NavList:

Martin, you wrote:
"The practical experience would also include systematic errors..."

The least-squares analyses and the entire setup of this problem assume zero systematic error. So you're comparing apples and oranges. If you have sights where you believe that the "lion's share" of the error is systematic (meaning that it is an identical amount for each sight), then the technique you are describing is fine, and it works. But then you have to ask the question: why do we have systematic error?? Can't we fix that?! Usually we can, and that should be our first priority before we even take our sights, but not always.

The secondary problem with the method you describe is that it has become an object of gross misuse. It applies to systematic error, which can usually be eliminated by proper adjustment of a sextant or other techniques. This angle bisecting method (or constant offset methods) is appropriate for systematic error if, for some reason, we have to leave systematic error uncorrected, but it absolutely does not apply to the general case of random error in our sextant sights. If there's random error, then we account for it by calculating the "least squares fix" and the "least squares error ellipse" Navigators have been clutching at straws for decades using nonsensical methods for dealing with the "triangle" of a three-body fix and avoiding the known solutions.

The "least squares" problem was solved and solved without doubt decades ago. It did not catch on universally despite the fact that the "least squares fix" has been included in the calculation section at the back of every edition of the Nautical Almanac for nearly forty years. Why is that? In part it's because this is after the "disco era" watershed... the era when celestial navigation became fossilized. In part, it did not catch on because it is explained... sold... in a way that sounds difficult and without any transparent explanation. And in part it has failed because navigators have been taught to adopt bad methods. And yes, using the bisector method (or constant offset method) is a "bad method" since it applies exclusively to systematic error. Another popular "bad method" sold to navigators is the "danger corner" theory. By this method, the long pointy end of a thin triangle of three sights is the one that a navigator should report as the position "fix" most worthy of attention if there is a "danger" in that direction. This is wrong and literally nonsense on multiple levels.

From my perspective the major problem with the "least squares solution" is that it is usually delivered with maximum mathematical baggage without simply delineating the results and without providing any simple tricks that a practical navigator can apply founded on the least squares results. That's the position fix problem that I am trying to "fix".

Finally, you should never expect that any "least squares fix" will match your GPS position, except by rare random chance. That would be "magical thinking". There's no magic here. The goal is to avoid poor estimates of the fix and the error ellipse within the context of the normal accuracy of celestial navigation. Navigators who see a long thin triangle from three sights will often apply the "systematic error" methodology like you did to make it go away --problem solved, the triangle is "corrected" away! Or barring that, they will put their best estimate of the fix way out in the middle of the long thing triangle, near the "center of mass" as they see it. These methods are just wrong. In many cases they can turn a reasonable error of one nautical mile into an error of two or three nautical miles.

Let's not forget the grumpy solution to the issue that says, "yeah, so what? It ain't that accurate any way! Who really gives a damn if it's accurate to five miles or not?!" But this is usually just an excuse to avoid breaking bad habits. And it's usually explicitly in conflict with the rest of the work that a celestial navigator does while working up a fix. If a navigator thinks it's important to read out and apply the altitude correction for the Sun to the nearest tenth of a minute of arc, then why would that same navigator throw away a method for estimating the least squares fix to the nearest tenth of a mile? Again, for some navigators, the "bad habit" is a beloved habit. They'll never give it up. Can't fix that! But we, as intelligent navigators, can still learn something new. ...Or we hope we can, right?

Frank Reed
Clockwork Mapping / ReedNavigation.com
Conanicut Island USA