NMO correction & stretch

Normal move-out correction is one of the most consequential operations between loading the data and imaging it, second only to getting the geometry right. Geometry QC is the real foundation: if the trace coordinates, offsets, and fold are wrong, every later step (NMO included) silently inherits the error (garbage in, garbage out). With the geometry correct, NMO is the operation that puts velocity into the stack. 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 beneath a single homogeneous layer has exactly, and beneath a layered earth approximately (for offsets up to about the reflector depth), the travel time

where $t\_0$ is the zero-offset time, $x$ is the source-receiver offset, and $V\_{\\mathrm{NMO}}$ is approximately the RMS velocity $V\_{\\mathrm{rms}}$ down to that reflector. NMO correction moves each sample from $(t, x)$ to $(t\_0, x)$ by solving for $t\_0$.

When $V\_{\\mathrm{NMO}}$ equals the true $V\_{\\mathrm{rms}}$ the reflection flattens; when it is off, it does not. In Figure 3.2 you set the NMO velocity as a multiple of the true $V\_{\\mathrm{rms}}$, or switch it to one constant, and choose the stretch mute; then you read how far the far live trace lands from flat, how much of the stack survives and what the stretch does to the wavelet.

The figure opens with the velocity 5 % slow: at 0.95 × $V\_{\\mathrm{rms}}$ reflector 2 is over-corrected. Its far live trace, at 1300 m, lands 27 ms early, the event curls up, and the stack peak falls to 41 % of the near-trace amplitude. At 1.00 it is flat to within 1 ms and the stack keeps 100 %. Now switch the applied velocity to one constant, the 2050 m/s of reflector 2, and turn the stretch mute off. Reflector 2 is flat, but at 3000 m reflector 1 lands 275 ms late (the velocity is too high for it, so it is under-corrected and droops, a frown) and reflector 4 lands 201 ms early (the velocity is too low for it, so it is over-corrected and curls up, a smile). One $V\_{\\mathrm{NMO}}$ flattens one event; production processing picks a velocity that varies with $t\_0$.

2. The three regimes

• $V\_{\\mathrm{NMO}}$ too low. The NMO correction moves samples up further than it should (over-correction). The far-offset tips curl upward, making a smile, and the stack smears. To fix it, raise the velocity.
• $V\_{\\mathrm{NMO}}$ correct. The event flattens, and the stack adds coherently.
• $V\_{\\mathrm{NMO}}$ too high. The NMO correction moves samples up too little (under-correction). The far offsets still curve downward, just less than before, making a frown, and the stack is only partly coherent. To fix it, lower the velocity.

3. NMO stretch, the hidden cost

The NMO operator is not a rigid shift. It is a time-variable operator: a reflection at $(t\_0, x)$ maps from input time $t(x)$ to output time $t\_0$, and the wavelet gets stretched in the process. The stretch is

It is worst at far offset and shallow times. For a constant velocity the dominant frequency falls to $f/(1 + \\Delta t\_{\\mathrm{NMO}}/t\_0)$: a 30 % stretch turns a 30 Hz wavelet into 23 Hz, and a 50 % stretch into 20 Hz. Where the velocity rises with depth the loss is larger, because the deeper half of the wavelet is pulled up by less than the shallower half. In Figure 3.2, at the right velocity with a 30 % mute, the farthest live trace of reflector 2 (1500 m, stretched 29 %) peaks at 17 Hz against 25 Hz at 100 m, below the 19 Hz the constant-velocity formula gives. You are losing frequency content exactly where you most need it.

Standard practice is a stretch mute: zero the samples whose stretch exceeds a threshold, typically 30 %. Tighten the mute in Figure 3.2 and the hatch in (b) grows over the far-offset, shallow corner of the gather; at the right velocity a 30 % mute keeps 7 of 30 traces at reflector 1 (out to 700 m) but all 30 at reflector 4. The stretched wavelets are gone, and so is any signal that lived in that corner. Processors accept the trade.

4. Picking the NMO velocity for a whole gather

Since one $V\_{\\mathrm{NMO}}$ can only flatten one event, production processing picks a piecewise-linear function $V\_{\\mathrm{NMO}}(t\_0)$, low at the top of the section and higher as time increases, that flattens every reflector. Pick it with the far offsets live: in Figure 3.2 a velocity 5 % low looks flat, within 3 ms, when a 5 % mute leaves only the offsets out to 500 m. The picking itself is done with semblance analysis, the subject of Section 3.3.

5. What a processor watches for

• Smile / frown direction reveals whether $V\_{\\mathrm{NMO}}$ is too high or too low. A smile (far offsets bend upward, over-corrected) means $V\_{\\mathrm{NMO}}$ is too low, so raise it; a frown (far offsets bend downward, under-corrected) means it is too high, so lower it. Watch the convention flip in migration (Section 5.2): there a smile means the velocity is too high, because over-migration, not over-correction, is what swings energy up.
• Amplitude loss at the target horizon after stacking says residual NMO error remains, re-pick velocity.
• Stretch-muted zone width at far offsets and shallow times is a direct reflection of mute strategy. Too aggressive a mute leaves the shallow section with too few offsets to stack.

6. The assumption set, made explicit

The NMO hyperbola is exact for a single homogeneous layer over a flat reflector. Beneath a real earth it is a good approximation when:

• Layers are flat and horizontal,
• Earth is isotropic,
• Offset is short relative to reflector depth (offset-to-depth ratio $x/z \\lesssim 1$) when the overburden is layered,
• Velocity is constant within each layer.

Figure 3.2 builds each reflection as an exact hyperbola so that it can isolate NMO and stretch; its offsets reach six times the depth of reflector 1, far past where a layered earth stays hyperbolic. Violate any of those conditions in real data and the NMO equation develops higher-order terms. Section 3.4 covers the anisotropy case (VTI layers), Section 3.5 covers residual-NMO and higher-order moveout.

Where this goes next

Section 3.3 answers how you actually pick $V(t\_0)$: 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.