Surface-consistent deconvolution

Part 2, Pre-Processing Foundations

Learning objectives

Extend the Section 2.4 surface-consistent decomposition from statics to wavelets
Set up the log-spectrum equation $\log A_{ij} = \log s_i + \log r_j + \log h_k + \log g_l + \log n_{ij}$ and solve it by least squares at every frequency, with the constraints that fix what the data cannot see
Explain why one operator per shot, receiver and offset makes the wavelet the same along the line, where one global operator cannot
Apply the framework to amplitude balancing, not just wavelet shape

One global deconvolution operator assumes the wavelet is the same on every trace. On land it is not: shot 7 fires a little stronger than shot 8, the geophone at 1875 m sits loose in sand, and the ground under each shot point and each receiver shapes the spectrum its own way, so the wavelet varies across the survey. Surface-consistent deconvolution describes that variation with one term per shot, one per receiver and one per offset, and builds from them an operator for every trace.

1. The multiplicative model

Model the amplitude spectrum of trace $ij$ , recorded from shot $i$ at receiver $j$ , as the product of four surface-consistent factors and a residual:

A\_{ij}(f) = s\_i(f)\\, r\_j(f)\\, h\_k(f)\\, g\_l(f)\\, n\_{ij}(f)

The shot term $s\_i$ depends only on the shot, the receiver term $r\_j$ only on the receiver, the offset term $h\_k$ only on the offset bin $k$ , and the earth term $g\_l$ only on the CMP bin $l$ ; $n\_{ij}$ holds whatever is not surface consistent. The earth term belongs to the CMP bin, not to the trace: a term with its own value on every trace would give one unknown per equation, and nothing could be solved.

Take logarithms and the product becomes a sum:

\\log A\_{ij} = \\log s\_i + \\log r\_j + \\log h\_k + \\log g\_l + \\log n\_{ij}

This is the same linear decomposition we met in Section 2.4 for residual statics, and the same least-squares solve.

2. Live decomposition

The figure below records a 2D land line: 24 shots every 100 m into 48 receivers every 50 m, a rolling split spread out to 575 m, 491 traces with 13 missing. Each shot has its own strength, each geophone its own coupling resonance, the far offsets lose more high frequency, and the earth has flat layers with a bright spot at 0.5 s under 1050 to 1350 m. The method sees only the recorded traces. Plate (a) maps their log amplitude at one frequency by shot and receiver, (b) shows what is left once the recovered station terms are removed, (c) compares the recovered shot and receiver terms with the true ones, and (d) shows the near-offset section before or after the operator you choose.

Plate (a) is striped. Every trace from one shot shares that shot’s strength, so a shot runs along its row, and every trace at one station shares its coupling, so a station runs down its column. At 30 Hz the recovered shot terms span 10.6 dB and the receiver terms 3.0 dB. Before any operator, the wavelet differs from trace to trace by 3.7 dB (RMS over 10 to 80 Hz). One global operator leaves that at 3.7 dB, because it applies the same filter to every trace; the operators built from each trace’s recovered shot, receiver and offset terms take it to 0.7 dB. Whitening has a price in both cases: the S/N falls from 25 dB to 19 dB with the global operator and to 17 dB with the per-station ones.

Two switches show where the method can go wrong. Leave the CMP term out of the model and the bright spot has nowhere to go: the shots and stations above it absorb its extra amplitude, their operators turn it down, and it measures +6.5 dB in (d) against +9.9 dB in the earth itself (+9.3 dB with the CMP term in). Drop the constraints and the fit is as close as before (a residual of 0.08 dB at 30 Hz), yet the station terms sit 3.8 dB from the truth instead of 0.03 dB: the data only see the sum of the terms, and the split has become arbitrary. Give 8 % of the traces (39 of them) a gain of their own and they fit no term: 72 % of their gain stays in the residual, which rises to 1.43 dB at 30 Hz, and the residual map in (b) points at them; the rest leaks into the terms, and the station terms drift to 0.51 dB from the truth.

3. From amplitudes to operators

For amplitude balancing, the decomposition of the trace amplitudes already gives you what you need: divide each trace by its shot and receiver factors, and the amplitude becomes consistent along the line. This runs in nearly every land production flow, usually under the name “surface-consistent scaling.”

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

\\log|A\_{ij}(f)| = \\log|s\_i(f)| + \\log|r\_j(f)| + \\log|h\_k(f)| + \\log|g\_l(f)| + \\log|n\_{ij}(f)|

Solve at each frequency independently and recover the shot, receiver and offset spectra. The decomposition gives amplitude spectra only, so the operator takes its phase from the minimum-phase assumption: cascading the shot, receiver and offset operators on each trace compresses the wavelet as spiking deconvolution does, while allowing the wavelet to vary from station to station, provided the wavelets are minimum phase.

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 corrects 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 recovers most of the bad station’s loss, but not all of it. In the figure, the loose geophone at 1875 m raises the spread to 4.0 dB and the per-station operators bring it to 1.0 dB, yet the loose station’s own traces keep 4.9 dB of wavelet error against 0.7 dB for the rest: its weak high band lies partly under the 1 % prewhitening floor. A lower floor recovers more of it (2.1 dB at 0.1 %), at a cost in S/N.
Offset-dependent wavelet. The wavelet broadens at far offset from attenuation along the longer path. An offset-bin operator removes the average offset-dependent change within the design window; the progressive loss down the trace still needs inverse-Q filtering.
Graceful degradation. A very noisy shot biases only its own term, so its traces get a poorer operator while every other trace is untouched. A global operator is robust to one bad shot, which is diluted in the mean of all 24, but it cannot correct any shot.

5. Setting up the least-squares solve

Number the unknowns: $N\_{\\mathrm{shots}}$ shot terms, $N\_{\\mathrm{rcvs}}$ receiver terms, $N\_{\\mathrm{off}}$ offset terms, and one earth term per CMP bin. Write one equation per trace. The system is large but sparse: each row has only four non-zero entries. Solve it with conjugate gradients (Section 0.9). The system is rank deficient, so add constraints: a constant can move between any two families of terms without changing the data, and because offset and midpoint are both built from the shot and receiver positions, linear and quadratic trends along the line can trade between the offset, CMP and station terms. The figure removes the mean, linear and quadratic parts from the shot and receiver terms and the mean from the offset terms.

Run this at every frequency in the seismic band. The operator for trace $ij$ is the minimum-phase inverse of $|s\_i(f)\\, r\_j(f)\\, h\_k(f)|$ times the survey’s mean wavelet, with a little white noise added for stability; apply it by multiplication in the frequency domain, which is convolution in time.

6. Practical notes

Regularize. Add smoothness penalties on shot-to-shot and receiver-to-receiver variation of the operators. Bad data on one isolated shot should not produce a wildly different operator from its neighbours.
QC the factors. Plot the shot and receiver terms along the line, and map the residual. Outliers usually correspond to known acquisition issues: a cable change, a weather pause, a reshot. Traces that break the model show in the residual, as they do in (b).
Iterate with statics. Decon, then residual statics, then decon again with the improved gathers. Typically 2 passes.
Compute cost. The per-frequency solve costs $O(N\_{\\mathrm{traces}})$ per conjugate-gradient iteration. That is small next to imaging, which is why the method runs in nearly every land flow; the real effort goes into QC of the recovered terms.

The one sentence to remember

Surface-consistent decon takes logarithms of trace spectra, splits them by least squares into shot, receiver, offset and CMP terms at every frequency, and builds an operator for each trace from its own shot, receiver and offset terms; keep the CMP term in the model, or the geology goes into the station terms.

Where Part 2 goes next

You have walked the pre-processing chain: reformat and QC → trace editing and amplitude recovery → refraction statics → residual statics → noise attenuation → deconvolution. The three decon methods you met (spiking, predictive, surface-consistent) are a toolbox, not a cascade: a production flow usually applies a single prestack decon pass chosen for the area, sometimes followed by a post-stack spectral-whitening pass for extra resolution. Which method, and how the whole flow is ordered, is always tailored to the problems of the prospect; a sequence that works beautifully in one area can fail in a more complex one. The operators you have just built set the band of the output, so Section 2.9 closes Part 2 by measuring that band before and after every step. Part 3 then 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.