Q-compensated imaging

Real earth is not a perfect elastic medium. Every wave cycle loses a small fraction of its energy to heat — intrinsic attenuation — 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 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:

• $Q = 20–50$: shallow unconsolidated sediments, heavy-oil reservoirs, gas clouds (heavily attenuating).
• $Q = 50–150$: consolidated sediments (sand, shale).
• $Q = 200–500$: 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 widget

Left panel: the source Ricker wavelet (teal), the attenuated wavelet after propagating through Q for time t (yellow), and the Q-compensated wavelet (red dashed). Right panel: the corresponding amplitude spectra on the same color scheme. Drag the sliders to watch:

• Lower Q → faster attenuation, spectrum collapses to low frequencies, wavelet broadens in time.
• Longer t → same effect: more travel = more cumulative attenuation.
• Lower ε (tighter regularisation) → Q-compensation recovers more of the high-f spectrum but risks amplifying noise where the attenuated signal is small.
• Higher ε (looser regularisation) → compensation is stable but undercorrects; the output spectrum still shows some attenuation.

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 $\exp(32\pi/10) \approx e^{10} \approx 22000$. The attenuated signal at 80 Hz is, effectively, noise; multiplying noise by 22 000 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 $\text{damp} \gg \varepsilon$ (low f, short t, high Q), the operator reduces to the exact inverse and fully recovers the signal. At frequencies where $\text{damp} \ll \varepsilon$ (high f, long t, low Q), the operator flattens toward $\text{damp}/\varepsilon^2 \to 0$, gracefully rolling off without amplifying noise. $\varepsilon$ is the noise-to-signal ratio; in production it is set to match the measured background noise level of the data (typically 0.02–0.1).

5. When Q-compensation matters

• Deep targets under heavy overburden. Shallow gas clouds, heavily-weathered shales, or thick salt sections all attenuate the passing wavefield. 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, faking a negative AVO gradient on every reflector under the lossy zone.
• Pre-stack inversion inputs. The spectrum of the pre-stack gather must match across offset and depth for simultaneous inversion to converge. Q-compensation flattens the spectrum so inversion can focus on the contrast of interest.
• 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.

Q-compensation is also inherently directional: source-to-subsurface and subsurface-to-receiver paths both attenuate. A full imaging flow compensates both legs.

7. Pitfalls

• Wrong Q. If the assumed Q is too low, compensation over-boosts high frequencies; too high, it under-corrects. Spectral analysis on pilot sections is essential.
• ε too aggressive. 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; many pedagogical treatments (including this widget) omit it for clarity.

Where this goes next

§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 "A" (intercept) term 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.