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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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