The Kraken is an ancient giant sea monster that originated in Norwegian folklore in the twelfth century. In 1752, the Bishop of Bergen, Erik Pontoppidan, described it as the largest sea monster in the world with a width of one and a half miles. When the Kraken attacked a ship, it wrapped its arms around the hull and capsized it. The crew would drown or be eaten by the monster. The Norwegians thought that fish were attracted by the Kraken and therefore, when a fisherman had a good catch it was said that he had successfully fished over the Kraken without waking it.
The seismic source is hopefully not as horrifying as the mythological sea-creature, although when asleep, we believe it, like the Kraken, attracts fish. When it shoots, it does not. Therefore, it is of interest to understand sound propagation in the sea.
During airgun shooting, the sound level in the sea measured far away from the seismic vessel depends not only on the seismic source parameters but also on the water layer propagation channel. Numerical modeling studies are valuable for providing rough estimates of plausible scenarios for the transmission of seismic energy in the water column. Luckily, there are several standard acoustic propagation models available to model sound propagation in range-dependent ocean waveguides. One of these is a computer program – nicknamed, appropriately enough, Kraken.
The Kraken program models so-called normal modes – complex solutions of wave propagation which at first sight may look as scary as the octopus-like Kraken. But, normal mode theory (see box later on this page) is an efficient tool to model the acoustic wavefield trapped within the sea layer, and can be used to understand the sound level that marine life is exposed to when it is several kilometres from a seismic vessel.
Normal Modes
There are only a few relatively simple oceanic waveguides which allow us to obtain a closed analytical form of the solution describing sound propagation at long distances from the source. Pekeris (1948) is regarded as one of the pioneers in this field. Consider a water layer with thickness D and water speed c with source at depth zs and receiver at depth z. The pressure is effectively zero at the sea surface; the reflection coefficient is R=-1. It is well known that for plane wave incidence on the sea floor beyond the critical angle, there is a perfect reflection with an accompanying phase shift. This reflection can be represented with an equivalent reflection having R=-1 at a virtual pressure release interface displaced a distance below the sea floor. Therefore, when we study long-range sound propagation in a water layer over a real sediment, the simplest waveguide is the homogeneous water layer that has interfaces with vanishing pressure at the upper and lower boundaries.
The sound pressure is the sum of the pressures in the modes. For source at depth zs and receiver at depth z, the pressure is given as (Jensen et al., 1994; Medwin, 2005):
where A depends on the source power, Ρ is the ambient density, and the summation is over all allowed modes m=1,…,M with real propagation wavenumbers; M increases with increasing frequency; am is the modal excitation. The rest of the symbols are defined further down this page. This normal mode expansion of the field in the waveguide is referred to as the Pekeris model and is useful to understand the basic principles of normal modes. Each term in the series has simple trigonometric depth dependence of sinusoidal form.
There are two major observations to be made: first, the amplitude decreases with square root of distance r, as one would expect for a cylindrical spreading wave trapped in the sea layer. Second, while the source has a single frequency, the dependence of pressure on distance and depth is very complicated because each of the mode components has different dependencies on distance on range and depth.
In real waveguides, absorption losses in the sea floor cause the sound pressure to decay faster than 1/Πr. These losses, as well as others, can be included in the empirical mode attenuation rate δm.
Often, however, we find that we lack sufficient information to perform realistic numerical modeling. Therefore, we frequently end up with a simple comparison using logarithmic plots to judge whether the signal decays as –10log(r) or –20log(r) or more. We will use a recent example from a seismic survey where the water depth is between 40 and 50m as an example.
Measured RMS (root mean square) amplitude (solid red line) and maximum amplitude (solid black line) versus offset. For comparison -20, -40 and-60 log(x) straight dashed lines are fitted to the data.
Normal mode solutions for first mode (m=1; solid black line), first and second (m=1 and m=2; solid red line) and the three first (m=1, m=2 and m=3; solid blue line) for the Pekeris model; water layer 100m, source depth 6m, receiver depth 50m, 100 Hz frequency.
Shallow Water Example
Normal mode solutions (including 5 modes) for two different sea layer thicknesses for a water layer of 100m (black solid line) and 200m (red solid line). The source depth, receiver depth and frequency as before. There is a significant reduction in acoustic amplitude level with depth, approximately 15-20 dB. Looking at the complexity of the solutions as more modes are included, it is easy to understand why normal mode solutions have Kraken-like behaviour.The Pekeris model has cylindrical attenuation with distance, which means that the signal attenuates as 10log(r). In GEO ExPro Vol. 8 No. 5 we reported that transmission loss in a waveguide such as the sea, with constant speed of sound, follows a spherical spreading law (20 log(r)) at short distance and cylindrical spreading at longer distance. A combination of the two spreading laws gives for distances r greater than the ocean depth D (in meters), the asymptotic loss behavior 20 log (D) + 10 log (r/D).
Water depth at source versus offset for the field data.The sound pressure is the sum of the pressures in the modes, as given in the main text.
Deploying airguns. Source: Martin Landrø
To illustrate the possible huge difference between simplified models (where attenuation effects are neglected) and reality, we will use a field data set where we have measured 585 shots for 12 seconds each shot and where a geophone located at the seabed records the shots from a seismic vessel at distances from 30m up to nearly 15 km. The water depth at each source location is shown in the figure on page 30. A comparison between the root mean square (RMS) (measured for a window from 0-12 seconds) is shown together with the maximum amplitude as a function of offset. For comparison we have inserted attenuation proportional to -20, -40 and -60 times the logarithm of r. For distances up to 3–4 km, we observe that the -20log(r) damping curve fits reasonably well with the observed data. Furthermore, there is practically no deviation between the maximum amplitude curve and the RMS-curve up to this point. However, for offsets larger than 3–4 km, we observe a distinct difference between the maximum amplitude curve and the RMS-curve. The latter has a slope corresponding to -40log(r) and the former is closer to -60log(r). This means that for this example the attenuation of the seismic data is significantly stronger than -20log(r) for offsets greater than 3-4 km.
We notice that the water depth changes gradually from 50 to 40m for offsets larger than 3 km. It is hard to judge how much a change in the water depth will influence the field data. However, based on the equations given above, the maximum amplitude should increase rather than decrease if we assume the simple isovelocity model. Therefore, it is likely to assume that this change in slope for the measured data is not caused by this change in water depth, but rather that it is caused by attenuation effects. Unfortunately, this data set does not contain offsets above 15 km, so we are not able to check if the -60log(r) behaviour continues for larger offsets.
60log(r), which is observed between 4 and 15 km, corresponds to attenuation close to one over the cube of the offset. This indicates that the attenuation at far offsets (larger than 4 km) is severe. More experiments and larger offsets are needed to check if this behavior is common for shallow water depths.
