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    Re: Systematic error and its resolution
    From: Peter Fogg
    Date: 2007 Apr 7, 16:04 +1000

    Geoffrey Kolbe wrote:
    "In conclusion, I was able to get reliable fixes
    far more quickly by moving away from the
    traditional fix using three objects about 120°
    apart with its resultant cocked hat, and adopting
    a strategy of taking sightings on four objects,
    each near one of the Cardinal points. I know that
    if the box formed by the position lines is
    substantially square, the errors are far more
    likely to be dominated by systematic rather than
    random errors. With a tradition
    three-position-fix you never know if random or
    systematic errors are dominating, so you can
    never be sure just how accurate your fix is."

    Four position lines are generally preferable to three but I don't want to become sidetracked down that path today – so far I have been carefully avoiding stepping into that topic!

    However; to deal with the resolution of systematic error involving four position lines it may be an appropriate moment to introduce the expression of this correction in the same manner as my principal source on the topic, a text for surveyors:
    "Field Astronomy for Surveyors" by GG Bennett and JG Freislich, published by the New South Wales University Press in 1979; ISBN 0 909465 93 2.

    It proposes the construction of circles that touch the position lines; the improved fix lying at the centre of the circle, the radius expressing the extent of error. The practical effect is the same as the dotted lines method.

    It is possible to have four enclosing position lines that are all tangent to a circle within them – one with four right angles and sides of equal length, for example. But the chances of coming across such a shape in the practice of nav are presumably few. A more typical shape is shown below.

    "… by bisecting angles or drawing bisecting parallel lines etc., construct a circle of best fit, i.e. one where the distances from the circle to the adjacent position line conform to the principle of Least Squares" (Bennet & Freislich).

    [At this point the text goes on to remind the reader that each position line is presumably derived from an analysis of a number of observations of the same body taken successively, a topic already addressed by me on this forum in the context of the resolution of random error.]

    So a circle of best fit of the shape is constructed with the improved fix, free of systematic error, lying at its centre, the radius expressing the extent of error.

    Geoffrey again:
    "… research revealed that
    Royal Air force navigators used exactly the same
    strategy to ensure accurate fixes when using a
    bubble sextant. Too, surveyors of old who had to
    rely on a transit theodolite to determine their
    position by stellar observations, also used the same strategy."

    Indeed they did. The technique was the one explicitly taught for this purpose.

    Incidentally Geoffrey, I've enjoyed following your trips across the Sahara and been impressed at how well you have tracked your position, using a bubble sextant.

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