Tuesday, September 8, 2009

Seismic dispersion

All of seismology is based on waves and a primary property of any wave is the velocity (v) at which it travels. This is related to wavelength (L) and frequency (f) through v = f L. This shows that as the frequency changes, so does the wavelength in just such a way that their product is preserved as the constant velocity. But it is important to note that velocity itself is not a function of frequency, a situation termed nondispersive wave propagation. As the frequency is ramped up, the wavelength drops, and the waves always travel at the same speed. This is the case with waves in unbounded ideal gases, fluids, and elastic materials.

Porous media is another matter. Wave speed is a function of material properties (matrix and fluid) and environment variables (pressure, temperature, stress). Luckily for us, in the low frequency range (0-100 Hz) of surface seismic data, the velocity does not depend on frequency to within measurable tolerance. However, as frequency ramps up to sonic logging (10-20 KHZ) and bench top rock physics (MHZ) the wave speeds do become dispersive (classic paper is Liu et al., 1976).

This is the classical meaning of the word 'dispersion', velocity is a function of frequency. Here we will take a more general definition that includes any wavefield property, not just speed. Examples will include velocity, of course, but also attenuation, anisotropy, and reflection characteristics. We could also lump all of these things into the name 'frequency-dependance', but 'dispersion' is already out there with respect to velocity and it seems better to go with the shorter, more familiar term.

I am a bit embarrassed to admit that I made a strong point to a colleague (Jack Dvorkin, I believe) a few years ago about his use of 'dispersion' for something other than frequency-dependent velocity. I think he was talking about attenuation. Anyway, my tardy apologies because I have arrived at the same terminology.

It is curious that so much of classical seismology and wave theory is nondispersive: basic theory of P and S waves, Rayleigh waves in a half-space, geometric spreading, reflection and transmission coefficients, head waves, etc. Yet when we look at real data, strong dispersion abounds. The development of spectral decomposition has served to highlight this fact.

We will distinguish two kinds of dispersion. If the effect exists in unbounded media then we will consider it to be 'intrinsic' and thus a rock property that can be directly measured in the lab. On the other hand, if the dispersion only presents itself when layering is present then we will term it 'apparent', this case being responsible for the vast majority of dispersive wave behavior in the lower frequency band of 0-100 Hz.

To make some sense of the seismic dispersion universe, we will break down our survey into the traditional areas of acquisition, processing, and interpretation.

It is a fascinating and sometimes challenging topic. We will not seek out mathematical complexities for their own sake. Rather we will gather up interesting and concise results, presented in a common notation, and dwell more on the physical basis, detection and modeling tools, and especially the meaning of dispersive phenomena.

Reference:
Liu, H.-P., Anderson, D. L., and Kanamori, H., 1976, Velocity dispersion due to anelasticity; implications for seismology and mantle composition, Geophys. J. R. astr. Soc., 47, 41-58.

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