Predictive deconvolution

Spiking deconvolution of Section 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: the same wavelet and reflectivity, delayed by the two-way time through the water column and flipped in polarity at the sea surface, repeating at a fixed period.

1. The predictable-error idea

Given a trace s\[n\], design a filter p\[k\] of $N$ coefficients (an operator length $L\_{\\mathrm{op}} = N\\Delta t$) that predicts s\[n\] from earlier samples s\[n-\\alpha-k\], where $\\alpha$ is the prediction gap (or lag) and $k = 0, 1, \\dots, N-1$. The prediction error

is the part of the trace that cannot be predicted from its history. The operator looks back over a window of lags from $\\alpha$ to $\\alpha + L\_{\\mathrm{op}} - \\Delta t$, so anything that repeats with a period inside that window cancels; anything new (primaries below the multiples, noise, unpredictable reflectivity) is preserved.

Setting $\\alpha$ to one sample ($\\alpha = \\Delta t$) recovers spiking decon. Placing the window over the multiple period targets that multiple. Two settings, one operator.

2. Reading the autocorrelation

A periodic event marks the autocorrelation at the lag equal to its period. For a water-layer train the mark is a trough, not a peak, because the sea surface flips every copy against the one before it. In the figure below, 180 m of water gives a period $L = 2h/V\_{\\mathrm w}$ = 240 ms. The sea floor, with reflection coefficient $R$ = 0.40, and a reflector 404 ms below it are each followed by a train whose amplitude is multiplied by $-R$ at every bounce. The autocorrelation of the trace has its trough, $-0.38$, at 240 ms and a smaller peak at 480 ms. The peak near 404 ms, at 0.44 taller than the trough, is a cross-term between the sea floor and the deep reflection, not a period: read the lag from the physics as well as from the plot.

The figure opens with a 100 ms gap and a 60 ms operator. Their window, 100 to 156 ms, misses the period, and the multiples do not move against the primaries (0 dB). Slide the gap to 200 ms: the window now covers 240 ms, the prediction in (b) captures the train, and the multiples fall by 26 dB against the primaries, which keep 100 % of their amplitude. The primaries survive because nothing one period earlier predicts them. You never set an amplitude: the Wiener operator learns it from the autocorrelation. The gap did not have to equal 240 ms; the window only had to contain it. Take the gap down to one sample and the same operator becomes a spiking filter that shrinks every wavelet, the primaries keeping only 30 % of their amplitude. Stretch the operator to 320 ms instead and its window swallows the 404 ms cross-term: it predicts part of the deep reflection from the sea floor, the attenuation drops to 10 dB and the primaries keep 87 %, the too-long-operator failure noted below.

3. How to find the lag

• Autocorrelation. Compute the autocorrelation $r\_k$ over a window containing the multiple train. The strongest secondary extremum after lag 0 (a trough, for water-layer multiples) gives the multiple period. Check it against the physics, because cross-terms between reflections also make peaks. This is the universal approach.
• Physics. For water-bottom multiples in marine data, the period is $L = 2h/V\_{\\mathrm w}$, where $h$ is the water depth and $V\_{\\mathrm w} \\approx 1500$ m/s. If you know the bathymetry, you know the period to a few samples.
• Peg-leg multiples. Peg-legs are primaries that have made one extra bounce between a shallow reflector and the free surface. They appear at the primary time plus $2z/V$, the two-way time of the shallow layer. If the sea bottom is at 0.15 s two-way time, every primary gets a peg-leg 150 ms later. A deep reflection can reverberate on the way down or on the way up, so its train grows as $(k+1)(-R)^k$ (Backus 1959), and the operator needs coefficients at both $L$ and $2L$: in the figure, switching to both legs drops the attenuation from 26 to 11 dB.

4. Practical parameter picking

• Prediction gap, from the autocorrelation or the physics, so that the window of lags from $\\alpha$ to $\\alpha + L\_{\\mathrm{op}} - \\Delta t$ contains the period.
• Operator length. With $\\alpha = L$, a little longer than the wavelet. With $\\alpha$ at the second zero crossing of the autocorrelation, long enough that $\\alpha + L\_{\\mathrm{op}}$ passes the period. 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 $\\varepsilon$, the same stabilization as spiking decon; 0.1 to 1 % is typical.
• Mind the offset. The multiple period is exact only at zero offset; at far offsets it shrinks with angle and the periodicity breaks down. Apply predictive decon to near offsets, in the $\\tau$-$p$ domain, or after stack, where the trace approximates zero offset.

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: deepen the water from 180 to 225 m in the figure and the old operator removes nothing. A per-trace lag is needed; SRME (Section 4.2) handles this more robustly.
• Amplitude-time-variant multiples. Peg-legs have different amplitudes depending on the transmission path. Adaptive subtraction (Section 4.4) is preferred.
• Inter-bed multiples. Not tied to the free surface; they need internal-multiple prediction, data-driven (inverse scattering series) or model-based (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

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.

Where this goes next

Section 2.8 turns to surface-consistent deconvolution, applying the same surface-consistent decomposition we used for residual statics (Section 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.