# NavList:

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

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Re: Resolution of systematic error
From: Gary LaPook
Date: 2007 Mar 01, 02:21 -0800
Gary LaPook wrote:

I don't get it, there seems to be something wrong with your analysis.

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

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.

Gary LaPook

Peter Fogg wrote:

There is a method for the resolution of systematic error that I understand was first proposed by the German geodesist Lehmann.

In the example below, the lines represent portions of position lines (aka LOPs). The arrows point towards the sub-stellar point or, to put that another way, indicate the direction of the azimuth.

The arrows should, of course, form a right angle with the position line. This drafting inaccuracy is not important for our purposes; the arrows just indicate direction. The small circle indicates (again; roughly) the fix position.

If all arrows point in the same direction (all pointing out or all pointing in) it means the azimuths used cover an arc of less than 180 degrees. This is less than ideal, and one reason is that if there is any systematic error the actual position will lie outside the triangle.

Lehmann (as have others since) proposed a simple method for establishing a position that has a higher probability of being correct.

Shadow (dotted) lines are drawn towards the direction indicated, parallel to and at the same distance from the position lines.

Then lines are drawn that pass through the intersecting position lines and the intersecting shadow position lines.

These blue lines meet at a common point: the centre of a circle whose radius, touching the original position lines, indicates the extent of the systematic error.

Note that the distance of the shadow lines from the position lines is not relevant, although this distance must be consistent and the shadow lines must be parallel with the lines they are derived from (again, this freehand diagram is a little crude – try drawing it as accurately as possible on graph paper to check whether it works).

The point of intersection of the blue lines is the improved fix position. With systematic error and azimuths that cover less than 180 degrees the actual position will ALWAYS lie outside the triangle.

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