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The following courtesy of:
	The Navigation Foundation  (Tel. 301-622-8448)
	P.O. Box 1126, Rockville MD 20850
	The Navigator's Newsletter, IssueTwenty-Four, Spring 1989


Dan Hogan
    dhhogan@earthlink.net     dhhogan@delphi.com

*************
The Personal Equation, Variation in the Dip Correction, Random Error in
Sextant Observations
by John M. Luykx

The two major sources of error in celestial observations are a) the
observer's personal error in actually measuring the altitude of the
celestial body with sextant and b) the variations in normal dip caused by
changes in refraction due to sea/air surface temperature.

Personal Error
	It is known that the technique, quality and accuracy of observation
will vary from observer to observer. Each individual observer, however, can
determine the quality of his observations by conducting a comprehensive
series of tests over, generally a long period of time and with several
celestial bodies. Such tests will provide siginificant results when the
observer is satisfied that:
	a. the sextanr I.C. is correctly computed
	b. the sextant was in correct adjustment
	c. the observer's height of eye is accurately known
	d. the sextant was not "canted" during observation
	e. a "good" observation was made
	f. the most accurate sight reduction method was employed
	g. no computational errors were made

A series of tests under the above conditions should be conducted by the
observer:
	a. with different celestial bodies
	b. at various altitudes
	c. under various conditions of weather and sea
	d. at different places

	The altitudes obtianed during the tests should then be compared with
computed values obtained by computer, sight reduction tables or mathematical
computation. The difference between the two (observed and computed values)
may be considered the personal error and applied with sign reversed to
future observations as a personal correction.

Dip Error
	The dip correction (height of eye) tabulated on the inside front
cover of the Nautical Almanac is based on the following formula which
incorporates a standard value for atmospheric refraction;
		D = 0.97 x SQRT h

	Where D = minutes of arc
	            h = hight of eye in feet

	Significant variations in dip error caused by non-standard
refraction may be experienced when large differences between sea and air
temperatures at the surface occur. Because non-standard refraction is
probably the single most important source of error in sextant observations,
a considerable amount of research has been conducted to determine the mean
values of refraction (used in the Dip tables), the conditions under which
values differ from the mean and the value of such differences.

	The results of this research indicate generally that:

	a. When surface air temperature is less that the suface  water
temperature, the value		of dip is greater than normal.

	b. When surface air temperature is greater than the surface water
temperature, the 		value of dip is less than normal.

	When significant variation in sea/air surface temperature exists,
corrections to the standard value of dip in the Nautical Almanac may be
applied as follows (h. of e. 10'-30'):

	1) For every ten degree decrease of thr surface air temperature
below the value of the surface water temperature, the value of the tabulated
dip is to be increased by 1 minute of arc.

	2) For every ten degree increase in the surface air temperature
above the value of the surface water temperature, the value of the dip is to
be decreased by 1 minute of arc.

	Dip meters of varing types have been developed during the past 100
years to measure the actual value of dip for normal heights of eye and for
any instant of time. These instruments are optical instruments which measure
the vertical angle from a point on the horizion through the observers's
zenith to the opposite point of the horizion. The angle subtracted from 180
and divided by 2 is equal to the actual value of dip at the time of
observation. An instrument of this type developed by Admiral Davies,
President of the Foundation, is described on pages 7-8 of issue 18 of the
Newsletter (Fall. 1987).

Random Error
	Individual observations by sextant taken on board a rolling, yawing,
pitching vessel often contain large values of random observational error.

	To improve the reliability of position lines from such sextant
observations, it is often recommended that a series of observations be taken
and the results averaged to reduce random error. The mean time of
observation and the mean value of altitude  for the series of observations
are then computed and used in sight reduction. Individual observations of
the series may be plotted on cross section paper (altitude verses time) and
a curve "faired" through a series of plotted points of observation. Unless
the observations were made at or near the time of meridian passage, the
faired curve would be nearly a straight line. Any point along the line may
be used as an observation.The slope of the curve of observations when
compared with the slope of a curve of altitude change, computed for the time
of observation will further indicate the accuracy of individual observations
in the series.

	The average of a series of observations was standard practice for
celestial observations taken from aircraft. Accurate and reliable mechanical
averaging devices for aircraft sextants [were manufactured] especially
during World War II by C. Plath in germany, Henry Hughes in England and by
Bendix and Kollsman in the United States. During World War II the Japanese
sextant manufacturer, Tamaya, developed a mechanical averger whic was
installed as an intergal part (not removeable) of a marine sextant of the
Gago-Coutinho type.

   

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