Linearity, reciprocity, stationarity

Everything downstream of the field, from sorting traces into gathers through NMO, stacking and migration to time-lapse differencing, leans on three properties of the earth: it responds linearly, it is reciprocal, and it does not change while we survey it.

Linearity

For the small strains of a seismic wave, away from the source, the earth’s response is linear. Double the source amplitude and every recorded amplitude doubles (double the source energy and amplitudes grow by a factor of $\\sqrt{2}$). Fire two sources together and the recording is the sum of their separate recordings. This superposition is why array outputs can be summed, why a measured source signature can be divided out, and why simultaneous-source acquisition can be deblended.

Reciprocity

Reciprocity says that swapping a source and a receiver leaves the recorded trace unchanged: the response at $\\mathbf{x}\_B$ to a source at $\\mathbf{x}\_A$ equals the response at $\\mathbf{x}\_A$ to the same source at $\\mathbf{x}\_B$. It holds in any linear elastic medium at rest, however heterogeneous, anisotropic or attenuating, as long as the medium is the same for both experiments; for vector sources and receivers the components swap too. In the figure, shot 1 fires at A and records at B, and shot 2 fires at B and records at A. Change what differs between the two shots and find which changes make the recordings differ.

With nothing changed between the shots, the two recordings agree to the last sample, even over a sea floor dipping 20°, where the ray reflects about 1.2 km up-dip of the midpoint (exercise 2). Reciprocity makes the trace from A to B a duplicate of the trace from B to A, which is why a survey can be sorted either way round (Section 3.7). It does not make every trace with the same midpoint see the same reflection point: that also needs flat layers, and dip smears common-midpoint reflection points up-dip. The one way to break reciprocity itself in the figure is to move the medium: a 1.5 m/s current over 3000 m of offset delays shot 2 by 4.0 ms, close to $2xu/V^2$ for offset $x$, current $u$ and water velocity $V$, the effect ocean acoustic tomography uses to measure currents.

Stationarity

Stationarity assumes the earth does not change between shots. On the timescale of a survey it is usually safe, but not everywhere: the speed of sound in the sea changes with temperature by several metres per second between seasons and even between sail lines, and tides change the water depth. In the figure the water warms by 0.4 % (6 m/s) between the shots, shot 2 arrives 5.6 ms early, and the swapped traces differ by an NRMS of 108 %: a shift of a sixth of a period is enough. Time-lapse (4D) surveys break stationarity on purpose, since the reservoir is meant to change, and must separate that change from every other one. Their repeatability is quoted as NRMS, and independent noise alone sets a floor under it (exercise 5).

Where the approximations break

Close to an explosive source the strains are large and the rock responds non-linearly (it crushes and cracks), so how the charge couples to the ground is hard to predict; beyond that zone the wave is linear again and acts like an equivalent elastic source. Field reciprocity fails when the swapped source and receiver are not truly equivalent (a buried charge against a surface geophone, different coupling, a vertical source against a horizontal sensor) or when the medium moves, like a current in the sea. And a time-lapse survey is non-stationary by design. Good acquisition engineers know which approximation they are riding on at every step.

References

• Aki, K., Richards, P. G. (2002). Quantitative Seismology (2nd ed.). University Science Books.
• Knopoff, L., Gangi, A. F. (1959). Seismic reciprocity. Geophysics, 24(4), 681-691.
• Kragh, E., Christie, P. (2002). Seismic repeatability, normalized rms, and predictability. The Leading Edge, 21(7), 640-647.
• Munk, W., Worcester, P., Wunsch, C. (1995). Ocean Acoustic Tomography. Cambridge University Press.
• Sheriff, R. E., Geldart, L. P. (1995). Exploration Seismology (2nd ed.). Cambridge University Press.
• Yilmaz, Ö. (2001). Seismic Data Analysis: Processing, Inversion, and Interpretation of Seismic Data (2 vols.). SEG Investigations in Geophysics 10.