We are usually taught in college that dispersion is not an issue in seismic data processing. Sure, we are told, when we try to match rock physics, sonic log, and surface seismic estimates of velocity we find discrepancies, but that is because we are passing through orders of magnitude difference in frequency. In the 10-100 Hz band of typical surface seismic data velocity is independent of frequency.

But that is a sloppy compression of reality. It is pretty nearly true for seismic body waves (P, S, and mode converted) moving around the far subsurface. In the near surface, however, velocities often show strong dispersion and the description is terribly inaccurate. Strangely, this is especially the case in marine shooting over shallow water. I say strangely because the speed of sound in water is independent of frequency to an exceptional degree, although it does depend on temperature, pressure, and salinity. It is only at immense frequencies, where wavelengths become vanishingly small, that sound speed begins to have any dependence on frequency. Yet, our standard 10-100 Hz data in shallow water leads to measured velocities well above and below the physical speed of sound waves in water.

This paradox arises because shallow water over an elastic earth forms a waveguide, bounded above by air and below by the seafloor. Like an organ pipe, sound gets trapped in the water layer and interferes to form a series of normal modes. Guitars and other stringed musical instruments are perhaps the most familiar example of such modes. The string is anchored at each end and can support a wave that spans the entire string, or harmonics of that wave that progressively scan one-half, one-quarter, etc. of the total string length.

The understanding of trapped or guided waves involves generalizing our usual concept of velocity. At a basic level, we think of a wave traveling a certain distance in a certain time and the ratio of these is the velocity of the wave. This definition works to determine the speed of sound in water at high precision as pressure, temperature, and salinity are varied. But now fix all these so that our lab measurement the speed of sound in water is, say, 1500 m/s. Fill an ocean with such water, make it a few tens of meters deep, set off an impulsive source, and listen with a sensor in the water a kilometer and a half away. We expect to observe the water wave arrival at one second (1500 m divided by 1500 m/s). This is the case for the highest frequencies in our data that have wavelengths (velocity divided by frequency) much smaller than the depth of the water column. In effect, they are not influenced by the seafloor. But lower frequencies have longer wavelengths, they feel the seafloor, tilting and jostling to fit in a water layer that looks increasingly thin as the frequency gets lower. In this regime, the concept of wave speed splits into two kinds of velocity, group and phase, neither of which is equal to the actual sound speed in water and both of which show dramatic, complicated variation with frequency. In other words, they are dramatically dispersive.

It is interesting that early work in quantum mechanics was also closely linked to phase and group velocity. In 1905 Einstein established that light particles, or quanta, had mass and other properties of matter. Twenty years later de Broglie flipped this around and asserted that matter had to have wave characteristics. The matter waves were investigated by means of a thought experiment in which a plane wave of frequency 'w' (omega) and wavenumber 'k'. For such a wave the speed is the phase velocity given by v=w/k. When this wave is summed with a second plane wave having slightly different frequency 'w+dw' and wavenumber 'k+dk'. The result is a low-frequency wave packet of frequency dw traveling at the group velocity 'dw/dk', and inside the wave packet the original wave is traveling with speed v=w/k. The picture is one of a wave packet moving at group velocity and and a monochromatic wave moving inside at a different speed. It is the group velocity that made physical sense in the case of matter waves, being simply the mechanical speed of the particle. The phase velocity is not so easy to understand, since it turned out to always be greater than the speed of light -- in apparent contradiction to the special theory of relativity.

Returning to the case of acoustic waves in a shallow water waveguide, we find mathematically similar phenomena. At a distance far from the source, compared to the water depth, the trapped waves form a spatial wave packet. The low frequency carrier wave travels at group velocity and represents the rate of energy transport by the wave field. The group velocity at high frequency is asymptotic to sound speed in water, then drops with decreasing frequency, until at low enough frequency it is no longer primarily controlled by the water layer but the elastic substrate. As it approaches a cur-off frequency, the group velocity is about equal to the Rayleigh wave speed of the seafloor.

As the wave packet is traversing this complicated velocity dispersion life cycle, the wave structures interior to the wave packet are traveling at the phase velocity. Phase velocity is also a strong function of frequency, but behaves differently. At high frequency, the phase velocity is, like group velocity, equal to sound speed in water. As frequency decreases, however, the phase velocity rises and always is greater than sound speed. We can think of this in terms of a plane wave front. At high frequency, this is vertical and represents the direct wave from the source. But as the frequency drops, the wave front tilts and receivers along the sea surface now are measuring the apparent velocity 'vw/cos(a)' where 'vw' is sound speed in water and 'a' is the propagation angle away from the horizontal. With lower and lower frequency, the wavefront has to lay down ever more to fit the increasingly long wavelength in the water column. As the cut-off frequency is approached, phase and group velocity meet once again at Rayleigh wave speed of the substrate.

Since about 1981 there have been processing tools to image phase velocity curves of the kind that develop in shallow water exploration. Park et al. (1998) found a scanning method that works will with real 2D or 3d data which is often irregularly sampled in space. Imaging of dispersive group velocity curves will be discussed at the 2009 SEG meeting in Houston in a paper by Liner, Bell, and Verm. The concept this: if we look at a single trace far from the source in shallow water shooting, a time-frequency decomposition of this trace will reveal that low frequencies are traveling slower than high frequencies, precisely the behavior we expect for group velocity curves. Quantitative investigation of the observed curves supports this interpretation.

From a data processing viewpoint, dispersive guided waves represent strong noise in the data. It is therefore a prime target for some kind of filtering technology. But phase and group velocity curves also posses valuable information about elastic properties of the sea floor, particularly shear wave speeds that are difficult to estimate otherwise. In principle, every shot record could be used to estimate laterally a varying shear wave model for use in converted wave exploration.

Next... Dispersion and Processing: Attenuation and Anisotropy

## Monday, September 21, 2009

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