The Accuracy Triangle       Back to Terry's Articles

A means for precision chart plotting

What precision is needed when plotting positions on a chart? Mariners have lived with the ‘inaccuracy’ of celestial navigation so you may argue that "four miles is close enough". Which is probably true on a passage far from land when a daily fix is done to maintain the DR. But when close to land something better is required for finding those magic out-of-the-way spots, that hidden pass, or for confidence that you are safely navigating among reef-strewn waters in little or no visibility. Navigating with confidence can avoid turning a pleasure cruise into a nightmare of frustration and worry, not to mention the potential for disaster.

With the advent of GPS, formerly undreamed accuracy in position is now continuously available to the mariner. Even with Selective Availability giving the civilian user degraded performance, our thirst for an exact position has fostered the development of DGPS. And now, the discovery is that the charts we have relied upon for so long are sometimes grossly inaccurate in positions as shown by the GPS. I have experienced differences in latitude and longitude of up to one mile on the most current charts in areas such as the Solomon Islands. This has made navigation tricky where reefs and look-alike islets abound with narrow passages between them, and ‘eyeball navigation’ is required due to the lack of navigational aids.

Charts are usually accurate for the displayed information relative to the other information displayed on that chart. And ‘reported discrepancies’ are commonly shown until confirmed or denied prior to the next update. The basic inaccuracy stems from errors in latitude and longitude as drawn on that particular chart. The precise position of a feature on the earth’s surface is shown only to the degree of accuracy available to the survey method. When the amount of error ‘offset’ is known at the time of publication, a marginal note may give the latitude and longitude corrections required to make GPS-derived positions correspond with the positions on that particular chart.

After finally getting underway in April 1998 to continue our cruising life aboard VALHALLA, our Fuji 32 Ketch built in 1976 but rebuilt over the past few years in Guam, I now have the time to catch up on my reading of back issues of Ocean Navigator. Two items of correspondence and one full-length article relate directly to the subject of difference between charted and satellite-derived positions. The first related article, in Issue #82, Lessons to be learned from BVI groundings, by William H. Trafors, reports the loss of vessels in the British Virgin Islands which may have been due to a difference between a GPS-derived position and the vessel’s assumed position on a chart. The second article, in Issue #84, More thoughts on grounding in BVI, by Phillip Jones, offers thoughts on selecting the correct datum (other than the GPS standard of WGS-84) as a means of correcting the mismatch in positions. A comprehensive treatment was presented in Issue #87, The roots of chart accuracy, by Nigel Calder. Nigel says "Apart from the fact that the astronomically derived starting points are often seriously in error (sometimes by miles—according to the British Admiralty, the worst discrepancy, which is in the South Pacific, is seven miles), there are frequent surveying errors on these older charts (imprecisely measured angles between features or poorly calculated distances, etc.)." So even when the datum used for a particular chart is entered in the GPS; there is no guarantee that the positions will match. The ‘offset’ of these positions is still a function of the accuracy available to the surveyors when the chart information was collected and subsequently used in compiling the chart.

There are continuing world-wide efforts to correct the data base from which charts are drawn and national efforts to update navigational charts are becoming more expensive each year, and also suffer decline from shrinking budgets. (A good example of this difficulty was covered in Ocean Navigator Issue #76; Charting plan focuses on commercial areas, by Tim Queeney) But the average cruising sailor is still faced with either old charts, charts without correction data, or charts that are known to have some correction for satellite-derived positions but just don’t always seem to work out right. Ironic, when you think about it … new technology has brought us problems along with solutions.

Getting ‘in sync’

Interestingly, my first GPS (still my primary one) was purchased in 1992 and cost $1200. It has a feature that doesn’t exist on the ‘latest and greatest’ $200 backup (three months old), which has plenty of bells and whistles such as a tracking display. The feature of the (now obsolete) Micrologic Explorer GPS that serves me so well is it’s ability to determine the ‘offset’ for any particular chart and apply that offset until you tell it otherwise. Conversely, if you know the offset by comparing a GPS reading with a known position on a chart, you can insert it and the displayed positions will now match the chart that is in use. It will continue to apply these offsets until they are cancelled. Each time the unit is turned on, a warning is displayed to remind you that offsets are being used. So getting the GPS ‘in sync’ with the chart you are using is a first step in avoiding the pitfalls from differing positions.

Even without a GPS that corrects the offsets automatically, once you know the offsets in latitude and longitude, it is a matter of carefully adding or subtracting the offsets before plotting a position. And note that I say ‘carefully’. As you know from the exercise of correcting a compass using TVMDC, it is easy to make mistakes between the need to add or subtract. A simple sketch often helps keep you straight.

An important thing to remember, though, is that the precision you obtain in plotting the latitude and longitude of your known position on the chart determines how accurate the ‘offsets’ are calculated, either manually or automatically. Also, it would be nice to take advantage of the ‘three decimal places’ accuracy of the GPS (recognizing that Selective Availability makes that third decimal place suspect).

Precision chart plotting

The precision you obtain in plotting will certainly be a function of the scale of the chart and the latitude and longitude divisions printed on that chart. But the cartographers haven’t done you many favors in that regard. A random sampling of charts will show that the minutes of latitude and longitude are divided in several ways depending, I suppose, on the whim or style of the cartographer (or the policies of his or her office). Most charts I’ve encountered use tenths of a minute, where each smallest division equals 6 seconds (0.1-minute spacing). At least these are in decimal format so that eye interpolation is the easiest. But estimating the space between divisions by eye can still take a lot of practice to come anywhere near a state of accuracy that matches or takes advantage of the GPS precision. Other charts, however, use divisions of 5, 10, or 15 seconds of a minute. Plotting in a ‘decimal’ format on these charts is a frustrating exercise of conversion, often leading to mistakes and at least being time consuming.

 The technique

The Accuracy Triangle. A graphical technique I have developed allows at least one more decimal point of accuracy in eye interpolation and simultaneously provides a ‘decimal scale’ for those charts still using divisions in seconds of a minute. Once constructed on a chart, it remains for future use. With this technique, when plotting, for example, a GPS position of "2 degrees 12.852 minutes", it can be done with confidence, to at least 2 degrees 12.85 minutes. With normal eyeball interpolation the best you would expect is 2 degrees 12.8 minutes.

The technique uses the principal of proportionality of sides of a triangle. I construct and use an Accuracy Triangle on both the latitude and longitude sides of a chart in the following fashion.

Constructing the Accuracy Triangle. (See Figure 1) Choose any one-minute division on the latitude or the longitude scale of the chart. (Though in practice both are constructed, I will use the latitude scale only in this example) I call this ‘one-minute division’ the Scale Line. Draw a line at any convenient angle (90 degrees works fine, especially if you use the intersection of a printed latitude line with the latitude scale, in which case the line is already drawn for you). Measure a distance of 10 units along this line, which I call the Division Line. (I find that ten centimeters is convenient, allowing any distance to be read quite accurately to 0.5 mm on a small metric ruler, equal to 0.005 minutes of latitude or longitude. With practice, this can be reduced by half to 0.0025 minutes). Now draw a line from the end of the 10 units (centimeters in my case) at the intersection point with the Division Line to the other end of the Scale Line (the one-minute division on the latitude scale), completing the Accuracy Triangle. I call this newly constructed line the Baseline.

Using the Accuracy Triangle. (See Figure 2) Assume you have used a set of dividers to measure the latitude of some position above the 22 degree north latitude line. Place the dividers on the Scale Line and mark this point. Using the parallel rulers, construct a line parallel to the Baseline to intersect the Division Line. (In practice, the line isn’t drawn, only the intersecting point on the Division Line is determined). I now use the metric ruler to determine the distance along the Division Line to be 6.45 millimeters, which equates on this chart to 0.645 minutes of latitude. The desired north latitude is therefore 22 degrees 0.645 minutes. Even with the latitude scale divided into tenths (as in this example) the expanded scale of the Division Line, with the Metric Ruler, is easier and more accurate to read.

 Is it worth the effort?

The next time the depthsounder alarm goes off when you are in 20 fathoms of water based on your chart position or, hopefully not, you come to a sudden and expensive stop, you may find it has some merit <grin>.

Back to top