The Layered Earth

What a Wave Can and Cannot See

The earth is layered at every scale a geologist cares about, from the millimetre laminae in a turbidite to the tens of metres of a reservoir zone. A seismic wave does not resolve all of it. A wave responds to the average properties of everything much smaller than its own wavelength, and it resolves only the boundaries that are a good fraction of a wavelength apart. So the first question about any layered rock is not what the layers are but how big they are compared with the wave that is looking at them. Below that threshold the layers stop being separate reflectors and start being a single material with its own effective velocity, density, and, as the next section shows, its own directional character.

Wavelength is just velocity divided by frequency, $\lambda = V / f$. For a rock at 3 km/s the three instruments that measure it live decades apart in frequency, and therefore decades apart in the scale they can see. A surface seismic wave at 30 Hz carries a wavelength of 100 metres. A sonic log near 10 kHz has a wavelength of 0.3 metres, thirty centimetres. A laboratory ultrasonic pulse at 1 MHz has a wavelength of 3 millimetres. Four and a half orders of magnitude in frequency become four and a half orders of magnitude in resolving scale.

The Same Lamination, Three Verdicts

Now put a real rock under those three rulers: a sand-shale sequence laminated on a scale of tens of centimetres, alternating beds each a fraction of a metre thick. To the laboratory pulse, with its 3 mm wavelength, each bed is a vast slab and the core plug you cut may sit entirely inside one lamina; the pulse sees a single lithology. To the sonic log at 0.3 m, the beds are comparable to the wavelength, so the tool half-resolves them and reports a jagged, bed-by-bed velocity as it passes through. To the seismic wave at 100 m, the entire package of laminae is a hundredth of a wavelength deep; the wave cannot distinguish sand from shale within it and treats the whole interval as one effective medium with one averaged velocity. Three instruments, one rock, three genuinely different answers, and none of them is wrong.

Why This Needs a Theory

The consequence is sharp. When beds are far below the seismic wavelength, we cannot model the interval by picking one bed and calling it representative, nor by simply averaging the velocities as if the wave took a straight run through each in turn. The wave feels the layers as springs in series and in parallel at once, and the right average depends on how the layers are stacked relative to the direction of travel. That is a specific piece of physics with a specific answer, and it does something a single homogeneous rock never does: it makes the effective medium faster along the layering than across it. The rock becomes anisotropic purely because it is layered. Turning a stack of thin isotropic beds into the one anisotropic medium a seismic wave actually sees is the Backus average, and it is the star of the next section.

References

• Backus, G. E. (1962). Long-wave elastic anisotropy produced by horizontal layering. Journal of Geophysical Research, 67(11), 4427-4440.
• Mavko, G., Mukerji, T., & Dvorkin, J. (2009). The Rock Physics Handbook (2nd ed.). Cambridge University Press.