Wednesday, November 22, 2017


Someone has to care about the details of marine navigation and weather, or things slip by, which might one day show up and cause confusion, and we don't want confusion. Indeed, a hallmark of good navigation and seamanship is clarity in communications. Today, we ran across a gold nugget of doublespeak: MSLP.

We have published a book called the Mariners Pressure Atlas, which contains pressure statistics that are difficult to find, despite their great value for weather tactics in tropical storm prone waters of the world. The book contains global plots of the mean values of the sea level pressure, called mean sea level pressure  (MSLP) patterns (isobars) along with the standard deviations (SD) of these pressures on a month to month basis. The SD are a measure of the variation of the pressure we can expect purely from a statistical spread around the mean value. Sample sections are below.

These are the mean sea level pressures (MSLP) in this part of the world in July. Below are the SD values.

For example, in the tropics with a MSLP of 1012 mb and an SD of 2.0 mb, we know that an observed pressure of 1010 is one SD below the mean and a pressure of 1008 is 2 SD below the mean. When we observe an average pressure of 1007 mb, we are 2.5 SD below the mean. That takes on special meaning when we look at the probabilities.

In other words, the probability of normal pressure fluctuations being down 2.5 SD is 0.6%. A pressure that low is almost certainly (99.4%) not normal fluctuation—that is the approach of a tropical storm! The wind will for sure not warn of that at this point, and maybe the clouds on the horizon might not either, but there is indeed a tropical storm headed your way.

This powerful storm warning technique was well known in the late 1700s, early 1800s when ships carried accurate mercury barometers, but unfortunately with the advent of aneroids, by1860 or so this knowledge slipped away because they were not accurate enough then to do this job.  Eventually even the textbooks stopped talking about absolute pressures and just started preaching up or down, fast or slow, which is useless for this type of long range storm forecasting.  Now we have accurate barometers (including accurate aneroids), which is why we rejuvenated this classic method... mentioned in Bowditch, but sadly without a link to the crucial MSLP and SD data.

There are spot values of mean sea level pressures in Appendix B of the Coast Pilots.

But that is not the point at hand. What we are dealing with in the above is the mean value of the sea level pressure, which we periodically see abbreviated MSLP in weather and navigation documents.

Now, however, look at these weather maps from Australia and Canada.  The UK Met Office also uses the MSLP notation to describe a surface analysis map.

Now we have an all new meaning of MSLP.  This cannot be the mean sea level pressure we discussed above;  these are the actual values of the pressure at sea level at the valid map time. What is going on here is they are not calling the reference plane "sea level," which we often see, i.e., sea level pressure (SLP), but instead they are calling the reference datum "mean sea level."  MSLP in this context is the same as SLP.

Here is an example of aviation weather (METARs) using SLP; it is also used in some numerical model outputs.

Here is another example that shows the fluidity of the terminology.  The ECMWF defines

"MSLP is the surface pressure reduced to sea level." 

So they know that "sea level" is the same as "mean sea level," but they choose to help us make our point!

It is not unreasonable to tack on the "M"; the sea level does change with the tides (to a good approximation mean sea level, MSL, is halfway between MLW and MHW), not to mention that it varies with the pressure above it (called the reverse barometer effect). In fact, MSL is an even more complex concept, but in ways that do not at all effect our use of it as a pressure reference. For present context, this just reminds us to think through the terms we use.  We might note that on nautical charts building, towers, lights, bridge clearances are referenced to MHW, but spot elevations on the land and elevation contours are actually referenced to MSL.

We thus have in common navigation conversation both:

     MSLP = M - SLP,   being the mean value of the sea level pressure, and

     MSLP = MSL - P,   being the pressure at mean sea level.

If you found this abbreviation in some context of your work, and then went to a navigation or weather glossary to look it up, chances are probably only 50% that the glossary will come back with the appropriate answer for your inquiry. Put another way, you will not find an official glossary that has both definitions; they will have one, or the other.

... which I thought we should document, so no one gets the impression we are sitting around the office all day working on trivial matters.

PS. Just ran across this at the Navy site (FNMOC):

Sea Level Pressure (MSLP): The model-estimated pressure reduced to sea level. Units are in millibars; contour intervals are 4 mb. 

Maybe the "M" stands for "model-estimated"?     

Tuesday, November 7, 2017

Global Warming and Tropical Cyclone Statistics

We happen to be updating some of our training materials today and thought to check the latest stats of tropical storms and hurricanes... there is much talk these days in the news about the various implications of global warming, including affects on tropical storms. Partial results are shown below.

These storms may be getting more severe on average, and maybe wandering off to higher latitudes more often, we have not checked that. All we did is compare what Bowditch reported in 1977 compared to what they report in the brand new 2017 edition as to the total number of systems.  In 1977 there were not many convenient sources of this data. Now we have all the detail we could ever want about every system, and we get it directly from the primary sources, but it is not clear that the new Bowditch data might need updated itself.

The statistics shown below are all data up to 1977 compared with the latest systematic study in the 2017 Bowditch, described as 1981 to 2010. You can click the pic for better view.

The values are average number of incidents per month. "S" is number of tropical storms, meaning sustained (> 1 minute) winds ≥ 34 kts. "H" is number of hurricane-force systems, also sustained.

Note that the storms include the hurricanes... all hurricanes start out as storms.  So of the 12.1 storms per year average in the North Atlantic, only 6.4 of these on average proceeded to become hurricanes. (If you happen to look at the 1977 Bowditch data, they used a different convention on presenting this information; we regrouped that early data to make this comparison.)

The North Atlantic region (including Caribbean and Gulf of Mexico) definitely has more storms (we have about 29% higher chance of seeing storm force winds), but there is a slightly lower chance according to this data that these become hurricanes—but this still leaves us with slightly more hurricanes than earlier, about 20%.

But these are statistics. We could have 10 hurricanes this year (as we did), then 2 the next year, and we are back on the average of 6. The question is, how likely is that, just 2? If we have 10 next year as well, we better have zero the next year, just to approach the average.  In short it could be that these Bowditch stats need to include more recent data, ie 2011 to 2017.

For example, here are the recent data from the NHC.

For the North Atlantic, over the past 7 years (not included in Bowditch 2017) we see 13.8 and 7.1, which is higher than 12.1 and 6.4, but not that much.

Over the past 7 years we see higher numbers for East Pacific: 15.9 and 9.4 compared to 16.6 and 8.9, which is about the same... but both notably higher hurricanes than in 1977 (15.2 and 5.8).

With our check of the recent data we can compute the standard deviations (SD), which are:

East Pacific:  15.9 ( 4.6) and 9.4 (3.0)
North Atlantic: 13.8 (4.8) and 7.1 (3.4)

We do not have a lot of data here, but these are large SDs, which means we can expect large variations of these numbers from year to year. Below is the distribution of events if the variation is indeed random.

This means that 68% of the values should be within 1 SD of the mean, or we can look at it as shown  below

With, say, 7 hurricanes per year with an SD of 3 it means that there is only roughly 16% chance of having 4, or put the other way, there is also a 16% chance of having 10, but if we have 13 events (2 SD above the average) then we are down to 2.3% chance by random, which raises more the issue of looking toward trends.  It would be nice if we had the SD for the 2017 Bowditch data. That was not given in the book, but it is fairly easy to look up the actual values and compute it as we did here.

Without an in depth analysis, it seems we can likely rely on the numbers in 2017 Bowditch values, with the awareness that these do appear to be rising slightly still, beyond that 2010 data sample.  Other notable changes can be seen in the other zones.

It is likely a more interesting study for climatologists to look at severity, but this has little interest to mariners, i.e., we would obviously treat a 150 kts forecast the same as we would a 115 kts... but we might want to keep an eye on storm size. With all the data that is available, one could do a very precise study as a home project on, say, the average area covered by 34-kt winds from inception up to hurricane strength, and then the area covered by 50-64 kts and then >64 kt winds after that.

The other study would be how far north do they go, and indeed how long do they last.  If you have a student with a science project on the horizon, this is very easy data to get online and the analysis would be a good exercise in using numbers.  Furthermore, this has much value to mariners and we cannot count on anyone without a maritime interest in putting these specific values together. (If a student is interested they can call us and we will help.)

Check out the 2017 Bowditch, Chapter 39. They have very good coverage of tropical systems, that even include QR-codes to go directly to the various Regional Specialized Meteorological Centers (RSMC) that do the job of our National Hurricane Center for other tropical cyclone zones.

If you plan to be sailing in a hurricane zone, a mandatory reference is:

Noting especially the Mariners 34-kt Rule and the Mariners 1-2-3 rule on storm track uncertainty.  I would also like to think that our own book would be helpful

Here is a sample of the 2017 Bowditch's extensive use of QR-codes, which is pretty techy,  but all the links in the pdf are interactive in the first place.

Wednesday, November 1, 2017

Compass Bearing Fix — An Overview

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 2x2.7 = 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.