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I just spent the last couple of hours reading the entire set of messages in
this thread, at the Nav-L archives.  Thanks to that, and to the message Alex
just sent, I have revised my summary.  How does this look now?

Jim Thompson
jim2@jimthompson.net
www.jimthompson.net
Outgoing mail scanned by Norton Antivirus
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Averaging Sights

Sextant sights are subject to a variety of errors that lead to imprecision
and inaccuracy. This is a serious concern for a navigator who is relying on
celestial navigation to find his or her position at sea, making comfortable
sleep difficult. One way to deal with random observational error is to
average a set of several sights, and then to plot the average time and
altitude. See also Bowditch article 1609, and the Nav-L thread "averaging"
in October 2004.

Always apply basic principles to a run of sights:

1. Use 3-5 sights taken within a few minutes, a minute or less between
sights.
2. Use raw sextant observations, before applying corrections.
3. Inspect the set of sights for consistency and discard obvious outliers,
or sights that fail to increase and decrease in time or altitude reasonably
smoothly. It helps to plot the sights on a graph of altitude over time; or
build a table, use arithmetic to calculate the change in time and altitude
between sights, and then inspect the table for outliers.

And then, to average a run of acceptable sights:

4. Average time by adding up the minutes and seconds and dividing by the
number of sights, then adding that average to the whole hour.
5. Average altitude by adding up the minutes and seconds and dividing by the
number of sights, then adding that average to the whole degrees.
6. Work out the DR position of the average time based on vessel speed and
course, and use the latitude and longitude of that position to reduce the
average and plot the resulting azimuth and intercept.

Beware:

Simple arithmetic averaging improves precision in most cases by taking out
random error, but beware some pitfalls:

1. Averaging does not take out bias errors caused by problems in the sextant
or observer.

2. A celestial body's altitude changes in a non-linear fashion. But,
fortunately for navigators, altitude non-linearity in most of the celestial
window is smaller than our ability to detect it with an observational device
like a hand sextant, which in the hands of most of us allows a precision of
about than 0.5' of arc. The change in altitude over short time durations of
5-10 minutes is so nearly linear that for our purposes we can assume it is
linear. The important exception to this is when the body passes through the
observer's meridian at high altitudes, over about 60?-75?. The change in
altitude is significantly non-linear in that case. For runs of sights of
typical bodies taken up to five minutes, nonlinearity introduces a
systematic error of only 0.1 to 1.0 nautical miles.  The error is in the
lower half of that range under 60? of altitude, and greater than 20? from
the meridian. In the very worst case scenario, 2 minutes before and after a
body goes through the zenith, the error introduced by averaging is up to 30
nautical miles, but this is a very rare situation.

In the usual sight-taking altitude-azimuth window, navigators can average
their sights, comfortable that they are taking out random error without
introducing a signficant error caused by non-linearity in the way the
altitude changes during sight-taking. Averaging should not be used in
certain circumstances. I obtained these rules from a long thread about
averaging on the Nav-L list in October 2004. If these rules are followed,
then the systematic error owing to non-linearity in the change of altitude
over 5 minutes will be less than 1' of arc, or 1 nautical mile (0.5' arc and
0.5 NM if the navigator uses the 60? and 20? limits):

1. Use arithmetic averaging if the body is lower than 60?-75?.
2. If the body is over 60?-75?, then use averaging only if the azimuth is
>10?-20? from the meridian, and do not use simple arithmetic averaging if
the body is closer to the meridian at those high altitudes.
3. Use arithmetic averaging only if all sights are obtained within about
about 5-10 minutes.

Using Computers Instead of Averaging

With modern programmable caculators, handheld computers and laptops, it is
very easy for navigators to reduce every sight individually and then plot
all the reduced sights, rather than average a run of raw observations and
plot just the average sight. The navigator can then plot the 3-5 acceptable
(consistent) reduced sights for each body and graphically find the best
single intercept between them. Following this method, the navigator does not
have to take sights on the same body before moving to another body, but can
shoot bodies as the opportunity arrives, coming back to earlier bodies to
take additional sights if the horizon is still good.

And as Gary LaPook wrote on Nav-L during the averaging thread, "As long as
we are talking about the St. Hilaire method (computing an azimuth and
intercept from an assumed position) we should remember that it was developed
as an easy method of laying down the "Sumner Line," now called an LOP,
requiring only one computation. The original Sumner method required
computing two time sights, twice as much work. With programmable calculators
it is now just as easy to do the two computations and lay down the LOP
without measuring an azimuth or intercept or using an assumed position. You
simply choose two longitudes, one east and one west of your DR, and the
calculator calculates the latitude where the LOP crosses those longitudes.
You prick those latitudes on the chart and draw a straight line between
them."

The disadvantage of the computer methods is the risk of "blunder", of
entering a wrong data element into the computer, so that the reduced sight
ends up being wrong, even though the raw observation might have been
excellent. The risk of such blunders increases with the number of sights
reduced, especially for a fatigued navigator in a sailboat with a small
crew.

But there are many similar ways to think outside the box when so much
computational power is available to modern celestial navigators.

   

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