Dead reckoning and estimated position

How it works

The estimated position starts from a known position (a fix) and what you know of the boat's movement. If you use only the course steered and the distance through the water (from the log, or speed times time), you get the DR (dead reckoning) position. If you add the tidal stream, you get the EP (estimated position), the position where you most probably are. The DR is 'where I would be with no tide'; the EP is 'where I am, from my best data'.

First step: the true course through the water (water track). Start from the compass course and apply deviation and variation (see the compass procedure). If there is leeway because of the wind, add or subtract it depending on the side the wind comes from. Example: you steer 077°(M), leeway 5°, so 082°(M); variation 7° W → 075°(T). This is the direction in which you plot the distance run. For the DR you stop here; for the EP you go on.

Second step: the distance through the water. Read the log at the start and at the end (from 10.2 to 15.0 = 4.8 NM), or speed × time (see the speed procedure). Plot it from the starting position along the water track. The point you find is the DR. Then add the tidal stream: the set is the direction the stream is going towards (not from), the drift is the distance the stream makes in the time: stream rate × time. With 2.0 knots for 1 hour the drift is 2.0 NM. From the DR, plot the set for the drift: the tip is the EP.

The line from the start to the EP is the ground track (COG); its length divided by the time is the speed over the ground (SOG). They are different from the course and speed through the water. On the chart the symbols are: one arrow for the water track, two for the ground track, three for set and drift; the EP is a triangle, the DR a cross or a tick on the line, the fix a circle with a dot.

As an alternative to plotting, the numerical calculation. For each leg (distance D, true course C): difference of latitude in minutes = D × cos C; departure in miles = D × sin C. Add the components. Difference of longitude in minutes = departure / cos(mean latitude), with mean latitude = average of departure and arrival. Example, from 50°00'.0 N 004°00'.0 W, 060°(T) for 12 NM: dlat = 12 × cos 60° = 6'.0 N; departure = 12 × sin 60° = 10.4; with mean latitude 50°03' (cos 0.642) dlong = 16'.2 E. DR: 50°06'.0 N 003°43'.8 W.

Now the stream: set 150°(T), 1.5 knots for 2 hours, drift 3.0 NM. dlat = 3.0 × cos 150° = −2'.6 (south); departure = 3.0 × sin 150° = 1.5. Add to the water track: dlat = 6.0 − 2.6 = +3'.4 N; departure = 10.4 + 1.5 = 11.9. Mean latitude about 50°01'.7: dlong = 11.9 / 0.642 = 18'.5 E. EP: 50°03'.4 N 003°41'.5 W. The ground track is about 074°(T) for 12.4 NM, so SOG about 6.2 knots over 2 hours. The calculation and the plot on the chart should give the same position within a tenth of a mile.