# NavList:

## A Community Devoted to the Preservation and Practice of Celestial Navigation and Other Methods of Traditional Wayfinding

**Position Lines and a Systematic Error**

**From:**George Bennett

**Date:**2003 May 28, 17:21 +1000

Further to my previous message about position
fixing, which was probably a bit terse I offer the following additional
comments.

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Consider the situation where a constant
systematic error( dh) is present in the observed altitudes.

First of all assign arrows, indicating azimuth, to
all the plotted position lines as described previously.

Case 1. If two bodies have been observed it is not
possible to effect a solution for all three unknowns (Latitude, Longitude
and Systematic Error) but we can find the a locus of our position as
follows: Draw a circle to touch the position lines such that either
both arrows point towards the centre of the circle or both point away.
This can only be done in two of the four areas formed by the intersection.
It is obvious that there are an infinite number of circles that can be
drawn, and as the centres of the circles are the position of fix, the locus of
the fix is along a line joining all those centres. This line of course
bisects the angle between the position lines and extends on either side of
the intersection.

Case 2. A fix from three bodies.

(a) Provided observations have not been made
in an arc of less that 180 degrees the solution
lies within the triangle formed by the position lines. Draw a circle within the
triangle to touch each line. This can be done without trial and error by
bisecting the internal angles of the triangle. It will be seen that
either All the arrows point toward the centre of the circle or ALL point
away.The centre of the circle is the fix and the radius of the circle is dh
(same scale as intercepts).or

(b) When observations have been made within an arc
of less that 180 degrees the fix lies outside of the triangle in one of the
three areas surrounded by the three position lines. We will need to construct a
circle to touch each one of the three lines in one of those areas. Without
drawing circles imagine where they might lie and choose the area where
either All the arrows point inwards or ALL point away from the area.
If you are unsure then sketch the three circles lightly and examine the
directions of the arrows. Having decided on the area construct a circle to touch
each line. This may be done by bisecting the angles between adjacent position
lines.Then proceed as in Case (a).

Case (3).When
more than three bodies have been observed plot all the
position lines with their associated azimuth arrows towards their ssp's.
Draw auxiliary position lines at a distance (D) on the side either ALL
towards their ssp's or ALL away from their ssp's. Choose a distance such that
the auxiliary position lines are well away from the original position lines. If
the observations have been made with reasonable accuracy and there are no
mistakes in the calculations ALL the arrows should now point in or All
point away from the figure formed by the auxiliary position lines. By a process
of trial and error draw a circle to touch as closely as possible these
lines. The small distance between each line and the circumference of the circle
represents the error of observation. I call this
process 'eyeball least
squares" ie you are trying to get the sum of the squares of the errors to be a
minimum. The centre of the circle is the position of fix. The systematic error
in the altitude is the difference between the radius of the circle and the
distance (D). If this difference is very small it is unlikely that a systematic
error in altitude was present.

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A rigorous solution to fixing position from
multiple timed sextant altitude observations has been published by C de Wit in
the British Journal of Navigation. The solution requires the operator to make 'a
priori' estimates of all possible errors which are then taken into account in a
rigorous least squares solution. My personal view is that such rigour is not
warranted when we are usually dealing with a small sample and error
estimates are very uncertain. What I have described is a convenient
approximation to the correct model, which has been shown by long experience to
be quite satisfactory for marine navigation.

I will be leaving for Europe (from OZ) tomorrow and will be away for the
whole of June. Therefore I cannot participate in the Group's activities. I will
be keen to see if there is any interest in the subject on my return.

George Bennett.