A Community Devoted to the Preservation and Practice of Celestial Navigation and Other Methods of Traditional Wayfinding
From: Frank Reed
Date: 2019 Aug 3, 19:34 -0700
Bill Lionheart, you wrote:
"I used the cocked hat in navigation as a simple example, and quote the Admiralty Manual of Navigation's suggestion of taking the corner nearest danger as a practical example of NOT assuming the least squares solution."
But this is just navigation "custom" and that's exactly how it's described in the 1938 vol.3 of the Admiralty Manual. It's important to know what's "customary" in the history of navigation, but that doesn't mean it's logical or that it makes sense in the 21st century.
This procedure is usually described in examples of "cocked hats" from piloting sights, and in that context it makes some sense in a certain specific class of possible error: those cases when there is suspicion of a navigational "blunder" --not normal random "noise" in the observations but an actual mistake. For example, you shoot bearings on three shore points which you think you have identified, and you're expecting the resulting lines of position to cross with no more than a quarter mile separating them based on prior experience. But when you plot, you get a triangle that's three miles across. It's evidence that something had gone very wrong. One likely explanation is that one of your shore points was misidentified. You should throw out one line of position leaving a two-bearing fix, and if you have high confidence in any two of them, you can do that. But suppose you don't know which one is bad. The sensible choice is to pick the two-bearing fix among the three that will put you nearest danger as you move ahead (not the point nearest the rocks or reef or whatever now, but the point nearest future dangers as you project ahead). This is a method of choosing which line of position to toss out when you have a strong reason to believe that one of them is simply bad data.
Contrast this with the case where there is no evidence of a blunder. You shoot three star sights. You know for certain that you have the right stars. You expect something like a minute or arc random error in each sight. When you plot your lines, you find that they make a long, thin triangle with a short side that's, let's say, 0.7 miles long and long sides that are 5.0 and 5.2 miles long because the latter two sights were from stars on nearly opposite azimuths. Everything is consistent with random error. This triangle is consistent with good observations containing a little normal random error. And in this case your fix should be quite close to the short side. You could also draw an error ellipse around that fix. The two corners of the triangle on the short side would be inside that error ellipse, while the third crossing point would be well outside of it. There's nothing special about that third point, and no navigation should be based on it, even if that point is close to danger (or future danger when projected). Further, we can think about this in terms of the confidence that could be placed in the three two-body fixes that make up the triangle. In the triangle I've described, the two two-body crossings on the short side have crossing angles near 90°. Those would be considered "good" two-body fixes taken individually, by themselves. Meanwhile, the crossing at the third corner has a rather shallow crossing angle and it would be counted as a "weak" fix by itself with a long narrow error ellipse . So by "averaging" the two high-confidence two- body fixes with the low-confidence two-body, we can see again that the best fix lies close to the short side.