# NavList:

## A Community Devoted to the Preservation and Practice of Celestial Navigation and Other Methods of Traditional Wayfinding

**Re: Exercise #14 Multi-Moon LOP's**

**From:**George Huxtable

**Date:**2009 May 3, 15:31 +0100

Frank Reed wrote- "most of these software packages do exactly the same thing: they use the least squares solution for multiple sights provided in many sources, including every copy of the Nautical Almanac for some twenty-odd years. " Indeed, that's true. The Nautical Almanac provides a routine by which an increasingly good fit can be made to a series of observations by a least-squares process, under the heading "Position from intercept and altitude by calculation". What the almanac omits is any estimate of the errors involved in that process. That omission is made up for, in "AstroNavPC and compact data", from her majesty's nautical almanac office (in my now-outdated edition 2001-5, the error ellipse estimation procedure is described in para 7.5). He adds "In a (typical) rapid-fire fix, the error ellipse will be aligned with its long axis perpendicular to the mean azimuth of the sights (quite comparable to lat/lon at noon, by the way). The error in this direction is given approximately by (3/2)*S/(sqrt(N)*A) where S is the standard deviation of the observations themselves, N is the number of sights, and A is the range of azimuth in the sights (azimuth expressed as a pure number, "in radians")." That may perhaps be so, but Frank gives no reference or reasoning to back it, for us to check for ourselves. It's somewhat disconcerting to find that the multiplier constant is uncertain by a factor of nearly 2 (0.8 to 1.5). Does that expression take account of the uniform spacing of the observations over the time-span, as Frank proposes? How would it differ, then, if instead, half were closely grouped at the start of that time-span, and the rest at its end? That was a question that's been asked but not yet addressed. and continues- "So take Jeremy's moon sights in his "Exercise #14". First, we need the standard deviation of observations. Jeremy has posted quite a few cases, and he gets pretty good sights, with s.d. around 0.5 minutes of arc. In this particular set, N is 11, A is 0.1 (5.5 degrees is a tenth of a radian). So the standard deviation "cross-range" would be 2.3 nautical miles. Pretty good." Not quite that good. Indeed, Jeremy's observations show little scatter; my own estimate of the standard deviation, in [8062] was somewhat less than Frank's. It shows what can be done from a big-ship, in good conditions. Taking Frank's error formula at face value, if you refer to the data-sets in Jeremy's linked file, at http://www.fer3.com/arc/imgx/f1-Rapid-Moon.pdf the range of azimuths is 4.4 degrees, not 5.5, which makes the factor A 0.077, not 0.1, making the calculated scatter 30% greater, or now, 3 nautical miles. So our position ends up as somewhere within a bracket 6 miles or so either side of the estimated position. Not a marvellous result, when you take the high precision of the original observations, and the fact that it called for 11 such observations to be made. Alternatively, if, after just a single such observation, Jeremy had simply waited a few hours, then taken one additional Moon altitude, at a moment to suit himself, the uncertainty should have been within a mile. Or instead, he could have immediately taken a sight of another object in a different direction. So, hardly a saving of effort, this "rapid-fire fix". Instead, a labour-intensive method for discarding most of the inherent precision achievable from celestial observation, and ending up with a second-rate result. Frank has presented this particular observation as an example of his proposed method; indeed, the only example he has offered so far. Was this a typical situation, then? Far from it. Jeremy's own words, in [6066] explain "that is why I shot the moon, as the azimuth is changing rapidly, even away from the time or meridian transit" It was selected, deliberately, for enhanced effect. And indeed, that was the case. In the 7min 48 sec of the observation set, the Moon's azimuth changed by 4.4�, or at a rate of 34� per hour, mainly because it was rather high in the sky. In most circumstances, a typical rate of change is more like 15� per hour, corresponding to the spin rate of the Earth. In which case, the errors that we've seen in this example would be more than doubled. Frank had a dig at me, in [8145], when he wrote- "I reminded the group of an interesting case where someone on NavList actually tried this out for himself, rather than just pontificating and declaring it impossible from his armchair, as you have done, George. It was an EXAMPLE of the sort of results you can expect in the real world." It's the responsibility of the proponent of such a notion to rise from his own armchair and try it out to show results that prove its worth, but Frank has not bothered to do so. It's left to others to point out the weaknesses in the optimistic gloss, which is all we've been offered so far. George. contact George Huxtable, at george@hux.me.uk or at +44 1865 820222 (from UK, 01865 820222) or at 1 Sandy Lane, Southmoor, Abingdon, Oxon OX13 5HX, UK. --~--~---------~--~----~------------~-------~--~----~ Navigation List archive: www.fer3.com/arc To post, email NavList@fer3.com To unsubscribe, email NavList-unsubscribe@fer3.com -~----------~----~----~----~------~----~------~--~---