large compared to the wavelength. Te corresponding expression for the pressure has a similar exponential decay with depth. Te particle velocity represented by these equations can be represented as circular (elliptical if the water depth is shallower) particle orbits. Te dispersion relation for ocean waves


depends on the water depth. However, if we assume large water depths, it is simply given as ω2


= gk where g is the gravitational acceleration (9.8 m/s2), ω is


the circular frequency (ω = 2π/T) and k is the wave number. From the dispersion equation we can estimate the period of a sinusoidal ocean wave if the wavelength is known, since k = 2π/λ. It is clear that disturbances caused by the ocean wave will decrease with depth, and hence it might be a good idea to tow seismic streamers at large depths. Typical time periods (T) for ocean waves are more than a


Figure 5.15: Smoothed measurements of wave height (black solid line) at the Gullfaks field in 2012 and corresponding modelled wave heights by assuming a quadratic relation (red solid line) between wind velocity and wave height. The average wave height is 2.7m. (a=0.025 s2


/m; b=1 m).


second; sitting on a beach we can usually count eight seconds between each wave. Tis means that the noise both on the pressure component and the velocity component will be outside the typical frequencies (4–100 Hz) of interest for a seismic experiment. However, as we are striving for lower and lower frequencies (below 3 Hz – see Section 5.1) the noise caused by slow variations in particle velocity caused by ocean waves close to the streamer might influence our data. Te influence from ocean waves depends strongly on the streamer depth; for streamer depths greater than 20m, which is typical today, the typical particle velocity caused by the ocean wave height shown in Figure 5.16 is less than 0.1 m/s and very low frequency (0.2 Hz). Compared to typical towing velocities for streamers (2.5 m/s), 0.1 m/s is small. Since both the pressure variations and the velocities caused by water waves decay exponentially with depth, the direct influence is assumed to be minimal for normal seismic frequencies (above 4–5 Hz). Tis exponential decay rate is controlled by the streamer depth (z) divided by the wave length (equations 1 and 2 on previous page). During heavy storms it is known that


ships crossing the North Sea via, for instance, the Dogger area, which has shallow water ranging from 20 to 60m, get sand on deck, indicating that the waves have sufficient energy to lift sand from the sea bed to the surface. Close to shore this phenomenon is well known, as a thick layer of sandy foam can be observed on the beach after a storm.


5.3.2 Wind Speed and Wave Height Correlation


Te close relationship between wind speed and ocean wave height is well known. For fully developed ocean waves it is commonly assumed that the wave height increases with wind velocity squared, and a recent example showing


203


wave data from the Gullfaks field in the North Sea is shown in Figure 5.15. We observe a surprisingly strong correlation between


the smoothed average wave height and the wind speed. Furthermore, we observe that the ocean waves for some periods lag behind the wind: first the wind calms down and then the waves. Te average wave height measured at the Gullfaks field in 2012 was 2.7m. Pierson and Moskowitz derived an empirical formula for the energy of a fully developed sea state as a function of frequency, known as the Pierson-Moskowitz spectrum, which showed that the peak wave energy decreases as the wind velocity increases. Another crucial parameter that is used to describe ocean


waves is wave steepness, s, which is equal to the wave height divided by the wavelength. If s is larger than 1/7, the wave will break. Tis value might vary significantly. Typical average values for the North Sea are between 0.06 and 0.006 (Torsethaugen, 1993). Tis means that for the Gullfaks example shown in Figure 5.15, assuming a steepness of 0.06, the average wavelength is 40m, corresponding to a period of 5 seconds. One


Figure 5.16: The decay curve for particle velocity versus depth. Here we have used the exact dispersion relation without assuming that the water depth is much larger than the wavelength. The vertical velocity at the seabed (60m) is actually zero; the deviation is caused by the logarithmic plot. The period of this wave is 5 seconds, corresponding to a frequency of 0.2 Hz.


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