Dead reckoning foundations
Dead reckoning predicts where the aircraft is by advancing a known position with direction, speed, wind and elapsed time.
What dead reckoning uses
A DR calculation starts from a confirmed position or fix. It then uses heading, true airspeed, wind velocity and time to estimate the aircraft's new position. Because the result is computed rather than observed, it is an estimated position. Each new fix should be used to restart the calculation and prevent old errors from continuing forward.
Direction terms
| Term | Meaning | Usual reference |
|---|---|---|
| Heading | Direction in which the aircraft's fore and aft axis points. | True, magnetic or compass north |
| Required track | Ground path intended for the flight. | Normally drawn on the chart |
| Track | Direction of the path over the ground. | True or magnetic north |
| Track made good | Actual track achieved between two known ground positions. | Measured from the plot |
| Course | Often used for intended track, but its meaning varies between publications. | Use heading and track when precision matters |
| Wind velocity | Direction from which the wind blows and its speed. | Written as direction and speed, such as 240 degrees at 30 kt |
Speed terms
True airspeed is the aircraft's speed through the air mass. Groundspeed is the speed of the aircraft over the Earth's surface. In still air they are equal. A headwind reduces groundspeed, a tailwind increases it, and a crosswind changes track unless the heading is corrected.
Heading is the direction in which the fore and aft axis of the aircraft is pointing.Oxford ATPL Book 10, chapter 7
Dead reckoning is navigation based on airspeed, heading, wind, groundspeed and elapsed time.R.K. Bali, Air Navigation, chapter 9
The triangle of velocities
The triangle of velocities is a vector picture of three simultaneous motions: aircraft through air, air over ground, and aircraft over ground.
The three vectors
| Vector | Direction | Magnitude | Book arrow convention |
|---|---|---|---|
| Air vector | Heading | TAS | One arrow |
| Wind vector | Direction towards which the air moves | Wind speed | Three arrows |
| Ground vector | Track | Groundspeed | Two arrows |
The air vector begins at the origin and ends at the air position. The wind vector is placed head to tail from the end of the air vector. The ground vector joins the original start point to the final end point. This closes the triangle.
Wind direction must be reversed on the picture
A reported wind direction is the direction from which the wind blows. A wind reported as 240 degrees at 30 kt moves the air mass towards 060 degrees. The drawn wind vector therefore points towards 060 degrees and has a length representing 30 kt.
Velocity and displacement triangles
A velocity triangle uses knot values. A displacement triangle uses distances travelled during one common time interval. The shape is identical because every side has been multiplied by the same time. For a 30-minute interval, divide every speed by two to obtain the corresponding displacement in nautical miles.
Drift and wind correction angle
Drift describes the angular movement from heading to track. Wind correction angle applies the same angular size in the opposite sense when selecting a heading for a required track.
Drift right and drift left
Look forward along the aircraft. If the track lies to the right of heading, drift is right or starboard and is treated as positive. If track lies to the left, drift is left or port and is treated as negative.
For heading 090 degrees and track 097 degrees, drift is 7 degrees right. For heading 030 degrees and track 026 degrees, drift is 4 degrees left.
Wind correction angle
Wind correction angle has the same magnitude as drift but is applied in the opposite direction. To maintain a desired track, point the aircraft into the wind. If an uncorrected heading would produce 7 degrees right drift, apply 7 degrees left correction to the required track to obtain the heading.
| Observed relation | Drift | Correction for a required track |
|---|---|---|
| Track right of heading | Right or starboard, positive | Heading must be left of required track |
| Track left of heading | Left or port, negative | Heading must be right of required track |
| No crosswind component | Zero | Heading and track coincide |
Keep the north reference consistent
Heading, track and wind direction must use the same north reference before they are combined. A true track cannot be added directly to a magnetic heading or a compass heading without first applying variation and deviation as required.
Closing the triangle: find track and groundspeed
When heading, TAS and wind velocity are known, the triangle closes to reveal track and groundspeed.
Scale-drawing method
- Choose a speed scale, such as 1 mm for 1 kt.
- Draw the air vector on the heading with a length equal to TAS.
- From its end, draw the wind vector downwind with a length equal to wind speed.
- Join the original start point to the final end point.
- Measure the direction of this ground vector for track and its length for groundspeed.
Component method
Use the downwind direction for the wind vector. Track is the quadrant-correct direction whose east and north components match the resultant.
Oxford worked example
Heading is 000 degrees true, TAS is 100 kt and wind is from 240 degrees at 30 kt. The wind moves towards 060 degrees. Its components are approximately 26 kt east and 15 kt north. Add these to the air components of zero east and 100 kt north. The ground vector is about 26 kt east and 115 kt north.
Groundspeed is the square root of 26 squared plus 115 squared, approximately 118 kt. The calculated track is about 013 degrees true. Oxford's scale drawing reads approximately 012 degrees true, the same result within normal plotting accuracy.
Opening the triangle: find heading and groundspeed
When the required track, TAS and wind are known, the unknown air vector must be aimed into wind so that the resultant ground vector lies on the required track.
Opening rather than closing
Draw the required-track line from the origin. Draw the wind vector separately in its downwind direction. The heading is found by placing an air vector of the known TAS so that the air and wind vectors combine to end on the required-track line. This is often called opening the triangle.
Crosswind and along-track components
Bali worked example
Required track is 120 degrees true, TAS is 180 kt and wind is from 340 degrees at 50 kt. Relative to the required track, the wind contributes about 32 kt of crosswind and 38 kt of tailwind. The correction angle is the inverse sine of 32 divided by 180, about 10 degrees.
The heading is therefore about 110 degrees true, 10 degrees into wind. The along-track part of TAS is about 177 kt and the tailwind adds about 38 kt, producing about 216 kt groundspeed. These are the values in Bali's worked set.
Wind components, groundspeed and time
Resolving wind along and across the track explains what the wind does to both direction and speed.
Resolve the wind
If the reported wind is within 90 degrees of the nose, the along-track component is a headwind. If it is within 90 degrees of the tail, it is a tailwind. The crosswind component determines the crab angle needed to maintain track.
Worked component example
A 40 kt wind is 30 degrees from the aircraft's nose. The headwind component is 40 multiplied by cosine 30 degrees, about 35 kt. The crosswind component is 40 multiplied by sine 30 degrees, 20 kt. The sign of the crosswind identifies whether it comes from the left or the right.
Beam-wind distinctions
| Condition | Consequence |
|---|---|
| Wind 90 degrees to required track | The aircraft must crab into wind. The TAS component along track is less than TAS, so groundspeed is less than TAS. |
| Wind 90 degrees to heading | Air and wind vectors are perpendicular. The resultant ground vector is the hypotenuse, so groundspeed is greater than TAS. |
Time and distance
At 150 kt groundspeed, 300 NM takes 300 divided by 150, which is 2 hours. Use groundspeed, not TAS, because the distance is measured over the ground.
ETE, ETA and fuel
Estimated time en route is the route distance divided by groundspeed. Add the ETE to the time overhead the departure point to obtain ETA. Fuel used equals the planned hourly fuel flow multiplied by time, with the required reserve added separately. For 240 NM at 160 kt, ETE is 1 hour 30 minutes. A departure time of 0835 UTC therefore gives an ETA of 1005 UTC.
Averages over unequal legs
An arithmetic mean is misleading when the legs take different times. Weight TAS, groundspeed or wind component by the time spent on each leg. In Bali's three-leg example, distances of 450, 200 and 720 NM combine with groundspeeds of 385, 280 and 228 kt. The leg times are 1.16, 0.71 and 3.15 hours, so total distance 1370 NM divided by total time 5.02 hours gives about 272 kt mean groundspeed. The same time weighting gives about 289 kt mean TAS and a 17 kt mean headwind.
Air position, DR position and fix
A DR plot separates the aircraft's motion through the air from the movement of the air mass over the ground.
Three positions
| Position | How it is obtained | What it represents |
|---|---|---|
| Air position | Advance the last confirmed position by heading and TAS for elapsed time. | Where the aircraft would be in still air |
| DR or ground position | Apply the wind displacement, or advance by track and groundspeed. | Estimated position over the ground |
| Fix | Determine position from visual, radio, radar or satellite observations. | Confirmed position within the accuracy of the source |
Worked half-hour plot
From a fix, fly heading 090 degrees true at 120 kt TAS with wind from 360 degrees at 30 kt. In 30 minutes the air position is 60 NM east of the fix. The wind blows towards 180 degrees and moves the air mass 15 NM south. The DR position is therefore 60 NM east and 15 NM south of the fix.
The resultant ground distance is the square root of 60 squared plus 15 squared, about 62 NM. Over half an hour this is approximately 124 kt groundspeed. Track is about 104 degrees true, giving about 14 degrees right drift.
The complete DR plot
A useful DR plot records not only the answer, but also which vector, time and north reference produced it.
Build the plot in order
- Mark the last confirmed fix and its time.
- Draw the heading and TAS vector for the elapsed time to obtain air position.
- From air position, draw the downwind displacement for the same elapsed time.
- Mark the endpoint as the DR position and label the estimated time.
- Join the original fix to the DR position to show track and ground distance.
- Divide ground distance by elapsed time to confirm groundspeed.
Track made good
Once a later fix is obtained, join the two confirmed ground positions. The direction of that line is track made good. Its length divided by the actual elapsed time gives groundspeed made good. The difference between required track and track made good is track error, which is treated fully in the 1 in 60 chapters.
Error growth
DR errors accumulate because an error in heading, TAS, wind or time moves the estimated position, and the next estimate is then advanced from that displaced point. A fresh fix breaks the chain. Longer intervals without a fix create a larger region of uncertainty.
Bali gives a practical rule that the radius of a DR circle of error may be taken as about 5 percent of TAS when expressed as distance after one hour. If a Doppler-derived track and distance are carried forward from the last fix, the estimated uncertainty may be drawn as an ellipse whose dimensions are about 0.1 percent of distance along track and 0.2 percent across track. These are planning approximations, not guarantees of position accuracy.
| Input error | Effect on the DR plot |
|---|---|
| Heading error | Produces an increasing cross-track displacement. |
| TAS error | Changes the length of the air vector. |
| Wind-direction error | Rotates the wind-displacement vector. |
| Wind-speed error | Changes the length of the wind-displacement vector. |
| Time error | Scales every displacement used for that interval. |
Final discipline
- Use the same time interval for every side of a displacement triangle.
- Use TAS with heading and groundspeed with track.
- Reverse reported wind direction before drawing its vector.
- Label true, magnetic or compass directions explicitly.
- Replace an estimated DR position with a fix as soon as reliable position information is available.
Operational adjustment and recovery
Progress can be checked mentally by comparing expected and observed landmarks, estimating distance and direction from the planned line, and revising ETA. A small timing error may be corrected by a modest speed change. A corner can be cut to gain time, while a deliberate dog-leg can lose time. Any change must remain compatible with terrain, controlled airspace, fuel and the cleared route.
If uncertain of position, use a logical recovery sequence: maintain safe altitude, widen the visual search, compare prominent features with the chart, use available radio or GNSS aids, and communicate early. Climbing may improve visual and radio range when conditions and airspace permit. An urgency or distress call is appropriate when safety demands it.
Choosing the plotting chart
On a Mercator chart, a rhumb-line track is straight and distance is measured against a nearby meridian. On a Lambert chart, a short DR track may be treated as straight and distance is measured using the chart scale or a nearby meridian. On a polar stereographic chart, convergency changes direction rapidly, so a short initial track is a better local approximation than extending one direction indefinitely.