“"It is useless to attempt to reason a man out of a thing he was never reasoned into." ? Jonathan Swift”

Navigation Using Dead Reckoning

The traditional process for dead reckoning (DR) is graphical. See the American Practical Navigator's chapter on Dead Reckoning for more details and technical information about navigating using dead reckoning. However if you try our way, you can convert your DR information directly to Latitude and Longitude.

Warning -- Use these tables and methods at your own risk. They have been tested however we aren't responsible for their misuse or any errors that might cause an accident. Results checked against Ed Williams Great Circle Calculator.

Tools

There are some basic measurements you need to make to track your position.

  • Compass with known correction factors and magnetic variance.
  • Method for tracking your speed
  • Ability to achieve an occasional fix (GPS, Celestial, Landmarks, Buoys, etc.)
  • And as a navigation aid, your own copy of the DR tables

Rules for DR

  1. Plot Course and Speed every hour

  2. Plot all changes in course and/or speed.

  3. Plot after every Fix

  4. Plot after every line of position (Celestial Navigation)

Trick for Speed Measurements

If your instruments have failed, use the following formula and something that floats.

Speed (kts) = 0.6 * Distance / t

For example, say your boat is 30 feet long from bow to stern.  Drop a small piece of wood (yes, this is like a Dutchmans Log or Chip Log) in the water at the bow.  Start a stopwatch.  When the wood passes the stern stop the watch.  The elapsed time is "t" and the "Distance" is 30 ft.  From this you can approximate your speed.  Try this several times in different conditions and compare to your instruments and refine your technique before you need it.

Using The DR Tables

Assuming you're familiar with the basics of DR, we'll jump into the meat of the tables.  The goal is to covert the distance you've traveled on a course to Latitude and Longitude.

You'll also need your own copy of the DR tables.

The tables are quite simple, but you'll need a calculator.  The procedure will be explained in detail, however these are the basic steps:

  1. Using your true heading find and record your Lat Factor and Lon Factor.  Make sure you note the correct sign (positive or negative).

  2. Using your latest Fix, take the degrees from your Latitude and find the scale factors for that latitude for both the Latitude and Longitude.

  3. Multiply the distance you've traveled on that heading by the respective two factors for Lat and Lon to get the change in minutes of your latitude and longitude.

Let's take a closer look at the tables as they are used in the above procedure.  The first set of tables will convert your course into two factors, a Lat Factor and a Lon Factor.  Look at the following except from the DR tables:

True HD Lat Fact Lon Fact True HD
      (Reverse Signs)
0 1.00 0.00 180
2 1.00 -0.03 182
4 1.00 -0.07 184
6 0.99 -0.10 186
8 0.99 -0.14 188

How you use this table is by finding your heading and noting the Lat Factor or Lon Factor.  For example, say your heading (once corrected to true) is 8°.  Lat Fact=0.99 and Lon Fact=-0.14.  If your heading is 188°, just reverse the signs on the factors so Lat Fact=-0.99 and Lon Fact=0.14.

The next step is to scale the distance you've traveled according to the Latitude of your last Fix.  This can be done with the second table as shown in the following excerpt:

Column NM per 1° of Lat/Long are not used in the DR calculation 
(n or s) NM per 1° NM per 1° 1' per NM 1' per NM
Latitude of Lat of Long of Lat of Long
0 59.71 60.11 1.005 0.998
1 59.71 60.10 1.005 0.998
2 59.71 60.07 1.005 0.999
3 59.71 60.03 1.005 1.000
4 59.71 59.96 1.005 1.001
5 59.71 59.88 1.005 1.002

Using the degrees of latitude from your Fix, you can move across the table to the last two columns to get the scale factor to convert your distance traveled to minutes of Latitude and Longitude.  (The NM per 1° of Lat or Long is just for your reference.)

Finally, you compute your change in Latitude and Longitude by multiplying the respective portions together as follows:

Change Lat = Distance Traveled * Lat Fact * Lat Scale
Change Lon = Distance Traveled * Lon Fact * Lon Scale

Example

LAST FIX:  34°44.6' N  118°23.3' W
Course 288° magnetic (local variation 12° east, no compass deviation) Note: some GPS units try to remove variation automatically.
After travling at 4.3 knots for 45 minutes what is our new position?

SOLUTION


Correct the compass course  ("east add") 288 + 12 = 300° T
Compute distance traveled = 4.3 * 45/60 = 3.23 nm.
from table: LAT FACT (for 300°) = 0.50 (reverse the sign)
from table: LON FACT (for 300°) = 0.87 (reverse the sign)
from table: LAT SCALE (for 34°) = 1.002
from table: LON SCALE (for 34°) = 1.203
Change LAT = 3.23 * 0.5 * 1.002 = 1.62
Change LON = 3.23 * 0.87 * 1.203 = 3.38
Answer = 34° 46.22' N and 118°26.68' W
(compare to 34°46.2169' N and 118°26.6955' W without rounding errors for an overall error of 0.013nm)

Remember you're working in degrees and minutes!  For example 1°30' - 0°45' has be calculated like 0°90' - 0°45'.  Likewise adding 1°30' + 0°45' = 2°15'  and NOT 1°75'.

Explanation of How the Tables Were Made

The first set of tables are generated by using a normalized right triangle of 1 nm at a true heading of θ degrees.  The angle is varied through the four quadrants (360 degrees) and the resulting Latitude and Longitude distances are computed with the proper sign applied depending on the heading.  This allows the theoretical distance traveled in Latitude and Longitude to be computed for any distance, just by multiplying by each factor by the real distance traveled.

The theoretical distance then needs to be mapped onto the curved Earth's surface.  This is where the second table is used.  The second table scales the theoretical distance to fit the curved distance of the Earth at the appropriate Latitude.  The following formulas were used (from American Practical Navigator Java Scripts):

Distance of 1degree Latitude at
               LATd = (111132.92-559.82*COS(LAT*PI()/180*2)
                       +1.175*COS(LAT*PI()/180*4)
                       -0.0023*COS(LAT*PI()/180*6))
                        /(12*5280*1.15077945)*39.370079  in nautical miles

Distance of 1degree Longitude at
               LONd = (111412.84*COS(LAT*PI()/180)
                       -93.5*COS(LAT*3*PI()/180)
                       +0.118*COS(5*LAT*PI()/180))
                        /(12*5280*1.15077945)*39.370079 in nautical miles

(NOTE THAT PI()=3.14159265358...  and LAT=latitude in degrees)

Then to compute the scale factor of minutes per nm, the above equations were divided into 60 to generate minutes per nm which are the last two columns in the tables used for the DR calculation.

So by just knowing your speed and distance and multiplying through the two tables the change in Latitude and Longitude can be computed in minutes, then added to the last fix.