Surface-consistent deconvolution

Part 2 — Pre-Processing Foundations

Learning objectives

Extend the §2.4 surface-consistent decomposition from statics to wavelets
Set up the log-amplitude equation log A_ij = log s_i + log r_j + log e and solve it by least-squares
Explain why a separate operator per shot and per receiver is more robust than one global operator
Apply the framework to amplitude balancing, not just wavelet shape

One global deconvolution operator assumes the wavelet is the same on every trace. Reality: a slightly different source strength on shot 7, a slightly under-coupled geophone at receiver 23, different ground impedance under different shot points — the wavelet varies across the survey. Surface-consistent deconvolution absorbs that variation by building a separate operator for each shot, each receiver, and each offset bin.

1. The multiplicative model

Model the amplitude (or spectrum) of trace ij — from shot i to receiver j at offset h_ij — as the PRODUCT of four factors:

A_{ij} = s_i \cdot r_j \cdot h_{h_{ij}} \cdot e_{ij}

Shot term s_i depends only on shot, receiver term r_j only on receiver, offset term only on the offset bin, and everything else (including the true earth response) lives in e_ij.

Take logarithms — the multiplicative model becomes ADDITIVE:

\log A_{ij} = \log s_i + \log r_j + \log h + \log e_{ij}

This is the same linear-decomposition template we met in §2.4 for residual statics. Same math. Same least-squares solve.

2. Live decomposition

The widget below simulates a small survey with 8 shots and 10 receivers. You control whether per-shot and per-receiver variations are present, and how much residual noise muddles the observation. The right-hand bar charts show the recovered estimates compared to the TRUE factors (dashed teal ticks).

Toggle both variations on at 5 % noise: the left heatmap is chaotic, the right heatmap is nearly uniform at 1.0 — and the bar charts match their true values to within a few percent. Crank the noise to 30 % and the estimates wiggle; turn everything off and the operator recovers identity, as it should.

3. From amplitudes to operators

For amplitude balancing, the decomposition above already gives you what you need: divide each trace by its shot and receiver factors, and the amplitude becomes consistent. This is run every production flow, often as “surface-consistent scaling.”

For deconvolution, apply the same logic to the SPECTRUM at every frequency:

\log|A_{ij}(f)| = \log|s_i(f)| + \log|r_j(f)| + \log|h(f)| + \log|e_{ij}(f)|

Solve at each frequency independently, recover the shot-side and receiver-side amplitude spectra, and design a per-shot and per-receiver deconvolution operator. Two operators per trace cascaded gives the same compressive effect as spiking decon but tolerates wavelet variation.

4. Why it is worth the extra step

Variable source coupling. Shot 5 fires in a damp ditch, shot 6 in a dry roadbed. The wavelet is different. A per-shot operator fixes shot 5 without touching shot 6.
Variable receiver coupling. A geophone planted loose in sand produces a different wavelet than one hammered into clay. A per-receiver operator fixes the bad station.
Offset-dependent wavelet. The wavelet broadens at far offset from Q attenuation. An offset-bin operator corrects it without needing a separate Q-compensation pass.
Graceful degradation. If one shot has huge noise, only its operator is hurt; global operators would smear it across the whole survey.

5. Setting up the least-squares solve

Number the unknowns: N_shots per-shot, N_rcvs per-receiver, N_offbins per-offset, plus one earth term per CMP bin. Write one equation per trace. The system is huge but sparse — each row has only four non-zero entries. Solve with conjugate gradients (from §0.9), add mean-zero constraints on shot and receiver terms to break the inherent rank deficiency (one constant can be shuffled between shot, receiver, and earth terms without changing the observation).

Run this at every frequency bin in the seismic band. The operator for trace ij is then the inverse of s_i(f) · r_j(f); apply in the frequency domain (convolve with IFFT in time).

6. Practical notes

Regularize. Add smoothness penalties on shot-to-shot and receiver-to-receiver variation of the operators. Bad picks on one isolated shot should not produce a wildly different operator than its neighbors.
QC the factors. Plot shot- and receiver-factor maps spatially. Outliers usually correspond to known acquisition issues — a cable change, a weather pause, a reshot.
Iterate with statics. Decon, then residual statics, then decon again with the improved gathers. Typically 2 passes.
Compute cost. The per-frequency solve is O(N_traces) per iteration with CG and sparse setup. Production codes exploit this, but the method is not cheap — it is only worth the cost when the global operator is visibly biased.

**The one sentence to remember**

Surface-consistent decon takes logarithms of trace amplitudes, decomposes them into per-shot, per-receiver, per-offset, and earth terms by least-squares, and builds a separate deconvolution operator per trace — the same decomposition that underlies residual statics and amplitude balancing.

Part 2 closes here

You have walked the full pre-processing chain: reformat + QC → trace editing + amplitude recovery → refraction statics → residual statics → noise attenuation → spiking decon → predictive decon → surface-consistent decon. Every step is either a repair operation on the raw traces or a separation operation that pulls signal from noise via a transform-domain trick. Part 3 moves to the velocity work that every imaging algorithm depends on.

References

Robinson, E. A., Treitel, S. (2008). Digital Imaging and Deconvolution. SEG.
Yilmaz, Ö. (2001). Seismic Data Analysis (2 vols.). SEG.
Claerbout, J. F. (1976). Fundamentals of Geophysical Data Processing. McGraw-Hill.
Treitel, S., Robinson, E. A. (1966). The design of high-resolution digital filters. IEEE Trans. Geosci. Electron., 4, 25.