2.1 Dead Reckoning (DR) & Water Track Principles

2.1 Dead Reckoning (DR) & Water Track Principles

1. Definition and Principles of Dead Reckoning

Dead Reckoning is the technique of determining a vessel's projected position by advancing a known, verified prior position (a Fix) using strictly:

1. The True Course Steered through the water (compass heading corrected for deviation and variation), and
2. The Distance Run measured by the ship's log (or speed through the water multiplied by elapsed time).

The defining attribute of a Dead Reckoning position is that it deliberately ignores environmental displacements:

• DR ignores wind leeway: It assumes the hull travels strictly along its fore-and-aft centerline.
• DR ignores tidal stream and current: It assumes the surrounding body of water is completely stationary over the seabed.

Because coastal waters are rarely motionless and surface vessels continuously experience wind force, a DR position represents an idealized position through the water medium rather than a definitive geographic spot on Earth. A DR is a useful baseline; an Estimated Position (EP) also allows for leeway and tidal stream over the same interval.

2. Water Track vs. Ground Track: STW vs. SOG

Navigators must maintain a strict distinction between movement through the water and movement over the seabed:

• Speed Through the Water (STW): Measures the velocity of the hull relative to the immediate water flow. STW governs rudder authority, keel lift, and vessel handling.
• Speed Over Ground (SOG): Measures actual geographic speed over the seabed.

For instance, a yacht motoring at $6.0\text{ kt}$ STW directly against a $2.0\text{ kt}$ tidal stream achieves an SOG of only $4.0\text{ kt}$. Conversely, with the same stream astern, STW remains $6.0\text{ kt}$ while SOG increases to $8.0\text{ kt}$. Plot DR using log distance along the true heading steered, before allowing for leeway or tide. A leeway-corrected water track is used in constructing an EP; do not substitute SOG for log speed.

3. Marine Log Mechanics, Errors, and Calibration

Marine speed logs are subject to fouling, boundary layer turbulence, and heel effects:

• Fouling: Marine growth on paddlewheel impellers creates friction, causing logs to under-read.
• Heel Angle: Flow dynamics change as the hull heels, often introducing asymmetric errors between tacks.
• Boundary Layer: Water friction along the hull skin can distort sensor readings at varying speeds.

Calibrate the log using its manufacturer's procedure. For a simple measured-distance check, choose calm water with negligible current, maintain steady speed, and compare the known distance with the log distance. Under those conditions:

$\text{Log Factor} = \frac{\text{Actual Surveyed Distance}}{\text{Recorded Log Distance}}$

$\text{Corrected Distance} = \text{Recorded Log Distance} \times \text{Log Factor}$

4. Distance Calculations & Mental Arithmetic

The distance run along the water track is calculated using:

$\text{Distance (NM)} = \text{Speed (knots)} \times \frac{\text{Time (minutes)}}{60}$

The 6-Minute Rule

Because 6 minutes equals $\frac{1}{10}$ of an hour ($0.1\text{ hr}$): $\text{Distance in 6 minutes (NM)} = \frac{\text{Speed in knots}}{10}$ At $6.8\text{ kt}$, a yacht travels $0.68\text{ NM}$ every 6 minutes. Over 18 minutes ($3 \times 6\text{ min}$), distance run equals $3 \times 0.68 = 2.04\text{ NM}$.

5. Standard RYA Chartwork Conventions for DR

To ensure universal clarity on nautical charts, RYA standards specify:

1. Track Line: Drawn as a clean pencil line from the starting fix along the True Course Steered.
2. Keep the Vectors Distinct: The DR course line follows the true heading. A separate leeway-corrected Water Track is marked with one arrowhead ($>$).
3. Course Notation: The three-digit True Course is written above the line (e.g., 249°T).
4. Distance and Speed: Distance run and STW are written below the line (e.g., 9.6M / 6.4kn).
5. DR Symbol: Placed at the terminus as a small cross ($+$) or a tick drawn perpendicular across the track line, annotated with the four-digit time (e.g., 0930 DR).

6. Comparison: DR vs. Estimated Position (EP) vs. Fix

7. Worked Navigational Scenario

Scenario: At 0800, a vessel departs a fix $2.0\text{ NM}$ South of St Catherine's Point ($50^\circ 34.5'\text{N}, 001^\circ 17.8'\text{W}$). Compass course steered is $254^\circ\text{C}$. Deviation is $3^\circ\text{W}$, and local Variation is $2^\circ\text{W}$. The log reads $14.0\text{ NM}$ at 0800, and $23.6\text{ NM}$ at 0930. Plot the 0930 DR.

Solution:

1. Distance and Speed:
   • Elapsed time: 0800 to 0930 = $1.5\text{ hours}$.
   • Distance run: $23.6\text{ NM} - 14.0\text{ NM} = 9.6\text{ NM}$.
   • STW: $9.6\text{ NM} / 1.5\text{ hr} = 6.4\text{ kt}$.
2. Course Conversion:
   • Magnetic Course: $254^\circ\text{C} - 3^\circ\text{W (Dev)} = 251^\circ\text{M}$.
   • True Course: $251^\circ\text{M} - 2^\circ\text{W (Var)} = 249^\circ\text{T}$.
3. Plotting:
   • Draw the DR course line on $249^\circ\text{T}$ from the 0800 Fix, before applying any leeway.
   • Measure $9.6\text{ NM}$ on the latitude scale adjacent to $50^\circ 30'\text{N}$.
   • Mark a tick at $9.6\text{ NM}$, labeled 0930 DR.
   • Write 249°T above the track and 9.6M / 6.4kn below.