Pre-stack gathers for simultaneous inversion

Part 7's four previous sections each fix one amplitude hazard: AVO-preserving pre-processing (Section 7.1), true-amplitude migration (Section 7.2), Q-compensation (Section 7.3), near-offset conditioning (Section 7.4). The result is a pre-stack gather whose amplitudes reflect geology rather than acquisition or operator geometry. This section shows what simultaneous inversion actually does with that gather, and the QC every production flow applies before shipping the inversion output.

1. Partial angle stacks, the inversion ingest

A pre-stack gather is information-rich but noisy; an inversion run on raw pre-stack traces would spend most of its effort fitting noise. Production flows instead build a small number of partial angle stacks, each averaging a range of incidence angles, which raises the signal-to-noise ratio by roughly $\\sqrt{N}$ when $N$ traces with incoherent noise are summed. Typical decomposition:

• Near stack: 0\\text{-}15^\\circ. Dominated by the intercept $A$, the normal-incidence reflectivity $R\_0$.
• Mid stack: 15\\text{-}30^\\circ. Sees both $A$ and $B$; the range where the offset dependence starts to be visible.
• Far stack: 30\\text{-}45^\\circ. Carries the largest gradient contribution ($\\sin^2\\theta \\approx 0.25\\text{-}0.5$), so it is where an AVO anomaly departs most from the near stack.

Some projects use four or five stacks for finer angle resolution; beyond that each stack gets noisier while adding little, because the reflectivity over 0\\text{-}45^\\circ has only two or three independent parameters. Simultaneous pre-stack inversion ingests all stacks together and outputs elastic volumes: $I\_{\\mathrm P}$, $I\_{\\mathrm S}$ and $\\rho$ at every time sample of every trace.

2. Reading three stacks

Below, a cap shale over a sand is recorded as an NMO-corrected angle gather, stacked into near, mid and far thirds of the usable angles, and read the way an interpreter reads it. You will flatten the gather, swap the pore fluid between brine and gas, and then damage the gather on purpose to see which readings survive.

The figure opens on a Class III gas sand with $+10$ ms of residual moveout left at 45^\\circ. The far pick is $-0.24$ instead of $-0.27$, so the fitted gradient $B$ reads $-0.03$ where the flat gather gives $-0.13$. Drag the moveout to zero and the top of the sand settles $0.26$ below the wet trend in (d), while the same sand filled with brine sits $0.08$ below it: the fluid shows as a departure from the trend, not as a quadrant. A balancing error does the same damage without any timing error: a $-20$ % gain on the far angles alone turns this Class III reading into Class IV (exercise 4).

The table in (e) is what a three-term inversion returns from the same three picks. On the flat Class III gather it gives $\\Delta I\_{\\mathrm P}/I\_{\\mathrm P} = -0.45$ against a true $-0.44$, but $\\Delta\\rho/\\rho = -0.32$ against $-0.17$: with a contrast this large the linearised equations are strained, and density, whose term is small until the angles are large, absorbs the misfit. Take the far angles away from the Class II gas sand (exercise 5) and the density error grows from $0.04$ to $0.24$ while $\\Delta I\_{\\mathrm P}/I\_{\\mathrm P}$ barely moves.

3. Class interpretation at a glance

• Class I (large positive $A$, negative $B$): tight, high-impedance sands or carbonates. The positive top-of-reservoir amplitude decreases with angle and may reverse at far angles.
• Class II ($A$ near zero, either sign; strong negative $B$): a gas sand with almost no normal-incidence contrast. It is nearly invisible on the near stack and strong on the far stack. When $A$ is slightly positive the reflection reverses polarity with angle (Class IIp).
• Class III (negative $A$, negative $B$): the bright-spot gas sand, a direct hydrocarbon indicator that brightens with offset. Historically the most exploited AVO target.
• Class IV (negative $A$, positive $B$): a low-impedance gas sand under a stiff, high-velocity cap rock such as a hard shale, siltstone or carbonate. The bright negative near-stack amplitude dims with angle, so far-stack attributes can miss it.

4. Simultaneous inversion, what it actually computes

Given partial stacks and a low-frequency model (from tomography or FWI), simultaneous pre-stack inversion solves for the elastic attributes at every time sample of every trace by minimising the model-data misfit across all angles simultaneously:

where $d\_\\theta$ is the partial-stack data at angle $\\theta$, $W\_\\theta$ is the wavelet extracted for that stack, $R\_\\theta(I\_{\\mathrm P}, I\_{\\mathrm S}, \\rho)$ is the Aki-Richards reflectivity predicted from the elastic attributes, and $\\lambda\\,\\mathcal{R}$ is the regularisation. The optimisation is heavily regularised (lateral smoothness, well-tie constraints, low-frequency anchoring) to keep the solution geologically plausible.

The primary outputs are:

• P-impedance ($I\_{\\mathrm P} = V\_{\\mathrm P}\\rho$): the intercept is the normal-incidence reflectivity, $A \\approx R\_0 = \\tfrac{1}{2} \\Delta I\_{\\mathrm P}/I\_{\\mathrm P}$, so $\\Delta I\_{\\mathrm P}/I\_{\\mathrm P} \\approx 2A$. Useful for lithology discrimination.
• S-impedance ($I\_{\\mathrm S} = V\_{\\mathrm S}\\rho$): obtained jointly with $I\_{\\mathrm P}$; shear stiffness is nearly blind to pore fluid, so $I\_{\\mathrm S}$ separates fluid effects from matrix effects.
• Density ($\\rho$): the third parameter, poorly resolved unless the gather carries reliable angles beyond about 35^\\circ. The ratio $V\_{\\mathrm P}/V\_{\\mathrm S} = I\_{\\mathrm P}/I\_{\\mathrm S}$ follows from the first two outputs and is the main fluid indicator.

Fluid and lithology classification is done by cross-plotting the elastic attributes against wells and applying a rock-physics template that maps $(I\_{\\mathrm P}, V\_{\\mathrm P}/V\_{\\mathrm S})$ clusters to geological classes.

5. Pre-stack gather QC pack

Before handing the gather to the inversion engine, production flows apply one final QC pack:

1. Trim statics. Residual moveout from imperfect NMO or migration is corrected by cross-correlating each trace with a pilot (the stacked trace) in sliding time windows and shifting it. Even a few milliseconds at the far angles bias the gradient: in the figure above, +10 ms at 45^\\circ (+2.5 ms at 30^\\circ) cuts the fitted $B$ of the Class III gas sand to a quarter.
2. Angle decomposition. Convert offset to reflection angle using the velocity model; discard samples with angles above the mute threshold (typically 40\\text{-}45^\\circ).
3. Amplitude balancing across stacks. Normalise each partial stack to consistent RMS amplitude; compensates for residual geometry effects.
4. Spectrum matching across stacks. Apply shaping filters that bring each stack to a common reference spectrum (the far stack loses high frequencies to NMO stretch and longer paths), or else extract a separate wavelet per stack.
5. Wavelet extraction. Estimate the source wavelet per angle from well ties; typically a single average wavelet is acceptable for short angle ranges.
6. Well-tie QC. Cross-correlate synthetic from each well's log with the partial stacks at the well location. Correlation > 0.7 is typical; lower flags that either the well-log processing or the gather conditioning needs work.
7. Low-frequency model tie. Ensure the low-f absolute impedance from the velocity model is consistent with well-tied impedance.

6. The end-to-end QI chain

1. Pre-processing with AVO preservation (Section 7.1): no AGC, proper spherical divergence, SCA balancing.
2. True-amplitude migration (Section 7.2): Kirchhoff / beam / one-way WE / RTM with the correct weights.
3. Q-compensation (Section 7.3): frequency-domain Wiener-regularised deconvolution of the attenuation operator.
4. Near-offset conditioning (Section 7.4): mute or reconstruct the contaminated first degrees of offset.
5. Angle decomposition + partial stacks (Section 7.5): collapse to 3 or 4 partial stacks.
6. Wavelet extraction + well-tie QC.
7. Simultaneous inversion → $I\_{\\mathrm P}$, $I\_{\\mathrm S}$ and $\\rho$ volumes.
8. Rock-physics classification → fluid and lithology volumes.

Each step has a characteristic failure mode that Part 7 has catalogued. A QI product that fails interpretation is almost always traceable to one of them.

Part 7 closes here

You have the full QI processing toolkit: amplitude-preserving pre-processing (Section 7.1), true-amplitude migration (Section 7.2), Q-compensation (Section 7.3), near-offset conditioning (Section 7.4), and partial-stack + inversion QC (Section 7.5). The output of this chain is an elastic-attribute volume that reservoir engineers and geologists can treat as a rock property map. Part 8 extends the workflow into time, how to process repeated surveys so you can track reservoir changes from production, injection, or depletion.

References

• Castagna, J. P., Backus, M. M. (1993). Offset-Dependent Reflectivity. SEG.
• Russell, B. H. (1988). Introduction to Seismic Inversion Methods. SEG.
• Yilmaz, Ö. (2001). Seismic Data Analysis (2 vols.). SEG.
• Hampson, D. P., Russell, B. H., Bankhead, B. (2005). Simultaneous inversion of pre-stack seismic data. SEG Technical Program Expanded Abstracts, 1633-1637.
• Fatti, J. L., Smith, G. C., Vail, P. J., Strauss, P. J., Levitt, P. R. (1994). Detection of gas in sandstone reservoirs using AVO analysis: a 3-D seismic case history using the Geostack technique. Geophysics, 59, 1362-1376.
• Rutherford, S. R., Williams, R. H. (1989). Amplitude-versus-offset variations in gas sands. Geophysics, 54, 680-688.
• Castagna, J. P., Swan, H. W. (1997). Principles of AVO crossplotting. The Leading Edge, 16, 337-342.