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    Re: R: Problems with AstronavPC
    From: George Huxtable
    Date: 2004 Feb 17, 21:35 +0000

    Fedirico Rossi wrote about the equations for fix error in AstronavPC-
    >I've read your email and it proved useful in clarifyng some aspects of
    >the problem, still one thing is still obscure to me: the formula to
    >calculate the variance S reads:
    >S = F - DdB - EdL cos B(F)
    >I guess that D and E are the same that appear in the formulas to
    >calculate lat and long of fix while B(F) is the latitude of fix.
    >What about dB and dL? Are they the difference respectively in latitude
    >and longitude between the fix and the initially assumed position? If so,
    >what is F?
    Sorry, I didn't compare the texts, in the Almanac and in AstronavPC,
    closely enough. Thanks to Federico for seeking clarification.
    In the Nautical Almanac version, F isn't defined, because it's needed only
    for the error estimate, which is in the AstronavPC version only.
    The missing equation is-
    F = p(1)*p(1) + p(2)*p(2) + ...  [the sum of the squares of the intercepts]
    where brackets are written here, instead of subscripts.
    As Federico surmised, D and E are the same as in the Almanac section 11.
    The text of the AstronavPC equivalent of section 11, which is numbered 7.4,
    is effectively the same as in the Almanac, with the addition of the
    definition of F, until you get to "where the number of terms in each
    summation is equal to the number of observations." Then 7.4 continues as
    "As a check verify that A + C = n where n is the total number of
    observations included in the solution.
    Calculate G = A * C - B * B
    Then an improved estimate of the position at the time of fix L(I) , B(I) is
    given by
    L(I) = L(F) + dL     and B(I) = B(F) + dB
    where dL= (AE - BD) / (G cos B(F)) and dB = (CD - BE) / G
    Calculate the distance d between the initial estimated position L(F) , B(F)
    at the time of fix and the improved estimated position L(I) , B(I) in
    nautical miles from-
    d = 60 * square root of ( (dL * cos B(F))squared + (dB)squared)
    If d exceeds about 20 nautical miles set L(F) = L(I), B(F) = B(I) and
    repeat the calculation [note that this involves recalculating the
    intercepts and azimuths- George] until d, the distance between the previous
    estimate and the improved estimate, is less than about 20 nautical miles.
    It is possible but not advisable to start the iterations with a position
    that is in a different hemisphere. Provided L(I) is kept in the range
    -180deg to +180deg and B(I) in the range -90deg to +90deg the solution in
    most cases will begin to converge after a few iterations."
    =============== end of quote.
    Some further comments from George-
    The text above isn't entirely clear to me.
    As I pointed out earlier, the final paragraph describes choice of starting
    values for L(I) and B(I), but those are not values that you initially
    choose, they are the improved long and lat. The chosen starting values are
    L(F) and B(F). It seems most likely to me that the final paragraph intended
    to refer to those quantities
    The size of dL and dB will shrink at every iteration, and if iteration is
    continued, well past the suggested 20-mile limit then both dL and dB would
    become arbitrarily small. The equation for S in 7.5 is-
    S = F - DdB - EdL cos B(F),
    That calculation for S appears to include an allowance for the
    "improvement" stage of the last iteration. If reiteration was continued
    until dL and dB were allowed to shrink to negligible proportions, then we
    would get convergence toward
    S = F, where F is the sum of squares of the intercepts.
    I hope this is somewhat clearer now. Sorry to have complicated matters by
    omitting vital components.
    contact George Huxtable by email at george@huxtable.u-net.com, by phone at
    01865 820222 (from outside UK, +44 1865 820222), or by mail at 1 Sandy
    Lane, Southmoor, Abingdon, Oxon OX13 5HX, UK.

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