Monday, March 18, 2024

A New Revolution in Barometers

We have worked for many years promoting the use of accurate pressure in marine navigation, which had literally fallen out of all standard texts on marine weather twenty years ago. The word "barometer" was barely mentioned. We would see occasionally that a falling barometer means bad weather, but nothing more, and certainly nothing about how fast it must fall for bad weather. And all of these books state—they are all still in print—that the value of the pressure does not matter; it is just a question of rising or falling, fast or slow, but never with any numerical values.

Accurate pressure was crucial in the late1700s and early 1800s when much of global marine weather was first learned and understood with the aid of accurate mercury barometers used at sea. But they were unwieldy and difficult to use and happily set aside with the development of aneroid barometers in the mid 1800s. That revolution took place without the full recognition that with the great convenience of the aneroids came a notable loss of accuracy over the higher and lower ends of the dial, which typically matter the most in routing decisions—a fact that has followed aneroid use into modern times. Thus began the doctrine that only the change in the pressure matters, not its actual value.

Now it remains as it was then: only the high-end, expensive aneroid units can be counted on for accurate pressures over the full range we care about in marine navigation. I would venture to guess that most barometers on vessels today are there primarily for traditional reasons, and not referred to for routing decisions.

We began our goal to change that with the first edition of Modern Marine Weather and had gone into the interesting history of how this came about in The Barometer Handbook. Both books show how important it is to know accurate pressure to evaluate numerical weather predictions that we ultimately rely on for routing. 



Accurate pressure is also often the fastest way to detect a change in the weather or the movement of a High pressure system we are carefully navigating around. Responding to the motion of a High is often a key decision for sailors in an ocean crossing.

In the tropics, where the standard deviation of the seasonal pressure is just a couple millibars (mb), we can know from accurate pressure alone whether or not a tropical storm is approaching—and we can know this before we see notable changes in the clouds or wind. Needless to say, we navigate in such waters primarily based on official forecasts and tropical cyclone advisories, but an accurate barometer gives us early notification that forecasted storm motions are on time, early, or late. On the other hand, any loss of wireless communications makes the barometer even more important.


In the hurricane zone between Panama and Hawaii, we would expect a July pressure of about 1012 mb, with a standard deviation of 2 to 2.8 mb.  A measured pressure of 1007 mb (2.5 standard deviations below normal) has only a 0.6% chance of being a statistical variation and a 99.4% chance of being an early tropical storm warning.  This type of analysis does not work at higher latitudes because the standard deviations are much larger.

Pressure statistics needed for this type of analysis are included in our Mariners Pressure Atlas.


We developed a sophisticated electronic barograph that was quickly adopted by the NWS for use on the voluntary observing ships  (VOS). We later sold that product to another company.


To further support the use of accurate pressure, we became the US distributor for the state of the art Fischer Precision Aneroid Barometer, used by those who want the best of the best in a mechanical unit, including the Navies, Coast Guards, and Weather Service vessels around the world, including the US. Fischer is one of the last sources for accurate, hand-made aneroid barometers.

To follow up on that, we developed both a free Marine Barometer app and low-cost Marine Barograph app for iOS and Android mobile devices. 


In short, we have worked on barometers for over 20 years now, but I felt we still did not have the unit that could have the biggest impact on marine navigation, which is what lead to the development of the Starpath USB Baro.

Not all vessels can invest in the high-end units. The mobile apps, while providing a convenient backup that can indeed broadcast pressure data to a navigation program, still rely on a device that must be charged and protected. Also running it full time does put a strain on the phone's battery life.

The New Revolution

Our goal was to develop a barometer that was first and foremost highly accurate and dependable, plus we wanted it to be easily portable. Finally, we wanted to produce it at a low enough cost to be attractive to all mariners, even those using it as a backup. For mariners we also need the output signals to be in the NMEA standard to match navigation electronics and software.

The result is the Starpath USB Baro for $49, which includes a metal transport case. It can be read in any Navigation program, or use our free USB Baro app for Mac or PC.

In stock and ready to ship from the link above.

Below shows how the pressure appears in three popular navigation programs. Video setup procedures for each are shown in the link above.


We can compare this with official pressure data from the West Point Lighthouse (NDBC WPOW1), which is 1.6 nmi from where the USB Baro data were accumulated.


The red square marks the data corresponding to our measurements with the USB Baro. We can now overlay that data with what we measured, as shown below.


So, we see that with this simple device we have access to the same pressure data that NOAA relies on to make their official forecasts and numerical weather predictions.  

The difference between1023.0 mb indicated in the Lighthouse value and the 1017.2 mb observed in our office can be accounted for to the tenth of a mb, because of the elevation of the USB Baros compared to the sea level data from NOAA.  All of the Nav apps used offer the option to incorporate this offset so the instrument reads sea level pressure directly. Our free Marine Barograph apps made for the USB Baro also have that option.

Our Guarantee

If you have now a common aneroid barometer and then compare what it reads with the known accuracy of the USB Baro over a pressure variation of 30 mb or so, you will be very pleased to own the USB Baro. 

You will either show that your aneroid is accurate, effectively calibrating it, which otherwise costs $195, or you will learn that you did indeed need a more accurate source of pressure for your boat or home.

Thursday, March 14, 2024

Special Uses of the Star Finder and Sight Reduction Tables

The 2102-D Star Finder is essentially a hand-held planetarium designed for mariners to assist with celestial navigation. It can be used to plan the best sights as well as its main function which is to identify stars or planets whose sights have already been taken. We have devoted a short book to the many uses of this powerful tool called The Star Finder Book.

Sight reduction tables are permanent mathematical solutions to the Navigational Triangle that form the backbone of celestial navigation carried out in the traditional manner using books and manual plotting — as opposed to modern solutions using computers or calculators with dedicated cel nav apps.

There are several styles of these tables, popular versions are called Pub 229, Pub 249, and the NAO Tables, a copy of which is included in every Nautical Almanac. Complete free copies are available online as free downloads, which are good for practice, but do not make sense for use underway because any device that can read the files can also support a cel nav app that does the full process, sights to fix.

All sight reduction tables, regardless of format, do the same thing. You enter the tables with three angles and come out with two angles. We enter with the declination (dec) and local hour angle (LHA) of the object sighted and the assumed latitude (a-Lat) of the observer, and we come out with the angular height of the object (Hc) and its direction (Zn) as seen from the assumed position.

Put in plainer terms, the Almanac tells us where the sun and moon, stars and planets are located at any time of the year, and the sight reduction tables tells us what the height and bearing of any one would be as seen from any latitude and longitude.... or it tells us the object is below the horizon at that time and place.

The Star Finder does exactly the same thing, but with less accuracy. We look up in the Almanac a number that tells us how to set up the disks for the time and latitude we care about, and then we read the Hc and Zn of the celestial objects from the blue templates.

With that background, I want to point out that either of these tools can also be used to answer more non-conventional cel nav questions such as one that is part of our Emergency Navigation course. Part B of question 6 on quiz 5, asks us what are the conditions that lead to the sun's bearing changing with time at a rate of 45º per hour or faster?

This comes up in the context of using the "Eskimo Clock Method" to get bearings from the sun based on the local time of day, which makes the assumption that the sun's bearing moves along the horizon at the rate of 15º per hour. That condition, we show in the course,  requires the peak height of the sun at noon (Hc) to be less than 45º, which leads to the nick name "Eskimo clock," because at high latitudes the sun is always low.

Here we have a more specific related question, but it can be solved with the Star Finder or with sight reduction tables. 

We know that fast bearing changes means the object is very high and the fastest change will occur when the object passes overhead or near so. Consider sailing at lat 15º N during a time when the declination of the sun is also about N 15º (first few days of May).  [Note latitudes get the label following the value; declinations get the label preceding the value.] In this example, the sun will bear near due east (090) all morning and then change to near due west (270) in a matter of minutes as it passes overhead.  The question we have is, how do we specify the conditions that will lead to this bearing change being ≥ 45º/hr? It will have to be high, but it won't have to cross over head.

This could be worked from any latitude in the tropics, but we stick with 15º N, and look at the star Alnilam (declination about S 1º, which corresponds to the sun's declination in Sept, 19th-21st).  Below is the Star Finder set up for the time Alnilam crosses our meridian bearing due south.



Alnilam crosses the meridian bearing due south (180) at a local hour angle of Aries equal to 84.5º.  We see that the height of the star as it crosses is 74º, which we would expect in that we are at 15N and the star is at S1 so the zenith distance (z) is 15+1 = 16º which makes the Hc (90-z) = 74º. 

The rim scale corresponds to time at the rate of 15º/hr, so 30 min later (LHA Aries = 91.0º (84.5+7.5), we see that the star has descended very slightly but now has moved west.


Thirty minutes later the bearing is 205º, or 25º to the west of 180º. Thus if we imagine this star to be the sun in late Sept, viewed from 15º N, we would see its hourly change in bearing at midday to be about 50º per hour.  This is a bit faster than the exercise asked for, but we could experiment around for a closer answer.

We can also do such studies with sight reduction tables, such as Pub 249. We enter the tables with a-Lat = 15º and dec = 1º. With these tables we do not use North or South labels but just specify if they are both north or both south or is one north and one south. The former condition is called Same Name; the latter is called Contrary Name.  We have Same Name in this example.

We will also start with LHA = 0º, which means the sun is crossing our meridian (bearing 180), and like wise look at 30 min later with LHA = 7.5º. We could look at LHA = 15º, exactly one hour later, but the rate of bearing change at LHA 352.5º to 007.5º, as it crosses our meridian, is a bit faster than the full hour on either side.


In Pub 249, each Lat has a set of pages, LHA is on the side of the page, and declination is across the page. The tabulated values are Hc, d, and Z. The d-value is how much the Hc changes with 1º of declination — for Alnihlin, dec = S 1º 12', so we would reduce the tabulated Hc by 12/60 x 60 = 12' for precise values of Hc, but we can neglect Hc for present study.)

At meridian passage the bearing is 180º, then 30 min later (LHA=7.5), we see the body dropped from 74º high at the meridian to about 72º 20' at which time the relative bearing (Z) is about 154º, from which using the rule provided (Zn=360-Z) to find the new bearing of 206º, which agrees with what we found from the Star Finder.

These terms and procedures become more familiar with a full study of cel nav, but we hope the brief discussion of the principles show how these tools might be used for other questions. Note that LHA is defined as how far west of you the body is, so as it approaches from the east it has large, increasing LHA, which goes from 358, 359, 360, 1, 2, 3 as it crosses the meridian.









***


Wednesday, December 13, 2023

How to Remember the Equation of Time


On Valentine’s Day, February 14, the sun is late on the meridian by 14 minutes (LAN at 1214); three months later, it is early by 4 minutes (LAN at 1156). On Halloween, October 31, the sun is early on the meridian by 16 minutes (LAN at 1144); three months earlier, it is late by 6 minutes (LAN at 1206).

These four dates mark the turning points in the Equation of Time. You can assume that the values at the turning points remain constant for two weeks on either side of the turn, as shown in Figure 12-7. Between these dates, assume the variation is proportional to the date.


There is some symmetry to this prescription, which may help you remember it:

14 late three months later goes to 4 early

16 early three months earlier goes to 6 late

but I admit it is no catchy jingle. Knowing the general shape of the curve and the form of the prescription, however, has been enough to help me remember it for some years now. It also helps to have been late sometimes on Valentine’s Day! An example of its use when interpolation is required is shown in Figure 12-7.

The accuracy of the prescription is shown in Figure 12-8. It is generally accurate to within a minute or so, which means that longitude figured from it will generally be accurate to within 15′ or so.


This process for figuring the Equation of Time may appear involved at first, but if you work out a few examples and check yourself with the almanac, it should fall into place. If you are going to memorize something that could be of great value, this is it. When you know this and have an accurate watch, you will always be able to find your longitude; you don’t need anything else. With this point in mind, it is worth the trouble to learn it.

Also remember that the LAN method tells you what your longitude was at LAN, even though it may have taken all day to find it. To figure your present longitude, you must dead reckon from LAN to the present. Procedures for converting between distance intervals and longitude intervals are covered in the Keeping Track of Longitude section below.

For completeness, we should add that, strictly speaking, this method assumes your latitude does not change much between the morning and afternoon sights used to find the time of LAN. A latitude change distorts the path of the sun so that the time halfway between equal sun heights is no longer precisely equal to LAN. Consider an extreme example of LAN determined from sunrise and sunset when these times are changing by 4 minutes per 1° of latitude (above latitude 44° near the solstices). If you sail due south 2° between sunrise and sunset, the sunset time will be wrong by 8 minutes, which makes the halfway time of LAN wrong by 4 minutes. The longitude error would be 60′, or 1°. But it is only a rare situation like this that would lead to so large an error. It is not easy to correct for this when using low sights to determine the time of LAN. For emergency longitude, you can overlook this problem.

In preparing for emergency navigation before a long voyage, it is clearly useful to know the Equation of Time. Generally, it will change little during a typical ocean passage. Preparing for emergency longitude calculations from the sun involves the same sort of memorization required for emergency latitude calculations. For example, departing on a planned thirty-day passage starting on July 1, you might remember that the sun’s declination varies from N 23° 0′ to N 18° 17′ and the time of LAN at Greenwich varies from 1204 to 1206. Then, knowing the emergency prescriptions for figuring latitude and longitude, you can derive accurate values for any date during this period.

This article is taken from Emergency Navigation by David Burch

Monday, December 4, 2023

Great Circle Distance — The Three Options

The great circle (GC) route is the shortest distance between two points on the globe, so we must always keep it in mind when planning an ocean crossing, even if we do not end up following that route. 

The GC route is defined by cutting the earth with a plane that goes through the departure (A), the destination (B),  and the center of the earth (C). That plane cuts the earth in half, and the points A and B lie along a circle (a great circle) whose circumference is the circumference of the earth, and the track along that line from A to B is called the great circle route.  If the plane does not go through the center of the earth, you also get a circle where it intersects the earth, but its circumference will be smaller than that of a great circle.




Distance along a great circle is measured  in nautical miles, which is a unit that was invented for just this purpose. Namely, the full great circle spans 360º, and each degree is 60', so a nautical mile (nmi) is defined as the length of 1 arc minute (1') along the circumference of a great circle of the earth. 

This is very convenient for navigation if we consider the great circle between the north pole, earth center, and south pole, which is a meridian of longitude. Arc minutes along this great circle are minutes of latitude.  Thus a navigator knows immediately if they are to sail from Cape Flattery, WA at about Lat 48 N to San Francisco at about Lat 38 N, they must go 10º of Lat or 600 nmi. Every 1' of Lat = 1 nmi.

There are other implications of this definition that are integrally related to the topic at hand.  For one, this assumes the earth is a sphere... which is not too radical an idea, having been known — or believed to be true — by every educated person on earth except Christopher Columbus for over a thousand years

As it turns out, the earth is not a perfect sphere, it is squashed a bit at the poles, as we might slightly compress a beach ball into more of a doorknob shape. Consequently a nautical mile cannot be simply defined as 1' of Lat, because the length of 1' of Lat changes slightly with latitude on this non-spherical shape. That simple definition is reserved for the less precise term sea mile, which is defined as 1' of Lat at a constant Lon. But nautical mile is the official international unit of global navigation so it has to have a definition, and that was given to it 1929: 1 nmi = 1852 meters, exactly.

That definition then tells us what we mean by spherical earth, based on the geometry of a circle. Namely, the circumference (c) of a circle = 2 𝜋 x radius (r) of the circle. Thus we have for spherical earth, c = 2 𝜋 r = 360 x 60 x 1.852 km, or solving for r:

r (spherical earth) = 360 x 60 x 1.852 /(2 x 3.141) = 6,367.9 km.

Thus we are at the first of three types of great circle distance computation, which is assume the earth is spherical with a radius of 6,367.9 km, which makes 1' on the circle = 1 nmi and we can use spherical trigonometry to compute the great circle distance (d) between point 1 and point 2, namely:

Cos(d) = Sin(Lat1) x Sin(Lat2) + Cos(Lat1) x Cos(Lat2) x Cos(Lon2 – Lon1).

This formula can be solved with an inexpensive trig calculator, and indeed this is the solution we would see in many calculators or apps, especially those that are largely celestial navigation oriented, because cel nav assumes the earth is a sphere as defined above.

If we use this method to compute the GC distance between San Francisco (37.8N, 122.8W) and Tokyo (34.8N, 139.8E) we would get 4,473.61 nmi.

But it is not just cel nav apps that use this equation. The Bowditch computations also assume this same 1' = 1 nmi spherical earth, and present the same value.

Besides cel nav focused apps, some chart navigation apps, officially referred to as electronic charting systems (ECS), also use this spherical earth solution, such as Rose Point's Coastal Explorer. We might call this traditional radius, the cel nav radius (6,367.9 km).



But if we open another popular ECS like qtVlm, and ask for the GC distance between these two points we get a different answer, 
namely 4,476.62 nmi. 


We see essentially the same answer in OpenCPN.




It is not just qtVlm and OpenCPN (two popular free ECS),  other computer or mobile nav apps might show this answer for these two points.

...that is, unless we are looking at a GPS chart plotter app or a handheld GPS unit with routing options, such as the Garmin GPSmap 78 shown below. 



In this case, we get a still different value of this same "great circle distance," namely 4,486.7 nmi. 

We also see this value in the ECS TimeZero.



In short, we have three values for the "great circle" distance between SF and TKY, and the one we get depends on how or who we ask. The differences in these example spans 13.1 nmi — and this, in an age where we pride ourselves with a GPS that gives our position accuracy to about a boat length or two (± 0.01 nmi).

Navigator's do not like inconsistent information, and will usually stop to figure out the source of the discrepancy. This note is intended to help with that.

The three values we noted were presented in increasing accuracy, which is tied to the shape of the earth that was used to compute the value. In most cases, these differences do not have a practical affect on navigation, but it is good to know if something is working right or not, and to understand what we see.

Type 1.  SF to TKY = 4,473.61 nmi. Spherical earth with 1' = 1 nmi. This solution is used in cel nav and other apps, as noted. Earth radius used is 6,367.9 km. The cel nav radius.

Type 2. SF to TKY = 4,476.62 nmi. This is what we would see in selected ECS that want to improve on the accuracy by using an improved earth radius. 

An improved earth shape is more of an oblate ellipsoid (doorknob), which can be approximated with a new spherical earth, but now using the average of the polar and equatorial radii, as shown. This improved method still computes the distance as a spherical earth, but uses this slightly smaller average radius of 6,371.0 km. This can be called the WGS84 average radius.


WGS84 earth dimensions. Keep in mind the scale. The equatorial bulge (7 km)  is just 0.1% of the radius; the depression of the poles (15 km), just 0.2%. The earth is actually pretty spherical.


Type 3.  SF to TKY = 4,486.7 nmi. Is in principle the most accurate solution as it uses not assume a spherical earth shape, but computes the distance along the surface of an oblate ellipsoid, the size and shape of which we get from the geodedic datum we have selected, such as WGS84. We will get this (Type 3) solution in most apps or hardware that lets us choose the horizontal datum, such as any GPS unit, hand-held or console chart plotter. This choice is actually an important thing to check in your GPS to be sure it matches your nautical charts; most should default to WGS84.


We also get this geodetic or ellipsoidal solution for "great circle" distances in several popular computer based ECS, such as TimeZero.


Google Earth will also give this value, but for other locations you may get different results as they may use different datums for different locations, which we do not seem to have control over. (The same is true, by the way, for the elevation data set or model it uses for different parts of the world. It is likely the best we can conveniently come by, but we will not know the details.)


Numerical values of these distances can be checked online with the
Jack Williams calculators.


These values can be used to determine what type of computation your device is doing. Use Departure = (37.8, -122.8); Destination =  34.8, 139.8). 
Then check for the GC distance between them.

4473.6 means spherical earth using the cel nav radius (6,367.9 km)
4476.6 means spherical earth using the WGS84 average radius (6,371.0 km)
4486.7 means a WGS84 ellipsoidal computation

A consequence of a true ellipsoidal computation means a nominal, long-distance great circle estimated position depends on which way you are headed. Consider starting from the equator at 130 W and traveling 50º N versus 50º E. Sailing along the surface of a spherical earth, the distance you travel would be the same in both directions, namely 3,000 nmi. But sailing on the surface of an oblate ellipsoid, this is not the case. You have a smaller radius going toward the pole than you do going along the equator. Going north you sail 2,991.8 nmi but sailing east you go 3,005.4 nmi.





For completeness, let me add a 4th solution!  One that goes in the other direction: not striving for high precision, but looking for a solution that can be done with a plastic device that still works if soaking wet, after falling off the nav station and getting stomped on by numerous crew members' wet boots.

Great Circle Solutions with the 2102-D Star Finder