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    Instrument Error estimation: An Experiment
    From: Phil Sherrod
    Date: 1999 Mar 13, 2:09 PM

    I enjoyed the message posted by Daniel K. Allen on 12 Mar 1999 regarding
    his experiments to determine the accuracy of his sextant observations.  I
    performed a similar experiment on 20 Feb 1999, but for a somewhat
    different purpose.
    
    I own a surplus, Kollsman, periscope bubble sextant of the type that was
    used by the Air Force up until the 1980's.  The device has about a
    one-foot-long periscope tube where the light enters near the top, then it
    is reflected by a tilting mirror down into the body of the device where
    the image is superimposed on a bubble and then projected out the
    eyepiece.  It has a knob with an attached mechanical, digital dial to set
    the angle of tilt of the mirror, and it has another knob which selects
    one of 8 internal sun shades of different colors and densities.  The
    digital readout is sufficient to provide an altitude measurement to about
    the nearest 1/2 minute of arc.  There is also a built in mechanical
    averaging device that allows you to take 2-minute readings which are
    automatically (mechanically) averaged over the interval.  The sextant has
    connections for a battery pack to illuminate the internal bubble, but
    other than illumination (which isn't required during daylight use), the
    device is totally mechanical, requiring no electrical power.  It is a
    very nice piece of mechanical engineering.
    
    The purpose of my experiment was to determine the best estimate of the
    index error of the instrument.  To do so, I took 29 observations
    of the lower limb of the Sun over the course of a day.  I simultaneously
    took GPS position readings and averaged them to determine a good estimate
    of my true position.  For maximum accuracy, I had the sextant resting on
    a solid platform while making the measurements.  The timing was done with
    a digital watch synchronized to NBS standard time obtained through a
    high-speed Internet connection.
    
    The method that I used to determine the best estimate of the Instrument
    Error (IE) is somewhat unusual.  What I did was perform a non-linear
    regression analysis to fit the calculated altitudes with the observed
    altitudes after correction for refraction and semi-diameter.  For those
    of you who are not familiar with non-linear regression, it is very
    similar to the more common linear regression, except a non-linear
    function is being fitted to the data rather than a linear (straight line)
    function.  The "best estimate" of the parameters of the regression are
    those values that minimize the sum of the squared deviations, where
    'deviation' in this case is the difference between the observed altitude
    (after corrections for refraction and semi-dimameter) and the calculated
    altitude.  For my regression problem, the only parameter whose value was
    being estimated was the Instrument Error (IE).
    
    Using this technique, I was able to calculate that the best estimate of
    the instrument error for my sextant is +2.3' (i.e., you must add 2.3' to
    the observed altitude, so I believe this would be "off the arc").
    Applying this correction to my observed values, the worst observation was
    off by 8.8', and the average of the absolute values of the errors was
    3.07'.  This average error is about twice as large as what Daniel Allen
    reported when using his Tamaya Jupiter sextant, but not bad for a novice
    using a $90 surplus unit.
    
    For those of you who are interested in studying the analysis in more
    detail, I am attaching the source code and data for the non-linear
    regression analysis.  Although this program looks a lot like a C program,
    it is actually the programming language used to describe a function to
    be fitted by the NLREG nonlinear regression program.  If anyone cares to
    run the program, you can download a shareware version of the NLREG
    program from http://www.sandh.com/sherrod/nlreg.htm
    
    Phil Sherrod
    
    /*
     *  NLREG nonlinear regression analysis to estimate Instrument Error.
     *  Sights of the LL of the Sun taken 20 Feb 1999 using a bubble sextant.
     *  The location was 36 deg. 00.12' N, 86 deg. 50.70' W
     */
    Title "Sun sights 20 Feb 1999";
    Variables Time, H;  /* Input variables whose values are read as data */
    Parameter IE;   /* IE (Instrument Error) is parameter to be calculated */
    Double Ho,Ha,Hc,refract,dec,GHA,LHA;  /* Work variables */
    /* latitude and longitude (determined from averaged GPS reading) */
    Constant latitude = 36.0029;  /* (North) */
    Constant longitude = 86.8453;  /* (West) */
    /* Apply Index Error (note, IE is being computed by analysis) */
    Ho = H + IE; /* Value of IE will be computed by NLREG to minimize errors */
    /* Apply correction for refraction */
    refract = (0.0167/tan(Ho+7.31/(Ho+4.4)))/60.;
    Ha = Ho - refract;
    /* Apply correction for semi-diameter */
    Ha += 16.2/60;
    /* Estimate Declination and GHA around 18:00 GMT */
    dec = -10.88833 + 0.015*(time-18.00);
    GHA = 86.565 + 15*(time-18.00);
    /* Compute Local Hour Angle */
    LHA = GHA - longitude;
    /* Compute altitude */
    Hc = asin(sin(latitude)*sin(dec) + cos(latitude)*cos(dec)*cos(LHA));
    /* Here is the function whose error is to be minimized by adjusting IE */
    Function Hc = Ha;
    /* Plot the observed and calculated altitudes */
    Splot xvar=time,yvar=Ha,yvar2=Hc,connect;
    /* Write the values to a file */
    Output to "iecalc.out" Ha,Hc,residual;
    /* Here comes the data:
     * Time (GMT, decimal hours), Observed altitude (decimal degrees) */
    Data;
    14.54  21.653
    15.21  28.130
    15.87  33.710
    16.52  38.078
    16.73  39.333
    17.63  42.467
    17.86  42.677
    17.90  42.753
    17.96  42.883
    17.99  42.803
    18.01  42.833
    18.03  42.817
    18.05  42.850
    18.08  42.783
    18.13  42.838
    18.19  42.820
    18.24  42.717
    18.37  42.513
    18.50  42.342
    19.71  37.150
    19.75  36.833
    20.55  30.800
    21.27  24.233
    21.32  23.610
    21.35  23.317
    21.49  21.750
    21.53  21.500
    21.56  21.217
    21.62  20.530
    
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