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Re: Error ellipse proportions
From: Frank Reed
Date: 2017 Jun 11, 11:34 -0700

Tom Sult, you wrote:
"I'm generally happy if I get within 5 miles of GPS. Now make the rhombus with 5 mile wide corridors."

Sure. That's right --except it is not a rhombus. It's an ellipse. If under a specific set or rather trying circumstances (which you described as "small sailboats in racing conditions in a good blow"), you find that your typical error level is +/-x miles, well then now you also know how long the ellipse should be for a pair of LOPs crossed at some angle dZ. For the unobvious cases (anything near 90° crossing angle is obviously a circle), you have the simple rule that the ratio is dZ/114 when dZ is given in degrees. Consider a practical example: you shoot the Sun at 9:00 and get a line of position. Some time later, when its azimuth has shifted south by two points (22.5°), you shoot it again and plot a second line of position. So where is your boat? Your fix is where the lines cross, and your error ellipse extends along the bearing between the two sights (which should be obvious) and has a width that is 22.5/114 shorter than its length. Or flipping that ratio, the ellipse is five times longer than it is wide. You go ahead and decide how wide it is based on your experience. You now have a mathematically legitimate means of assessing how long it should be as well. If that isn't useful to you, then you're not really navigating. Maybe it's just entertainment with no practical value?? ...not that there's anything wrong with that! But if you are navigating, then an error ellipse is the second most important thing you get from your observations. The lines of position are a means to an end and should be thrown out. The result of the process is the fix and the error ellipse.

Also, while I'm at it, it's important to specify just what you mean by your "error limits" or "error bars" on your sights. You have said here and in other posts that you're comfortable with +/- 5 miles in the rather difficult conditions that you described. Do you get better results under easier conditions? What sort of error bars do you see when you take practice sights from land under excellent conditions? And when you quote these numbers, are you thinking of them as "1 s.d." ranges, implying that roughly two-thirds of sights are within those limits, or do those error bars correspond more to "2 s.d." ranges, implying that 95% of sights are inside those limits? Also, are those ranges on "either side" of the known position, or is that the full-width of the error band around the known position? Any of these options are valid and reasonable, but we can't compare at all without being specific. If one navigator says "my fixes are within +/- 1 mile under good conditions" and another navigator says "my fixes are within 5 miles in typical conditions", they could actually be describing exactly the same error expectations. Regardless of the specific personal choices, each navigator can use the same rule, as described already, to get the ratio of length to width for the error ellipse in a fix with two lines of position.

Frank Reed

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