Predictive deconvolution

Part 2 — Pre-Processing Foundations

Learning objectives

Explain the prediction-error view of multiples: predict the trace at lag L, subtract
Read a trace autocorrelation to identify the multiple period
Pick prediction lag and operator parameters for water-bottom and peg-leg multiples
Distinguish predictive decon from spiking decon and know when to use which

Spiking deconvolution of §2.6 collapses the source wavelet toward a spike. Its cousin, predictive deconvolution, uses the same Wiener framework for a very different goal: remove events that can be predicted from earlier parts of the trace. The classic target is a water-bottom multiple — same wavelet, same reflectivity, just delayed by the two-way-time through the water column, appearing over and over at a fixed period.

1. The predictable-error idea

Given a trace s[n], design a filter p[n] of length L_op that predicts s[n] from earlier samples s[n − α − k], where α is the prediction lag and k = 0, 1, …, L_op−1. The prediction error

e[n] = s[n] - \sum_{k=0}^{L_{op}-1} p[k]\, s[n - \alpha - k]

is the part of the trace that cannot be predicted from its history. Anything periodic with period α cancels; anything new (primaries below the multiples, noise, unpredictable reflectivity) is preserved.

Setting α = 1 recovers spiking decon. Setting α = multiple period targets that multiple. Two settings, one operator.

2. Reading the autocorrelation

A periodic event produces a bump in the autocorrelation at the lag equal to its period. The trace below has primaries at 0.40 s and 1.00 s and a matching multiple train 240 ms later (amplitude 0.75 of the primary). The autocorrelation has a clear secondary peak at 240 ms — that is the signature of the multiple.

Slide the prediction lag to 240 ms (where the teal and red lines coincide in the autocorrelation plot) and adjust the subtract-amp to 0.75. The output panel goes nearly flat at the multiple locations: the prediction captured them, and subtraction removed them. The primaries (marked “P”) are preserved because they are not predictable from earlier samples.

3. How to find the lag

Autocorrelation peak. Compute R_ss over a window containing the multiple train. The highest secondary peak after lag 0 gives the multiple period. This is the universal approach.
Physics. For water-bottom multiples in marine data, α = 2 · water depth / v_water. If you know the bathymetry, you know α up to a few samples.
Peg-leg multiples. Peg-legs are primaries that have made one extra bounce between a shallow reflector and the sea surface. They appear at the primary time plus 2 · shallow reflector depth / v. If the sea bottom is at 0.15 s TWT, every primary gets a peg-leg 300 ms later.

4. Practical parameter picking

Prediction lag — from the autocorrelation bump or physics.
Operator length — a few samples longer than the wavelet duration. Too long absorbs primary energy.
Design window — containing the multiples, NOT containing just primaries (otherwise the operator does not have multiple energy to learn from).
White noise ε — same stabilization as spiking decon. 0.1–1 % typical.
Apply to gathers, not stacks. Multiples are periodic in the trace domain but their NMO velocity differs from primaries; on a stack, they are already NMO-flattened and harder to separate.

5. Where predictive decon fails

Variable water depth. Marine data over variable bathymetry — the multiple period changes across the survey. A single lag does not work; per-trace lag is needed. SRME (§4.2) handles this more robustly.
Amplitude-time-variant multiples. Peg-legs have different amplitudes depending on the transmission path. Adaptive subtraction (§4.4) is preferred.
Inter-bed multiples. Not surface-related; require model-based prediction (Part 4).
Short period vs primary duration. If multiples arrive before the primary wavelet fully dies, predictive decon damages primary amplitudes.

6. Spiking vs predictive — side by side

Spiking decon (α = 1): compress the wavelet to a spike. Primary tool to sharpen resolution.

Predictive decon (α = multiple period): remove a specific multiple family. Primary tool to attenuate water-bottom and peg-leg multiples before NMO / migration.

The two are often applied in sequence: predictive decon to kill multiples, then spiking decon to sharpen what is left. Each reduces the complexity the other has to deal with.

**The one sentence to remember**

Predictive decon predicts the trace at a chosen lag and subtracts the prediction; the lag is read from the autocorrelation bump, and α = multiple period is what separates it from spiking decon.

Where this goes next

Section §2.8 closes Part 2 with surface-consistent deconvolution — applying the same surface-consistent decomposition we used for residual statics (§2.4) to the design of a trace-by-trace decon operator. One source-side operator per shot, one receiver-side per receiver, one offset-side per offset bin, one trace-residual. Robust when the wavelet varies across the survey.

References

Robinson, E. A. (1957). Predictive decomposition of seismic traces. Geophysics, 22, 767.
Treitel, S., Robinson, E. A. (1966). The design of high-resolution digital filters. IEEE Trans. Geosci. Electron., 4, 25.
Robinson, E. A., Treitel, S. (2008). Digital Imaging and Deconvolution. SEG.
Wiggins, R. A. (1978). Minimum entropy deconvolution. Geoexploration, 16, 21.
Yilmaz, Ö. (2001). Seismic Data Analysis (2 vols.). SEG.