Near-offset conditioning for inversion

The intercept $A$ of an AVO fit $R(\\theta) = A + B\\sin^2\\theta$ is the extrapolation of the reflection coefficient to normal incidence ($\\sin^2\\theta = 0$). That extrapolation is anchored by the smallest-angle samples in the gather, typically the near-offset traces. If those traces are noisy or missing, $A$ becomes poorly constrained. Far-offset traces drive the gradient $B$ and fix $A$ only by extrapolation, so removing the near offsets turns $A$ into the end of a long lever arm whose error grows with the gap. This section is about the production strategies for dealing with contaminated or missing near-offset data.

1. Why near-offsets are often compromised

• Direct arrivals. The direct wave from source to receiver arrives earliest and loudest on near-offset traces. Even after muting, residual direct-wave energy contaminates the shallowest reflections.
• Ghost notches. Marine streamers towed at depth $h$ also record a ghost reflected from the sea surface. Its delay $2h\\cos\\varphi/V$ is longest near vertical incidence, so on near traces the notches at $f = nV/(2h\\cos\\varphi)$ sit lowest, inside the useful band, and distort the amplitudes of reflections there.
• Surface-related multiples. Short-period water-bottom multiples survive demultiple worst on near traces, where their moveout is almost the same as the primaries'. After NMO they sit on the reflection as a coherent residue.
• Source-generated noise. Near-field source effects, cavitation and source-generated scattered energy are strongest on the closest receivers. (The bubble pulse is different: it is part of the far-field signature on every trace, and designature handles it.)
• Acquisition gaps. A streamer with a 150 m towing offset has no near-offset receiver. The first 150 to 300 m of offset is simply missing.
• Land data. Ground roll, air blasts, and receiver coupling artefacts dominate short offsets even after filtering.

2. Four strategies on contaminated near offsets

The figure builds one flat reflector, shale over a sand of a chosen AVO class, whose true amplitude follows the three-term curve $R = A + B\\sin^2\\theta + C(\\tan^2\\theta - \\sin^2\\theta)$, and picks it on 58 NMO-corrected traces from 150 to 3000 m (2.9° to 45°). Inside a near zone $\\theta < \\theta\_{\\mathrm{cut}}$ the traces carry a coherent residue and stronger random noise. Set the contamination and choose a strategy. Each strategy fits the two-term line $\\hat A + \\hat B\\sin^2\\theta$ on 300 seeded noise draws, so (c) shows the bias of $\\hat A$ and its scatter side by side, and (d) shows where the answer lands among the AVO classes.

• None. Do nothing: fit every pick, the contaminated ones included. Because the near picks sit near $\\sin^2\\theta = 0$, their errors pull $\\hat A$ directly, and the least-squares fit tilts to compensate, so $\\hat B$ is pulled the other way. A coherent residue biases both; random noise widens the scatter of both. With the default Class I sand and a residue of +0.03 on the traces inside 12°, $\\hat A$ is biased by +0.010, 11 % of $A$.
• Mute. Exclude the near picks and fit only the far ones. The residue is gone, but $\\hat A$ is now an extrapolation from $\\theta\_{\\mathrm{cut}}$, and a two-term line cannot follow the curvature of the true $R(\\theta)$: the default mute leaves a bias of −0.008, and widening the zone to 25° raises it to −0.018, with a variance 13.8 times that of the same picks at normal incidence.
• Far fit. Fill the near picks from the mute's own fit and refit the whole gather. The filled picks lie exactly on the far fit's line, so the refit returns the muted answer to every digit: filling the gap with the far-offset fit's own predictions adds nothing.
• Rebuild. Rebuild the near traces from independent information, such as neighbouring CMPs or an $f$-$x$ or sparse-Radon interpolation across offset, then fit the full gather. The figure's Rebuild is idealised: each rebuilt pick is the true near amplitude plus an interpolation error plus whatever fraction of the residue survives the reconstruction, so with none left it is an upper bound on what reconstruction can do. With an interpolation error of 0.008 and no residue left, the bias falls to −0.004, the part that no two-term fit can remove; leave half the residue in and it becomes +0.003. Reconstruction wins only when the rebuilt samples carry information the far offsets do not, and only as far as they are clean.

The size of these errors matters for interpretation. Pick the Class IIp sand, $A$ = +0.015: with None, the residue lifts $\\hat A$ across the boundary and (d) reads the reflector as Class I, while muting keeps it in Class IIp.

3. Other production conditioning steps

• Trace reconstruction. If traces are physically missing (acquisition gap), reconstruct using $f$-$x$ prediction, sparse Radon, or matching-pursuit interpolation. Keep the reconstructed amplitudes honestly flagged downstream.
• Direct-wave muting. Pick the direct-arrival onset per shot and mute above it. Keep the mute gentle; over-muting into the reflection energy is worse than leaving a thin direct-wave tail.
• Ghost deconvolution. Design a minimum-phase ghost operator from the streamer depth and apply before NMO. Restores amplitudes in the spectral notches.
• Surface-consistent amplitude balancing. Decompose amplitude anomalies into shot, receiver, offset, and CMP terms; remove shot and receiver (acquisition), keep offset and CMP (geology).
• $\\tau$-$p$ filtering. Slant-stack the near-offset gather, mute the low-velocity, large-$p$ noise (ground roll, direct arrival), inverse-transform.

4. When near-offset conditioning matters most

• Simultaneous pre-stack inversion (Section 7.5) relies on well-sampled angle gathers to decompose into elastic attributes. A coherent residue on the near offsets biases $I\_P$; random noise there widens its uncertainty.
• Intercept-based fluid detection. A gas-brine contact often shows as a bright-spot change in $A$. A noisy $A$ masks the contrast.
• AVO gradient vs intercept cross-plots. A 5 % bias on $A$ shifts the whole cross-plot population, flipping classification for many reflectors near the boundary between classes.
• Shallow targets in deep-water marine have few near-offset traces relative to the target depth; those few matter disproportionately.

5. When it matters less

• Long-streamer marine with good geometry: many near-offset traces, each contributing independent information. Random noise on a small fraction of them averages down, though a coherent residue does not.
• Gradient-dominant interpretation: for targets where the signature is the gradient (Class II gas sand, for example), $\\hat B$ is the primary deliverable and the near offsets matter less.
• Far-offset-only inversion: some workflows deliberately discard near offsets (e.g., for steep-dip imaging); AVO intercept is then not a deliverable.

Where this goes next

Section 7.5 closes Part 7 with the pre-stack gather, the combined output of the QI-grade processing flow. How to build a gather that is ready for simultaneous AVO inversion: the angle decomposition, the noise floor, the Q-normalisation across offsets, and the end-to-end QC that validates the gather before the inversion sees it.

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.