Stacking, RMS, and interval velocities

Part 3 — Velocity Analysis & NMO

Learning objectives

Distinguish interval, RMS, stacking, and average velocities — and remember which is which
Compute V_RMS from a stack of interval velocities and times
Apply Dix’s equation to recover V_int from two V_RMS picks
Recognize why NMO velocity ≈ V_stack ≈ V_RMS only at short offset and flat layers

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_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_avg(t₀). Total distance divided by total time — Δz/Δt summed to depth of a reflector. Useful for depth conversion.
RMS velocity V_RMS(t₀). Root-mean-square of interval velocities weighted by layer time:

V_{\text{RMS}}^{2}(t_n) = \frac{\sum_{k=1}^{n} V_{\text{int}}^{2}(k)\,\Delta t_k}{\sum_{k=1}^{n}\Delta t_k}

Stacking velocity V_stack(t₀). The velocity you put into the NMO equation to flatten reflections before stacking. For flat horizontal layers and small offsets, V_stack ≈ V_RMS.

2. Dix’s equation — recovering the interval velocities

You pick V_RMS values from seismic velocity analysis — that is what §3.3 does. From two RMS picks at times t₁ < t₂, Dix’s formula extracts the interval velocity in the layer between:

V_{\text{int}}^{2}(t_1, t_2) = \frac{V_{\text{RMS}}^{2}(t_2)\,t_2 - V_{\text{RMS}}^{2}(t_1)\,t_1}{t_2 - t_1}

This is the round trip. Interval velocities build up RMS; Dix inverts to recover interval. The widget below demonstrates it numerically — the column “V_int (Dix recovered)” matches the input V_int to the integer.

3. What the two curves on the profile mean

The velocity-vs-time plot on the right shows:

Teal stairstep — interval velocity. One horizontal segment per layer, jumping to the next layer’s velocity at each boundary.
Amber curve — RMS velocity accumulated through the layer stack. It always lies between the minimum and maximum interval velocity, and it changes smoothly because it is an RMS average.

Slide any layer velocity and watch the amber curve recalculate. A big change at shallow depth moves the entire RMS curve below it; a big change deep moves only the deep portion.

4. Why the RMS caveat exists

The NMO equation t²(x) = t₀² + x²/V_stack² is only exact for small offset-to-depth ratios in flat horizontal layers. For larger offsets or for dipping reflectors, V_stack diverges from V_RMS. Part 3 will tackle each source of error:

§3.2 — the NMO hyperbola itself.
§3.3 — how to pick V_stack by maximizing semblance.
§3.4 — when anisotropy (VTI) makes the hyperbola non-hyperbolic.
§3.5 — residual-velocity and higher-order-moveout corrections.
§3.6 — tomographic inversion for the full interval velocity field.

5. Quick numerical sanity check

With two layers of V₁ = 2000 m/s, Δt₁ = 0.5 s and V₂ = 3000 m/s, Δt₂ = 0.5 s:

V_{\text{RMS}}^{2}(1.0) = \frac{2000^{2} \cdot 0.5 + 3000^{2} \cdot 0.5}{1.0} = \frac{2{,}000{,}000 + 4{,}500{,}000}{1.0} = 6{,}500{,}000 \Rightarrow V_{\text{RMS}}(1.0) \approx 2550\text{ m/s}

And the Dix inversion at (0.5, 1.0): V_int² = (2550²·1.0 − 2000²·0.5) / 0.5 = (6,502,500 − 2,000,000) / 0.5 = 9,005,000 ⇒ V_int ≈ 3000 m/s. The round trip is exact.

**The one sentence to remember**

Interval velocity is the rock; RMS velocity is the time-weighted root-mean-square up to a given reflector; stacking velocity is what you put in the NMO equation; Dix’s formula converts RMS back to interval.

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_stack and watch reflection hyperbolae flatten (or over- / 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.