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    Re: Cocked hats, again.
    From: Dave Walden
    Date: 2007 Mar 16, 22:03 -0700

    
    Below is a short FORTRAN program to do Monte Carlo simulations of a
    three point LOP.  It is rough and uncommented.  I provide it as is
    since there seems to be quite some interest at the moment.  I have
    found it, and other versions and modifications, very helpful as others
    have said recently such a program might be.  It's late, and I apolgize
    in advance if I mis-speak below.
    
    It assumes, without loss of generality, the true location is at x=0,
    y=0.  sd is the standard deviation of altitude uncertainty.  For each
    Monte Carlo run, for each LOP it finds the altitude error based on a
    random selection from a distribution of errors. h45 for example.
    (NORRAN is the subroutine.  The choice of random number generators can
    require some attention, as many may realize.)  It checks if the true
    location is inside the cocked hat, itest=3.  It finds the center of
    the cocked hat, the a,b,c,d,e,g calculation from the Nautical
    Almanac.  It calculates the distance from the true location to the
    center of the cocked hat, dist.  It accumulates some variables to
    provide statistics on this distance as well as x and y distance from
    the true location. It calculates the length of the side of this cocked
    hat, len, via CALL CHECK.  It calculates the number of times the point
    is with a cirle of radius of .5887*len of the true location.  It
    writes out dist, len, and itest.  By analyzing this file, one can
    calculate probabilites of such things as small cocked hats that do not
    contain the true location, how often large cocked hats do contain
    cocked hats, how often cocked hats of various sizes occur, etc.
    
    These quantities can be calculated analytically and I have found the
    Monte Carlo results to tend towards these results as the number of
    runs increases.  (A million only takes a minute on an old, slow PC.)
    
    In the sample output, next, one sees that of the 100,000 runs for
    25,230 cases, the true location was inside the cocked hat.
    Approaching 25%.   The average x and y errors are -.0012 and .0008
    respectively.  Approaching zero as expected.
    
    
     ii 100000
      number less than   0.37134999 *len away ratio=  26119
     in, inside cocked hat=  25230
     ii,sumsq,sum 100000  0.331236064  0.510192156
     sd  0.26634568
      sum, sumsq,sqrt( sumsq- sum**2)  0.510192156  0.331236064
    0.26634568
     xsum,xsumsq,sqrt(xsumsq-xsum**2) -0.00127442868  0.16588138
    0.407283396
     ysum,ysumsq,sqrt(ysumsq-ysum**2)  0.0008370672  0.165351331
    0.406633288
    sh-3.00#
    
    
    
    The first few lines from the output file,
    
    
        0.057159    1.233867   3
        0.724052    0.217705   1
        0.387911    0.322774   1
        0.608533    0.352897   1
        0.635789    0.097189   1
        1.136409    1.546760   1
        0.175892    0.142276   1
    
    
    show for example, a case where the distance from the center of the
    cocked hat to the true location is .057 min (or naut miles) and length
    of the sides of the cocked hat is 1.233 min and the true location is
    inside the cocked hat, itest=3.
    
    
    
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