently give : i.e. just one LOP as depicted yesterday. Anything else seems to me HIGHLY SUBJECT TO CAUTION.
Only in high open oceans, and far away from everything, would I give such Fix precedence over to my DR position. Unless I am totally lost without any DR whatsoever, in which case ... better "a little something" than a "big nothing" ... But in such a case I would have to be be prepared for a big position update sooner or later."
Sure. This is ten sights in a row over twenty minutes with typical standard deviation of the sights of somewhat more than a minute of arc. You said you would have to be prepared for a big position update. Ok. Just keep shooting. Suppose I had continued what I was doing yesterday morning, taking an altitude every two minutes. We're not particularly interested in the changing error of the fix in the direction towards the Sun's azimuth. That's good enough after a single sight. But what happens to the uncertainty in position perpendicular to that? The output from any position calculation is not a point. It's an error ellipse. With ten sights over twenty minutes, we end up with a long thin ellipse oriented with its long axis perpendicular to the mean azimuth of the Sun. It's that long axis that we want to improve. If we continue shooting every two minutes, first, we get an improvement proportional to sqrt(N) where N is the total number of sights just due to statistical averaging. We also get an improvement that will usually be at least roughly proportional to N since the spread in azimuths of the individual sights is proportional to the number of sights (for sights at a constant time interval, like two minutes). So that means that the long axis of the ellipse is shrinking at a rate of approximately N^(3/2). If the long axis of the ellipse is 49 n.m. after ten sights, then after twenty sights, it will be 17 n.m. After thirty sights (an hour's worth), it would be just 9 n.m. And that's the 95% certainty ellipse. The 1 s.d. ellipse is half that size.
I don't see this as particularly practical since it requires a computing device of some sort. And if we have that, then we miaght as well just turn on the GPS, unless it's just for fun (which is fine, too). For a small-boat navigator, if you shoot four Sun sights at the beginning of the hour, average for one LOP, and then shoot four sights at the end of the hour, average for another LOP, and cross, you would only have about twice the uncertainty of the procedure above. Of course the traditional LOP plotting procedure provides no error ellipse so the comparison is obscured. The only semi-practical case worth considering is some automated system. Imagine an automated sextant shooting altitudes a hundred times faster (every 1.2 seconds). While the azimuth range obviously won't be affected, the statistical sqrt(N) factor would be ten times bigger and the position would converge ten times faster. You could expect a position accurate to 9 n.m. at 95% confidence level in just six minutes. Now suppose that this auto-sextant can measure altitudes with four times better resolution than my "octant" data example. That wouldn't be an unreasonable level of accuracy to expect. That improvement applies in direct proportion to the error ellipse so now we're down to a little over 2 n.m. for the length of the error ellipse perpendicular to the Sun's azimuth in just six minutes. That's a pretty good fix in a short period of time. And yes, I realize that an automated system like that is a long, long way from traditional navigation, but at least it would still be celestial! :-)
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