NMO correction & stretch

Part 3 — Velocity Analysis & NMO

Learning objectives

State the hyperbolic NMO equation and explain each term
Predict how over- and under-correction bend a gather (frown vs under-flattened)
Apply a stretch mute and explain what it throws away and why
Recognize why one V_NMO can only flatten one event at a time when V varies with depth

Normal move-out correction is the single most important operation between loading the data and imaging it. It takes a CMP gather whose reflection events curve down with offset (hyperbolae) and flattens them so that stacking adds coherently. Get the velocity wrong and the stack gets weaker; get it very wrong and the stack carries artifacts shaped like the uncorrected hyperbolae.

1. The NMO equation

A reflection from a flat horizon at depth under a layered earth has travel time

t^{2}(x) = t_{0}^{2} + \frac{x^{2}}{V_{\text{NMO}}^{2}}

where t₀ is the zero-offset time, x is the source–receiver offset, and V_NMO is the (approximately) RMS velocity down to that reflector. NMO correction moves each sample from (t, x) to (t₀, x) by solving for t₀.

When V_NMO = V_true, the reflection flattens. When it is off, it does not.

The true velocities are 2000 m/s (shallow event) and 2700 m/s (deep event). At V_NMO = 2000 the shallow event is perfectly flat but the deep event is still curving. At V_NMO = 2700 the deep event is flat but the shallow event is over-corrected and bends upward (“frowns”). One V_NMO flattens one event; production processing picks a velocity that varies with t₀.

2. The three regimes

V too low. The NMO correction moves samples up further than it should. Events bend the wrong way — the tips curl upward, making a frown. Stacking smears.
V correct. Event flattens. Stacking adds coherently.
V too high. The NMO correction moves samples up too little. Events still curve downward, just less than before. Stacking partially coherent.

3. NMO stretch — the hidden cost

The NMO operator is not a rigid shift. It is a time-variable operator: a reflection at (t₀, x) maps from input time t(x) to output time t₀, and the wavelet gets stretched in the process. The stretch factor is

\text{stretch} = \frac{t(x) - t_0}{t_0}

It is worst at far offset and shallow times. A 30 % stretch means a 30-Hz wavelet becomes 23 Hz after NMO; a 50 % stretch pushes it to 20 Hz. You are losing frequency content exactly where you most need it.

Standard practice: apply a stretch mute — zero out samples where stretch exceeds a threshold (typically 30 %). Slide the mute slider in the widget to 30 % and watch the near-offset shallow samples vanish on the right panel. The stretched wavelets are gone; so is any signal that lived in that region. Engineers accept the trade.

4. Picking V_NMO for a whole gather

Since one V_NMO can only flatten one event, production processing picks a piecewise-linear V_NMO(t_0) function — low at the top of the section, higher as time increases — that flattens every reflector. The picking itself is done via semblance analysis, which is the subject of §3.3.

5. What a processor watches for

Smile / frown direction reveals whether V is high or low. Smile (event curves up at near offsets) means V is too high; frown (event bends upward at far offsets) means V is too low.
Amplitude loss at the target horizon after stacking says residual NMO error remains — re-pick velocity.
Stretch-muted zone width on the near-offset shallow section is a direct reflection of mute strategy — too aggressive and the shallow image is missing at small offset.

6. The assumption set, made explicit

The NMO hyperbola is exact only when:

Layers are flat and horizontal,
Earth is isotropic,
Offset is short relative to reflector depth (x < 2z, roughly),
Velocity is constant within each layer.

Violate any of those and the NMO equation develops higher-order terms. §3.4 covers the anisotropy case (VTI layers), §3.5 covers residual-NMO and higher-order moveout.

**The one sentence to remember**

NMO flattens reflections by solving t² = t₀² + x²/V² in reverse; the right V flattens one event, all events need a piecewise V(t₀), and the correction stretches the wavelet at near-offset shallow times enough that a mute is standard.

Where this goes next

Section §3.3 answers how you actually pick V(t₀): the semblance plot — a 2D map of (velocity, time) where bright bands mark the velocities that flatten the gather best. It is the processor’s primary velocity-picking tool.

References

Dix, C. H. (1955). Seismic velocities from surface measurements. Geophysics, 20, 68.
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.