Friday, April 19, 2019

Finding UTC of LAN with the Starpath Custom Sun Almanac

We published a short book last year that is intended to be the bare minimum celestial navigation training that will serve as a backup to a loss of GPS. In keeping with the backup concept, the book is presented in such a way that it is totally self-contained, which means the book also covers how to use the simple Davis Mark 3 sextant. This is a small device, for under $50, that could be used to safely circumnavigate the world.

Our promise is that this book could be opened up and read for the first time when it was actually needed, and it would be adequate to teach how to take the sights, and then find your position from them using only data in this small book—a large part of which is a custom Sun Almanac designed to make the position solutions especially easy.

The techniques taught in this book are finding Lat and Lon from "noon sights" (local apparent noon,  LAN) as well as Lat by Polaris in the Northern Hemisphere. What we forgot to include in this first printing was a way to determine an efficient time to start taking the sights near midday. LAN occurs when the sun crosses our meridian, bearing due north or due south at its peak height in the sky.

When the sun is less than halfway up the sky at noon, we can approximate its motion along the horizon as a bearing change of 15º per hour.  So if we want to start about 30 min before the sun reaches the horizon, we would start taking sights when the sun was bearing about 173 T.  After that first round of sights, we will know the time of LAN and the next day we can fine tune the starting time.

When the noon sun is much higher in the sky it is more difficult to predict its bearing change rate as it approaches noon, and as we head off toward the tropics the sun will indeed be much higher at noon, so it is valuable to have a systematic way to predict the time of LAN to plan around. The custom Sun Almanac offers an easy way to do this. In fact, it is easier than the standard methods we use when teaching the "full cel nav" course.  The full theory is in the picture below from our textbook Celestial Navigation: A Complete Home Study Course.

As the earth rotates toward the east, the position on earth directly below the sun (its geographical position, GP) moves west at the rate we are rotating, namely 360º of Lon in 24h = 15º of Lon per hour.

The custom Sun Almanac tells us the longitude of the GP (called its Greenwich hour angle, GHA) every hour of every day. So we can go into Sun Almanac on the right day to see which whole hour of UTC has the sun's GHA just east of our location.  Then we subtract that GHA from our DR-Lon to see how far it has to go (as an angle) to get to us, and then we just covert this angle to time at the rate we are turning, which can be derived from 15º = 1 hr, 1º = 60 min/15 = 4 min, etc.

                      1º = 4 min

                      1' = 4 sec.

Here are two practice problems from our textbook (page 33.)

Example 1.  July 25 from DR-Lon = 122º 18'W.

DR  = 122º 18'   =  121º 78.0'
20h = 118º 21.7' = -118º 21.7'
            diff =    3º 56.3'

3º = 3º x 4m/1º                 = 12m
56.3' = 56.3' x 4s/1' = 225.2 s =  3m 45s
                            SUM = 15m 45s

Final time is 20h 15m 45s, which we would round to 2016 UTC, as we never need to know these times more precise than that, not to mention that the DR position that it is based on has some uncertainty.

This is the UTC of LAN observed from 122º 18' W. If we wanted the watch time (WT) of the event for a watch set to zone description (ZD) + 7, then we would have to back out that ZD.

Recall the definition of ZD, which comes from this equation UTC = WT + ZD, where WT is the watch time being used for navigation. Thus if the ZD = +7 and the WT = 1200, then the UTC = 1900. Likewise if we know the UTC is 1900 on a watch with ZD = +7, then the WT = 1200. 

In the above example, LAN = 2016 UTC would correspond to WT = 1316 for a watch set to ZD = +7.

Example 2. Find  UTC of LAN on October 27 viewed from 136º 10.5' E.

This is an eastern longitude. In east longitudes the GP of the sun is still moving west with increasing time, so the longitude it crosses gets smaller as it passes across the eastern half of the globe. But that does not matter to us, because, unlike longitude, GHA does not decrease on the eastern half of the globe. GHA is defined as 0º to 360º, measured west from  Greenwich.  Our job is to determine what GHA does 136º 10.5' E correspond to. Starting at Greenwich we head west to 180º W (the dateline), and then proceed from 180º E to 136º 10.5' E, or we cover 180º - 136º 10.5'  =  179 60 - 136 10.5 = 43º 49.5', which is what we must add to 180º to get the GHA equivalent of this eastern longitude. The answer is 223º 49.5'.

Now we are back to solving the problem just as we did for western longitudes.

 "DR" = 223º 49.5'E 
  02h = 214º  1.7'  
 diff =   9º 47.8'

9º = 9º x 4m/1º                  =  36m 00s
47.8' = 47.8' x 4s/1' =    191.2s =  3m 11s
                              SUM = 39m 11s

We add this to 02h to get 02h 39m = 0239 UTC,

If we need to convert back to WT, we refer to the definition: UTC = WT + ZD, or with ZD = -9, we WT = UTC +9h = 1139 WT. 

For those who might like extra practice, you can use the USNO sun data computer (Form B) to randomly select locations and dates to compute UTC of LAN (they call it "sun transit"), and compare that with what you get from the custom Sun Almanac.

In principle the answer depends on the year, but our custom Sun Almanac averages out yearly differences, but even with this, the time of LAN found from the these tables will always be right to well within one minute.

Monday, April 15, 2019

New PDF Edition of a Classic NWS Pub... With Related Notes on FTPmail

There is a new edition just out, dated April 11, 2019 of the long standing publication, now called 

We refer to this document in our teaching by its file name "rfax.pdf" rather than its title for a couple reasons. First, even though the content and title of the book have changed over the years, the filename since it was first put online has not changed; you can always find this with a search for "rfax.pdf." 

This is now a free PDF, but back before internet it was a must-have nav pub for ocean navigation. In those days, in keeping with its earlier title (below), the pub included the voice broadcast schedules as well as radio fax schedules, including valuable global diagrams of the forecast regions. 

Oldtimers with memory of ocean sailing before the internet will recognize this book that everyone had on board. For historical interest, old versions are online.

Secondly, beyond containing all the radiofacsimile schedules from around the world, it has also for many years now included the best of all of the official references to the FTPmail service offered by NOAA. This part of the publication has been hidden in this pub since it was added some years ago by not being mentioned on the title page or bookmarks. It was at the end of the document as "Appendix B," but the only bookmark was to Appendix A-8, without a title. 

Granted, if you did read to the end of the Table of Contents you would see it listed as Appendix B. In this new edition, the FTPmail instructions are in the same place, and they have minimized the confusion by removing any reference to any appendix in the bookmarks.

Appendix A is also valuable for mariners including contact info for all branches of NWS as well as a detailed list of all weather resources online. 

Appendix A and B are major resources, not implied by the title of the document, so we just have to know they are there... which you would know from our weather book as we have stressed the value of this reference of years.

I am hoping that if anyone from NOAA sees this note they might consider reissuing this latest version of rfax.pdf to include a mention of these important references in a subtitle of the document (such as "and Related NWS Resources"), and add bookmarks with real titles.

Also if they do make a new one, I hope they will also update the section at the end on the Voluntary Observing Ship (VOS) information, including the links to Port Meteorological Officers (PMO) listed. PMOs are the key liaison between participating ships and the weather service, and their personnel and contact information in this April 2019 issue is a year or two out of date. Having that data wrong is incongruous in that the book starts out on the first page of text praising the VOS program and how important it is to mariners. 

There is another subtle reason modern mariners have an interest in this rfax.pdf document, even if they do not receive weather maps by rfax, but instead get their maps by email request. What we learn from this document is which nations actually produce synoptic weather charts. If such maps exist, they are the products of professional meteorologists, and not just reprints of a GRIB presentation of a pure model output. As such, they add much to our confidence in forecasts compared to using pure model data alone.

Once we know these maps exist, we can go about finding out how to get them by email request. We have compiled a list of Southern Hemisphere sources, and we can find international sources in the FTPmail folder on line—see, for example, the file called otherfax.txt.

The rfax.pdf also tells us when these maps are available for email request, as they must certainly exist at the time the rfax version is broadcast.

One possible reason why these valuable FTPmail instructions in rfax.pdf are not being stressed as much as I think they should be, is the NWS is working on new publications devoted to FTPmail.

Historically, the most often referred to instructions for FTPmail is a file called FTPmail.txt, which is pretty much unchanged since the system was developed many years ago. It is an introduction to conventions, along with a list of links to more detailed product lists, i.e., otherfax.txt. 

One version of the developing new instructions can be found at a link at the Ocean Prediction Center (OPC) under menu: INFORMATION / Receiving Products / FTPMail: Products via Email, which is a more convenient version of FTPmail.txt.

Another version is attributed to the VOS. Find this one at the OPC menu: INFORMATION / Product Information / Product Guides / Marine Radiofax Chart FTP Email Procedures.  This one has the valuable feature of showing not just a list of map file names, but also includes samples of each of the maps. But this document is maps only, whereas FTPmail can be used to obtain just about any product made by the NWS, including all reports and observations, and such important ones for offshore as the Metarea forecasts and tropical storm advisories.  Also, none of the files are actually linked in this presentation.

In short, both of these alternative discussions of FTPmail are better than the simple FTPmail.txt, but neither one of them, nor both together, is as good as the set of hidden instructions in the back of rfax.pdf, which stand out first because they provide live links to the data and second they specify the broadest range of products, directly without reference to other documents.

For that reason and others mentioned, it seems navigators will want to download this file and store it on one of their their devices they will have when underway.

Wednesday, April 10, 2019

Starpath News — April 2019

• Tracking Jacob Adoram 
We are in contact with  former Starpath Student Jacob Adoram as he rows from Seattle to Cairns AU. He left on July 7 and now, after some 6,000 miles of rowing, he is just over 1,000 miles from Cairns. To make this event into a marine weather study project we have created a series of Google Earth overlays so you can see live conditions of wind, current, pressure, and sea state. We have an ongoing series of short articles introducing short video discussions of his progress. Start at the first post to get an overview of what we are doing, and check the index of overlays.  Even disregarding the factor there is a row boat somewhere around where we are covering, it is very interesting to see the ASCAT live satellite wind measurements.  For a jumpstart, if you have Google Earth installed you can click here to load the KML file

Our original goal was to try to get live measurements of ocean currents to compare with the RTOFS and the OSCAR predictions. This has not produced much fruit. First these two models do not often agree (they are totally different kinds of products) and second it is difficult for such a large boat with so little control to measure accurate boat speed and compass heading to compare with COG and SOG from GPS. The difference between these two vectors is the current we want to measure. In cases where both models did agree there was notable current (ie > 1 kt) and more or less aligned with the wind, then we do indeed see him make more progress than he would without the current, but that is about it.

On the other hand, we have had much more success in testing new simulated radar reflectivity forecasts of the FV3 GFS model. We discuss this new breakthrough in marine weather technology in Modern Marine Weather  (see pages 112 to 115), but now we see it live. Namely we can forecast to the hour when he should be in squalls and how severe they are using a GRIB format of the combined reflectivity,  and then compare that to his observations. These are easy to measure events so the data are good. We have just started this study but it does look promising in the cases we have had.  We use the program LuckGrib to download the FV3GFS data and display the reflectivity

• New Regiment of the Pole
In keeping with one of our unpunished mottos "Always old; always new," we have revamped the late 16th century celestial navigation tool called a Regiment of the Pole. This will be written up in  more detail shortly in our Navigation Blog. It is a way to read from the orientation of stars near the pole the correction to a Polaris sight needed to find latitude.  We had originally developed a prescription for this in the mid 80s for the book Emergency Navigation, and now we are fine tuning that one. It is the same general principle, but with a relatively easy addition we get more accurate results.  This new method is used in our new book GPS Backup with a Mark 3 Sextant.  With this rule and a sextant, you can find your latitude in the Northern Hemisphere to within a few miles on any date without any further tables.

• Weather Trainer Live Updates
Over the past few months we have updated Weather Trainer in several ways. We have updated the content and reorganized several of the sections.  We also now have the Mac and PC versions working in the same manner, and we have changed the general structure of this cloud based service and training tool. The initial activation period remains at one year, but at the end of one year the subscription does not end if users have an active starpath web card of any kind, which includes being registered in any course. If you sign up for a second course after the weather course, for example, then the Weather Trainer is active for another year at no charge. After your web card expires, you could come back in a year or two and reactivate a web card for as low as $17 and have full access to the Weather Trainer to check out all the latest resources.

• Looking into live online lectures
We have done a few short test runs on offering live online lectures. This looks promising and we will be pursuing that shortly. We will announce the courses in the Discussion Forums and in our Facebook page.

 You could call it experimentation… or you could call it tinkering.
We have experimenting with use of pitot tubes for anemometers. The idea came from a new product from Ireland that apparently works well at least for stronger winds.  It is nicely called the Wind Urchin.  This is too big for the mast head, but I do not see why it could not be made much smaller and then we read honest 3d wind direction and correct vector speed.  We now have a real pitot tube and a differential barometer for testing. All seems to work very simply, but to get accurate values at low wind speed will be challenging. The first application of this type we know about is in an observatory on Mt. Washington.

We are also experimenting with a new style of artificial horizon suitable for use with stars. It has a lot of promise but needs some work. Seems it is best tested during daylight with a low daytime moon so testing times are limited. As spring approaches we also gain the number one factor for good cel nav studies of any kind, namely warm weather!

Monday, March 11, 2019

How to Fold a Chart

Ask a dozen cruising mariners how best to fold and store charts, and you will likely get a dozen different answers. So with that background, we boldly go on and proclaim what the best method is!

First though, we should say that if you have just a few charts, it really doesn't matter much how you keep them, it will always be easy to find what you want. The crucial issue of chart storage comes into play when you have a lot of charts, because then the situation can get very much out of hand in just one long trip or a season of day sailing with multiple charts.

The long tested solution is to fold them chart-side in, blank white paper side out, either once or twice, depending on the chart size. Most charts take two folds. Then label the corner with the chart number, as shown in Figure 12.26-1. It is best to use a consistent size and style of lettering. We’ve found that a bold Sharpie pen is ideal for this.

Figure 12.26-1. Best method for folding charts: chart-side in, labeled on the folded corner with chart number and perhaps name. Stacked in numerical order. From our text Inland and Coastal Navigation.

Then arrange the charts in numerical order and store them flat somewhere, preferably in a sealed garbage bag or other waterproof wrapping. Under a settee cushion might work, or under a bunk mattress.

The virtue of this method of folding and stacking is that when you are looking for a chart in the stack, each chart presents only one corner. Folded this way, you have a crisp, uniform presentation for all charts. If you try to fold them right side out and use the nice large chart numbers printed in each corner, then each chart will present four sheets to you as you search the stack, and they will not be uniform at all. It should in principle not be the case, but sometimes charts must be found in a hurry, and this is the method that best solves the problem.

In the old days, we would get a free printed chart catalog from your local chart dealer... but printed chart catalogs are now a thing of the past. We want one of these catalogs so we can index what charts we have on board, but now we have to get a PDF of one and print it to get the pages we need. Find a link to the chart catalog pages at Any voyage of a few days could take many charts—at least in the old days before electronic charts. Here is an example headed to Maui, which would call for two catalogs Pacific Coast and Pacific Islands. We can print the pages we need and highlight the charts we have on board. Put the catalog pages into plastic page protectors, then they go in with the charts. The HI section might look like the following.

This type of visual overview of what you have on board is very helpful, especially on long voyages on inland or coastal waters. On a long run on inland waters your index will be referred to frequently. You might use your catalog to find the ones you need for the day, pull them all out, and stack them on the chart table. When done, they can be refolded and inserted in proper numerical sequence back into your main stack for the trip back.

For a boat with a lot of charts, I cannot stress how important this operation is. On most new boats I ever sailed on, the very first thing that had to be done was spend the afternoon organizing the charts. This procedure usually reveals multiple copies of identical charts, as well as important ones that are way outdated. If you find duplicates, you might want to note the chart date in the label, and then decide what to do about them later on. In any event, they will be easy to find after this operation.

If you ever visit any Navy, NOAA, or USCG vessel that does not have a 4 by 5 foot chart locker, which can store them all unfolded, then you will almost certainly see the charts folded in the manner described above. In fact, even ships with chart lockers often use this system, which is how I first learned of this technique—buying a stack of old charts from a Navy shipyard. The technique has since been confirmed from other sources and personally tested for many years.

What now postdates my early learning and practice of this method is the advent of echarts. Headed to HI from Seattle, for example, there is no reason at all that your nav program should not have stored every chart from the states of WA, OR, CA and HI. They are free, and much easier to download by state than individually. I would also take both RNC and ENC of the ones I anticipate actually using. The others are just bail out options if something goes wrong.

Furthermore, there is also no reason at all that PDF versions of all the charts you might ever conceivably need in any contingency are not also stored in your phone or tablet. Electronics are vulnerable, but if put the chart sets in many devices onboard, you are fairly likely to have one when needed.

This allows us to be more judicious in choosing our relatively expensive paper charts ($19-$25; echarts are free), which might then reduce our selection to a bare minimum needed to cover the unlikely circumstance that all electronics are lost.

Recall, too, in the contingency mood, the valuable exercise we have in our course on making a working chart sketch from the description of a waterway found in the Coast Pilot.

Not sure we need a video to show this process, but there is one....

Wednesday, February 27, 2019

Star Names

For navigational purposes, stars have two naming conventions: proper names such as Canopus and a Greek letter designation called the Bayer system such as Alpha Carinae, meaning the alpha star of the constellation Carina. The alpha star means most dominate, or if all are about the same brightness, the first one in a logical sequence of numbering, as in the Big Dipper, which goes alphabetically from Alpha Ursa Majoris (Dubhe) at tip of the cup to Eta Ursa Majoris (Alkaid) at the end of the handle. Officially the constellation name in this system is in the Latin genitive form (belonging to), but Alpha Ursa Major would be adequate for record keeping and communications.

Science fiction readers will likely know of Alpha Centauri, which navigators call by its proper name Rigil Kentaurus, because this is the nearest star to our solar system and perhaps an early one to visit or receive visitors from.

Figure 1. Sample of star names. The red labels have been added. Selected stars called navigational stars are assigned  unique numbers in the Almanac.

The USNO star chart above (from the support page of our cel nav text) shows how these names are used in the Almanac, sometimes using one name form and sometimes another. We see this also in the star charts from the Almanac, a sample of which is below.

Figure 2. Sample star chart from the Almanac.

Figure 3. Greek alphabet

Comparing the alphabet with Figure 1, see the points mentioned above. In Bootes, there is one dominant star and it is alpha, and the other letters are more or less random.  In the Big Dipper the stars are similar in prominence, so they are labeled sequentially along the figure. In Leo, the brighter Regulus  (magnitude 1.4) is alpha and Denebola (magnitude 2.1) is beta. The brightness of stars is specified in terms of magnitude (see Brightness of Stars and Planets.)  All magnitude-1 stars  and most magnitude-2 stars have proper names. Less bright stars typically do not.

Note on Constellations
We often think of constellations as groups of stars, but that is not the full picture. The full sky is divided up into sections with straight boundaries (SHA and Dec), much as a state is divided up into counties.  The full globe of the sky is divided into 88 constellations, which means that every point in the sky is in a constellation, even if there are no stars near it.  If you look up any constellation in the wiki it will include a nice vector image of the constellation boundaries. A sample is shown below.

Figure 3a. Outline of the constellation Ursa Major (white area).  See these for any constellation
 in the wikipedia.

Navigational Stars
There is yet another classification of stars that is crucial to navigators. The 57 stars listed on the daily pages are called navigational stars. They are also on the Index to Selected Stars in the back of the Almanac (p. xxxiii). These are bright stars, magnitude 1 or 2, but they are not chosen by brightness alone, but rather selected uniformly around the sky so that several of them will be in view at all times from any location. Polaris is notably not on that list, though it is used routinely for navigation and certainly qualifies as a "navigational star."

Figure 4. Index to Selected Stars—the navigational stars. 

This list has been on page xxxiii  of the Almanac for many years, which is easy to remember, and it is also printed on the yellow cardstock bookmark that comes with the Almanac.  Listing the stars by number and alphabetically is a handy feature of this list.  Cel nav apps, including our own StarPilot, often use these star numbers for a quick way to select a star.

There are another 116 stars listed in the back of the Almanac that do not show up on the daily pages. That full list (174 stars) includes the navigational stars, and is presented in a unique manner, illustrated in Figure 4. The first half of the year (January to June) lists the stars by their Bayer designation ("Greek-letter name"), whereas from July to December they are listed by their proper names. This can be an important nuance to know when it comes to some star ID questions. All navigational stars have a proper name.

Figure 5. Sections of two facing pages of the star list at the back of the Almanac.

Here we learn, for example, that the brightest star in the sky, Sirius, is also known as Alpha Canis Majoris, the alpha dog in the Dog constellation. It has still another name—you guessed it: the Dog Star.

Stars in this list are sorted according to increasing SHA. If you know the star but not its SHA, check the star maps in the Almanac or the USNO version linked above, which plots star locations on a Declination-SHA grid.

This list provides more stars for navigation but it is very rare that we need one from this list. The navigational stars usually carry us across the ocean. Indeed, any ocean can typically be crossed using the same few stars with the aid of a planet or two.  See a real example in Hawaii by Sextant.

 How to Pronounce a Star's Name
My answer would be, just about however you want to.  These are often complex names; many evolved from the Arabic, although Arabic speaking persons might not recognize the present form of the word. Alnilam, Alioth, Altair, are examples, as is, by the way, Alcohol. In fact, many of the prominent stars have many names, or many spellings for the same name, although there is now an official list of names and spellings as of June, 2018.  You will see that most of these were only formalized in 2016 or 2017.

The wiki has good presentations of stars organized by constellations, which include detailed maps of the constellation. Standard pronunciation guides are complex, so here is a starting point for the navigation stars, plus a few others we use in the course.  There are more alternatives than I have listed here.

Name                Pronunciation
Acamar               AH-kuh-mar
Achernar             AK-er-nar
Acrux                A-krucks
Adhara               ad-HAR-a
Al Nair              all-NAYR
Aldebaran            al-DEB-ah ran
Alioth               AL-lee-oth
Alkaid               AL-kade
Alphecca             al-FECK-ah
Alpheratz            AL-fer-rats
Altair               AL-tair
Ankaa                ANG-kah
Antares              an-TAIR-ease
Arcturus             arc-TOUR-russ
Atria                AH-tree-a
Bellatrix            BEL-la-trix
Betelgeuse           BET-el-jooz
Canopus              can-OH-pus
Capella              kah-PELL-ah
Caph                 KAF
Castor               CASS-ter
Deneb                DEN-ebb
Denebola             de-NEB-oh-la
Dubhe                DOOB-huh
Elnath               EL-noth
Eltanin              EL-ta-nin
Enif                 EEN-if
Fomalhaut            FO-mal-oh or FOH-mal-owt
Gacrux               GAK-kruks
Gienah               JEEN-ah
Hadar                HAH-dahr
Hamal                HAM-al or hah-MAHL
Kaus Australis       KOWSS ow-STRAH-liss
Markab               MAR-kab
Megrez               meg-REZ or MEG-rez
Menkar               men-KAHR
Menkent              men-KENT
Merak                MER-ak
Miaplacidus          mee-a-PLASS-id-uss
Mintaka              MIN-ta-ka
Mirfak               MERE-fak
Mizar                MYE-zahr
Nunki                NUN-kee
Peacock              pea-COCK
Polaris              poe-LAIR-is or poe-LAHR-is
Pollux               POL-lucks
Procyon              PRO-see-on
Rasalhague           RAHS-al-haig
Regulus              REG-you-luss
Rigel                RYE-jel
Rigil Kentaurus      RYE-jel ken-TAW-russ
Sabik                SAH-bik
Saiph                SAFE
Scheat               SHEE-at
Schedar              SHED-er
Segin                SEG-in
Shaula               SHOWL-a
Sirius               SEER-ee-us
Skat                 SKAHT
Spica                SPEE-ka or SPY-ka
Suhail               soo-HALE
Vega                 VEY-ga or VEE-ga
Zubenelgenubi        zoo-BEN-el-je-NEW-bee

Zubeneschamali       zoo-BEN-ess-sha-MAH-lee

My suggestion is choose a way to say it and then whenever you do say it, say it bold and clear. Anyone hearing who might have thought it was done another way would then pause, and likely take your word for it. It would be difficult in many cases to argue what is right.

For more, the definitive study of star names is the classic Victorian text  Star Names Their Lore and Meaning by Richard Hinckley Allen, which is online, being a remarkable work of Bill Thayer, formerly of University of Chicago.  There are also tons of used print copies online for a dollar or two, being the Dover edition from the early 60s. We have a copy of that one which we scanned, OCRed, and cleaned up to make an ebook long before Bill Thayer's work and also long before Google got into the practice of scanning all books.

The best stargazing program for Mac or PC is the amazing free one called Stellarium. Has a short learning curve, but is a remarkable and beautiful tool.  You can read digital values of height and bearing for cel nav studies as well.

Saturday, February 23, 2019

Check Assumed Positions After Plotting Cel Nav Fix

A question about DR came up in our cel nav course that took us back to a couple basic points. One is the difference between manual cel nav solutions using tables and plotting sheets compared to doing sight reduction and position fixing with a computed solution using a computer or mobile app. We discuss most of these differences in our textbook, but realized today that one important difference is not stressed in the book, although it is covered in many places in our online course.

The issue at hand is evaluating a fix based on the dimensions of the plot itself. In other words, we look at the lengths of all the lines involved. There are two kinds of lines on the plot of a cel nav fix: the lines of positions (LOPs) themselves, and the lengths of the azimuth lines that run from the assumed positions to the LOPs (the a-values). A sample is shown in Figure 1.

Figure 1.  Crucial lengths defined.  If any of these are approaching 60 nmi long, we should use the fix for a new DR and redo the sight reduction.

The crucial lengths are the a-values, a1 and a2, as well as the lengths along the LOPs from the fix to the azimuth line (L1 and L2).  Chances of long lines is enhanced when the DR is about halfway between two whole latitudes, because we must choose one of these for the assumed latitude. 

We make the assumption in standard manual plotting procedure that the circle of equal altitude near our position can be approximated as a straight line (the LOP), and that the azimuth lines, which are segments of a great circle, can be approximated by a straight line as well.  Both of these approximations can break down if any of these lines gets too long.

Below is a new addition to textbook Section 11.24, followed by examples and background of this concern.

Evaluate Assumed Positions in Manual Sight Reductions

Before we start to evaluate a full set of sights for optimum accuracy, we should pause at the end of the sight reduction to check that our basic assumptions are in order for the sights at hand.  How we do this depends on how we are doing the sight reduction. If we are using a computer or nav app for sight reduction and position fixing,  this step can be skipped completely, because the location of the assumed position (AP) does not matter when computing a fix. With calculator solutions, we generally use our DR at the fix time as the basis of the computations, and even if the DR is way off, the programs will work around this either by solving for intersecting circles of position, or iterating lines of position (LOPs) automatically, as explained in the manual approach below. In any event the choice of AP is not a concern for computed solutions, but we must consider this when doing manual solutions using plotting.

When doing sight reduction by hand using tables and manual plotting, we must check that none of the lines leading to our plotted fix are too long. This can happen with manual sight reduction as we must choose an AP based on the minutes part of the GHA and our DR Lat and Lon. We can always choose the AP to be within 30' of the DR, but even then with various configurations of body bearings and APs, we can end up with large a-values. If the DR position was wrong by a lot, meaning the distance between the DR position and the fix found using the APs chosen properly for each sight, then the a-values can get even much larger.  In this case the distance between the fix and the azimuth line measured along the LOP can get large as well. The lines that can get too long are illustrated in Figure 1, above.

We might consider that any crucial line in our plotted fix that is over (or approaching) 60 nmi should be considered too long for best results, meaning they may violate our basic assumptions in the plotting. Underlying the manual plotting solution is the assumption that the LOP itself is a valid straight-line approximation to a segment of a circle of equal altitude (see textbook Section 10.6), and we assume that the azimuth line is a segment of a great circle, even though we are plotting it as a straight rhumb line. These approximations can break down when the lines get too long. The consequences of this also depends on the direction they are oriented, but a generic filter on the lengths should catch all cases.

The solution to long lines on the plot, is to plot the fix in the normal way, then read that Lat and Lon and call that the new DR for these sights.  Then do the sight reduction again using this DR, which will call for new APs, and then the lines will all be shorter, and the fix you get will be more accurate.

In most cases of routine cel nav—see Hawaii by Sextant for examples—we can proceed as normal, and will not find any excessively long lines, but if we do, we can fix it. On the other hand, if we suspect ahead of time that our DR could be wrong by over 40 miles or so, then we might do a quick 2-LOP fix to check the lines and find a new best DR to use before any further analysis.

The instructions to Pub 229 include a Table of Offsets for curving the LOPs to help with this correction. That table shows corrections of several miles for lines L1 or L2 of just 45 miles. Errors due to long a-values are more subtle and depend on the azimuths; they occur when a straight line approximation to the azimuth line diverges from the curved great circle tract between AP and GP.

In summary, any of the crucial lines in a fix plot approaching 60 nmi long should call for getting a new DR from the fix and redoing the sight reduction. Errors of several miles can occur without this precaution. Such errors are larger for high sights, above 70º or so, and line lengths can get enhanced when  DR Lat is about half way between two parallels.

More Background on this Topic

This section goes into more detail to look at how these errors actually come about and the sizes of them in a few examples.  At this point, you can say "I know the rule now, and I will use it as needed," and then skip this section!  

Below is a sample of the Pub 229 Table of Offsets and how it is used to make the corrections for curved LOPs.

Figure 2.  Table of Offsets from Pub 229 and their example of how they are used.

We see from the Table that for a sight that has Ho about 73º, an LOP line length (L1 or L2 in Figure 1) of 45 nmi (45') would cause an error of 1 nmi (1') on the LOP position.  At 60 nmi, the error would be much larger. All in all, it is likely faster to just get a new DR leading to shorter lines than to correct the individual LOPs this way, and with that in mind, we generally do not cover the use of the Offsets in our course.

Here is a way to approximate this correction if you might care to.

Figure 2a.  We see the 0.9' at 72º in the Pub 229 table, and then can estimate that at 100 nmi, the correction would be about 4.6'. ( To be redrawn when we can. )

Errors due to large a-values are a bit more subtle and there is no easy table for the correction. The situation is illustrated in the graphics below, starting with cel nav data from the USNO.

Figure 3. Cel Nav data from

The Mars and Markab sights are plotted below in OpenCPN, using a neat feature of that program that lets us plot a line segment as rhumb line or great circle, or both.

 Figure 4. Geographical positions (GP) of two bodies showing the great circle track to them from the assumed position (AP). 

Here we see an interesting example of spherical geometry.  From this AP at this time, we would actually be looking just north of west to see a star that is a long way south of us!  The direction we look to see a star is called its azimuth, 279.8 T to Markab in this example. This azimuth generated in the sight reduction process is numerically the same as the initial heading of a great circle (GC) track from the AP to the GP.  Below we expand the section near the AP.

Figure 5. Deviation of a rhumb line along the initial heading of a great circle track.

In the case of Markab, we would plot the azimuth line in direction 279.8, and then plot the LOP perpendicular to that line. However if the a-value is ~60 miles or so long, this line is some 4 nmi off the GC track as computed in OpenCPN. This, in itself, would not matter, but the offset will effectively rotate the LOP by some small amount. In this simple picture the rotation would be arctan (4/60) which is about 4º—but this example overestimates the effect. We get a better feel for the effect by breaking the GC track down into smaller steps, which is easy to do with a program like StarPilot.

Figure 6. Start of the great circle track to Mars from the AP in 0.25º of Lon steps (~ 12 nmi).

In this example we see that an a-value of 85 nmi would yield an LOP that is rotated by 1º off the true orientation.  This could lead to a relatively large fix error if the lines L1 or L2 were also large.  On the other hand, a quick redo of the sight reduction using new DR removes this error.

A real example
As a practical example that includes a bit of each of these long-line errors, we look at  Problem 8 from the real sight data included in Hawaii by Sextant (HBS).

Figure 7. Sample sights from HBS.

In this voyage and the documentation of it, we did a running fix between two sun lines taken at 1102 and 1334, which led to the fix labeled FIX 1334.  For now, we do not care about running fixes and just treat these two sights as if we were dead in the water.  That would lead to the fix marked with a red circle, where no line was advanced.  That is the fix we are studying in light of the fact that L2 from the 1334 sun line is indeed over 60 nmi long.

In the next picture we used the 1334 fix to get a new DR and then did the sight reduction again, using Pub 229 (without any offsets). That is shown below with the new APs.

Figure 8. Redo of Problem 8 sight reductions with new DR position.

The red data are the original plotting using original APs shown in Figure 7. The green plotting is the new fix using the previous fix position as a new DR. You can view both work forms to see where all the numbers come from.

This correction moved the fix by 3.5 nmi, which is a significant amount if we are striving to get the best possible cel nav fix.


To follow up on the point that these long-line problems do not arise in a computed solution, we look at a computed solutions using the raw data from the initial sights, including the inaccurate DR.

Here is the raw sight data of the two sights and the computed LOPs 
using 1802 DR 36º 31' N, 133º 28' W. Times in UTC
July 10, 1982, HE = 9 ft, IC=0, 
18:02:02, Hs 49º 24.5',   a = 21.5' T 098.0
20:34:46, Hs 74º 42.0',   a = 10.7' T 158.0

For a fix at 36º 27.8' N, 133º 01.6' W.... which is what we got by plotting, but only after one iteration of the AP.

In other words, again, if you compute the fix with a standard navigation program you do not have to worry about this factor.  The crucial feature of a proper cel nav program is it will iterate the results automatically that we are illustrating here manually, meaning it gets a fix from the user input DR, then replaces the DR with the fix, then computes the fix again to see if it changes, and it if did, it repeats again till there is no change. Alternatively, a program can intersect the two circles of position directly and not use any DR at all.


As a closing note on this topic, in the informational section at the back of the Nautical Almanac, called Sight Reduction Procedures: Methods and Formulae for Direct Computation, in Section 11, Position from intercept and azimuth by computation, they address the topic we covered here. 

They propose that the distance to be monitored (equivalent to our fix to AP) should be less than 20 nmi and if not, change AP to fix and recompute.  They can afford to make the distance smaller because they are computing the solution, as we noted that all programs do.  When doing it by hand, we can stick with the "anything approaching 60 nmi" for our trigger.