Refraction & tomographic statics

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. Statics 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, with $v\_w = 800$ m/s and $v\_r = 3500$ m/s, trace A’s reflection arrives about 10 ms earlier than trace B’s ($10\\,\\text{m}\\times(1/800 - 1/3500)\\,\\text{s/m} \\approx 9.6$ ms), purely because of the near surface. That 10 ms offset is a static. When you stack CMP gathers, statics that vary within the gather smear reflections. Statics that vary from bin to bin shift the stacked traces against each other: jitter at short wavelengths, false structure at long ones.

In the figure below you roughen the base of a weathered layer under 48 receivers and watch one flat reflection jitter in (d). Then you estimate the statics, from the first breaks of two refraction shots or from the reflection itself, and apply the estimate in (e). Statics only remove the jitter: the reflection in (e) is flat because NMO has been applied as well, and turning NMO off shows the smooth hyperbola underneath.

At the figure’s opening setting, 5.5 m RMS of thickness variation under the receivers gives 5.3 ms RMS of receiver static. That scatters the reflection by 5.9 ms and cuts the stack to 50 % of a single trace, and the figure opens with nothing corrected. Choose Refraction: statics estimated from the first breaks bring the scatter down to 0.6 ms and the stack back to 99 %, and the estimate misses the model’s statics by only 0.5 ms RMS. In the fourth exercise, with a 2000 m wavelength and 8 m of variation, residual statics leave the scatter at 7.5 ms. They compare traces with each other, so they cannot see the smooth part of the statics: 5.7 ms RMS of it stays, which across one gather passes for a velocity error and along a stacked line would be false structure. Nor can they see the 31 ms bulk delay, which NMO turns into another 1.5 ms RMS of moveout.

2. The refraction-statics workflow

1. Pick first breaks on every trace. The first-arriving energy is the refracted head wave through the consolidated refractor ($v\_r \\approx 2500$ to $4500$ m/s).
2. Fit a line $t(x) = x/v\_r + t\_i$ to the first-break travel times, where $t\_i$ is the intercept time.
3. Residual per trace = observed first-break minus fitted straight line. This residual is the sum of the shot and receiver delay-time anomalies (plus any refractor dip); they become statics only after conversion (item 5).
4. Decompose the residuals into per-shot and per-receiver delay times. With reversed shots, one at each end of the line, the plus-minus method does this directly, and it is what the figure uses: the minus time $t\_A - t\_B$ has slope $2/v\_r$, and the plus time $t\_A + t\_B - t\_{AB}$ is twice the receiver’s delay time. On production data the same split is a surface-consistent least-squares problem (Section 2.4).
5. Convert and apply: turn each delay time into a thickness and a static (below), then shift every trace by its shot static plus its receiver static. The reflection hyperbolas become smooth.

3. Two pieces of math to know

Weathered-layer static: a ray crossing a weathered layer of thickness $h$ and velocity $v\_w$ near one station arrives late, compared with the same path through the refractor, by $t\_s = h\\left(\\frac{1}{v\_w} - \\frac{1}{v\_r}\\right)$. For $h = 20$ m, $v\_w = 800$ m/s and $v\_r = 3500$ m/s that is $25 - 5.7 \\approx 19$ ms per station; a trace carries a shot static plus a receiver static. 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

where $\\theta\_c = \\arcsin(v\_w/v\_r)$ is the critical angle. The intercept $t\_i = 2z\\cos\\theta\_c/v\_w$ is the sum of a shot delay time and a receiver delay time, each $z\\cos\\theta\_c/v\_w$. Each delay time gives the thickness under its station, $z = t\_d v\_w/\\cos\\theta\_c$, and that thickness gives the static $t\_s = z\\left(\\frac{1}{v\_w} - \\frac{1}{v\_r}\\right)$.

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: station-to-station variations at the scale of the receiver spacing, caused by the very shallowest material (loose sand, soil, surface conditions). Refraction statics remove most of them; residual statics (Section 2.4) clean up what remains.
• Long-wavelength statics: slow variations over distances longer than the spread, caused by changes in weathering thickness and velocity along the line. Residual statics cannot see them, because they shift a whole gather at once, and uncorrected they appear as false structure. Refraction statics recover them when the near surface behaves as a layer over a refractor; where velocities vary laterally or the layering is complex, tomographic statics do better.

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 (Section 0.7 and Section 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. Uncorrected 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 (Section 2.4) → re-pick velocity → re-pick residual statics. Two or three passes is normal; land data with rough topography may need more.

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.
• Cox, M. (1999). Static Corrections for Seismic Reflection Surveys. SEG.
• Palmer, D. (1981). An introduction to the generalized reciprocal method of seismic refraction interpretation. Geophysics, 46, 1508-1518.
• Hagedoorn, J. G. (1959). The plus-minus method of interpreting seismic refraction sections. Geophysical Prospecting, 7, 158-182.