Attenuation and Q from acquisition view
Learning objectives
- State the Q-attenuation law: amplitude ∝ exp(-π f t / Q)
- Explain why high frequencies attenuate faster than low
- Estimate the usable bandwidth at target depth given source bandwidth and Q
- Connect Q to source choice (broadband vs narrow-band) and receiver bandwidth requirement
The earth is a low-pass filter that varies with travel distance. The mechanism is intrinsic attenuation, parametrised by the quality factor Q.
The law
For a wave at frequency f travelling for time t through a medium of quality factor Q:
At fixed Q, doubling either t or f doubles the magnitude of the exponent — which squares the attenuation factor. High frequencies die first. A 60 Hz component at t=2 s, Q=50 retains exp(−π·60·2/50) ≈ exp(−7.54) ≈ 0.05% of its original amplitude — essentially gone. The 10 Hz component at the same t and Q retains exp(−1.26) ≈ 28%.
Why this drives source and receiver choice
If your target is at 3 s two-way-time and Q ≈ 100, the usable bandwidth at the target is roughly the source bandwidth scaled by exp(-π f · 3 / 100). That’s a 3 dB loss at ~30 Hz already. So if you want 60 Hz signal from your target, you’d better put a lot more 60 Hz energy at the source than you’d need at the receiver floor. And your receivers must respond flat across the band you care about.
Q-compensation in processing (§7.3 of the Processing textbook) inverts this filter — it boosts the attenuated high frequencies. But it cannot recover energy the source never put in. Plan for Q at the acquisition stage.
References
- Aki, K., Richards, P. G. (2002). Quantitative Seismology (2nd ed.). University Science Books.
- Yilmaz, Ö. (2001). Seismic Data Analysis: Processing, Inversion, and Interpretation of Seismic Data (2 vols.). SEG Investigations in Geophysics 10.
- Sheriff, R. E., Geldart, L. P. (1995). Exploration Seismology (2nd ed.). Cambridge University Press.