Pre-stack gathers for simultaneous inversion

Part 7's four previous sections each fix one amplitude hazard: AVO-preserving pre-processing (§7.1), true-amplitude migration (§7.2), Q-compensation (§7.3), near-offset conditioning (§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 and thereby improving signal-to-noise per input by $\sqrt{N}$. Typical decomposition:

• Near stack: 0–15°. Dominated by the intercept A = normal-incidence reflectivity.
• Mid stack: 15–30°. Sees both A and B; the pivot range where AVO starts to be visible.
• Far stack: 30–45°. Dominated by the gradient B; where the AVO anomaly is most visible.

Some projects use four or five stacks for finer angle resolution; beyond that the SNR benefit per stack is outweighed by the complexity of inversion. Simultaneous pre-stack inversion ingests all stacks simultaneously and outputs elastic attribute volumes ($I_p, I_s, \rho$ per pixel).

2. The widget

Left panel: three bars for the partial-stack averages of $R(\theta) = A + B\sin^2\theta$ with the user-chosen (A, B). Right panel: the A–B cross-plot with Rutherford-Williams class regions shaded; the current (A, B) point sits in the region that classifies this reflector. Slide A and B to see how the partial stacks and the classification co-vary.

3. Class interpretation at a glance

• Class I (large pos A, neg B): tight sands or carbonates — the top of the reservoir is a positive contrast that darkens with offset. Common for low-porosity producing sands.
• Class II (small pos A, strong neg B): classic gas-sand signature where A flips sign across the contact — the most valuable exploration AVO anomaly.
• Class III (neg A, neg B): bright-spot gas sand — a direct hydrocarbon indicator that brightens with offset. Historically the most-exploited AVO target.
• Class IV (neg A, pos B): slow reservoir — rare, indicates unusual lithologies (coal seams, overpressured shales).

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 elastic attribute perturbations at every time–offset pixel by minimising the model-data misfit across all angles simultaneously:

where $d_\theta$ is the partial-stack data at angle $\theta$, $W$ is the wavelet, and $R_\theta(I_p, I_s, \rho)$ is the Aki-Richards reflectivity predicted from the elastic attributes. 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_p = V_p · ρ): related to A via $A \approx \tfrac{1}{2} \Delta I_p/I_p$. Useful for lithology discrimination.
• S-impedance (I_s = V_s · ρ): obtained jointly with I_p; constrains fluid vs matrix effects.
• Density (ρ) or V_p/V_s ratio: the remaining degree of freedom. Fluid sensitivity lives here.

Fluid and lithology classification is done by cross-plotting the elastic attributes against wells and applying a rock-physics template that maps $(I_p, V_p/V_s)$ 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:

• Trim statics. Residual moveout from imperfect NMO or migration is corrected by cross-correlating adjacent traces and shifting. Sub-sample precision needed.
• Angle decomposition. Convert offset to reflection angle using the velocity model; discard samples with angles above the mute threshold (typically 40–45°).
• Amplitude balancing across stacks. Normalise each partial stack to consistent RMS amplitude; compensates for residual geometry effects.
• Spectrum matching across stacks. Q-compensate each stack to the reference spectrum so that partial stacks differ only in AVO, not bandwidth.
• Wavelet extraction. Estimate the source wavelet per angle from well ties; typically a single average wavelet is acceptable for short angle ranges.
• 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.
• 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

• Pre-processing with AVO preservation (§7.1): no AGC, proper spherical divergence, SCA balancing.
• True-amplitude migration (§7.2): Kirchhoff / beam / one-way WE / RTM with the correct weights.
• Q-compensation (§7.3): frequency-domain Wiener-regularised deconvolution of the attenuation operator.
• Near-offset conditioning (§7.4): mute or reconstruct the contaminated first degrees of offset.
• Angle decomposition + partial stacks (§7.5): collapse to 3 or 4 partial stacks.
• Wavelet extraction + well-tie QC.
• Simultaneous inversion → $I_p, I_s, \rho$ volumes.
• 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 (§7.1), true-amplitude migration (§7.2), Q-compensation (§7.3), near-offset conditioning (§7.4), and partial-stack + inversion QC (§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.
• Pratt, R. G. (1999). Seismic waveform inversion in the frequency domain, Part 1. Geophysics, 64, 888.