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    Re: The rapid-fire fix
    From: Frank Reed
    Date: 2009 Apr 28, 23:20 -0700

    John Karl, you wrote:
    "I don't understand the sight reduction of your "quick fix". You start with 
    one (averaged) altitude at 0930 and get the longitude by a time sight.  But 
    longitude by a time sight requires some other info, normally the latitude, 
    which is not known.  So what am I missing here in your discussion??
    Then you get the latitude form the two sights at 0910 and 0950.  I do 
    understand how this is possible because we now have two pieces of 
    information, two H's and the time interval."
    Whoa there! :-) I wasn't trying to suggest that this was how you would work a 
    rapid-fire fix today. Merely that there was a historical set of sights that 
    you could "map" the problem onto and to point out that even a 19th century 
    navigator had the tools to work certain special cases of statistical sights, 
    somewhat like the "rapid-fire fix" that I'm talking about.
    Incidentally, the requirement for a latitude in an "old navigation" time sight 
    was usually met with the DR latitude, and the best way to avoid any 
    sensitivity to error in that latitude was to shoot the Sun near the Prime 
    Vertical. But again, that's just a historical side-note --not relevant to 
    this rapid-fire fix.
    You added:
    "I don't understand your explanation of "using the rather complicated methods 
    found in most 19th century navigation manuals, we would get latitude from the 
    two altitudes and the time interval between them."  Are you referring to 
    something like I just discussed?"
    Yes. Just so. This particular problem was very popular among the 
    mathematicians of nautical astronomy back in the day. There are a number of 
    solutions. One of them, the preferred solution in Bowditch in mid-century, 
    was the topic of one of the presentations in Mystic in June, 2008.
    And you wrote:
    "On the other hand, there is explicit method in my book on pp 78-79, eqs 7.5.  
    These are a simple, understandable, application of the familiar H and Zn 
    equations.  The method uses only one more trig equation than two St. Hilaire 
    LOPs.  And it gives both Lat and Lon explicitly, without requiring tables or 
    the 0930 calculation."
    That's mathematically almost identical to crossing two LOPs. You can use that 
    with any pair of sights. But I ask you, John: if someone gave you eleven 
    altitudes of the Moon taken every couple of minutes in close succession, what 
    would you do with them? I'll give you a big hint. You mention the solution in 
    your book peripherally just before describing the method leading up to your 
    set of equations 7.5.
    And incidentally, what kind of error ellipse would you put around a fix 
    derived from your equations 7.5? A disadvantage of a direct solution like 
    that one is that there is no indication during the calculation that you may 
    have a pair of LOPs that don't yield a good fix. By contrast, a standard LOP 
    plot of two sights does at least show plainly that the two sights may be 
    yielding nearly identical positional information (from two objects in the 
    same or opposite azimuths, in other words). The standard LOP plot doesn't 
    help much in that sense when we get to a dozen sights or more. In fact, the 
    plot looks more confused and less accurate while the actual fix derived is 
    increasingly accurate with more sights. So what we need is a set of equations 
    that gives a least squares solution for the fix (and I should add that this 
    is a running fix) from multiple sights and ALSO gives some error ellipse on 
    the resulting position. After all, what we want to know is not just the best 
    estimate of the position, but also a reasonable estimate of the amount of 
    trust that we should have in that position. Such solutions are available and 
    have been available in many software solutions for celestial navigation for 
    years and years. But because it's hidden in the code, the significance of 
    these solutions is very much un-appreciated.

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