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 bounce off the free 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. Surface-based tricks fail, so the multiple has to be predicted from the subsurface: either from a model of the overburden (this section) or data-driven from the recorded primaries of the generating layers (Jakubowicz 1998; the inverse-scattering-series attenuator of Weglein et al. 1997). The prediction is then subtracted adaptively.

1. The inter-bed geometry

In Figure 4.5 you predict a pegleg from a model of two layers, subtract it from the recorded trace, and read how much of it is left. Put errors into the model, one at a time, and find out which kind of error the subtraction cannot forgive.

In plate (a), $R\_1$ is a shallow reflector at 0.4 s (two-way time) and $R\_2$ a deeper one at 1.0 s. The primary of $R\_2$ goes down to $R\_2$ and back, arriving at 1.0 s. The pegleg inter-bed multiple (dashed) goes down to $R\_2$, up to the underside of $R\_1$, down to $R\_2$ again and back to the surface. Compared with the primary it adds one extra round trip through the interval between the two reflectors, whose two-way time is $t\_2 - t\_1$ = 0.6 s. So the multiple arrives at:

Its amplitude is $-R\_1 R\_2^2 (1 - R\_1^2)$: two reflections off the deep reflector $R\_2$ sandwich one off the underside of $R\_1$, whose coefficient is $-R\_1$ and flips the polarity, and $1 - R\_1^2$ is the loss in transmission through $R\_1$ on the way down and back up. With $R\_1$ = 0.20 and $R\_2$ = 0.30 that is −0.017 against +0.288 for the primary of $R\_2$, so in plate (b) the multiple is a small trough at 1.60 s. In this toy it lands where no primary exists, so you can see it alone; in real data it lands on top of deeper primaries, which is what makes it dangerous (switch on the target primary to see this). The cubic 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. The inter-bed period is $t\_2 - t\_1$, which changes from one pair of layers to the next, so the trace has no single global period.
• SRME relies on the identity $d = p + r\_0\\,(p \\ast d)$ (Section 4.2), where the second term comes from bounces at the free surface. Inter-bed multiples have no surface bounce, so the identity does not apply.
• Radon works when primary and multiple have different NMO velocities. The stacking velocity of an inter-bed multiple is usually only slightly lower than that of primaries arriving at the same time, so the moveout difference is too small for Radon to separate them.

3. Model-based inter-bed prediction

The recipe, given an overburden velocity and reflectivity model:

1. Model the primaries. Forward-model the trace with primaries only; call this the predicted primary train.
2. Identify scattering points. In the predicted primary train, locate each strong reflector time and amplitude.
3. Generate the inter-bed train. For every pair $R\_i$ above $R\_j$, add a multiple at $2t\_j - t\_i$ with amplitude about $-R\_i R\_j^2$ (quadratic in the deeper reflector). More generally, every triplet with $t\_i < t\_j, t\_k$ gives a first-order multiple at $t\_j + t\_k - t\_i$ with amplitude about $-R\_i R\_j R\_k$. Repeat for all significant combinations.
4. Apply wavelet. Convolve the spike train with the estimated source wavelet.
5. Adaptive subtract. Use Section 4.4 to absorb amplitude and phase errors between prediction and observed.

4. Model-quality sensitivity

Start from the default of Figure 4.5: the model’s $R\_2$ is 10 % too strong. Because $R\_2$ enters the pegleg twice, the prediction is $(1.1)^2 = 1.21$ times the multiple, and raw subtraction attenuates the multiple by only 13.6 dB (−10 % leaves 14.4 dB; +30 %, a factor of 1.69, only 3.2 dB; −30 %, 5.8 dB). With an exact model the result in the window is exactly the primaries, and the readout shows >50 dB. Now switch to a one-tap adaptive filter: a pure scale error is absorbed, and the attenuation returns to >50 dB.

Timing is different. Make the amplitudes exact and the model’s interval velocity 0.6 % too slow: the predicted pegleg lands 7.2 ms late, and with this 26 Hz wavelet raw subtraction adds energy (−1.5 dB). Plate (d) shows the raw curve falling through 0 dB at a misfit of about 6 ms. A one-tap filter cannot shift the prediction and reaches only 0.4 dB; an 11-tap filter, whose lags span ±10 ms, can, and reaches 43.1 dB, but no short filter reaches a misfit much beyond its taps. Long filters have their own cost: switch on the target primary 10 ms under the multiple and the 11-tap filter shifts the prediction onto the target and subtracts that too.

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 (Section 4.1), eliminate surface-related multiples data-drivenly with SRME (Section 4.2), kill the residuals via Radon demultiple (Section 4.3), clean up with adaptive subtraction (Section 4.4), and predict the pure inter-beds from the subsurface (Section 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

• Jakubowicz, H. (1998). Wave equation prediction and removal of interbed multiples. SEG Technical Program Expanded Abstracts, 1527.
• 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.