Residual velocity & higher-order moveout

After the semblance picks of Section 3.3 you have a velocity function $V(t\_0)$ that flattens the gather to first order. There is still work to do. Residual moveout remains because the earth is not the single hyperbola the first pass assumed: small velocity errors, mild anisotropy, dip and heterogeneity within the gather. This section covers the second pass that cleans them up.

1. Where residuals come from

Even when your first-pass velocity is good to ± 50 m/s, you are not done. Sources of remaining residual moveout:

• Limited aperture. With a short spread, a higher $V$ with a larger $t\_0$ fits almost as well as a lower $V$ with a smaller $t\_0$, so the pick is noisy; at long offsets the trade-off is between $V\_{\\mathrm{NMO}}$ and $\\eta$ instead.
• Coarse picking grid. Picks every 100-200 ms leave slow-drifting residuals.
• Anisotropy. Layered and anisotropic rocks bend the far offsets away from the hyperbola. Even with $\\eta$ picked, the fourth-order term rests on only the far offsets and carries its own residual.
• Dipping layers. For a single homogeneous layer the moveout stays hyperbolic even with dip, but the stacking velocity rises to $V/\\cos\\phi$ (Levin, 1971), so events of different dip in one gather cannot share one velocity.
• Heterogeneity. Lateral velocity variation inside the bin.

Each of these is small, a few to a few tens of milliseconds, but once the misalignment approaches a quarter of the dominant period (10 ms for a 25 Hz wavelet) the traces start to cancel in the stack. Fixing them recovers real signal-to-noise and sharpens the image.

2. Residual velocity picking

The second pass runs a residual semblance analysis: the same algorithm as Section 3.3, but over a narrow $\\Delta V$ range centred on zero. The question is not “what is $V$” but “what correction does my current $V$ need?” A typical range is ± 200 m/s. Pick the bright spot, add the correction, iterate.

In Figure 3.5 the first-pass velocities are a little slow and assume a hyperbola. You add a residual velocity $\\Delta V$, picked on the residual semblance in (d), then an anellipticity $\\eta$ for the far offsets, and read what is left in (e), trace by trace, in milliseconds.

The figure opens on the first pass, and both reflections curl up. Reflector 2 lands 42 ms early at 3000 m, and its stack peak is 46 % of its near-trace amplitude. Of those 42 ms, 26 follow the hyperbola fitted to its offsets out to the depth, the part $\\Delta V$ removes, and 16 are a hook beyond it that grows like $x^4$, the part $\\eta$ removes. Drag $\\Delta V$ onto the bright spot of reflector 2 in (d) with $\\eta$ still at zero and the gather gets much flatter, but the velocity has absorbed the hook: reflector 1 now runs late in the middle and early at the far end. Raise $\\eta$, pick $\\Delta V$ again, and the loop converges on one pair that leaves both reflectors flat to within 1 ms, with the stack peak of reflector 1 back to 100 % of its near-trace amplitude from 38 % after the first pass.

3. Higher-order moveout (HOMO)

Long offsets show a non-hyperbolic signature. In a layered or VTI earth it is described by the anellipticity $\\eta$ in the moveout of Alkhalifah and Tsvankin (1995), which Figure 3.5 uses:

The approximation holds at small offset, and it shows the sign: a positive $\\eta$ makes $A$ negative, so after a hyperbolic correction the far offsets arrive early and curl up. The hook shows only where the offset is well past the depth: cut the spread in Figure 3.5 to 1000 m and reflector 1 stays inside a quarter period for every $\\eta$ from 0 to 0.20, so a short spread can neither measure $\\eta$ nor much need it. Some processors fit $A$ directly as a general higher-order (HOMO) term. HOMO is the practical cousin of $\\eta$: it does not care which parameters are the physical source, it just fits the residual. That is useful when you cannot afford a full VTI analysis or when anisotropy is heterogeneous.

4. The iterative loop

Production velocity analysis runs two to four iterations of:

1. Pick $V\_0(t)$ on every CMP (Section 3.3).
2. Apply NMO, run residual statics (Section 2.4) on the corrected gathers.
3. Re-pick residual $\\Delta V(t)$ with shorter gates.
4. Add residual statics again (they shift slightly after the $\\Delta V$ correction).
5. Pick $\\eta$ or HOMO if needed.
6. Stop when residuals fall below a threshold, typically about 2 ms per pick.

Each pass makes the next one easier: velocity picks are more reliable because statics are smaller, and statics are cleaner because velocity is closer. Two iterations usually capture most of the improvement; three or four reach production quality.

5. When to stop

• Residual pick RMS is < 2 ms across a test of CMPs.
• Stacked section amplitude at a target horizon plateaus between iterations.
• Post-stack coherence does not improve further.

If residuals remain large after four passes, something is structurally wrong: overlooked anisotropy, missed statics, or lateral velocity variation too rapid for a gather-by-gather approach. The next step is Section 3.6: tomographic inversion that solves for the full velocity field simultaneously.

Where this goes next

Section 3.6 closes Part 3 with tomographic velocity inversion, a fundamentally different approach where instead of iterating picks per CMP you solve for the full $V(x, y, z)$ velocity volume simultaneously, using ray paths through the model and observed travel-time residuals to update the model.

References

• Taner, M. T., Koehler, F. (1969). Velocity spectra, digital computer derivation and applications of velocity functions. Geophysics, 34, 859.
• Levin, F. K. (1971). Apparent velocity from dipping interface reflections. Geophysics, 36, 510.
• Alkhalifah, T., Tsvankin, I. (1995). Velocity analysis for transversely isotropic media. Geophysics, 60, 1550.
• Yilmaz, Ö. (2001). Seismic Data Analysis (2 vols.). SEG.
• Dix, C. H. (1955). Seismic velocities from surface measurements. Geophysics, 20, 68.