# NavList:

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

**Systematic Error (LOPs revisited)**

**From:**Peter Fogg

**Date:**2003 May 28, 15:51 +1000

Some time ago what turned out to be the biggest controversy of my experience of the Nav. List erupted, with the claim by George Huxtable that the fix position was 3 times more likely to lie outside 3 intersecting position lines (Lines of Position, or LOPs in Americanese) than it was to lie within them. I was mightily impressed by the efforts different people put into this discussion; diagrams drawn, computer programs written, websites set up, people 'reluctantly converted'. All good stuff. To recap briefly (and hopefully accurately enough): The claim was that the fix position cannot lie along any LOP since one is a line, by definition of infinitely thin width, and the other a point without area. Since an infinitely fine point cannot occupy the same (non-existent) space as an infinitely thin line the fix position must lie on one side or the other, thus a 50% chance of either. When 2 LOPs intersect this becomes a 25% chance of it lying in any of the 4 quadrants. With 3 LOPs there are 3 of these intersections, thus a 25% chance of the fix lying inside the triangle and a 75% chance of it lying outside. I was intrigued by this 3 to 1 claim and followed the discussion with great interest and not a little scepticism. After a while I stated that it is an irrelevance. My reasoning was that if you take any triangle of LOPs or 'cocket hat' there is only one fix position that can be found by the doctrine of least squares. If you move these LOPs outwards to encompass any possibily of the fix position lying outside them they must be moved in proportion. You end up with a larger triangle and an identical fix position at its centre. Once the triangle is bigger than the earth the possibility of it encompassing the fix is 100% but the fix position hasn't changed. What would be useful is a method of calculating where, if not in the triangle, the fix might lie. And even better, a method of quantifying the error; putting a number to it; establishing a better fix position than the centre of the triangle. In what seems so far a little noticed posting by George Bennett on the 24th of May called 'Position Line Plots' a practical method for identifying and quantifying systematic error has been proposed: 'There is a technique ..... that allows the navigator to make a simple analysis and assessment of his work' I am still working my way through this posting but the implications for the LOP controversy already seem profound. This discussion will be easier with diagrams so please draw your own as we go along. Lets start with those 2 intersecting LOPs. If there is a systematic error then it will be either towards the direction of the 2 azimuths or away. So instead of a 25% chance of the fix lying in 4 quadrants there is a 0% chance of it lying in 2 opposing directions and 50% each for the other 2. In this case the fix position lies along the line bisecting the two 50% quadrants, passing through the intersection. It is still just a position line. For a fix we need more. With 3 LOPs it gets a little more complicated and a little more interesting. This is where I came in, prompted by a half-remembered claim in a book by George Bennett that with an azimuth spread of less than 180 degrees the fix position will lie outside the triangle. So, draw 3 LOPs with an azimuth spread of less than 180 degrees and add a direction arrow (or more as they get extended) to each indicating the azimuth. Use the same technique as above for each of the 3 intersections of 2 lines. You now have 3 bisectors with a common meeting point in the area away from the direction of azimuths, and the distance from this point to the 3 extended LOPs is the radius of a circle that touches each LOP. Where the bisectors meet is the indicated fix position. With any 'cocked hat' triangle, if all the azimuth arrows point outwards or all inwards then the fix position lies within the triangle. But with 1 in 2 out, or 2 in 1 out (as with an azimuth spread of less than 180 degrees) the fix position lies outside the triangle. The technique George Bennett describes, of drawing auxiliary LOPs some (large enough) distance from the originals, consistently either all towards the azimuth directions or all away, induces a large artificial systematic error. A circle is then drawn that touches each of the 3 auxiliary LOPs. The difference between the radius of this circle and the distance used for inducing the large systematic error is the indicated extent of the real systematic error. Apparently this technique can be used for any number of LOPs, and it seems that for greatest accuracy 3 is not the ideal number. But the more LOPs the more horribly confusing become the plots. Use different colours for different types of lines, and dotted lines for the bisectors, and any other method (thin and thick lines, etc) you can think of to avoid confusion. Bear in mind its only the plots which become complex, the base idea is brilliantly simple: a systematic error affects results systematically. Realizing this yesterday was like an Eureka! moment for me, I wanted to jump out of my bath and run naked through the streets. Fortunately enough the weather here is a bit too cool and wet for that at the moment. Speaking of confusion, I am only groping towards an understanding of this, and its implications, myself. I certainly don't set myself up as an expert on this fascinating technique. Good on ya, George