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    Re: It works - within limits.
    From: Arthur Pearson
    Date: 2002 Apr 11, 21:41 -0400

    This has been a great thread but I must make a request for some remedial
    education. I am generally familiar with St. Hilaire technique as the
    method by which I calculate an intercept and azimuth from by assumed
    position and use them to plot an LOP from a sight. I get the feeling
    from this thread that the formulas for a calculated fix solution given
    in the back of the Nautical Almanac are an extension of this
    methodology, but I am unsure where St. Hilaire begins and ends, and
    whether least squares was part of his contribution. Could someone make a
    short statement of the essence of his technique for the uninitiated?
    Also, Herbert states below that "When starting from a wildly wrong DR
    position, St. Hilaire will get you the right fix, albeit only after 2 or
    3 iterations. That's not surprising, because direct methods will not
    even need a DR."  What are the "direct methods".
    Apologies to those of you for whom this is old hat, my introduction to
    celestial was less theoretical than the discussion here so I am playing
    catch baseball (or cricket?).
    -----Original Message-----
    From: Navigation Mailing List
    [mailto:NAVIGATION-L@LISTSERV.WEBKAHUNA.COM] On Behalf Of Herbert Prinz
    Sent: Thursday, April 11, 2002 9:18 PM
    Subject: It works - within limits.
    To find GMT and our position simultaneously we need the observation of
    altitude of any two celestial bodies, the distance of the Moon from any
    body, and the time intervals between these three observations.
    The simplest case from a mathematical point of view is to measure the
    of the Moon and second body themselves (because they are needed anyway
    clearing the distance), and to measure all three quantities at exactly
    the same
    moment. One can cheat a little on the latter by bracketing the distance
    observation with the altitude observations and subsequent averaging.
    While it's true that the required altitudes can be computed, we also
    know that
    there is no free lunch. To use computed altitudes merely means that we
    already observed some (other) altitudes at an earlier stage (for the
    purpose of
    finding time and latitude). The solution by this method is, therefore, a
    Consider the case where double altitudes of the Sun, or some such method
    is used
    during the day to establish local time and latitude. If a vessel sailing
    from New
    York towards the Azores in and out of the meanders of the Gulf Stream
    Sun altitudes around 10:00 and 14:00 to establish local time and then
    takes a
    lunar around 20:00, the dead reckoning of longitude made good between
    the former
    and the later observations can easily be off by 10 nm, and hence its
    local time
    be off  by 40s. So the error in computed Moon altitude could be up to
    10' of arc,
    hence the error of computed parallax up to 10" of arc, which in turn
    translate to an error of as much as 20s in GMT or 5nm in longitude for
    the final
    fix. This is not much in the scheme of things. Most navigators were and
    will be
    happy to get GMT within a minute.  I am only emphasizing that this is an
    additional error that does not appear if altitudes for clearing the
    distance are
    measured directly and that cannot possibly be detected or eliminated by
    mathematical tricks.
    Bruce Stark soft-pedals the impact of DR error on the accuracy of the
    final fix
    in his message "It works", of April, 2. The reason why it works for
    Bruce even
    "when both GMT and longitude are wildly uncertain" is that in his
    example, he
    does not depend on measuring local time at all; he computes it from
    accurate data
    and only THEN shifts assumed GMT and assumed longitude in sync with each
    so as to not upset their relation (defining local time). Naturally,
    after 2
    iterations, one gets the correct GMT and longitude from the lunar
    distance. But
    this is tautological. In the real world, however, local time is only as
    good as
    your dead reckoning since the time you established it. The "wildly
    uncertain" DR
    does, indeed, not matter up to the moment where we start with the first
    observation for time. But any subsequent error in dead reckoning will
    have its
    inevitable effect on the resulting fix for GMT from the final lunar
    Of course, there is nothing special about lunars here. This is a general
    with the running fix. Take the standard timed altitude observation of
    two stars
    as an example. When starting from a wildly wrong DR position, St.
    Hilaire will
    get you the right fix, albeit only after 2 or 3 iterations. That's not
    surprising, because direct methods will not even need a DR. The same is
    true for
    a running fix, if and only if you are absolutely sure about your dead
    between observations. But if you screw up on the DR (e.g. by getting
    into a
    current) no method will tell you. All of them will result in the same
    wrong fix.
    There is another, minor problem with computed altitudes. I am not sure
    this has been mentioned already.  They rely on an unverified assumption
    there being ideal atmospheric conditions in the direction of the Moon
    and second
    object. But if there are unusual circumstances, the effect can result in
    up to
    12s error in GMT for every 0.1' deviation from standard refraction. If
    measures the altitudes, this error is automatically eliminated, at the
    cost of
    only the corresponding negligible positional error. Using computed
    altitudes is
    thus inherently less safe than measuring them.
    All numbers I gave are worst case scenarios that I could THINK of. I
    never lost
    time at sea and never had to depend on a lunar distance. So, I really
    don't know
    what I am talking about. All I have ever done were isolated experiments
    various kinds.
    Herbert Prinz

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