This topic is presented in several sections, you can skip to ones that might be of interest.
What is a compass bearing fix?
What vessels work best and where to stand
Compass choice
Choosing targets for the bearings
Practice bearing fixes in your neighborhood
Finding the most likely position from a three LOP bearing fix
Fix error due to a constant error in both bearings of a two-LOP fix
What is a compass bearing fix?
If the compass bearing to a lighthouse is 045M, then we can go to that light on the chart and draw a line emanating from it in the opposite direction (225M) and we know we are somewhere on that line. If we were to the right of that line, the bearing would have been smaller; on the left of the line, the bearing would be bigger. So we know we are on that line, but we do not know where on that line. That line on the chart is a line of position (LOP).
The next step is to find another identifiable landmark well to the left or right of that one, and take another bearing line and plot that one. The intersection of those two LOPs is a bearing fix. But we do not know much about that fix from these two measurements alone. The two lines will always cross at some position.
If the compass is wrong by some small amount, that fix will be wrong by some amount. The size of fix error depends on the compass error and the angle between the two targets. At the end of this overview there is a note on calculating the error in a two-LOP fix. You can use it to show that a 90º separation minimizes this error, but you do not gain much above 60º separation, whereas you lose fast in accuracy below a 30º intersection. As we shall see, we learn much more about the accuracy of our fix if we have 3 LOPs.
What vessels work best and where to stand
A position fix from 2 or 3 magnetic compass bearings is one of the basic piloting tools in marine navigation... at least for non-steel vessels. On steel vessels there is likely some disturbance of the compass, and that disturbance (deviation) will likely change from one place to another on the vessel. On ships this type of fix is carried out with gyro bearings, but the error analysis presented below will still apply.
Even on a non-steel power boat there can be issues that need to be checked if you are anywhere near the wheelhouse during the sights. If you have a favorite place to take sights, but are not sure about it, then stand there and take a bearing to a distant land mark as you slowly swing ship. If there is no deviation, then the bearing to the landmark will remain the same on all headings. If the bearing to the same target from the same location on the boat changes as we turn in a small circle, then we know we have a problem. On one boat we used for our Inside Passage training, we could take good bearings from the starboard door of the wheelhouse, but not the port side door. From the cockpit of a non-steel sailboat this is rarely an issue.
You can do this test this tied up at the dock as well. Just take bearings to several close targets and see if they cross at your location. That is essentially the process presented below, but we add some details, and we want to start off some place where external deviation is not an issue.
Compass choice
For the practice suggested later on you can use any compass you have. But thinking ahead to options for use underway, the first choice that comes to mind is the
"hockey puck" compass. This has been the bearing compass of choice for sailors for more than 30 years. It can be read to a half of a degree—which is not to imply the accuracy is that good, but we always want to start off with the best numbers we can read. A compass with index marks only every 5º can be used with practice, but it takes more concentration to interpolate the bearings to a degree.
Another excellent option is a good compass in a pair of binoculars. This has many virtues, not the least of which is you get a better view of the target. When personal vision is limited in twilight, which is a common issue, then this becomes a top choice for compass bearings. These cost anywhere from less than a hockey puck (~$120) up to to $600 or more for top of the line models.
There are many options for compasses these days. I have not surveyed the market in a long time. Electronic compasses would seem an ideal choice, but this choice takes special care. The primary issue for most of them is they are very sensitive to roll and pitch, so we need some way to be sure they are very level. Apps often show a bubble level or other graphic aid to insure it is level. Electronic compasses also typically offer some means of calibration for local deviation (rotating it in some prescribed pattern), but in a sense, this just adds to the mystery of the number we read. Some handheld GPS units include an electronic compass; these have the same issues mentioned. Which reminds us that a magnet glued to bottom of a floating card (magnetic compass!) is a pretty transparent tool—rather like the wheel when in comes to function and simplicity.
Choosing targets for the bearings
The first step is to choose the best targets when we have a choice. Ideally we want three targets 120º apart, so the goal is find three that best match that. If we have just two, then close to 90º is best, but with just two we do not get a real measure of our fix uncertainty, which can be as important as getting the fix itself. So we are concentrating on three sights. We do not really gain by taking more than three. It is better to take 4 or 5 sights each of 3 targets (1,2,3 1,2,3... not 111, 222...) than it is to take one or two sights of 10 different targets.
The targets should be as well defined as possible, i.e., sharp peak, rather than round peak, and as close to you as possible. For two equally good targets, equally well spaced, take the closer one. If you have your arm around a post with a light on it, the bearing to the light (using the other hand) could be totally wrong and you still get a good fix (i.e., you know where you are), whereas the bearing to Mt Rainer (90 mi off) is essentially the same from one side of Puget Sound to the other, so it is useless for navigation. Fixed aids are much superior to floating aids, because we know where they are. Also an obvious issue, the target we use must be identifiable on the chart... unless we are just using compass bearings to find distance off of that object, not caring what or where it is. That we can do with compass bearings alone, but that is another topic.
When moving we want to take the first bearing to the target whose bearing is changing the least with time (near dead ahead or astern), and the last being to the target whose bearing changes fastest (on the beam). We do this so we have the minimum DR run between sights for a running fix. This is not an issue for practice at home on land... unless you want to practice with running fixes using a bike or car, but that too is another topic.
Practice bearing fixes in your neighborhood
As noted, for this practice to learn the process, plotting, and analysis, it does not matter what compass you use. To drive that point home, we used an iPhone compass app for this exercise. We learn the process using any compass, and indeed our analysis should accommodate the good ones and the bad ones, providing we have multiple sights of each target.
With that background, we look at one way to practice using a Google Earth (GE) screen capture for a chart. We do not need actual coordinates for this, since we can print the picture and use plotting tools; but we do need the scale, which can be read from the GE ruler tool. Be sure to click the N button (top right of the GE screen) to get north, and also be sure the picture is flat (i.e., shift mouse roll). In lieu of printing and plotting tools, you can also do this with a graphics program as we have done here. Printing, however, offers better hands on practice.
On the other hand, if you want to import this image as an actual chart in, say, OpenCPN, then put a GE push pin near two opposite corners and record the lat-lon of these. Then you can use those locations to georeference the image in OpenCPN very easily with the WeatherFax plug-in (I need to make a video on that process.)
We used a compass app in an iPhone for this to show that you can use any compass. Not the best, but usable for this exercise. The targets were three telephone poles. I marked the base of their shadows as the locations. The green circle is where I was standing to do the sights.
Here are the data of the three sights, the averages of which are plotted above. These are expressed as True bearings, which is an option of the phone compass app. There are many free versions of these apps. The compass, like the barometer, inclinometer (heel sensor!), and other sensors in the phone do not have a native display. So to read any sensor we have to load a third-party app.
Note that the standard deviations of the actual bearing angles measured for each sight happened to be about what was estimated, but that is not really pertinent here, so long as they were not notably larger. It is some level of testimony for the phone app, which after all we just point at the target.
We see that even with this poor compass, we did get a triangle surrounding our actual position. So in one sense we can stop here, and you can use the above procedures to practice basic bearing fixes. You will soon learn that the averages of several sights in rotation are better than just taking three. Needless to say, this would be a great exercise to do from an anchorage on any day sail. Then you can use real charts or echart programs.
Finding the most likely position from a three LOP bearing fix
For those who want to pursue more details, we carry on to look into the accuracy of the fix and what point inside that triangle we might call the most likely position (MLP). This would have to be considered an advanced topic in navigation, and one that will not often be needed. It is for those circumstances were we want to do the best possible navigation with what we have to work with.
Once a triangle of LOPs has been plotted (often called a "cocked hat"), a common practice is to choose some center value of the triangle as the MLP, such as the intersection of the angle bisectors. This is better justified if the navigator is confident that the accuracies of the 3 lines are the same. If we have reason to believe the accuracies are not the same, or better put, that the uncertainties in the lines are not the same, then the centroid choice is not correct. To improve on that we must make some assessment of the uncertainty in each of the lines. That process and what we do with it is discussed below.
We have to first assume there is no local deviation that if present could cause a different error for each direction. In our land based practice from a fixed point, this would have to be something that rotates with us, like a wrench in our pocket, or steel screws in eyeglasses. A steel telephone pole on the corner would not really matter, as it would shift all sights the same amount. Furthermore, in principle, a compass app could detect this when we rotate the phone in the calibration mode, and correct for it. The pole would be just distorting the magnetic field where we stand, regardless of which way we are pointed, unlike on a boat where the disturbance rotates with the boat causing different errors on different headings.
Recall how the compass works. We are standing in a magnetic field that orients the compass card in the direction of that field—it has a magnet glued to its bottom side. Then when we turn the compass to take another bearing, the compass housing and index mark fixed to it rotates around the compass card, which itself is not moving. It swings about a bit, but goes right back to its original orientation, magnet pointing in the direction of the strongest magnetic field, which is called "magnetic north." Electronic compasses work in a different way but that same principle applies.
We have in the practice example a relatively good distribution of targets; not ideal, but not far off. We can fairly assume—as we must with most magnetic compass bearings—that we have a bearing uncertainty of not better than ± 1º. Even with a best possible magnetic compass, we have the uncertainty of not just the reading of it (this one showed whole degrees only; tenths would be better), but also we have uncertainty of magnetic variation when underway, and potential errors in plotting and reading the plots.
We can nail the variation issue at the
geomag web site. We have here 15.8º E, where I was standing, as of today. Underway you will have a larger uncertainty, unless you install the program
geomag on your computer or phone, which is easy to do. Many ECS programs do this automatically for you (OpenCPN has a plug in for this). Note you do not get this from GPS satellites. If your GPS is telling you variation, then the GPS unit itself has this program installed. The satellites tell it where you are, and the software in the GPS unit computes the variation for you.
So if we
optimistically assume a 1º uncertainty in each measurement, then we can use geometry to figure how much that offsets the LOPs near the place they intersected, which will depend on how far off they are... again, this is why we want close ones. We do an approximation here. We have an angle uncertainty and want to translate that into a lateral uncertainty—effectively, how wide is the line? One way to estimate the width uncertainty of a bearing line is to call it equal to the tangent of the bearing uncertainty multiplied by the distance off. This in turn can be well approximated with the small angle rule that is useful for many tasks in navigation, namely we assume the tangent of 6º is 1/10. (One application of the rule is, if i steer a wrong course by 6º I will go off my intended track by 1 mile for every 10 I sail; there are many applications.) This means that a working uncertainty in bearing lines can be estimated as
sigma = (target distance) /60.
We call this uncertainty "sigma," as it is on some level representing the standard deviation (often abbreviated with a greek letter sigma) we might expect among a series of sights to the same target. In the LOP of side 2 above the sigma from this reasoning is ± 1.3 yd. It is almost certainly larger than this, but for now the key issue is we use the same system for each of the three sights. That is, we could double that for each and it would not matter much. The key issue in this type of analysis is the
relative uncertainties of the sides. Below are the MLP data for these three sights.
Below is the triangle expanded showing the most likely position calculated from this data: the black dot inside the light blue ring.
The MLP is the red dot. The plot is scaled to the lengths of the sides, given above. The location is plotted relative to the bottom intersection. The green circle was just an estimate of where I was standing doing the fix before we did any analysis. You can solve for this MLP manually with a
form we have available, or use a free app (
MLP.exe) that computes the location based on the three sides and three sigmas. It is a simple computation that is easily done with a calculator. This work was largely motivated to analyze cel nav fixes, so it is discussed further in this note (
Analysis of a Celestial Sight Session). We will have the full derivation and other discussion online shortly. Below are screen caps from the app, which has a direct digital solution as well as an interactive graphic solution.
The light colored lines are marking the sigma values for each line, which we enter with the sliders on the left. The triangle is formed by dragging any corner. The black ellipses outline the 90% and 50% confidence levels. In this example we multiplied the sides by 10 to make a bigger triangle. (The MLP values shown reflect its location on the plot, which is marked in 20-unit steps.) The scale in this type of analysis does not matter. The main point here is that once you have a triangle, the MLP is not necessarily any of the conventional center points of the triangle, such as intersection of medians or bisectors. Each of these conventional center points can be compared using buttons on the bottom left of the app. The MLP depends on the shape of the triangle and on the sigmas, and on a fixed error if present.
The introduction of a fixed error that applies to all bearings complicates the analysis. First, with a fixed error the directions of the LOPs matter, which is why these LOPs show arrows on them. This is how we distinguish three LOPs at 60º apart from three at 120º apart, even though the triangles are identical! With no fixed error (as in this example) the arrows do not matter. Practice with the app will show that if the fixed error is larger than the random errors (sigmas), then the MLP will actually be outside of the triangle whenever the span of the bearing directions is less than 180º. Again, that is why we ideally want three bearings at 120º apart. Then a fixed (unknown) error will just make the triangle larger, but the MLP will still be located inside of the triangle. With our app you can play with various configurations to study this behavior.
Again, this is an attempt to get the very most out of our navigation measurements, which is not always needed. This requires extra analysis, and in particular needs realistic estimates of the uncertainty in each of the lines. This is always possible on some level, i.e., a compass bearing line on a chart will never be more accurate than ± 1º, and twice that is more likely. Nevertheless, even with these rather large uncertainties, we can obtain a final fix precision that is notably better than might be suspected based on the uncertainties in the individual lines—assuming we can make realistic assessments of these. Sometimes doable, other times not.
We also see numerically and graphically with the app what many navigators know intuitively. If one of the lines is notably better (smaller sigma) than the other two, then the fix will be on that line, and the other two just serve to determine where on that line.
Fix error due to a constant error in both bearings of a two-LOP fix
We can compute the error in a 2 LOP fix if there is a constant error in the bearing to each of them. For two targets a distance (D) apart that are separated by an angle (A) from your perspective, a bearing error (E) will cause the fix to move by an amount (fix error FE) given by
FE = D x sin(E)/sin(A)
For example, when the bearing error is 1.5º for two sights to targets 2.2 mile apart, that are separated by 45º, the fix will be in error by 2.2 x sin(1.5)/sin(45) = 0.6 nmi.
In the neighborhood practice example we have three pairs of 2 LOP sights. If we check 1 and 2, the fix error of using just those two (with a 1º compass error) would be
FE = 112 yds x sin(1º)/sin(326.5º - 192.8º) = 2.7 yds.
Note that E will always be a small angle, so you can solve this equation for E = 1º, and then just scale it. In the last example, if the E had been 2º, we would have got 2
x2.7 = 5.4 yds.
I might note that the solution for this error seems to have been wrong when it first appeared in the 1977 edition of Bowditch, and the rendering of it has just gotten worse with subsequent versions, leaving it pretty mangled up in the 2017 edition. It appears they copied it in 1977 from the 1938
British Admiralty Manual of Navigation, Vol 3, which itself somehow ends up with a formula that mixes up radians and degrees as well as nautical miles and arc minutes, and that seems to have been copied without noting what they had done.
You might wonder how it can be that the US Bible of Navigation can have this fundamental point wrong for so long? The answer from a colleague here at Starpath is: "Because you work on things no one cares about." (Maybe I am misinterpreting what is in Bowditch? Let me know and I will remove this.)
Below is a numerical example for two bearings (040 and 100) with an error of 5º using two targets that are 2.24 nmi apart, plotted in OpenCPN.
