Adaptive subtraction

Part 4 — Multiple Attenuation

Learning objectives

Explain why direct subtraction of a predicted multiple leaves residual energy
Derive the Wiener-filter adaptive subtraction equation as least-squares in a design window
Tune filter length and design-window location and see the residual quality change
Describe why adaptive subtraction is the universal clean-up step after SRME and Radon

SRME produces a multiple prediction that is close but not exactly right — the amplitude may be 5–30 % off, the phase may be shifted by a few milliseconds, and the wavelet shape is not quite the true multiple wavelet. Direct subtraction would leave a ringing remnant at every multiple location, which is why SRME always ends with an adaptive subtraction: fit a short filter to absorb the mismatch before subtracting.

1. The problem direct subtraction leaves

Observed: d = p + m (primary + true multiple). Predicted: m̂ ≈ m but off by some amplitude and phase. Direct residual:

r_{\text{direct}} = d - \hat m = p + (m - \hat m)

The m − m̂ term is small but non-zero. On a stack it appears as a ringing kopie of every multiple. The widget makes this concrete:

The prediction has amplitude 0.55 (truth is 0.70) and is phase-shifted by +15 ms. Direct subtraction leaves a residual visibly ringing at 1.0 s — the multiple location. Adaptive subtraction gets much closer to zero there.

2. The Wiener filter formulation

Find a short filter f of length L that minimizes the energy of d − f ∗ m̂ within a design window. Setting $\partial E/\partial f = 0$ yields

R_{\hat m \hat m}\,f = R_{d\hat m}

where R_m̂m̂ is the Toeplitz autocorrelation of the predicted multiple in the design window, and R_dm̂ is the cross-correlation of observed and predicted. Solve via Levinson or Gauss. Same math as spiking decon (§2.6), different goal.

Apply f outside the design window too: compute f ∗ m̂ over the full trace and subtract from the full d. The filter captured the amplitude/phase correction inside the window and hopefully generalizes.

3. Tuning the filter

Length. Short (5–10 samp) absorbs pure amplitude and single-sample phase shifts. Long (30–60 samp) captures richer wavelet-shape mismatches. Too long starts absorbing primary energy — over-subtraction.
Design window. Must CENTER on a region where the multiple dominates and the primary is weak. Sliding it to a region of pure primary will null the primary instead of the multiple.
Window width. Narrow catches local phase variation; wide averages. 100–300 ms is typical.
White-noise stabilization. Same trick as decon: add a small ε to the autocorrelation diagonal so the solve is numerically robust.

4. The production recipe

Predict the multiple (via SRME, predictive decon, or Radon).
Define design windows where the multiple dominates (picked manually or by time gates below strong reflectors).
Solve for f per window, per trace (or per gather, with a single filter shared).
Convolve f ∗ m̂ and subtract from data.
QC: check that the primary is preserved and the multiple is attenuated. Repeat with different windows if over-subtraction occurred.

5. Advanced variants

Time-varying adaptive. Different filter for every few hundred ms of the trace — handles non-stationary wavelet.
Pattern-matching adaptive. Instead of a linear filter, use a curvelet-domain or wavelet-domain matched filter. Deals with dispersion and dispersive phase errors in the prediction.
L1-regularized adaptive. Sparsity-promoting filter design; less prone to over-subtraction in areas where the prediction is weak.
Multi-trace adaptive. Fit a single filter to all traces in a CMP gather; enforces spatial smoothness of the correction.

6. The one failure mode to watch for

Over-subtraction. If the design window accidentally contains a primary, the filter will absorb it as if it were multiple. Result: the primary disappears along with the multiple, leaving a hole. Guard against this by picking windows below a strong reflector (so no primaries in the window) and by inspecting the residual for negative amplitudes at primary locations.

**The one sentence to remember**

Adaptive subtraction designs a short Wiener filter that minimizes residual energy in a window where the multiple dominates, then applies that filter to the predicted multiple before subtracting — it turns a rough prediction into a clean subtraction and is the last step of every SRME or Radon flow.

Where this goes next

§4.5 closes Part 4 with the hardest multiple class: inter-bed multiples that never reach the surface. SRME, Radon, and adaptive subtraction all fail for these because the surface-reflection leverage is absent — the only tool is model-based prediction through a known overburden.

References

Verschuur, D. J., Berkhout, A. J., Wapenaar, C. P. A. (1992). Adaptive surface-related multiple elimination. Geophysics, 57, 1166.
Berkhout, A. J., Verschuur, D. J. (1997). Estimation of multiple scattering by iterative inversion. Geophysics, 62, 1586.
Yilmaz, Ö. (2001). Seismic Data Analysis (2 vols.). SEG.
Claerbout, J. F. (1976). Fundamentals of Geophysical Data Processing. McGraw-Hill.