Stacking, RMS, and interval velocities

A single number called “velocity” in seismic can mean four different things. They are related, they have different uses, and they will cost you a morning of confusion if you mix them up. Part 3 opens with the vocabulary.

1. Four velocities you must know

• Interval velocity $V\_{\\mathrm{int}}(k)$. The actual propagation velocity within layer $k$. What the rock physics is. What FWI tries to recover. Varies layer by layer.
• Average velocity $V\_{\\mathrm{avg}}(t\_0)$. Total depth divided by total one-way time, $V\_{\\mathrm{avg}}(t\_n) = \\sum\_k V\_k\\,\\Delta t\_k / \\sum\_k \\Delta t\_k = 2z\_n/t\_n$ with two-way times. Useful for depth conversion, and never larger than $V\_{\\mathrm{rms}}$.
• RMS velocity $V\_{\\mathrm{rms}}(t\_0)$. Root-mean-square of interval velocities weighted by layer time:

• Stacking velocity $V\_{\\mathrm{stack}}(t\_0)$. The velocity you put into the NMO equation to flatten reflections before stacking. For flat layers and small offsets, $V\_{\\mathrm{stack}} \\approx V\_{\\mathrm{rms}}$.

2. Dix’s equation: recovering the interval velocities

You pick $V\_{\\mathrm{rms}}$ values from seismic velocity analysis (Section 3.3 shows how). From two picks at times $t\_1 < t\_2$, Dix’s formula extracts the interval velocity in the layer between:

Interval velocities build up $V\_{\\mathrm{rms}}$, and Dix inverts the build-up. On exact picks the round trip is exact, but a processor never has exact picks: the stacking velocities are fitted to a gather over a finite spread, and every pick carries some error. In the figure below you set four layers, and the stacking velocities are fitted to a ray-traced gather the way a processor would pick them. Start with exact picks on a 2000 m spread, then shorten the spread, add picking error and thin a layer, and watch how far Dix’s answer moves in (d).

3. What the figure shows

At the opening setting the picks carry no error at all, yet Dix puts layer 2 at 2340 m/s against the true 2300 m/s, 1.7 % too fast. Over a 2000 m spread the best-fitting hyperbola for reflector 2 has $V\_{\\mathrm{stack}}$ = 2070 m/s, 1.0 % above its $V\_{\\mathrm{rms}}$ of 2049 m/s, and Dix magnifies that difference. Shorten the spread to 500 m and Dix returns 2303 m/s. Plate (b), velocity against two-way time, shows:

• Stairstep: interval velocity. One vertical segment per layer, stepping sideways to the next layer’s velocity at each interface.
• Solid curve: $V\_{\\mathrm{rms}}$, accumulated through the layers above. It always lies between the smallest and largest interval velocity above that time, and it changes continuously, bending at each interface.
• Dashed curve: $V\_{\\mathrm{avg}}$, which never lies to the right of $V\_{\\mathrm{rms}}$; the two are equal only while the velocity is constant.
• Dots and dotted stairs: the picked stacking velocities and the interval velocities Dix recovers from them.

Raise layer 1 and every $V\_{\\mathrm{rms}}$ below it moves, although only one interval changed: a big change at shallow depth moves the whole curve below it, and a change deep moves only the deep part.

Dix divides by the time thickness of the layer, so it amplifies picking errors. For independent errors on the two picks that bound a layer, the relative error in $V\_{\\mathrm{int}}$ is $(t\_2 V\_2^2 \\varepsilon\_2 - t\_1 V\_1^2 \\varepsilon\_1)/((t\_2 - t\_1) V\_{\\mathrm{int}}^2)$, with $V\_1, V\_2$ the picked $V\_{\\mathrm{rms}}$ and $\\varepsilon\_1, \\varepsilon\_2$ their relative errors. In the figure’s thin layer, 0.10 s thick at 0.85 s, errors of 1 % on the picks move the interval velocity by about 4 %. Thin layers and deep layers are where Dix velocities need the most care.

4. Why the RMS caveat exists

The NMO equation $t^2(x) = t\_0^2 + x^2/V\_{\\mathrm{stack}}^2$ is, in a layered earth, a short-spread approximation (Taner and Koehler, 1969): it holds while offset is small compared with reflector depth. At larger offsets the moveout becomes non-hyperbolic, and a dipping reflector raises the stacking velocity to about $V/\\cos\\phi$, so $V\_{\\mathrm{stack}}$ departs from $V\_{\\mathrm{rms}}$. Lengthen the spread in the figure to 4000 m: reflector 2 then arrives 59 ms before its $V\_{\\mathrm{rms}}$ hyperbola, the fitted $V\_{\\mathrm{stack}}$ is 3.1 % above $V\_{\\mathrm{rms}}$, and Dix puts layer 2 4.8 % too fast. Part 3 will tackle each source of error:

• Section 3.2, the NMO hyperbola itself.
• Section 3.3, how to pick $V\_{\\mathrm{stack}}$ by maximizing semblance.
• Section 3.4, when anisotropy (VTI) makes the moveout non-hyperbolic.
• Section 3.5, residual-velocity and higher-order-moveout corrections.
• Section 3.6, tomographic inversion for the full interval velocity field.

5. Quick numerical sanity check

With two layers of $V\_1 = 2000$ m/s, $\\Delta t\_1 = 0.5$ s and $V\_2 = 3000$ m/s, $\\Delta t\_2 = 0.5$ s:

And the Dix inversion at $(0.5, 1.0)$: $V\_{\\mathrm{int}}^2 = (2550^2 \\cdot 1.0 - 2000^2 \\cdot 0.5)/0.5 = 9{,}005{,}000 \\Rightarrow V\_{\\mathrm{int}} \\approx 3001$ m/s. With the unrounded $V\_{\\mathrm{rms}} = 2549.5$ m/s the round trip returns exactly 3000 m/s: rounding the pick by half a metre per second already moved the answer by one.

Where this goes next

Section 3.2 presents the NMO equation itself and the stretch mute that comes with it. The widget there lets you set $V\_{\\mathrm{stack}}$ and watch reflection hyperbolae flatten, or over- and under-correct, on a CMP gather.

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.
• Yilmaz, Ö. (2001). Seismic Data Analysis (2 vols.). SEG.
• Sheriff, R. E., Geldart, L. P. (1995). Exploration Seismology (2nd ed.). Cambridge UP.