Residual velocity & higher-order moveout

Part 3 — Velocity Analysis & NMO

Learning objectives

Describe why residuals remain after first-pass NMO even for well-behaved data
Apply a ΔV residual correction on a first-pass NMO gather
Add a higher-order (4th-order) residual term and explain when it is needed
Sketch the velocity–residual–velocity iteration that converges velocity analysis to production quality

After §3.3 semblance picks you have a V(t₀) function that flattens the gather to first order. There is still work to do. Small residuals remain because the truth is more complex than the simple hyperbolic assumption: slight velocity errors, subtle dips, mild anisotropy, heterogeneity within the gather window. This section covers the second pass that cleans those up.

1. Where residuals come from

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

Velocity ambiguity at far offset. The NMO ambiguity (V high + small t₀ ↔ V low + big t₀) means your pick is noisy.
Coarse picking grid. Picks every 100–200 ms leave slow-drifting residuals.
Residual anisotropy. Even with η picked, the fourth-order term is a small-data-support estimate and has its own residual.
Dipping layers. The hyperbolic equation is exact only for flat layers; dip adds a cos(φ) term.
Heterogeneity. Lateral velocity variation inside the bin.

Each of these is small (a few ms), but together they halve the stack amplitude. Fixing them recovers real SNR and sharpens the image.

2. Residual velocity picking

The second pass runs a residual semblance analysis. Same algorithm as §3.3, but over a narrow ΔV range centered at zero: the question is not “what is V” but “what is the correction to my current V?” A typical ΔV range is ± 200 m/s. Pick the bright spot, add the correction, iterate.

After ΔV is applied, events that were still curving up or down become flatter. The widget below demonstrates the correction.

True residual is ΔV = +80 m/s and a 4th-order HOMO coefficient of 0.08. With both sliders at zero, the gather shows small residual moveout on both events — the first-pass NMO under-corrected the velocity and left a little non-hyperbolic tail. Set ΔV to +80 and HOMO to 0.08 — the events flatten and the status reads TUNED.

3. Higher-order moveout (HOMO)

Beyond the hyperbolic term and the VTI 4th-order term, real gathers sometimes show a residual non-hyperbolic signature that is not captured by η. Sources: heterogeneous overburden, long-offset ray bending, residual statics that couple into velocity. The fix is an additional fourth-order residual term:

t^{2}(x) = t_{0}^{2} + \frac{x^{2}}{V^{2}} + A \cdot x^{4}

where A absorbs whatever 4th-order signature is left. HOMO is the practical cousin of η — it does not care which parameters are the physical source; it just fits the residual. Useful when you cannot afford a full VTI analysis or when anisotropy is heterogeneous.

4. The iterative loop

Production velocity analysis runs two to three iterations of:

Pick V₀(t) on every CMP (§3.3).
Apply NMO, run residual statics (§2.4) on the corrected gathers.
Re-pick residual ΔV(t) with shorter gates.
Add residual statics again (which have shifted slightly after ΔV correction).
Pick HOMO if needed.
Stop when residuals fall below a threshold — typically ~ 2 ms per pick.

Each pass makes the next one easier: velocity picks are more reliable because statics are smaller; statics are cleaner because velocity is closer. Two iterations gets you 80 % of the way; three to four gets 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 3–4 passes, something is structurally wrong — overlooked anisotropy, missed statics, lateral velocity variation too rapid for a gather-by-gather approach. The next step is §3.6: tomographic inversion that solves for the full velocity field simultaneously.

**The one sentence to remember**

Residual velocity picking cleans up the hyperbolic leftover after first-pass NMO; HOMO absorbs the 4th-order leftover after that; the residuals ↔ statics iteration runs until stacks stop improving — typically 2–4 passes.

Where this goes next

§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.
Yilmaz, Ö. (2001). Seismic Data Analysis (2 vols.). SEG.
Dix, C. H. (1955). Seismic velocities from surface measurements. Geophysics, 20, 68.