Refraction & tomographic statics

Part 2 — Pre-Processing Foundations

Learning objectives

Explain what a static is and why the weathered near-surface layer causes them
Describe the refraction-statics workflow: first-break picking → weathered model → per-station delay → trace shift
Distinguish short-wavelength and long-wavelength statics and which tool addresses each
Recognize the signature of uncorrected statics on a stacked section (smeared or split reflectors)

A static is a fixed time shift applied to a trace to compensate for travel-time distortions that depend on surface location rather than subsurface structure. They come from the weathered near-surface layer — that top tens of meters of sand, soil, and broken rock sitting on top of the consolidated bedrock. The weathered layer has low velocity (~500–1500 m/s) and variable thickness, so the time a ray spends crossing it varies from shot to shot and receiver to receiver.

1. Why the weathered layer wreaks havoc

Imagine a reflection at 1.5 s arriving at two neighbouring receivers. If under receiver A the weathered layer is 15 m thick and under receiver B it is 25 m thick, trace A’s reflection arrives ~25 ms earlier than trace B’s, purely because of the near-surface. That 25 ms offset is a static. When you stack CMP gathers, statics that vary within the gather smear reflections. Statics that vary between bins smear the whole section.

The widget below shows a shot gather above an irregular weathered layer. Slide the statics standard deviation up — the first-break line develops wobble and the deeper reflection bumps up and down. The right-hand panel shows what happens after the statics have been picked and subtracted: smooth first break, flat reflection.

2. The refraction-statics workflow

Pick first breaks on every trace. The first-arriving energy is the refracted head wave through the consolidated refractor (v_r ≈ 2500–4500 m/s).
Fit a slope t(x) = x / v_r + intercept to the first-break travel times.
Residual per trace = observed first-break minus fitted straight line. This residual is the sum of shot static + receiver static + any refractor dip.
Decompose residuals into per-shot and per-receiver contributions (surface-consistent decomposition, see §2.4).
Apply corrections: shift every trace in a shot record by its shot static, and then by its receiver static. The reflection hyperbolas become smooth.

3. Two pieces of math to know

Weathered-layer traveltime: a ray travelling through a weathered layer of thickness h and velocity v_w spends time 2h / v_w (two-way). For h = 20 m, v_w = 800 m/s, that’s 50 ms. Wobble in h directly becomes wobble in your seismic.

Intercept-time refraction: for a flat refractor at depth z under a uniform weathered layer, the first-break time is

t(x) = \frac{x}{v_r} + \frac{2z\cos\theta_c}{v_w}

where θ_c = asin(v_w/v_r) is the critical angle. The intercept 2z cosθ_c / v_w is the static you need to remove.

4. Short-wavelength vs long-wavelength statics

When you decompose per-trace residuals into per-shot and per-receiver components, you typically see TWO scales of variation:

Short-wavelength statics: trace-to-trace variations at the same spatial scale as the receiver spacing. Caused by the very shallowest (loose sand, surface conditions). These are cleanly addressed by refraction statics.
Long-wavelength statics: slow variations over hundreds of meters. Caused by deep topography of the weathered-bedrock interface. Refraction statics do NOT recover these well — they leak into velocity and structural anomalies. Long-wavelength statics need tomographic statics or iterative refinement with velocity analysis.

5. Tomographic statics

Refraction statics assume a single refractor. When the near-surface is more complex — multiple low-velocity layers, karst, talus, buried river channels — you need tomographic statics: invert the first-break times for a full 2D or 3D near-surface velocity model, then compute theoretical travel times through it. The inversion is linear-algebra-heavy (§0.7 and §0.9), regularized for smoothness.

Tomographic statics are standard on land data in complex terrain (foothills, desert edges, glacial plains). They require hundreds of thousands of first-break picks — usually automated with hand-edits.

6. What uncorrected statics look like

Pre-stack: first breaks wiggle trace-to-trace; reflection hyperbolae are lumpy.
Stacked: amplitude loss where statics were random, or split reflections (same reflector appears twice) where statics jumped at a boundary. Long-wavelength statics look like structural undulations — fake highs and lows that are really just weathered-layer topography.
Downstream velocity picks: wrong. Residual statics contaminate velocity analysis, which contaminates imaging.

7. Order of operations

Statics are typically applied before deconvolution and noise attenuation, and they are iteratively refined: first-pass refraction statics → preliminary velocity → residual statics (§2.4) → re-pick velocity → re-pick residual statics. Two or three passes is normal; land data with rough topography may need more.

**The one sentence to remember**

Statics are fixed per-station time shifts from the weathered near-surface; refraction statics estimate them from first breaks, tomographic statics from a full near-surface model, and without either, reflectors smear.

Where this goes next

Section §2.4 covers residual statics — the final refinement after refraction statics and preliminary velocity picks. It uses cross-correlation across a gather to tease out the remaining small shifts, and introduces the surface-consistent decomposition that underpins statics, deconvolution, and amplitude work alike.

References

Yilmaz, Ö. (2001). Seismic Data Analysis (2 vols.). SEG.
Sheriff, R. E., Geldart, L. P. (1995). Exploration Seismology (2nd ed.). Cambridge UP.
Claerbout, J. F. (1976). Fundamentals of Geophysical Data Processing. McGraw-Hill.
Tarantola, A. (1984). Inversion of seismic reflection data in the acoustic approximation. Geophysics, 49, 1259.