Residual statics & surface-consistent decomposition

Part 2 — Pre-Processing Foundations

Learning objectives

Explain the residual-statics workflow: cross-correlate, pick lag, decompose, subtract, iterate
Derive the 4-term surface-consistent model: t_{ij} = t_src(i) + t_rcv(j) + t_offset(h) + t_residual
See why residuals matter for the stack even when they look small on a single trace
Apply the same surface-consistent framework to deconvolution and amplitude correction

After the refraction statics of §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 widget below has three panels. Left is a CMP gather with residual statics of standard deviation you control. Middle is the same gather after the residuals are subtracted. Right overlays the raw stack (orange) on the corrected stack (teal). Notice that as the residual magnitude grows, the raw stack shrinks dramatically even while the gather still “looks OK.” That amplitude loss at the reservoir horizon is what residual statics recover.

A useful rule of thumb: for a wavelet of dominant frequency f, a static of T/4 (quarter-period) applied incoherently halves the stacked amplitude. At 30 Hz, that is ~8 ms. So even 4 ms of residual standard deviation noticeably hurts the stack.

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.

R_{x,y}[\tau] = \sum_n x[n]\, y[n+\tau]

Pick the τ that maximizes R. Repeat for every trace. You now have a residual-lag estimate per trace.

3. Surface-consistent decomposition

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

The delay introduced at the SHOT location (the near-surface under the shot),
The delay at the RECEIVER location (the near-surface under the receiver),
A small residual-NMO delay that depends on source–receiver OFFSET,
Everything else, which we call the residual.

t_{ij} = t_{\text{src}}(i) + t_{\text{rcv}}(j) + t_{\text{offset}}(h_{ij}) + t_{\text{res}}(i,j)

where i indexes shots, j indexes receivers, and h is the offset. The first three terms are surface-consistent: they depend only on shot, only on receiver, or only on offset. The last term absorbs everything else.

4. Solving the decomposition

Set up a linear system: one equation per trace (N_shots × N_rcvs equations), unknowns are the three sets of terms (N_shots + N_rcvs + N_offsets unknowns). The system is hugely overdetermined — least-squares gives you a best-fit per-shot, per-receiver, and per-offset term.

Apply these three corrections (shift each trace by −t_src[i] − t_rcv[j] − t_offset[h]) and re-stack. Reflections become flat on the gather and sharp on the stack. Iterate a few times with velocity analysis for best results.

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

The same decomposition applies to far more than statics. Every linear-amplitude effect in the seismic chain can be written as

\text{observed}_{ij} = \text{shot}(i)\cdot\text{rcv}(j)\cdot\text{offset}(h_{ij})\cdot\text{subsurface}

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

6. Practical notes

Cross-correlation window matters. Short windows miss long-wavelength trends; long windows average over multiple events. A window containing 2–3 reflection periods is standard.
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. 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.

**The one sentence to remember**

Residual statics are a few-ms wobble picked by cross-correlation and decomposed into shot + receiver + offset components; they look small on a trace, they halve the stack if unfixed, and the same decomposition template reappears in deconvolution and amplitude balancing.

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.