Velocity model building for migration

Every method in Sections 5.4-5.7 is only as good as the velocity model you feed it. This section is the loop that makes that model good enough. The strategy is simple to state and hard to run: migrate with a trial velocity, measure residual moveout on common-image gathers, compute a velocity update, re-migrate, repeat until the CIGs are flat. Call it migration velocity analysis, reflection tomography, or just velocity model building: the inner loop is the same.

1. The residual-moveout formula

For a flat reflector at depth $z$ in a constant velocity $V\_{\\mathrm{true}}$, with zero-offset time $t\_0 = 2z/V\_{\\mathrm{true}}$, the arrival time at half-offset $h$ on a time-domain common-image gather (equivalently, the NMO-corrected gather) built with a trial velocity $V\_{\\mathrm{mig}}$ is

If $V\_{\\mathrm{mig}} = V\_{\\mathrm{true}}$, $t\_{\\mathrm{corr}}(h) = t\_0$ and the event is flat across offset. Otherwise it curves: downward at far offset if $V\_{\\mathrm{mig}} > V\_{\\mathrm{true}}$ (over-migrated), upward if $V\_{\\mathrm{mig}} < V\_{\\mathrm{true}}$ (under-migrated). The sign of the curvature gives the sign of the error and its size gives the size of the correction. In a depth gather the same error also moves the reflector: with $\\gamma = V\_{\\mathrm{mig}}/V\_{\\mathrm{true}}$ the imaged depth is $z(h) = \\sqrt{\\gamma^2 z^2 + (\\gamma^2 - 1)h^2}$ (Al-Yahya, 1989), so even the zero-offset depth is wrong by the factor $\\gamma$.

2. Flattening the gathers

The figure below depth-migrates one surface position of a three-layer earth with a trial velocity model you set, and picks the three reflections across offset the way a tomography workflow would. It opens with layer 1 8 % slow and layer 2 4 % fast. Read which way each event curls, then drag $V\_1$ toward flat, or press Apply update in (d) and let one tomography step move the whole model; press it again until the gathers pass the 2 m target.

At the opening setting reflector 1 curls up by 210 m at 1500 m of half-offset and is imaged at 923 m instead of 1000 m: the residual moveout and the depth error arrive together, both set by $\\gamma$. One tomography step from those picks moves layer 1 from 1840 to 2022 m/s, past the true 2000 m/s, because the update is linearised in $h^2$ and evaluated in the wrong model; three updates take the RMS residual moveout from 56 m to 0.9 m, under the 2 m target. Two warnings come with the loop. With only layer 1 slow, reflector 3 still curls up by 23 m at 1500 m although the two layers above it are right, so the overburden has to be fixed first. And flat is not proof: with offsets only to 500 m, a layer 3 that is 7 % fast leaves the gathers inside the target while reflector 3 sits 70 m too deep.

3. The tomographic update

Expanding the residual-moveout formula to first order about $V\_{\\mathrm{true}}$ gives

where $\\Delta V = V\_{\\mathrm{mig}} - V\_{\\mathrm{true}}$. The measured residual moveout $\\Delta t(h)$ therefore maps linearly to a velocity correction $\\Delta V$. In practice the derivative is evaluated in the current model ($V\_{\\mathrm{mig}}$ in place of $V\_{\\mathrm{true}}$), which is why the update is only approximately right and the loop has to iterate. In depth the same linearisation of $z(h)^2 = \\gamma^2 z^2 + (\\gamma^2 - 1)h^2$ turns the fitted curvature $a$ of $z(h) \\approx z\_0 + a h^2$ into $\\gamma^2 \\approx 1 + 2 z\_0 a$; that is the step the figure takes, reflector by reflector, before the Dix equation turns the corrected RMS velocities back into layer velocities. Production reflection tomography sets up the linear system $\\mathbf{G}\\mathbf{m} = \\mathbf{d}$ where $\\mathbf{d}$ is the measured residual moveout at many (CMP, depth, offset) points, $\\mathbf{m}$ is the correction to a gridded velocity model, and $\\mathbf{G}$ is the Fréchet derivative of travel time with respect to velocity. Solving this system by least squares with regularisation gives the next velocity model. Iterate 2-5 times until residual moveout stops shrinking.

4. The initial model: where do you start?

• Stacking velocities (PSTM or post-stack) + Dix. For time migration or gentle-dip depth migration, start with stacking-velocity-based RMS velocities converted to interval velocities via Dix's equation (Section 3.1). This is the fastest starting point but smooths over lateral variation.
• Log-constrained models. If wells exist, tie log-derived sonic velocities to seismic at the well and interpolate laterally. Best initial model when well control is available.
• Prior seismic. Previous surveys over the same area, if reconciled for geometry, provide a good starting model for newer acquisitions.
• First-break tomography. Use direct-arrival travel times (the first arrival on each trace) to invert the near-surface velocity. Essential for land data with statics problems.

5. Reflection tomography workflow

1. Migrate with current V.
2. Extract CIGs at regular surface positions (say every 100 m).
3. Pick residual moveout on the major events of each CIG, every 50 m or so in depth, often automated via semblance-like stacking across offset.
4. Build the tomographic matrix $\\mathbf{G}$ by tracing rays through the current V model to each CIG event.
5. Solve $\\mathbf{G}\\mathbf{m} = \\mathbf{d}$ for the velocity update $\\mathbf{m}$ by damped least squares (regularised with smoothness + well-tie constraints).
6. Update V, re-migrate, extract CIGs again. Compare residual moveout: if decreasing, continue; if stuck, review the picks.
7. Convergence: when the RMS residual moveout falls below a threshold (a few metres of depth on depth gathers, about 2 ms of two-way time) and the image stops changing visibly between iterations.

6. Beyond tomography, FWI

Full-waveform inversion (FWI) bypasses the pick-and-invert tomography workflow by minimising the full data misfit directly. Instead of just travel times, FWI matches the observed amplitudes and phases at every sample through an adjoint-state gradient method. It can recover velocity detail beyond ray resolution (thin layers, sharp velocity contrasts) and is the state of the art for sub-salt imaging. The cost is 10-100× tomography; the requirement is a starting model good enough that FWI does not get stuck in a local minimum. In practice, tomography builds the starting model, FWI refines it, and depth migration produces the final image.

7. How much model quality does each migration need?

• Post-stack time migration: needs stacking velocity accurate to ~5 % laterally.
• PSTM: needs RMS velocity accurate to ~2 %, laterally smooth.
• PSDM Kirchhoff: needs $v(x,z)$ accurate to ~1 % in depth; ray paths must be approximately right.
• Beam PSDM: similar to Kirchhoff but more robust to small model errors.
• One-way WE: needs a smooth $v(x,z)$; sharp contrasts handled by split-step/PSPI variants.
• RTM: needs $v(x,z)$ accurate to <1 % near reflectors; sensitive to sharp contrasts in the right places but can handle strong gradients that one-way cannot.
• FWI: its own iterative update; the starting model must predict travel times to within half a period at the lowest usable frequency (often a few per cent), and FWI then refines it to well under 1 %.

Part 5 closes here

You now have the full imaging pipeline: stack (Section 5.1), post-stack time migration (Section 5.2), PSTM (Section 5.3), Kirchhoff PSDM (Section 5.4), beams (Section 5.5), one-way WE (Section 5.6), RTM (Section 5.7), artefact QC (Section 5.8), and the velocity-building loop (Section 5.9) that ties it all together. Part 6 goes deeper on the velocity-building side: full-waveform inversion (FWI) replaces the pick-based tomography above with a full data-misfit optimisation, using the adjoint method to update every pixel of $v(x,z)$ against every sample of the recorded wavefield.

References

• Yilmaz, Ö. (2001). Seismic Data Analysis (2 vols.). SEG.
• Etgen, J., Gray, S. H., Zhang, Y. (2009). An overview of depth imaging in exploration geophysics. Geophysics, 74, WCA5.
• Virieux, J., Operto, S. (2009). An overview of full-waveform inversion in exploration geophysics. Geophysics, 74, WCC1.
• Tarantola, A. (1984). Inversion of seismic reflection data in the acoustic approximation. Geophysics, 49, 1259.
• Pratt, R. G. (1999). Seismic waveform inversion in the frequency domain, Part 1. Geophysics, 64, 888.