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    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.
    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.
    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.    
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