Inter-bed multiples & model-based prediction

All the previous sections of Part 4 leverage the free surface. Predictive decon works because WB multiples repeat at a fixed period. SRME predicts every surface-related multiple by convolving the data with itself. Radon separates primaries from multiples by their different post-NMO curvature, which comes from different velocities — which comes from the different ray paths surface multiples take.

Pure inter-bed multiples do none of this. They bounce only between subsurface reflectors; they never touch the surface. Their periodicity is irregular, their SRME data-self-convolution identity does not apply, and their velocities may be close to primaries so Radon cannot separate them. The only tool is model-based prediction — synthesize the inter-bed multiple using a known overburden model and subtract adaptively.

1. The inter-bed geometry

In the schematic, R1 is a shallow reflector at 0.4 s (two-way time), R2 is a deeper reflector at 1.0 s. The primary R2 (teal solid) is the ray S → R2 → R, arriving at 1.0 s. The pegleg inter-bed multiple (red dashed) is S → R2 → R1 → R2 → R. Compared with the primary, it adds exactly one extra round-trip between R1 and R2 — the interval whose two-way travel time is t(R2) − t(R1) = 0.6 s. So the multiple arrives at:

The key diagnostic: the inter-bed appears at a time where no primary exists, and its amplitude scales as R1² · R2 (two reflections off the shallow reflector sandwich one off the deep one). That cubed-reflectivity product is weak, but for strong contrasts — hard carbonate, volcanic sills, base of salt — it matters.

2. Why SRME / Radon / predictive decon fail

• Predictive decon needs a fixed periodicity. Inter-bed periodicity = 2·(t_R2 − t_R1), which varies with overburden geometry — not globally periodic in the trace.
• SRME relies on the d = p + p∗d identity where the second term comes from free-surface bouncing. Inter-bed multiples have no surface bounce — the identity breaks.
• Radon works when primary and multiple have different NMO velocities. Inter-bed multiples often share the same velocity as the primary they shadow (they traveled through the same overburden), so Radon cannot separate them.

3. Model-based inter-bed prediction

The recipe, given an overburden velocity and reflectivity model:

• Simulate primary and downgoing. Forward-model the trace assuming primaries only — call this the predicted primary train.
• Identify scattering points. In the predicted primary train, locate each strong reflector time and amplitude.
• Generate inter-bed train. For every pair (R_i, R_j), add a multiple at t(R_j) + 2(t(R_j) − t(R_i)) with amplitude proportional to (R_i)² · R_j. Repeat for all significant pairs.
• Apply wavelet. Convolve the spike train with the estimated source wavelet.
• Adaptive subtract. Use §4.4 to absorb amplitude and phase errors between prediction and observed.

4. Model-quality sensitivity

The widget’s most important pedagogical move is the Model amp error slider. At 0 % error the prediction matches the true inter-bed amplitude exactly and the residual at 1.6 s drops below the noise floor — the info bar reads >50 dB attenuation. At ±10 % error the predicted amplitude is wrong by a factor (1.1)² ≈ 1.21 (a 21 % mismatch), and attenuation collapses to ≈13 dB. At ±30 % error (factor 1.69) attenuation drops further to ≈4 dB — barely visible cleanup.

5. What you need for production inter-bed attenuation

• Migrated overburden. A post-migration image of the overburden layers gives the reflector times and approximate amplitudes.
• Well-to-seismic calibration. If you have wells, tie the seismic to the sonic log to refine reflectivity amplitudes.
• Iterative refinement. The first-pass inter-bed attenuation reveals previously-hidden primaries, which refine the overburden model, which improves the next inter-bed pass. 2–3 iterations typical.
• Compute budget. Per-trace full-wavefield modelling is expensive — often reduced to a 1D approximation per CMP for feasibility.

6. When inter-bed attenuation is worth the cost

• Strong shallow contrasts. A hard seafloor + volcanic sill combination produces massive inter-bed reverberation.
• Thin-bed reservoirs below strong reflectors. A reservoir at 2 s under a 1 s salt base is buried under inter-bed multiples that SRME cannot touch.
• AVO-quality deep targets. Inter-bed amplitude contamination invalidates AVO analysis on the underlying primaries.

Many projects skip inter-bed attenuation entirely — if the target is above the worst inter-bed layer, or if the contrast is small, other methods suffice. For hard carbonate or evaporite sections, it is unavoidable.

Part 4 closes here

You have the full multiple-attenuation toolkit: classify by kinematic signature (§4.1), eliminate surface-related multiples data-drivenly with SRME (§4.2), kill the residuals via Radon demultiple (§4.3), clean up with adaptive subtraction (§4.4), and model-predict the pure inter-beds (§4.5). Part 5 now moves into imaging proper — the migration family that takes the cleaned gathers from Parts 2–4 and produces the final subsurface image.

References

• Weglein, A. B., Araújo, F. V., Carvalho, P. M., et al. (1997). An inverse-scattering series method for attenuating multiples. Geophysics, 62, 1975.
• Berkhout, A. J., Verschuur, D. J. (1997). Estimation of multiple scattering by iterative inversion. Geophysics, 62, 1586.
• Verschuur, D. J., Berkhout, A. J., Wapenaar, C. P. A. (1992). Adaptive surface-related multiple elimination. Geophysics, 57, 1166.
• Yilmaz, Ö. (2001). Seismic Data Analysis (2 vols.). SEG.