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    Re: Resolution of systematic error
    From: Peter Fogg
    Date: 2007 Mar 5, 17:38 +1100
    Gary LaPook wrote:

    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

    This depends on the size of the triangle. I drew this up and found a smaller triangle displaced towards the substellar points (SSPs) but partially within the original LOPs and partially without. Without more information it is impossible to know whether it is wholly within or without or somewhere in-between. But I take your essential point; that in this case a smaller, presumably more correct triangle can be derived located in a direction towards the SSPs. 

    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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