Q-compensated imaging

Real earth is not a perfect elastic medium. Every wave cycle loses a small fraction of its energy to heat. This intrinsic attenuation is characterised by the dimensionless quality factor $Q$. Intrinsic attenuation is frequency-selective: high frequencies lose energy faster than low frequencies, because there are more cycles per unit time. Left unaddressed, this narrows the wavelet spectrum with depth, shifts its peak frequency downward, and (most importantly for QI) reduces deep amplitudes relative to shallow ones. Q-compensation restores the balance.

1. The Q-attenuation operator

where $S(f)$ is the source spectrum, $f$ is frequency, $t$ is the two-way travel time, and $Q$ is the quality factor. $Q$ is frequency-independent in the idealised constant-$Q$ model (a good approximation across the seismic band). Typical values:

• $Q$ from 20 to 50: shallow unconsolidated sediments, heavy-oil reservoirs, gas clouds (heavily attenuating).
• $Q$ from 50 to 150: consolidated sediments (sand, shale).
• $Q$ from 200 to 500: hard carbonates, basalts.
• $Q > 500$: metamorphic and crystalline basement (near-elastic).

At $Q = 50, t = 2\\ \\text{s}, f = 50\\ \\text{Hz}$: attenuation = $\\exp(-\\pi \\cdot 50 \\cdot 2 / 50) = \\exp(-2\\pi) \\approx 0.002$, a 54 dB loss. Even at a favourable $Q = 100$ the same frequency is down by 27 dB. The high-frequency end of the spectrum is the first casualty.

2. The figure

Figure 7.3 sends a Ricker wavelet down a constant-$Q$ earth and back, adds noise, and compensates it. It opens with weak regularisation: raise $\\varepsilon$ and watch the compensated half of the section in (d) trade noise for resolution.

At the opening setting ($Q$ = 60, $t$ = 2 s, noise 40 dB below the spectral peak of the source, $\\varepsilon$ = 0.002) the peak frequency of the arrival has fallen from 30 Hz to 15 Hz and compensation brings it back to 30 Hz, but the signal-to-noise ratio in the wavelet window falls from 22 dB to 3 dB. Above 42 Hz the arrival had already sunk under the noise, and no operator can pull the two apart there. Near $\\varepsilon$ = 0.03 the output is as clean as the recording, but its peak has slipped to about 25 Hz. Other things to try:

• Lower $Q$ or longer $t$: faster loss, a spectrum that collapses toward low frequencies, a broader and later wavelet. Only $t^{\*} = t/Q$ matters: halve both and nothing changes.
• Smaller $\\varepsilon$ (weaker regularisation): compensation recovers more of the high-frequency spectrum, up to a maximum gain of $1/(2\\varepsilon)$, and amplifies the noise floor where the attenuated signal has sunk below it.
• Larger $\\varepsilon$ (stronger regularisation): compensation is stable but undercorrects; above $f\_c = Q\\ln(1/\\varepsilon)/(\\pi t)$ the output spectrum still rolls off.
• Dispersion: with phase compensation off the band comes back but the wavelet still peaks 13 ms late (at $\\varepsilon$ = 0.01).

3. Naive inverse Q-compensation is unstable

The obvious inverse of the attenuation operator is

which exactly cancels the attenuation at any frequency. Trouble is, $\\exp(+\\pi f t / Q)$ grows exponentially. At $Q = 50, t = 2\\ \\text{s}, f = 80\\ \\text{Hz}$ the gain is $\\exp(32\\pi/10) = e^{10.05} \\approx 23\\,000$ (87 dB). The attenuated signal at 80 Hz is, effectively, noise; multiplying noise by 23 000 produces a "Q-compensated" spectrum that is almost entirely amplified noise at the high end.

4. Wiener-style regularisation

A stable Q-compensation replaces the exploding inverse with a regularised one:

At frequencies where $\\beta \\gg \\varepsilon$ (low $f$, short $t$, high $Q$), the operator reduces to the exact inverse $1/\\beta$ and fully recovers the signal. Where $\\beta \\ll \\varepsilon$ (high $f$, long $t$, low $Q$), it falls toward $\\beta/\\varepsilon^2 \\to 0$ and rolls off instead of amplifying noise. $\\varepsilon$ is the noise-to-signal amplitude ratio relative to the source peak, so $\\varepsilon^2$ is the power ratio. The operator never boosts by more than $1/(2\\varepsilon)$, reached where $\\beta = \\varepsilon$, at $f\_c = Q\\ln(1/\\varepsilon)/(\\pi t)$. In production $\\varepsilon$ is set from the measured background noise of the data, typically 0.02 to 0.1 (a gain cap of 14 to 28 dB). With dispersion the same form is applied to the complex earth filter $H$, as $\\overline{H}/(|H|^2 + \\varepsilon^2)$, which also undoes the delay; Figure 7.3 does exactly this.

5. When Q-compensation matters

• Deep targets under heavy overburden. Shallow gas clouds, unconsolidated near-surface sediments and heavily weathered shales attenuate the passing wavefield (salt, by contrast, has very high $Q$; the dimming beneath salt comes from the sediments above it). Deep reservoir amplitudes are dim without compensation.
• AVO through lossy overburden. Far-offset rays have longer travel paths than near-offset, so they accumulate more attenuation. Without Q-compensation, far-offset amplitudes are systematically low, biasing the AVO gradient negative on reflectors beneath the lossy zone.
• Pre-stack inversion inputs. Simultaneous inversion estimates a wavelet for each angle stack; attenuation that grows with offset and depth makes those wavelets vary within the inversion window and biases the inverted contrasts. Q-compensation reduces that variation.
• Time-lapse (4D) processing. Baseline and monitor surveys must have identical Q compensation to be subtractable.

6. Production implementation

Real Q-compensation has to deal with the fact that $Q$ varies in the subsurface: a single effective $Q$ applied uniformly is a crude approximation. Two standard approaches:

• Time-variant Q: derive $Q(t)$ from spectral-ratio measurements on pilot holes or VSPs, then apply the regularised inverse frequency by frequency, time by time.
• Q migration (Q-PSDM, Q-RTM): incorporate the attenuation operator into the migration itself. Each ray or wavefield carries its own accumulated $Q$ history. More expensive but handles lateral Q variation.

Both the downgoing and the upgoing legs attenuate. In the trace operator above, $t$ is the two-way time, so both legs are included; in Q-migration each leg is compensated along its own path.

7. Pitfalls

• Wrong Q. If the assumed $Q$ is too low, compensation boosts too hard and, with regularisation, reaches its gain cap at a lower frequency; too high, it under-corrects the signal while still spending its full gain on noise. In Figure 7.3 at $\\varepsilon = 0.01$, half the true $Q$ pulls the wavelet 16 ms early and ends the band near 22 Hz; twice the true $Q$ spends 34 dB near 88 Hz on noise. Spectral analysis on pilot sections is essential.
• $\\varepsilon$ too small. Setting $\\varepsilon$ too small produces the noise amplification the Wiener form was supposed to prevent. A good default: $\\varepsilon \\approx \\max(\\text{noise-floor}, 0.02)$.
• Ignored phase dispersion. Real Q-attenuation has a companion phase dispersion term (the Kramers-Kronig partner of amplitude attenuation). Production Q-compensation includes it, and so does Figure 7.3: with dispersion on, the attenuated wavelet arrives late and lopsided, and phase compensation restores its timing and symmetry.

Where this goes next

Section 7.4 covers near-offset conditioning: the zero-to-short-offset part of each CMP gather needs special attention because a clean near-offset trace underpins the intercept $A$ of the AVO fit. Production flows invest disproportionate effort in making the near-offset clean.

References

• Yilmaz, Ö. (2001). Seismic Data Analysis (2 vols.). SEG.
• Etgen, J., Gray, S. H., Zhang, Y. (2009). An overview of depth imaging in exploration geophysics. Geophysics, 74, WCA5.
• Virieux, J., Operto, S. (2009). An overview of full-waveform inversion in exploration geophysics. Geophysics, 74, WCC1.
• Claerbout, J. F. (1985). Imaging the Earth’s Interior. Blackwell.
• Kjartansson, E. (1979). Constant Q-wave propagation and attenuation. Journal of Geophysical Research, 84, 4737.
• Wang, Y. (2002). A stable and efficient approach of inverse Q filtering. Geophysics, 67, 657.