Velocity picking on semblance gathers

You have a CMP gather. You need to find $V(t\_0)$. Trying every velocity at every time by hand is hopeless, but a computer does it cheaply and shows the result as a semblance panel, also called a velocity spectrum: a map over $(V, t\_0)$ in which a point’s brightness says how well that velocity flattens the gather at that time. A bright spot means “this velocity flattens an event here.” Pick the right bright spots, from the top down, and you have $V\_{\\mathrm{rms}}(t\_0)$.

1. The semblance formula

where $t\_i$ is the NMO travel time at trace $i$ for the trial velocity $V$ and a time $\\tau$ inside the gate $W(t\_0)$, and $s\_i(t\_i)$ is that trace’s amplitude there, interpolated between samples. The numerator is the squared stack; the denominator is $N$ times the energy along the same curve. $S$ lies between 0 and 1: perfectly aligned signal gives 1, and incoherent noise gives about $1/N$, so the floor rises as the fold falls. Semblance ignores amplitude, which is why a weak but coherent event, or a lucky patch of noise seen by a few traces, can shine as brightly as a strong reflection.

In production the sums run over a short time gate $W(t\_0)$ of about one dominant period (20 to 60 ms), long enough to average across the wavelet and short enough to keep neighbouring events apart. The gather is also stretch-muted first, exactly as the NMO will mute it: without the mute a shallow event recorded to far offsets is stretched out of coherence there, and its peak drifts off its true time and velocity.

2. The figure

Figure 3.3 has five plates. (a) is a split-spread CMP gather, 30 traces from 100 to 3000 m, with four primary reflections. (b) is its semblance panel, computed with the same stretch mute as the NMO and hatched where fewer than four traces survive it; your picks sit on it, joined by the $V\_{\\mathrm{rms}}(t\_0)$ function they define. (c) is the gather after NMO with that function, (d) its stack, and (e) your $V\_{\\mathrm{rms}}$ picks with the Dix interval velocities between them. Apart from the Model line under the figure, nothing in it shows the true velocities except the grey reference stack in (d), which is labelled as an oracle.

The figure opens with pick 3 at 2070 m/s, slower than the bright spot at 1.30 s. The event it should flatten is over-corrected: at 2200 m, the farthest trace the 30 % mute keeps, it lands 70 ms early, and semblance at the pick is only 0.14 against 0.99 at 2240 m/s. Drag the pick onto the peak and the residual falls to a few milliseconds. Above the first pick and below the last, the function holds that pick’s velocity constant. The Scene switch adds multiples, noise and low fold, and Auto-pick finds local semblance maxima at least 100 ms apart, above a third of the panel’s peak and clear of the $1/N$ floor, with $V\_{\\mathrm{rms}}$ allowed only to rise.

3. What you see on a real semblance

• Primary reflections produce bright isolated spots at the correct $(V, t\_0)$. These are what you want.
• Multiples produce bright spots at lower velocity than the primaries at similar times. They are still hyperbolic, but at the primary velocity they stay under-corrected. Pick the higher-velocity spot (to the right), not the lower one, and let Dix check you: a multiple’s pick gives an interval velocity above it that is implausibly slow, and often imaginary.
• Coherent noise (ground roll, refractions, the direct wave) produces low-velocity streaks, mostly at shallow times, where the stretch mute removes much of it. Do not pick it.
• Low-fold patches produce weak, smeared semblance, and a patch of noise seen by only a few traces can line up by chance, so picks there are less reliable.

4. Supergathers and sort order

A single CMP rarely carries every offset you want, and the offsets it does carry are often sparse. So for velocity analysis you build a supergather: combine the traces from a handful of neighboring CMPs into one gather so the full offset range is densely, and redundantly, sampled. Duplicate offsets are welcome: they stack down random noise on the semblance. Where the velocity must be exact (pre-drill pore-pressure prediction along a planned well path, say), you keep the supergather tight, only CMPs next to the well, so you do not blur a real lateral velocity change.

How you sort the traces inside that supergather changes what a velocity error looks like:

• CMP, absolute offset. Order traces by absolute offset $|x|$, from zero outward, regardless of whether each was shot ahead of or behind the midpoint. The display is monotonic and tidy: a velocity error reads as a one-sided bend, the event drooping below $t\_0$ at far offset (velocity too high) or rising above it (too low), and you can read off the offset where the bend begins.
• CMP, signed offset. Order from the farthest negative offset (behind the midpoint) on the left, through zero, to the farthest positive offset (ahead of it) on the right, as Figure 3.3 draws it; some displays run the other way. Now an on- or off-velocity event is symmetric: flat when the velocity is right, a full smile or frown when it is not.

On the signed display the smile/frown of Section 3.2 applies directly: a frown (far offsets still below $t\_0$) is an under-correction, the velocity is too high, lower it; a smile (far offsets pulled above $t\_0$) is an over-correction, the velocity is too low, raise it. The counter-intuitive part is the direction, and it is worth pinning down: because the NMO shift scales as $1/V^2$, a slower velocity applies a larger correction, so it is the slow velocity that over-shoots into a smile. This is the mirror image of the post-migration convention, where a smile means the velocity was too high (over-migration); the two are exactly opposite, see Section 5.2.

Read the two sides separately near faults. When a midpoint sits on a fault, one side of the spread can image the footwall and the other the hanging wall, two different velocity histories at the same $t\_0$. A change that moves only the near traces, only the far traces, or only one side of a signed gather is a clue that you are picking across a fault. Cross-check the stacked section so every pick lands on the correct side of the fault for that time.

5. Picking strategies

• Top-down, shallow to deep. Start near $t\_0 = 0$ and work down. Shallow picks anchor the function; deep picks respond to the underlying velocity trend.
• Aim high in ambiguity. Between two plausible velocities at the same time, pick the higher one. Multiples spend longer in the slow shallow section, so they appear at a lower stacking velocity than the primary.
• Check with NMO. Display the gather at each candidate velocity. The best pick is the one whose NMO visibly flattens the event.
• Sparse is fine. You don’t need a pick every 50 ms; 10 to 20 picks over a 3 s section is normal for land data.

6. What automated pickers do

Automated velocity pickers scan the semblance for local maxima subject to constraints: monotonic $V$ with depth (usually), minimum separation between picks, minimum semblance threshold. They save a lot of time on clean data. In complex geology they miss, picking multiples as primaries, over-smoothing through a rapid change in $V$, or drifting into noise. A processor reviews every automated pick. At the figure’s default gate and mute, Auto-pick recovers the four primaries in the Clean, Multiples and Noisy scenes; in Low fold it misses reflector 1, which the mute leaves with too few traces, and takes a patch of noise at 1.92 s.

7. From picks to a velocity model

A picked $V(t\_0)$ at one CMP is one 1D profile. 3D velocity analysis picks $V(t\_0)$ at a grid of CMPs and interpolates between them, producing a $V(x, y, t\_0)$ volume. That volume is what goes into NMO, migration, and inversion.

Where this goes next

Section 3.4 breaks the hyperbolic assumption. Real earth is often anisotropic, in shale layers especially, and the exact moveout equation picks up a non-hyperbolic term controlled by a parameter called $\\eta$. When you pick a single $V\_{\\mathrm{NMO}}$ for an anisotropic layer, the far offsets stay uncorrected (the hockey stick) and deep reflectors are mis-imaged.

References

• Taner, M. T., Koehler, F. (1969). Velocity spectra, digital computer derivation and applications of velocity functions. Geophysics, 34, 859.
• Yilmaz, Ö. (2001). Seismic Data Analysis (2 vols.). SEG.
• Dix, C. H. (1955). Seismic velocities from surface measurements. Geophysics, 20, 68.
• Sheriff, R. E., Geldart, L. P. (1995). Exploration Seismology (2nd ed.). Cambridge UP.