Tuesday, September 15, 2009

Dispersion and acquisition

If we were to survey the universe of seismic sources in use today for production seismic data in the petroleum industry, it would reveal only a few serious contenders. Over the last 80 years or so, many seismic sources have been developed, tested, and tossed into the Darwinian struggle for market survival as a reliable commercial source. At present, there are three sources that account for the vast majority of data acquisition.

In marine seismic applications the airgun is ubiquitous. There are several dispersive effects related to airguns and airgun arrays, including ghosting and radiation patterns. Recall that we are using dispersion in a generalized sense meaning frequency-dependent phenomena, not just seismic velocity variation with frequency. The ghost is an interesting example of dispersion where the physical source interacts with the ocean surface to form a plus-minus dipole that is a strong function of frequency. For a given source depth, the radiated field can have one or several interference notches along certain angular directions away from the source. These show up in the measured seismic data as spectral nulls called a ghost notch. To further complicate the picture, ghosting occurs on both the source and receiver side of acquisition. The radiation pattern associated with an airgun array is an exercise in the theory of antenna design and analysis, again complicated by dipole characteristics due to ghosting.

For land seismic data, there are two major sources in use worldwide: explosives and vibroseis. The explosive source has, in principle, the weakest dependence on frequency. Certainly it has a bandwidth determined by shot characteristics and local geology, but is an approximately impulsive point source. A buried explosive shot will, like the marine airgun, develope a dipole nature due to ghosting. But this is often not as well-developed as in the marine case, likely due to lateral variations in near surface elastic properties and topography.

The other significant land source, vibroseis, has a host of dispersive effects. For a single vibe we can mention two fascinating phenomena, radiation pattern and harmonics. The theory of radiation for a circular disk on an isotropic elastic earth was developed by several investigators in the 1950's, most notably Miller and Pursey. They were able to show the power emmitted in various wave types (P, S, Rayleigh) ultimately depends only on the Poisson ratio. But even though the total power for a single vibe is not a function of frequency, in real world applications it is common to use a source array which will radiate seismic waves in a way that strongly depends on frequency.

A vibroseis unit injects a source signal (sweep) into the earth over the course of several seconds. The sweep is defined by time-frequency (T-F) characteristics and for simplicity we will consider a linear upsweep here (very common in practice). The emitted signal bounces around in the earth and is recorded by a surface sensor, the resulting time series being an uncorrelated seismic trace. Conceptually, when this uncorrelated time trace is transformed into the T-F plane by a suitable spectral decomposition method, we should see a representation of the sweep with a decaying tail of reflection energy. This is observed, but we also commonly see a series of other linear T-F features at frequencies higher than the sweep at any given time. These are vibroseis harmonics. Since the observed uncorrelated seismic trace is the summation of all frequencies in the T-F plane, these harmonics can interfere and distort the weak reflection events we are trying to measure.

The origin of harmonics can be understood in relation to human hearing. As first discussed by Helmoltz in the 1860's, when a sound wave interacts with the human hearing apparatus something very interesting happens. First we need to realize that away from any obstacle, a sound wave proceeds by vibratory motion of the air particles and this motion is symmetric (equal amplitude fore and aft). But when a sound wave encounters the ear it pushes against the eardrum which is a stretched elastic membrane with fluid behind. The amount of power in the sound wave is fixed, and that power will compress the eardrum (due to its elasticity) less than the sound wave will compress air. If we think of, say, a 200 Hz wave as a cosine, this interaction means the deflection will be asymmetric. It will be a waveform that repeats 200 times per second, but it will not be a symmetric cosine wave. How can something repeat 200 times per second and not be a pure 200 Hz wave? Helmoltz found the answer: it must a 200 Hz wave plus a series of harmonics (400 Hz, 800 Hz, etc.). The fact that the material properties of the ear impede the motion due to sound necessarily means that harmonics will be generated.

Now back to the vibroseis case, when the mechanical apparatus of the vibrator pushes down against the earth it is resisted by the elastic nature of the near surface. On the upstroke the motion is unimpeded, asymmetry develops, and harmonics are generated. All this happens despite some pretty amazing engineering in the system. With modern T-F methods, we can think up various ways to remove the harmonics by data processing the uncorrelated data traces. There is also ongoing discussion about how to use the harmonics rather than filter them out.

Next time.... Dispersion and processing

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