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Taking four stars for checking accuracy of fix - and "Cocked Hats"
From: Geoffrey Kolbe
Date: 2008 Aug 03, 08:43 +0100
From: Geoffrey Kolbe
Date: 2008 Aug 03, 08:43 +0100
Since, in this discussion, my name has been associated with a fix taken with four bodies at the cardinal points, I have a few observations to make. George wrote: > > ... it should be noted that whenever such a claim is made, that the > > true position is three times more likely to be outside rather than inside a > > cocked hat, it is (or should be) noted that any systematic errors have been > > first corrected for. Peter replied: >Sorry George, but this seems to be quite illogical nonsense. You >don't seem to have grasped the basic idea, and have thus tried to put >the cart before the horse. You CAN'T 'correct systematic errors >first' since their correction leads to a single point - a fix >position. How are you then going to go about correcting for your ... >whatever you like to call the other, erratic variety? > >This is WHY erratic error gets corrected at source, leaving an >assumption that what remains must be ... constant, or systematic, or >whatever you feel happy about naming it. My response: Why can't you correct for systematic error first? The point about systematic errors is that for any particular instrument, they can be identified and measured and thus accounted for in a measurement on that instrument. I have used this example before, but as it seems appropriate I make no apologies in using it again. Look at http://www.pisces-press.com/C-Nav/nav-plots/14%20March%20big.jpg and you will see a plot of an actual fix in the Egyptian Western desert, worked up with observations of bodies at (or close to) the cardinal points. There are a number of points to note: First, each of those LOPs is actually the result of five measurements on the body itself. Averages were taken of the five altitudes and times of observation - usually a minute apart - and then an Hc and Zn was calculated for the average time and altitude. Peter's method is to plot altitude against time and draw a straight line through the results. This amounts to much the same thing and - if the straight line is drawn correctly - my average should sit on a straight line fit through the results. Of course, there are statistically kosher ways to make straight line fits to data, but an eyeball fit actually works very well. As George has mentioned, taking averages of data to minimize random error in data does not get rid of the random error, as Peter seems to imply. It only reduces the random error to acceptable limits. (If the random error is still unacceptable, take more data points!) I would also caution against throwing out data. In my experience, I have ended up with a better Hc by keeping all the altitudes - even the "weirdo" ones. It is fairly easy to see if a succession of sights on the same body are giving good results. Given that the time between sights will be fairly constant, the difference in measured altitude should also be fairly constant. If I see it is wandering about for some reason, I take more sights and let statistics take care of whatever is getting in the way of me taking accurate sights. Second, it will be seen that the LOPs in the picture form a box, with the bodies observed always on the far side of the box. From this, I can deduce that the dominant error in the data is now a systematic error, which in this case is an index error due to the desert heat warping the sextant or index arm slightly. Given that the box is about 10 minutes of arc on a side, I can see that there is some 5 minutes of unaccounted for systematic error in the data. Now look at http://www.pisces-press.com/C-Nav/nav-plots/19%20March%20big.jpg for a fix taken a few days later, at our next campsite a little further North. This time I included the extra 5 minutes of systematic error which I found a few days earlier and the result is pretty much the single point fix Peter was talking about. So, to sum up, this is an example where I DID "correct systematic errors first" and where their correction did lead to a "single point fix". (Of course, where the pen is drawing a line which is about a minute wide, a "single point fix" should be taken in context.) It should be noted though, that the final fix was still a minute or so away from the actual position. In this case, it is not possible to determine if the residual errors are random or systematic. There are ALWAYS residual errors in data. Where random and systematic errors have been reduced to the precision of the instrument, it is not worth the effort to try and reduce these errors still further. With this instrument (an A-12 bubble sextant) an accuracy of 1 or 2 nautical miles on the fix is as good as it is going to get. On the subject of "COCKED HATS". In the past, I have always agreed with George that where there is no systematic error in the measured altitudes, the probability of the actual position being outside the cocked hat is three times higher than being inside. But my position has shifted slightly on this one. I wonder if I can induce George to come along with me. (In the following analysis, assume that there is no systematic error of any kind). First, consider a measured altitude on Polaris. On reducing the sight, I will end up with an LOP which will run East-West across the chart. What can I say about my actual position relative to this line? I can say that the probability of my position being on or near one point of the line is equal to that for any other point on the line. (In other words, I have no information regarding my longitude). I can say that the probability of my actual position being North of the line is 0.5, exactly the same as the probability of being South of the line. Now, suppose I take a sight on the star Canopus when it is on the meridian to the South. I reduce the sight and draw another LOP on the chart. This LOP also runs East-West across the chart, but I find that it is about 2 nautical miles (say) to the South of the Polaris LOP. What can I say about my position relative to this new Canopus LOP? Since it runs parallel to the Polaris line, I still have no new information about my longitude. The chance of me being on or near one point of this line is just the same as for any other point. What can I say about my position North or South of the line? Is the probability of my actual position being to the North of the Canopus LOP still 0.5, exactly the same as it being South of the line? I would now say no. Because the Polaris LOP sits to the North of the Canopus LOP, I have an independent piece of data which indicates that my position is actually more likely to be North of the Canopus LOP than South of it. Statistically speaking, this is a so-called "Bayesian" approach to statistical analysis. I have some a priori information regarding my actual position, which can inform my estimation of my position with respect to the Canopus LOP. Actually, there is no reason why I should have taken the Polaris sighting first. I could just have easily have taken the Canopus sighting first. That being the case, I can say by symmetry that the Canopus LOP can similarly inform, regarding my position relative to the Polaris LOP. Now, I can say that the probability of my position being South of the Polaris LOP is higher than the probability of being North of it. I will stop there as I am sure George will see where I am headed with this argument when applied to the "cocked hat" problem. And since the essence of the argument is already laid out, I need go no further to persuade George if he is willing to come with me this far. Geoffrey Kolbe --~--~---------~--~----~------------~-------~--~----~ Navigation List archive: www.fer3.com/arc To post, email NavList@fer3.com To , email NavList-@fer3.com -~----------~----~----~----~------~----~------~--~---