The distinction between intrinsic and apparent frequency-dependent seismic properties is nowhere greater than in the areas of attenuation and anisotropy. First a brief overview of the attenuation problem.
If we take a rock sample from a well core and test it in the lab, we are likely to find that there is some small amount of intrinsic attenuation, meaning the irrecoverable loss or conversion of wave energy into heat. This will yield the frequency-dependent wave energy loss as observed over the length of a 1 inch rock sample. In the band of surface seismic data, 5-100 HZ, this attenuation is relatively constant and exceedingly small. Extrapolating this minute intrinsic attenuation to a geologic section several kilometers thick will still predict only a modest loss of wave energy, so small is the intrinsic attenuation effect. But what is actually observed in field data? From VSP and other downhole measurements we can monitor the evolving wave field and estimate attenuation through the rock column. It is seen to be significant, much stronger than the lab results would suggest. The reason is layering. For layered media, the waves continuously undergo reflection, transmission, and mode conversion. All of these mechanisms conserve energy (no conversion to heat) and thus do not quality as attenuation in the intrinsic sense. But the observed down going wave field will rapidly loose amplitude due to these effects, and the amount of loss will be a strong function of frequency. High frequencies in the wavelet amplitude spectrum will erode much faster than low frequencies. On the other hand, interbed multiples cascade to reinforce the down going wave field as shown by O'Doherty and Anstey (1971). So the picture that emerges about apparent attenuation is not one of a rock property that can be measured in the lab, but a frequency-dependent attenuation field.
The term attenuation ‘field’ deserves some explanation. One of the great unifying concepts of physics is the field. There are many kinds of fields, but our view of apparent attenuation is that of a scalar field that associates with each point in space a number representing the attenuation at that location. Imagine someone does a 20 Hz calculation for total attenuation in a layered medium and comes up with a value at some location. If we were to go to that location in the earth and take a physical sample of the material then test it in the lab, we will only find the intrinsic attenuation and are left wondering about this total attenuation value. Now our someone redoes the calculation for 10 Hz and assigns a different total attenuation value to the same location. Again we go there, extract a sample and test in the lab. The results are of course the same, since the intrinsic attenuation (a rock property) has not changed. Yet waves of 10 and 20 Hz moving through our earth model will actually experience different levels of attenuation in line with the total attenuation calculation.
To summarize, the attenuation a wave will see at any given location is composed of two parts, the intrinsic attenuation of the material at that spot and a frequency-dependent apparent attenuation field due to layering effects. Furthermore, this attenuation field is not just dispersive (a function of frequency), it is also anisotropic (a function of propagation direction). Attenuation anisotropy is a current area of research (Zhu et al., 2007).
This naturally brings up the topic of seismic velocity anisotropy and how it depends on frequency. We will restrict our comments here to VTI-type anisotropy which is velocity variation with respect to the vertical axis in a horizontally layered earth. Unlike core plug attenuation measurements, lab-scale rock samples can show significant velocity anisotropy. Occurring on this fine scale, we would term this intrinsic anisotropy. It is not a function of frequency and represents a rock property like density or bulk modulus. Of the sedimentary rock types, only shale is seen to be significantly anisotropic at this small scale. The origin of this behavior is the alignment of clay minerals at the microscopic level. Several recent publications have shown that shale velocity anisotropy is ubiquitous. For sandstone and carbonate rocks, VTI behavior develops in proportion to shale content. So intrinsic anisotropy is predictable in the rock column at any particular location from knowledge of the shale volume, and this in turn can be determined from standard well logging analysis. This is the shale part of the VTI problem, the other part is layering.
Long before shale anisotropy was well-understood, there was a theoretical interest in waves traveling through layered media. There are many effects that arise from layering, we have already mentioned waveguides as a good example. But waveguides are a thick layer problem. Now we are discussing the effects of thin layers, meaning layer thickness is much smaller than the seismic wavelength. In fact, with respect to velocity you can think of a continuum of behavior as we go from high to low frequency. At very high frequencies, the wavelength is much smaller than the layer thickness and the waves see a homogeneous medium. At longer wavelengths the medium seems to be heteogenous or v(z). Finally, at very long wavelengths compared to the layer thicknesses, the material behaves like an anisotropic medium. The question is how to calculate the apparent anisotropic parameters of the layered medium as seen by very long waves. Backus (1962) solved this problem for the case where the layers are either isotropic or VTI, although he was hardly the first to work on it. He capped off 25 years of investigation on this problem by many researchers. Today, we have full wave sonic and density logs that give Vp, Vs, and density every half-foot down a borehole. Shale intervals can be detected by gamma logs, but there is no substitute for lab measurements on core to find intrinsic shale VTI parameters. This gives us all the raw material that Backus said we need, a thin layered elastic medium composed of some combination of isotropic (3 parameters) and VTI layers (5 parameters).
Armed with a layered model, Backus says we need to do a kind of averaging to find the equivalent medium. His theory showed that if we do the averaging with a suitable averaging length in depth, then as far as wave propagation was concerned the two models are the same. Let's be careful and precise about this. The original model consists of fine elastic layers with properties that vary arbitrarily from one layer to the next. Waves sent through such a medium can be measured at the top of the stack (reflected field) or at the bottom (transmitted field). Let's call these observations the original wave field. Now we do Backus averaging to come up with a new model that is smoother and more anisotropic than the original. We send the same waves through the new model and measure the field at top and bottom. What Backus said is this: If the averaging distance is small enough the original and new wave fields will be the same, even though the original and new earth models look quite different. That is, these two earth models are identical with respect to wave propagation. Here we should also make clear the distinction between intrinsic anisotropy due to shale layers and layer-induced anisotropy that can occur even when every individual layer is isotropic. The total anisotropy is a combination of the two.
So where does dispersion come into all this? It is buried in the thorny question of the averaging length. As the averaging length increases, the medium becomes smoother and more anisotropic, and the wave fields are only the same for long wavelengths or, conversely, low frequency. But it is only the layer-induced anisotropy that depends on the averaging length, shale anisotropy does not. This means that layer-induced anisotropy, and therefore total anisotropy, is dispersive.
As with attenuation, we come away with a concept that VTI-type anisotropy is a frequency-dependent field. A 20 Hz wave will see a different version of earth anisotropy than a 30 or 60 Hz wave. In principle, each frequency has a unique anisotropy and attenuation field. A challenge for seismic imaging in the future is to exploit this phenomena, perhaps leading to frequency-dependent anisotropy and attenuation estimation as descendents of today’s migration velocity analysis.
Next: Dispersion and Interpretation: Rough Surface Scattering
Backus, G., 1962, Long-wave elastic anisotropy produced by horizontal layering: J. Geophys. Res., 67, 4427--4440.
O'Doherty, R. F. and Anstey, N. A., 1971, Reflections on amplitudes: Geophys. ‘’Prosp.’’, Eur. Assn. Geosci. Eng., 19, 430-458
Zhu , Y., Tsvankin , I., Dewangan, F., and van Wijk, K., 2007, Physical modeling and analysis of P-wave attenuation anisotropy in transversely isotropic media, Geophysics, 72 , D1-7