2.3 Plotting an Estimated Position (EP) with Tidal Stream

2.3 Plotting an Estimated Position (EP) with Tidal Stream

1. Concept of an Estimated Position (EP)

An Estimated Position (EP) is the most accurate position calculable without external observations (such as visual bearings, radar ranges, or GNSS). It is constructed by solving the triangle of velocities:

$$\vec{\text{Ground Track}} = \vec{\text{Water Track}} + \vec{\text{Tidal Stream}}$$

• Water Track: The path and distance run through the water, combining course steered, magnetic corrections, leeway, and log speed.
• Tidal Stream Vector: The displacement of water over the seabed for the passage duration.
• Ground Track: The resultant vector over the seabed terminating at the Estimated Position.

While a Dead Reckoning (DR) position ignores wind and current, an EP incorporates both. It remains an estimate because leeway estimates, log calibration factors, and published tidal rates carry minor margins of uncertainty.

2. Anatomy of Tidal Streams: Set, Rate, and Drift

Scaling Drift for Fractional Hours

Tidal tables publish hourly rates in knots. When a run lasts for fractional hours, drift must be scaled proportionally:

$\text{Drift (NM)} = \text{Tidal Rate (knots)} \times \frac{\text{Elapsed Time (minutes)}}{60}$

If the tidal rate is $2.4\text{ kt}$ and run time is $45\text{ minutes}$ ($0.75\text{ hr}$): $\text{Drift} = 2.4\text{ kt} \times 0.75\text{ hr} = 1.8\text{ NM}$

3. Standard RYA Plotting Standards and Arrow Conventions

To maintain universal clarity, standard RYA symbology dictates:

• 1 Arrow ($>$): Water Track (course steered through the water with leeway applied).
• 3 Arrows ($>>>$): Tidal Stream Vector (direction of set and distance of drift).
• 2 Arrows ($>>$): Ground Track (resultant track over the seabed connecting starting fix to EP).
• EP Position Symbol: An equilateral triangle with a central dot ($\triangle$), annotated with the four-digit time (e.g., 1530 EP).

4. Step-by-Step Construction of an EP on the Chart

1. Plot Departure Fix: Prick off the starting fix, circle it ($\odot$), and label with time (e.g., 1400 Fix).
2. Lay Off True Water Track: Correct compass heading for deviation, variation, and leeway. Lay off this line from the fix.
3. Measure Water Distance: Prick off the log distance along the water track and mark with 1 arrow ($>$).
4. Lay Off Tidal Vector: From the terminus of the water track, lay off a line in the direction of the tidal Set ($^\circ\text{T}$).
5. Prick Off Drift: Scale drift ($\text{Rate} \times \text{Hours}$) along the tidal vector and mark with 3 arrows ($>>>$).
6. Mark EP & Ground Track: Draw an EP triangle ($\triangle$) at the tidal vector terminus with time. Draw a line from the departure fix to the EP with 2 arrows ($>>$) for Ground Track, measuring COG and SOG.

5. Multi-Hour and Multi-Leg Calculations

When passages span multiple hours, tidal streams alter direction and rate:

• Stepped Method: Plot each hour's water track and tide sequentially, deriving intermediate hourly EPs.
• Consolidated Method (Preferred): Plot all consecutive water tracks end-to-end, then apply consecutive tidal vectors joined head-to-tail at the final water track terminus. This yields the identical final EP with a cleaner chart.

6. Comprehensive Worked Navigational Scenario

Scenario: At 1400, a yacht departs a fix off Dartmouth fairway buoy ($50^\circ 19.5'\text{N}, 003^\circ 32.5'\text{W}$).

• Compass Course: $125^\circ\text{C}$, Deviation $2^\circ\text{W}$, Variation $3^\circ\text{W}$.
• Wind: North-northeast ($020^\circ$), Force 5. Leeway estimated at $6^\circ$.
• STW: $6.0\text{ kt}$ constant.
• Tidal stream: Sets $230^\circ\text{T}$ at $1.8\text{ kt}$.
• Target: Construct the Estimated Position for 1530 ($1.5\text{ hours}$ run).

Solution:

1. True Water Track:
   • $\text{MC} = 125^\circ\text{C} - 2^\circ\text{W} = 123^\circ\text{M}$.
   • $\text{TH} = 123^\circ\text{M} - 3^\circ\text{W} = 120^\circ\text{T}$.
   • Wind from $020^\circ$ strikes port side (pushes to starboard / add): $\text{Water Track} = 120^\circ\text{T} + 6^\circ = 126^\circ\text{T}$
2. Distances:
   • Run: $1.5\text{ hours}$ at $6.0\text{ kt} = 9.0\text{ NM}$ through water.
   • Tidal Drift: $1.8\text{ kt} \times 1.5\text{ hr} = 2.7\text{ NM}$ along $230^\circ\text{T}$.
3. Plotting:
   • Lay off $126^\circ\text{T}$ for $9.0\text{ NM}$ from 1400 Fix (mark 1 arrow).
   • From that terminus, lay off $230^\circ\text{T}$ for $2.7\text{ NM}$ (mark 3 arrows).
   • Mark the terminus with an EP triangle ($\triangle$), labeled 1530 EP.
4. Ground Track Results:
   • Connect 1400 Fix to 1530 EP (mark 2 arrows).
   • Measuring yields COG $\approx 143^\circ\text{T}$, Distance Over Ground $\approx 8.75\text{ NM}$, and $\text{SOG} = 8.75 / 1.5 \approx 5.8\text{ kt}$.
   • (Vector Verification):
     • Water Track: $X_1 = 9.0 \times \sin(126^\circ) = +7.28\text{ NM}$, $Y_1 = 9.0 \times \cos(126^\circ) = -5.29\text{ NM}$
     • Tidal Vector: $X_2 = 2.7 \times \sin(230^\circ) = -2.07\text{ NM}$, $Y_2 = 2.7 \times \cos(230^\circ) = -1.74\text{ NM}$
     • Net Vector: $X = +5.21\text{ NM}$ (East), $Y = -7.03\text{ NM}$ (South)
     • $\text{Distance Over Ground} = \sqrt{5.21^2 + (-7.03)^2} = 8.75\text{ NM}$
     • $\text{COG} = 180^\circ - \arctan(5.21 / 7.03) = 143.4^\circ\text{T} \approx 143^\circ\text{T}$
     • $\text{SOG} = 8.75 / 1.5 = 5.83\text{ kt} \approx 5.8\text{ kt}$

The cross-tide setting $230^\circ\text{T}$ deflected the vessel's track from $126^\circ\text{T}$ to $143^\circ\text{T}$ and slightly reduced progress over ground from $9.0\text{ NM}$ to $8.75\text{ NM}$ ($5.8\text{ kt}$ SOG).

7. Summary of RYA Plotting Symbols

8. Common Exam Traps in EP Plotting

1. Confusing EP with CTS: For an EP, both displacement vectors are known. Drawing water track then tide is the convention used here; reversing their end-to-end order gives the same endpoint. A CTS problem instead solves an unknown water-track direction.
2. Unscaled Tidal Rates: Always multiply tidal rate by elapsed hours to get drift in nautical miles.
3. Omitting Leeway: The water track must include leeway before applying the tidal vector.