# NavList:

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

**Re: Resolution of systematic error**

**From:**Peter Fogg

**Date:**2007 Mar 5, 17:38 +1100

Using your first diagram, you take the three shots and plot them and get the first diagram with a large cocked hat. You suspect a systemic error. You think it over and realize that you forgot to apply the IC to each of the sights. Your IC is plus 10 minutes so you are ten miles closer to each of the substellar points.

So, you adjust each of your LOPs in the direction of the ZNs by 10 NM each and end up with your second diagram with the dashed lines now illustrating the corrected LOPs.

In this case where a specific systematic error has been correctly identified your procedure to correct it seems sound.

You still have a cocked hat but it is smaller and its size it now determined only by random errors.

Assuming no other systematic error. You could then use these corrected LOPs (enclosing your smaller, corrected triangle) as shadow lines and then resolve residual systematic error by joining the intersection of the shadow lines and the intersection of the original LOPs. The point at which they meet is the position free of systematic error.

Your actual position will then be within the triangle, not outside

so I don't see how you get to the fix positioned in the third diagram, What am I missing?

Let's start at the beginning. What is a position line? It is the
position expressed as a line (for our purposes - really an arc that, extended,
completes a circle around the SSP). It is the position that has been derived
via a mathematical process. It is a rational artefact. It is the ONLY position that can be rationally derived from a specific
body at a specific altitude at a specific moment.

To say that this is not an expression of the position, that the position lies
at an indeterminate point that could be located anywhere at all in the whole
wide universe, except along the position line, is a bit of a furphy.

I do understand and appreciate the statistical 25% / 75% argument. However,
statistics are most useful navigationally, it seems, when employed to deal with
superfluous data. An example of this is the derivation of a fix from within
encompassing LOPs, where the sum of least squares can be used to derive the
centre of the shape.

Another way to put this is that statistics can be a good servant but a bad
master.

Moving along; two position lines resolve the inherent ambiguity (of a line
including an infinite number of position points) at the intersection where they
cross. This is the fix. Barring error, this is the position.

But what about error? (I hear you exclaim!)

Error can be divided into two camps: systematic and random.

**Random error**

There are useful practical methods to limit the effects of random error. *Eg*;
where there are only two position lines they should, ideally, intersect at an
angle close to 90 degrees. The more acute or obtuse the angle the more random
error will be magnified in the derived fix.

There are other examples. One of the most useful techniques I know of to reduce
random error is to make as many observations as possible of the same body over a five
minute period and then compare the slope of the body's apparent rise or fall
with the pattern of actual sights. Not to be confused with averaging.

The alternative is to adopt a single sight, a random event, without having any
idea of how valid it is; whether it contains error that can be rendered
apparent or is consistent with other sights of the same body taken just before
or just after. This technique will not resolve systematic error, but will
enable a useful analysis of the pattern of sights and the adoption of an
improved altitude/time set before sight reduction.

It seems to me more useful to eliminate random error as much as possible at
source, rather than worrying later about the resulting indeterminate nature of
a fix derived from random sights.

Surveyors used to use cel nav (that they called field astronomy) to derive
position from the stars. Although they used theodolites rather than sextants
the main reason they were able to establish a position to the nearest second of
arc (about 30 metres) is because of the attention to, *eg*; getting the
time right as well as work done to eliminate or reduce the effects of random
and systematic error. Such as comparing a pattern of sights with the slope. For
example.

For practical nav in a small boat or for any moving target great analysis is probably
not worth the effort, but as many contributors to this list work from a known
position on land and are mainly interested in what accuracy they can achieve,
some appreciation of these methods could be useful. While we're on that theme,
the traditional triangle formed from 3 LOPs is less than ideal. Four or more
position lines is better.

**Systematic error**

These are made up of errors in altitude and in time. Errors in time will
displace the most those LOPs whose orientation lies mainly north/south, and the
least those running east/west. One effect of this is that separate observations
for latitude during meridian passage, and observations for longitude along the
prime vertical, can yield particularly accurate results.

If two position lines are plotted, their intersection will reflect the
observer's position only if no systematic or random errors are present. If
systematic error is present the position will lie along the line that bisects
the angle between the two position lines.

In order to derive a point from this line at least one other position line is required, and an example of this has been given where the arc encompassing the three SSPs is less than 180 degrees of azimuth.

Although it is an interesting geometrical construction I am not sure that it signifies anything. You can see that if you reversed one of the ZNs, meaning that they no longer were contained within 180º, and drew your dashed lines, now with this one on the other side of the LOP, you will have just accomplished drawing the bisectors of the angles but with the lines now meeting in the middle of the triangle, your traditional fix. Again, I don't see how this construction signifies anything.

We may take this up in another post. Gary is on the right track.

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