Velocity model building for migration

Every method in §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 "vel-model building" — the inner loop is the same.

1. The residual-moveout formula

For a flat reflector at depth $z$ in constant-velocity $V_{true}$, the NMO-corrected arrival time on a CIG at half-offset $h$, given trial migration velocity $V_{mig}$, is

If $V_{mig} = V_{true}$, $t_{corr}(h) = t_0$ and the event is flat across offset. Otherwise it curves: upward at far offset if $V_{mig} > V_{true}$ (over-migrated in velocity terms), downward if $V_{mig} < V_{true}$. The curvature sign tells you the sign of the error; the curvature magnitude tells you how much to correct.

2. The widget

A CIG with a single reflector at 1000 m in $V_{true} = 2000\ \text{m/s}$, $t_0 = 1.0\ \text{s}$. The dashed teal line is the target: flat at $t_0$. The yellow/blue wavelet paint is the CIG event at its current trial velocity. Drag the slider to flatten it. The info strip reports direction and magnitude of the remaining error, the RMS residual moveout across offset, and the tomographic $\Delta V/V$ suggestion — what a single iteration of a first-order update would pick.

3. The tomographic update

For a Taylor expansion of the residual moveout formula around $V_{true}$:

where $\Delta V = V_{mig} - V_{true}$. The measured residual moveout $\Delta t(h)$ therefore maps linearly to a velocity correction $\Delta V$. 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. 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 (§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

• Migrate with current V.
• Extract CIGs at regular positions (say every 100 m laterally, every 50 m in depth).
• Pick residual moveout on major events in each CIG — often automated via semblance-like stacking across offset.
• Build the tomographic matrix $\mathbf{G}$ by tracing rays through the current V model to each CIG event.
• Solve $\mathbf{G}\mathbf{m} = \mathbf{d}$ for the velocity update $\mathbf{m}$ by damped least squares (regularised with smoothness + well-tie constraints).
• Update V, re-migrate, extract CIGs again. Compare residual moveout: if decreasing, continue; if stuck, review the picks.
• Convergence: when RMS residual moveout is below a threshold (typically ~2 ms for 3D depth migration) 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 details 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; starts from a model good to ~10 % and refines to <0.5 %.

Part 5 closes here

You now have the full imaging pipeline: stack (§5.1), post-stack time migration (§5.2), PSTM (§5.3), Kirchhoff PSDM (§5.4), beams (§5.5), one-way WE (§5.6), RTM (§5.7), artefact QC (§5.8), and the velocity-building loop (§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.