Turbulent Wake
The linear and angular momentum imparted to the water by the turning screws as well as by the wave drag and the drag of the hull gives rise to a turbulent wake in the track of this ship. Its width w increases very slowly with the distance x from the ship, Tunaley and Buller (1989) cite w~x^1/4 . The turbulence can persist for several minutes, thus a turbulent wake can be several kilometers long. The exact structure depends on the ship and the screws.

Imaging of Ship Wakes by the SAR of the ERS-1/2 Satellites
The European Remote Sensing satellites ERS-1 and 2 orbit the earth in 780 km height and among other sensors carry a synthetic aperture radar (SAR). The SAR operates at a frequency of 5.3 GHz (C-band microwaves) and has a ground resolution of 25 m.

In order to understand how ship wakes can be imaged by the ERS SAR, we have to take into account that the SAR senses only the short-scale roughness of the sea surface: The radar backscattering from the sea surface at incidence angles typical for SAR (i.e., 20° to 60°) is caused by short surface waves (Bragg waves) which travel in or opposite to the SAR antenna look direction (range direction) and which have a wavelength of l_B=l/2sinq , where l is the radar wavelength and q is the incidence angle. The backscattered intensity is proportional to the spectral energy of the Bragg waves (Valenzuela 1978). A common measure for the backscattered intensity is the normalized radar backscatter cross section (NRCS), which is represented in SAR images as a grey level. In the case of the ERS SAR, the radar wavelength is l=5.7 cm, the incidence angle is between 20.1° and 25.9° , thus the Bragg wavelength l_B is between 8.2 cm and 6.5 cm.

The waves of a Kelvin wake are much longer than the Bragg wavelength, but they modulate the Bragg waves in several ways. The modulation mechanism which is most important for spaceborne SAR imaging of ship wakes is the tilt modulation (Hennings et al., 1999): the short waves are tilted by longer waves, such that the local incidence angle as well as the local Bragg condition are modified which in turn changes the NRCS (see Fig. 2). A tilt towards the SAR results in an increase of the NRCS, and a tilt away from the SAR results in a reduction of the NRCS, whereas a tilt at right angles to the SAR look direction does, to first order, not affect the NRCS. Thus, range-traveling waves would appear brighter on their fore side and darker on their rear side in a SAR image, provided the wavelength is larger than the size of the SAR resolution cell. Waves traveling at right angles to the look direction, i.e., in azimuth direction, would not become visible by the tilt modulation. Since the NRCS enhancement due to a tilt of the water surface towards the SAR is greater than the NRCS reduction due to a tilt away from the SAR, both effects do not cancel each other if the size of a resolution cell is larger than the wavelength (i.e. if one pixel in the SAR image averages over several wavelengths, which is often the case for Kelvin wakes and ERS SAR). The waves of a Kelvin wake that travel in or opposite to the look direction can thus cause an NRCS enhancement even if they cannot be resolved individually, provided their slopes are large enough. In practice, mainly the cusp lines, where wave amplitudes and slopes are highest, show up as bright lines in SAR images when the cusp waves travel in or opposite to look direction. However, the visibility of the cusp lines becomes the lower, the higher the wind speed is.


Fig. 2: Tilt modulation of Bragg waves (adapted from Robinson, 1994)
The turbulent wake also modifies the Bragg waves in the following ways: (1) the turbulence acts like an additional viscosity (eddy viscosity) that dampens short surface waves (Milgram et al., 1993); (2) surface-active substances tend to be gathered in the turbulent wake as they are skimmed from the water column by rising bubbles, forming surface-active films that dampen short surface waves (Peltzer et al., 1992). The turbulent wake would therefore have a reduced NRCS and thus show up as dark in the SAR image. The investigation by Peltzer et al. (1992) has shown that the surface films persist for a longer time than the turbulence. In case of very calm conditions, however, the turbulent wake can be rougher than the surrounding sea due to the currents and eddies in the wake. Thus is would appear brighter than the background in the SAR image. When the sea surface is covered by extensive surface films (that dampen the short surface waves), the turbulence of the wake can break up these films. This would as well result in the turbulent wake appearing brighter in the SAR image than the background.

The backscatter from the ship itself need also be considered. Since a ship is mainly made of metal and has structures with right-angled corners, it is an excellent radar reflector and causes a very bright spot in a SAR image. If the motion of the ship has a radial component with respect to the radar, the corresponding spot in the SAR image appears shifted in flight (azimuth) direction. The reason is that the (moving) SAR obtains its azimuth resolution by using the Doppler shift of the returned signals, assuming targets at rest. A radial motion of a ship causes an additional Doppler offset which then results in the echo's being mapped to a "wrong" position in the image.

We shall now present examples from a set of 400 ship wakes in ERS SAR images that we have investigated. The ship wakes were identified as a combination of linear structures with a bright spot near one end. Fig. 3 shows 3 images where one cusp line (Kelvin arm) is visible. The second Kelvin arm is not visible because its cusp waves do not propagate in or opposite to range direction. In Fig. 3b, the azimuth (flight direction) displacement of the ship's echo (bright dot, upper left) is clearly visible. Note that in all 3 images, a dark turbulent wake is visible. Note also that in Fig. 3a and b, the whole sector between the turbulent wake and the Kelvin arm is slightly brighter than the background. The reason is that not only the cusp waves, but also the divergent waves of this half of the Kelvin wake have enough slope to produce more tilt modulation than the surrounding water surface. Since only 17% of all investigated wake images showed one or two Kelvin arms, it can be concluded that in most cases, either the look direction of the SAR is not favorable, or the wave amplitudes and slopes even in the cusp region are not sufficient to produce a measurable tilt modulation. The wave pattern was visible in only two cases. The wake angles we observed in the images which show a Kelvin arm are smaller than 20° in about 20% of cases. This cannot be caused by intermediate water depths, since this would cause the angle to widen. A possible reason is that in such cases, the cusp waves are not more pronounced than the divergent waves within the wedge, but the propagation direction of the cusp waves is less close to look direction than the propagation direction of certain divergent waves within the wedge. These waves would then cause a bright line within the wedge.


Fig. 3: ERS SAR images with Kelvin arms of ship wakes. (a) ERS-2, South of Singapore, 21 Oct 1997, 1544 UTC; (b) ERS-2, Straits of Malacca, 25 Apr 1996, 0337 UTC; (c) ERS-1, South China Sea, 9 Apr 1996, 0307 UTC. (Ó ESA 1996, 1997)