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Re: Averaging
From: Bill B
Date: 2004 Oct 8, 18:19 -0500

```I would ask Alex et al if using an average of sights over a short time, (say
4 minute span for this discussion) from a moving vessel, what will he use as
DR/EP/LHA/AP when reducing his sight(s)?

In the big picture, our goal is to get a fix on where we are/were at a given
point in time to use for navigation.  Hence we advance LOPs along the course
line for observations not taken simultaneously.  Even in a 4 minute period,
a sailboat making 7 kn at N 41d 41.7') could change position by >  0.6'
(.4666 nm x 1/cosine 41d 43.7'). Perhaps a small error for a cruising
sailboat, but not so for powerboat or nuclear aircraft carrier hauling along
at 40-50kn.

It strikes me that if the intervals between observations were equal, he
could in fact average Hs or Ho and time.  He could then use a position half
way between the position of his first sight and last sight.

The above implies symmetry.  This is where it gets tricky for this
liberal-arts major.

Let's assume that instead of 5 sites, one taken every minute for 4 minutes,
the following timetable (from the staring time, in minutes and seconds)
occurs :

Sight 1:    00:00
Sight 2:    00:21
Sight 3:    00:43
Sight 4:    02:35
Sight 5:    04:00

This is where I ask the mathematicians for help.  It looks to me that if one
simply averages the altitudes and times, the altitude and time will be
skewed toward the initial three sights.  In the case of a moving vessel, how
does one determine where to place their position for the average used for
reduction?  Perhaps apply a ratio between the average time and time span of
the sights, times the total distance from the position at sight #1 to sight
#5?

Second, Alex and my conversation was meant (by me a least) to include David
Burch's paper on fit slope method.  It is available by pointing your browser
at:

http://www.starpath.com/catalog/accessories/psextants_midpage.htm

Click in, "How to take plastic sextant sights" to bring up that PDF.  In
that PDF click on "Fit slope method."

Or email me off list and when I gather a number of requests I will send you
the PDF with my old dial-up connection.

In Summary, Burch looks at averaging intercepts from reduced sights using a
common AP. "All of this presumes we have a calculator or software so we can
sight reduce from the same assumed position a whole series of sights in a
short time. Note that you can indeed use the same position for multiple
sights with tables using Mike Pepeprday?s S-Table method..."

As a side bar, I did some trials over 4-minute periods at 1 minute intervals
(using Omar's Hc as Ho and Ho229 reduction to eliminate the human/sextant
variable)) and found I could average the resulting APs and intercepts
(Towards positive and Away negative) and be right on the money.  Again the
symmetry question lurks in my mind.

Then Burch examines plotting the Hs values to obtain the best value to
reduce.

His conclusion is that it is often difficult to determine if, or which
observation(s) should be thrown out.  He goes on to describe how to
determine the actual slope of the body rise or fall over a 4-minute period,
and then moving the computed slope to the plotted data for better results.
A method covered, he states, in older versions of Bowditch.

Looking at Alex's query, I submit the following::

1.  In the following, ignore acceleration, special relativity,
Dutch roll etc.
2.  For the sake or argument, ignore changes in the Sun's declination.

Assume we are in a vessel traveling west at 15d per hour.  Our LHA will
remain constant, and there will be no change (again ignoring declination) in
the Sun's altitude.  Graphed--a flat horizontal line with spot-on
observations.

Assume we remain at one fixed position.  Our LHA will change by 15d per
hour.  If we plot the Sun's altitude every 4 minutes, we will see a series
of straight lines of varying slopes that if connected will approximate an
arc.

Assume we are in a vessel traveling east at 15d per hour.  Our LHA will
change by 30d per hour, and the resulting 4-minute straight-line sections of
the arc will be approx. double the slope of the fixed position. This
4-minute slope line will be similar to an 8-minute slope line for the static
observer. (For example, slope of a line segment connecting 0d and 45d on a
circle, and 45d and 90 d will be different, as will be one connecting 0d and
90d.)

In the overview, a high speed derivation of the logic the Prince of Syria
used to conceived the International Date Line.

Therefore east or west components of the ships movement will affect the
slope of the line we are trying to fit, and observed altitudes.  To what
extent I am not certain, noise or significant?  I feel certain that the
vessel's movement and resulting DR/EP/LHA/AP and time at the observation
used for reduction must be taken into account before reduction and plotting
a LOP.  Otherwise, it will be *close*, but we will be playing slop pool
(Brownian motion at its best IMHO) instead of snooker.

As I consider Alex's question on boat motion, my current thinking (like the
weather, subject to change) is that the difference of the slopes will not
matter if the course/speed are constant.  A single or averaged series of
sights taken from the vessel moving west, a static vessel, or a vessel
moving east will all be valid IF...  The critical matter is determining the
vessel's location/LHA for the time/altitude used for reduction for the LOP
to reflect to relationship of the vessel to the LOP/fix..

Respectfully,

Bill

> Many manuals advise to take several sights
> in short sequence and then to average the result.
> and to reduce the average as one sight.
> (Chauvenet recommends at most 6, Russian manuals 3-5).
> The purpose is to increase accuracy,
> and, probably more importantly, to reject the sights
> with an evident "human error".
>
> Recently I was asked off-the-list by Bill Burchill,
> what is the effect of the motion of the vessel
> on these repeated sights
> (they are all taken from different positions!)
>
> As I did not find an answer in Bowdich (or Norie, or other manuals),
> I think this should be
>
> CORRECTIONS BY MORE EXPERIENCED navigators.
>
> a sailboat or even an aircraft carrier.
> (However, I imagine that this recommendation has to be modified
> for a navigator of a strategic bomber flying at 600 knots.
> I don't know much about the aerial navigation, but I suppose
> it had to be done by cell nav before GPS era, and if you are over an
> ocean).
>
> So, suppose your vessel moves at 10 knots, (10nm/hour=(1/6)mile/min
> =10"/min),
> and you take 5 altitude sights with the interval
> of approx. 1 minute. Then the maximal change in the altitudes
> due to your motion will be of the order of 50", =approx 1'.
> Thus it seems that on such a small interval, all your
> altitudes (those taken correctly) will lie on a straight line
> when plotted against GMT.
>
> This justifies the recipe: take the average of your altitudes
> and the average of your GMT, and reduce these averages,
> and the resulting position line will pass through your place
> at the average observation time.
>
> What is probably done in practice, is first plotting
> your altitudes against GMT, then looking for a straignt
> line which passes most closely through MOST of your dots,
> then discarding those observations which are far away from
> this line (this rejects crude human eroor),
> then averaging the times and the altitudes of the rest,
> and reducing this average.
>
> This recipe should indeed increase the accuracy,
> reject the sights with large "human error", and thus
> be more precise and more reliable than a single observation.
>
> Unless you are on the strategic bomber mentioned above,
> in which case the recipe does not seem to be applicable.
> (I have never seen a strategic bomber (or commertial aircraft)
> navigators manuals).
>
> Alex.

```
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