The AIS data that are used to assess surface currents in this study are: speed over ground (sog) for along-track currents and the difference between heading and course over ground (cog) for cross-currents.
A ship's propulsion creates a cruising speed (cs), relative to the surrounding water. The surface currents indicate the speed of the surrounding waters relative to the sea floor. In good approximation:
along current = sog - cs
Effects of wind and waves on the speed are neglected as for larger ships these are only noticeable under extreme weather conditions.
A basic assumption in the analysis is that during voyages in open waters the engine power, and therefore the cruising speed, is kept constant for longer times. Abrupt changes in power produce sharp changes in the ship's speed and are easily detected. In the analysis of along-currents such voyages are filtered out (see Data Preparation). Smooth speed changes are attributed to changes in the along-currents. Note that a slowly varying part of the current (on the time scale of ship passages) is absorbed in the estimation of the cruising speed and cannot be detected in this analysis, but tidal variations are clearly visible. The analysis of along-currents focuses on the assessment of the major tidal constants along the shipping lanes.
From the AIS datasets tidal constants were calculated with the help of a least square fitting procedure on the same longitude grid as used by the CMEMS model. The west-east lane covers 144 spatial grid points; the east-west lane is slightly shorter due to a change in heading near 0.7 degrees longitude: 134 grid points. To fit the tides the six most relevant constituents for the chosen area were selected: M2, S2, N2, K2, M4 and MS4.
Each AIS data point has a longitude, a time, a measured speed, and a passage number. From the time the tidal phase φₖ for constituent k can be calculated. The tidal current is then:
u(x_i, t) = Σ_k a_i,k cos(φ_k t) + b_i,k sin(φ_k t) (1)
with a_i,k and b_i,k the amplitude components of the tidal constituent k at grid point i. These twelve adjustable parameters are attributed to the grid point that is closest to the longitude of the measurement. The model vessel speed is then equal to u + cs_p, where the cruising speed cs only depends on the passage number p. The total number of adjustable parameters is therefore equal to twelve times the number of grid points plus the number of passages. This is more than fifty times smaller than the number of available measurements, so a reliable least square fit is expected. The formulation of the model fit is represented in a large sparse matrix and solved iteratively with the help of a scipy library routine.
For a comparison with the CMEMS model results the time-longitude grid with current speeds in the direction of the shipping lanes is interpolated to the time-longitude values of AIS measurements using a cubic method. The cruising speeds for each passage is determined as the median value of the measured speed minus the model along-current. The model speed for each passage is then constructed as the cruising speed plus the along-current. For a comparison of the tidal constants a harmonic analysis with a least squares fit for the selected constituents was applied to the model grid. This is not a full tidal analysis including nodal corrections but is consistent with the method with which the AIS tidal constants are calculated.
The analysis of cross-lane currents depends on the difference between course over ground and heading and the ship's speed over ground. In good approximation:
cross current = cs * sin(heading - cog) (2)
In practice, the unknown cruising speed cs is approximated by the measured speed over ground sog. Abrupt changes in cs, or sog, do not affect the reliability of the current assessment. Large changes in heading might have an effect but these are rare in open waters and are ignored here.
Heading is the most uncertain parameter. Not all ships provide a value (511 degrees means no data) and when provided, heading is given in integer values of degrees, whereas cog is given in tenths of degrees. This rounding means that the assessment of small cross-currents, such as on the shipping lanes, becomes less reliable. To further evaluate the potential of equation 2, an analysis is carried out along the track between Ouistreham and Portsmouth, where the strong currents in the Channel are cross-currents.
A tidal analysis of speeds during a passage is used to estimate the cruising speed of a ship. However, current variations that change over much longer time scales cannot be discerned. One way to assess persistent or slowly varying currents is to compare the median cruising speed going in one direction in a specified time interval with the median cruising speed in the opposite direction. Assuming that the median speed is independent of the course and that differences in cruising speeds can be filtered out as noise, the difference between the two median speeds equals twice the current. This method was tried on river data [3] with realistic results.
Next: Results.