A Community Devoted to the Preservation and Practice of Celestial Navigation and Other Methods of Traditional Wayfinding
From: Frank Reed
Date: 2019 Oct 29, 10:32 -0700
Calculate two points on each LOP. For any LOPs that need to be advanced, use standard DR to update the points.
But the point of modern celestial navigation, if we're being very careful, should be to produce an error ellipse: a fix and an uncertainty envelope around it. There are standard methods to get an error ellipse. These do not usually consider the declining quality of sights in a running fix. For older LOPs that have been advanced for longer periods of time, you should put "weights" on the sights determined by some decay factor dependent on the quality of the dead reckoning. A simple time-dependent weighting factor would be 2-t/T where T is chosen based on experience and conditions, but might typically be two hours. This puts a "half-life" on each sight equal to T hours (yes, e-t/T might be better form, but I'm doing this to make the math easy for an example).
For a simple example, suppose I get the latitude at noon. It's 38.8°N. This gives an E/W LOP. Then four hours later, I take a sight when the Sun is on the prime vertical, just for the sake of the example, which gives a longitude of 69.5°W. This is a N/S LOP. If I have been travelling south at 9 knots during those four hours, then I can advance my noon LOP pushing it south by 0.6° of latitude. Without weighting, your error ellipse is a circle, let's suppose with a radius of 1.0 nautical miles, centered on 38.2°N, 69.5°W. With weighting (and assuming T=2hrs), your first LOP has a weight of 0.25 while the second, fresh LOP has a weight of 1.0. How do you handle a quarter of a line of position?? It's easier to imagine if we just multiply the weights by four. The first LOP is equivalent to one common line of position while the second LOP is equivalent to four common lines of position. The result is an error ellipse centered on the same point, 38.2°N, 69.5°W, but now the error ellipse is "taller". No longer a circle, it's an ellipse with a long axis in the latitude direction (N/S) of 2.0 nautical miles and a short axis (E/W) of 1.0 nautical miles.
If you weren't even considering error ellipses, then count this as "just something to think about".