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Interpretation of Position Line Plots

The problem of deciding where the observer’s position lies and the detection and/or evaluation of a systematic altitude error from a position line plot is of interest and concern to most navigators. I offer the following theoretical and practical approach to those problems.

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If time altitude observations have been made to two celestial bodies, which are separated in azimuth by an angle, that is neither greatly acute nor greatly obtuse, the latitude and longitude of the observer’s position can be deduced, provided that the observations are not affected by  appreciable systematic errors.

Position lines, which are the locus of an observed altitude circle are constructed at a large scale on the representation of a small portion of the earth’s surface in the vicinity of the observer.

If a systematic error is present, the position of observation (fix) will be somewhere on the line bisecting the angle between the two such position lines. The truth of this statement will be understood later when the technique of evaluating the systematic error is dealt with.

When more that two bodies are observed, the semi-graphic process of solution is invaluable because all the resulting position lines can be considered as a whole.

If the observations are not affected by a systematic error in altitude a fix is selected, which is as close as possible to the position lines. Most text books describe this process.

The presence of a systematic error introduces a third unknown into the solution, the latitude and longitude being the other two. Therefore in the case of a three line fix there will be a unique solution to the position of the fix. Also from the navigator’s viewpoint it is desirable to work with redundant data so that mistakes can be isolated and some estimate of the quality of the fix be made.

There is a technique which can be used for fixes derived from three or more position lines that allows the navigator to make a simple analysis and assessment of his work. This procedure is dealt with in full in the book “Field Astronomy for Surveyors” by G G Bennett and J G Freislich, The NSW University Press, 1979. ISBN 086840 0807. Unfortunately the book is out of print. The steps in the solution are set out in Chapter 9 , The Simultaneous Determination of Latitude and Longitude, which is summarized here.

Step (1) Plot the position lines in the usual way and then mark each of them with an arrow that indicates the direction to the sub-stellar point* (ssp), which is the position directly below the body on the earth’s surface. *Also sub-solar, sub-lunar etc. This means that the arrows are in the directions of the computed azimuths.

Step (2) A line is drawn parallel to each position line at some convenient constant distance (D) away – some multiple of minutes of arc (miles). These additional lines (auxiliary position lines) must be drawn on the sides of  the position lines either ALL towards the ssp’s or ALL away from the ssp’s.

Step (3) A circle is now constructed, by trial and error, that touches each auxiliary position line as closely as possible. If the procedure has been followed correctly and (D) sufficiently big, it will be seen that the arrows will either ALL point towards the centre of the circle or ALL point away from it. The centre of this circle is the position of the fix and the distances between the circumference of the circle and each auxiliary position line is an estimate of the errors of observation.

Step (3) The presence, or absence, of  a systematic error in altitude is determined by the difference between the radius of the circle (R) and the constant distance (D) that was used to construct the auxiliary position lines. When R=D we may assume that the observations were not affected by a systematic error.

The process is best understood if an example of a multi-line fix is chosen and worked through using the above routine.

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As an illustration of the value of the above technique I can quote from a personal experience.

In July 1977, I was the navigator of a yacht making a passage from Noumea to Newcastle (Australia). On board were four men from the serving forces who were taking part in an “adventure training scheme”. During the week’s passage they were able to learn the rudiments of navigation. At twilight one evening, observations were made to a number of celestial bodies and later, position lines were calculated and plotted. One set of position line plots caused some concern because instead of intersecting closely the position lines were well removed from one another. After checking the calculations and putting arrows on each line and constructing auxiliary position lines, referred to before, it was seen that it would be possible to draw a circle to almost touch each line. All arrows pointed away from the centre of the circle, whose radius was nine minutes of arc. An error in the index correction of the sextant was suspected and verified after testing. In all probability the sextant was bumped when being carried. All the other adjustments were in good order.

This experience pointed up not only the desirability of always checking the index error of the sextant each time observations are made but the value of the plotting technique etc that has been described. “Internal” or “external” fixes are all taken care of.

George Bennett