Residual statics & surface-consistent decomposition

After the refraction statics of Section 2.3 are applied, the gather is mostly flat, but not quite. Small shifts remain, a few milliseconds per trace, from effects the refraction method could not capture: small-scale lateral variations, wavelet delays, ice or sand lenses. These are residual statics. They look small on any one trace, but stacking many of them incoherently destroys amplitude exactly where you need it.

1. The stacking penalty

The figure below is the centre CMP of a 40-station land line: 24 NMO-corrected traces with three flat reflections, each delayed by its shot static, its receiver static and a little jitter that no station term explains. Raise $\\sigma$ and watch the stack in (c) fall while the gather in (a) stays nearly flat. Then correct it two ways, with lags picked trace by trace by cross-correlation and with station statics solved from all the picks at once, and compare both with the true statics, which no processor knows. At the default setting the gather carries 3.7 ms rms of statics and still looks nearly flat, yet its raw stack keeps only 72% of the static-free peak; cross-correlation picks, 0.1 ms rms off the truth, bring it back to 100%.

A useful rule of thumb: random statics of rms $\\sigma$ scale the stacked amplitude at frequency $f$ by about $e^{-2\\pi^2 f^2 \\sigma^2}$, so a single frequency halves at $\\sigma \\approx 0.19/f$. A Ricker wavelet peaked at $f$ keeps $(1 + 2\\pi^2 f^2 \\sigma^2)^{-3/2}$ of its stack peak and halves a little sooner, at $\\sigma \\approx 0.17/f$: 5.8 ms at 30 Hz, the half-amplitude readout of the figure. A quarter-period $\\sigma$ (8 ms at 30 Hz) leaves only about 30%, and even 4 ms rms costs about 30% of the peak in theory; the figure’s centre gather, whose statics come to 3.7 ms rms at the default setting, loses 28%. More fold does not buy it back: the expected loss depends only on $\\sigma$ and $f$, and fold only narrows the scatter from one gather to the next, the band in panel (e).

2. How residuals are picked

After a preliminary velocity pick and NMO correction, the traces in a CMP gather should line up at every event. Cross-correlate each trace against a reference (a pilot stack, or a neighbor); the lag of the correlation peak is that trace’s residual.

Pick the lag $\\tau$ that maximizes $R$, refined between samples by a parabola, and repeat for every trace. The pilot is usually the CMP stack, stacked again after a first alignment. Panel (d) shows what can go wrong. A correlation has several lobes near the right one, and a wrong lobe can win: when the noise is heavy, or when large statics smear the pilot itself, the pick lands more than half a period off. That is a cycle skip, and the trace then stacks against the others. A static beyond the maximum lag is out of reach altogether: the picker stops at the limit or on whatever lobe the window holds.

3. Surface-consistent decomposition

A trace’s residual lag is not arbitrary. Physically, it is the sum of

1. the delay at the shot location (the near surface under the shot), $s\_i$;
2. the delay at the receiver location, $r\_j$;
3. a structural term for the reflector under the midpoint, $G\_k$;
4. a residual-moveout delay that grows with offset squared, $M\_k x\_{ij}^2$;
5. everything else, $n\_{ij}$.

where $i$ indexes shots, $j$ receivers, $k$ the CMP and $x\_{ij}$ is the offset. The first four terms are surface-consistent: each depends only on the shot, only on the receiver, or only on the CMP. Only $s\_i$ and $r\_j$ are statics; $G\_k$ is the geology, and $M\_k$ is a velocity error. The figure solves for $s\_i$, $r\_j$ and $G\_k$; its traces carry no residual moveout.

4. Solving the decomposition

Set up a linear system with one equation per live trace and unknowns $s\_i$, $r\_j$, $G\_k$ and $M\_k$, about $N\_s + N\_r + 2N\_{\\mathrm{CMP}}$ of them. There are many more equations than unknowns, yet the system still has a null space: a constant can move from every $s\_i$ to every $r\_j$, and a linear trend can trade between the station terms and $G\_k$, with the same fit. Least squares therefore needs constraints (zero-mean statics, damping), and statics with wavelengths longer than the spread are not resolved. Solvers such as Gauss-Seidel update one set of terms at a time from what the others leave unexplained.

Shift each trace by $-(s\_i + r\_j)$ and restack; the structural term stays in the data, and the residual-moveout term goes back into velocity analysis. Every shot and receiver appears in many traces, so the solve averages many noisy picks into one static per station. In the figure, station statics solved from the default picks in 3 iterations land 0.3 ms rms from the true ones, measured in the gauge where each set’s mean and a shared trend are zero; the jitter that no station term explains stays in the data. Iterate with velocity analysis for best results.

5. Why “surface-consistent” is a big deal

The same decomposition applies to far more than statics. Surface-consistent amplitudes, and the amplitude spectra used by deconvolution, factor the same way:

Take logarithms and the multiplicative model becomes additive, a linear system again. This is how surface-consistent deconvolution (Section 2.8) and surface-consistent amplitude correction work. One framework, many applications.

6. Practical notes

• Window and maximum lag matter. Correlate over several hundred milliseconds of strong reflections in the target zone, and cap the maximum lag near half the dominant period so a pick cannot jump a cycle, at the price of leaving any larger static out of reach. Statics with wavelengths longer than the spread are not resolved here; they belong to refraction or tomographic statics.
• Iterate with velocity. Residual statics depend on NMO being correct, which depends on velocity being correct, which depends on residuals being small. Alternate passes: residuals → velocity → residuals.
• Beware bias from the pilot trace. If the pilot has its own static error, every cross-correlation inherits it. A pilot smeared by large statics invites cycle skips as well (the figure’s third exercise shows it). Use a robust pilot (super-stack of many gathers) or the median of multiple pilot traces.
• Outliers ruin least-squares. Robust estimators (L1, Huber) are worth the extra cost on noisy land data.

Where this goes next

Section 2.5 turns to coherent noise: ground roll, swell, and multiples. Different signatures, different attenuators (f-k filtering, Radon, adaptive subtraction) but the same general recipe: transform to a domain where noise separates from signal, mute, transform back.

References

• Yilmaz, Ö. (2001). Seismic Data Analysis (2 vols.). SEG.
• Claerbout, J. F. (1976). Fundamentals of Geophysical Data Processing. McGraw-Hill.
• Sheriff, R. E., Geldart, L. P. (1995). Exploration Seismology (2nd ed.). Cambridge UP.