Linearity, reciprocity, stationarity

Physics prerequisites for acquisition

Learning objectives

State the three approximations we rely on: linearity, reciprocity, stationarity
Demonstrate reciprocity: the trace at B for a source at A equals the trace at A for a source at B
Recognise the conditions under which each approximation breaks (large-amplitude sources, anisotropy, time-variant velocity)
Use these approximations to justify CMP processing, receiver-side arrays, and 4D repeatability

Everything we do downstream — CMP gathers, NMO, stacking, migration — secretly rests on three approximations about the earth.

Linearity

The wave equation is linear to first order in amplitude. Fire twice the energy and you get twice the amplitude back. Fire two sources simultaneously and you get the sum of their responses. This is why arrays work at all — summing outputs is the same as one big wavefield — and why simultaneous-source acquisition can work (the deblending step relies on linearity).

Reciprocity

Swap source and receiver — the recorded trace is the same. This is a statement about the passive, symmetric medium. It’s what lets you treat many source–receiver pairs that happen to share a midpoint as redundant measurements of the same reflection point. CMP gathers are built entirely on this.

Stationarity

The earth doesn’t change between shots. The same reflection coefficient today will be there tomorrow. This assumption breaks for 4D (by design: you want the earth to have changed between baseline and monitor), and for very long acquisition where salt bodies creep or ice caps load and unload. In normal 3D it’s safe.

Where the approximations break

A nearby explosion is not linear (large pressure pulse, non-linear earth response near the source — hence “charge-depth coupling” being tricky). A strongly anisotropic medium is not fully reciprocal in the simple sense. And 4D is by design non-stationary. Good acquisition engineers know which approximation they’re riding on at every step.

References

Aki, K., Richards, P. G. (2002). Quantitative Seismology (2nd ed.). University Science Books.
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.