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    Re: Fix Maximum Probability Positions
    From: John Karl
    Date: 2013 Mar 24, 12:44 -0700

    Geoffrey: Thanks for your response. Here's my answers:

    From GK:
    In figure 1 of your paper, you show a circular area of uncertainty around your DR, which given no other information is reasonable enough. But in figure 5 you show an elliptical area of uncertainty, for which I can see no justification.

    From JK:
    We’re assuming that with no other info, the DR uncertainty grows equally in all directions. After the first DR position with the circular symmetry, Figure 2 starts with the oval narrow dark-blue line, its width uncertainty determined by the LOP’s accuracy (for CN), and its length uncertainty formed by the previous DR uncertainty’s size and shape (light blue). It grows all directions in size while retaining this the oval shape.

    From GK:
    Suppose, in figure 5, your course had been along the LOP1 so that DR1 was in fact on LOP1. What would your area of uncertainty around DR1 look like then? Since your original DR started out as a point (as shown in the figure) it would be circular, just as in figure 1.

    From JK:
    It would still be oval, with it’s long axis parallel the LOP1, because it grew from expanding equally it all directions from the oval narrow dark-blue section on LOP1. The uncertainty area would only be circular when the origin of the DR run is a true fix of two or more LOPs. So in Figure 3 all the DR uncertainty areas are oval.

    From GK:
    The point about advancing an LOP along the vector of your course and distance run is that it does not create a new DR position, as you attempt to do. It retains the LOP as an LOP, but makes use of the estimated course and distance run information in a reasonable way to re-locate the LOP.

    From JK:
    The estimation approach uses all available info, makes no unnecessary assumptions, and no contradictions. (This seems to be a difficult principle accept.) After a DR run with its associated uncertainties (in all directions), we are indeed at a new DR location.

    But the traditional advancing of the previous LOP to the DR position, and placing RFIX at its intersection with the new LOP, makes a very unreasonable assumption – it can even be viewed as a contradiction: As I’ve written many times this tradition assumes that our reckoning over the last run has a perfectly accurate component perpendicular to the advanced LOP, while at the same time, has a perfectly inaccurate component along that advanced LOP. Furthermore, the selection of these components is entirely dependent on the orientation of the advanced LOP, which obviously has no connection whatsoever with DR errors along the run between LOPs. Apparently, tradition has faith that the stars know how to command the drift and currents to force perfection on one component of our DR track, but not the other.

    From GK:
    Celestial navigation is not actually a method of navigation. "Celestial navigation" is a misnomer. "Celestial navigation" does not give you course steered, distance run, or your most probable current position, which is what "navigation" is about - having a good idea where you are at any given moment in time. Position fixes using celestial bodies are a way to get an independent check on your DR position. The navigation method used is actually dead reckoning first, last and always.

    From JK:
    Doesn’t “navigation” include all types, from GPS to depth contours, to spotting of land birds?

    From GK:
    It follows then that polluting your celestial fixes with where you arrogantly think you are is a dangerous way to proceed. It throws out the independent check.

    In your figure 5, if your LOP2 is good, then that should give you cause for concern that your course steered was not what you thought it was. There may well be a current flowing to the East which you had not suspected. You, however, are not treating CN as a check on your DR, you are using CN in combination with estimated course and distance run to create a DR, which - I respectfully submit - is dangerous.

    From JK:
    The EPRF does not pollute information, it uses all known info, and no more, i.e., it doesn’t invent any info (like the TRF does with its perfect perpendicular distance between the LOP and its advanced version). In the estimation approach when the DR is updated with a new LOP, it usually doesn’t fall on the DR location (very unlike the TRF, where without justification the advanced line is always plotted exactly through the DR location). It’s the EPRF that gets an indication of error (the distance between the new LOP and the DR location). In contrast, it’s the traditional running fix that ignores this independent check – and with that advanced LOP drawn exactly through the DR it’s certainly not independent of it!

    Furthermore, since the RFIX distance-to-the-DR location is largely determined by crossing angles of the LOPs, that measurement is pretty irrelevant for evaluating the DR error. So for these reasons, it’s the traditional running fix that does a darn poor job revealing the DR error.

    IN SUMMARY: The EPRF uses all available info, makes no unjustifiable assumptions, and makes no contradictions. The TRF makes assumptions that are clearly false, and gives unrealistic estimations (some zero, some hundreds of miles) in arbitrary directions of the dead reckoning errors (determined only by the stars).

    BTW, one traditional example of the EPRF is the old consecutive noon-sun latitude sights. They give consecutive parallel LOPs of latitude, with longitudes determined by dead reckoning. The updated ship’s position is plotted by dropping a line from the previous DR position perpendicular to the new latitude LOP. Nobody’s ever complained about that.

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