Seismic Reflections I

Flat-Layer Travel Time, NMO, and CMP Stacking

ESS 314 Geophysics · University of Washington

Week 3, Lecture 8 · April 15, 2026

Marine Denolle

By the end of this lecture…

[LO-8.1] Derive the normal-incidence reflection coefficient from boundary conditions; compute energy reflection coefficient
[LO-8.2] Derive the flat-layer hyperbola $t^2 = t_0^2 + x^2/V_1^2$ from the image-point construction
[LO-8.3] Apply NMO correction to a CMP gather; explain why stacking improves SNR by $\sqrt{N_\mathrm{fold}}$
[LO-8.4] State the RMS velocity definition; apply Dix equation to recover interval velocities
[LO-8.5] Interpret a semblance panel to pick stacking velocities

Before the Equations — Look at This

Five annotated sparker multi-channel seismic reflection panels (a–e) from the Cascadia accretionary wedge, offshore Washington, showing typical stratigraphic and deformational patterns: (a) inactive fold with flat postdeformational strata; (b) active fault-propagation fold with surface offset; (c) active fold with seafloor signature; (d) onlapping seismic facies at a sequence boundary; (e) stratal thinning and tilting onto an anticline indicating recent tectonic activity. Horizontal axis = distance along the survey line; vertical axis = two-way travel time (s), deeper downward.
Fig. 2 from Ledeczi et al. (2024), Seismica 2(4), doi:10.26443/seismica.v2i4.1158 · Licensed CC BY 4.0 · Reproduced unmodified
Discussion prompts: What is the horizontal axis? The vertical axis? Why do some layers appear bright and others dark? Why does the pattern change laterally?

Reading a Seismic Section

Orientation

  • Horizontal axis: distance along the survey line (km)
  • Vertical axis: two-way travel time (TWTT) in seconds — time increases downward; shallower reflectors plot at top
  • Each column of pixels = one seismic trace recorded at one location

Amplitude and polarity

  • Bright reflector → large impedance contrast between layers
  • Dark zone → gradational boundary, low contrast, or gradual velocity change
  • Polarity (light vs. dark wiggle) → sign of R=(Z2−Z1)/(Z2+Z1)R = (Z_2 - Z_1)/(Z_2 + Z_1):
    • positive RR if impedance increases with depth
    • negative RR if impedance decreases
Note: TWTT is not depth — converting it to depth requires knowing the P-wave velocity field (exactly what this lecture develops).

From a Single Point to the Cross-Section

Step 1 — Single interface, normal incidence:
A downgoing pulse hits a boundary → fraction RR of the energy reflects upward → arrives at the surface at two-way time t0=2h/Vt_0 = 2h/V.
One reflection = one amplitude value at one (x,t0)(x, t_0) pixel.

Step 2 — One vertical trace:
Stack all reflections from all interfaces at location xx → a time-series (one column of the image).

Step 3 — Sweep across the survey line:
Repeat for every source–receiver pair → mosaic all traces side by side → the 2D seismic section.

What controls brightness?

R=Z2−Z1Z2+Z1R = \frac{Z_2 - Z_1}{Z_2 + Z_1}

High-contrast boundary (e.g., sand–shale, water–rock) → bright.
Small contrast (clay–clay) → dark.

In the Cascadia image: the seafloor is the brightest reflector. Deeper, the décollement and sediment packages appear as bands of varying amplitude — each controlled by $R$ at that boundary.

What Makes a Bright Reflector? Acoustic Impedance

The brightness in the Cascadia image is controlled by acoustic impedance Z=ρ VPZ = \rho\, V_P — the resistance of a medium to wave propagation.

At normal incidence (boundary conditions: continuity of pressure + particle velocity):

R=Z2−Z1Z2+Z1T=2Z2Z1+Z2R = \frac{Z_2 - Z_1}{Z_2 + Z_1} \qquad T = \frac{2Z_2}{Z_1 + Z_2}

Energy fractions: R=R2\mathcal{R} = R^2, T=1−R2\mathcal{T} = 1 - R^2

Interface Z1Z_1 (MPa·s/m) Z2Z_2 RR
Sediment → limestone 3.5 6.0 +0.26
Sand → shale 4.8 4.2 −0.07
Crust → mantle (Moho) 13 20 +0.21
Most sedimentary contacts: $|R| = 0.01$–$0.15$ → only 1–2% of energy reflected. CMP stacking is essential to extract the signal.

From Impedance to Seismogram: The Convolutional Model

alt text: Four-panel figure: (A) coloured impedance depth profile with layer labels; (B) reflectivity spikes with signed R values; (C) 30 Hz Ricker source wavelet; (D) synthetic seismogram as wiggle trace with blue/red fill

$d(t) = w(t) * r(t)$  →  each reflector prints a copy of the wavelet, scaled by $R$ and shifted to its TWTT. Blue = positive $R$ (impedance increase); red = negative $R$.

Why Stack? The Signal-to-Noise Problem

Most sedimentary reflectors have ∣R∣=0.01|R| = 0.01–0.150.15: only 1–2% of seismic energy reflects at any one boundary — the rest propagates deeper or scatters as noise.

A single source–receiver trace is almost never enough to detect a reflector above ambient noise.

Solution: record the same reflection from NfoldN_\mathrm{fold} different source–receiver offsets. Signal adds coherently (∝N\propto N); noise adds incoherently (∝N\propto \sqrt{N}):

SNRstack=Nfold×SNRsingle\mathrm{SNR}_\mathrm{stack} = \sqrt{N_\mathrm{fold}} \times \mathrm{SNR}_\mathrm{single}

48-fold → 7×
96-fold → 10×
240-fold → 15×
"Aligning" traces from different offsets to the same reflection time requires understanding the geometry of offset travel time — the reflection hyperbola.

The Reflection Hyperbola

alt text: Two-panel figure. Left: acquisition geometry with source, four receivers and two-way ray paths to a flat reflector. Right: travel-time hyperbola with t0 horizontal dashed line and orange asymptote.

Image-point construction: reflect source through reflector.

t2(x)=t02+x2V12t^2(x) = t_0^2 + \frac{x^2}{V_1^2}

t0=2hV1t_0 = \dfrac{2h}{V_1}

t2t^2–x2x^2 is a straight line → slope gives V12V_1^2, intercept gives hh

NMO Correction

Normal moveout is the delay at offset xx relative to t0t_0:

ΔtNMO(x)=t02+x2VNMO2−t0≈x22 VNMO2 t0\Delta t_\mathrm{NMO}(x) = \sqrt{t_0^2 + \frac{x^2}{V_\mathrm{NMO}^2}} - t_0 \approx \frac{x^2}{2\,V_\mathrm{NMO}^2\, t_0}

NMO correction shifts each trace up by ΔtNMO(x)\Delta t_\mathrm{NMO}(x), flattening the hyperbola to t0t_0 — this is the alignment step that makes stacking work.

NMO stretch at large offsets distorts the wavelet. Traces beyond the mute zone (x/h≳1x/h \gtrsim 1–1.51.5) are discarded before stacking.

Acquisition: CMP Gather

alt text: Two-panel figure. Panel A shows a cross-section with five source-receiver pairs of different offsets all reflecting from the same point on a flat reflector. Panel B shows the resulting CMP gather of wiggle traces and the reflection hyperbola.

Every coloured pair shares the same common midpoint (CMP) → same reflection point on a flat reflector. NMO-correcting and summing these traces produces one stacked trace. Fold = spread / (2 × shot spacing). Modern marine: 120–240-fold.

RMS Velocity

For NN flat, horizontal layers with velocities ViV_i and two-way times Δti\Delta t_i:

Vrms,n2=∑i=1nVi2 Δti∑i=1nΔtiV_\mathrm{rms,n}^2 = \frac{\displaystyle\sum_{i=1}^{n} V_i^2\,\Delta t_i}{\displaystyle\sum_{i=1}^{n} \Delta t_i}

  • VrmsV_\mathrm{rms} replaces V1V_1 in the hyperbola for multi-layer media
  • Vrms≥V_\mathrm{rms} \geq any interval velocity above the interface (RMS > arithmetic mean for increasing-velocity profiles)
  • The NMO velocity measured from semblance = VrmsV_\mathrm{rms}

The Dix Equation

Recover interval velocity between two adjacent reflectors:

Vn=Vrms,n2 t0,n−Vrms,n−12 t0,n−1t0,n−t0,n−1V_n = \sqrt{\dfrac{V_\mathrm{rms,n}^2\,t_{0,n} - V_\mathrm{rms,n-1}^2\,t_{0,n-1}}{t_{0,n} - t_{0,n-1}}}

Key assumptions: flat, horizontal, isotropic layers — violations are addressed in Lecture 9.

Precision matters: a small error inVrmsV_\mathrm{rms} propagates strongly to VnV_n for thin layers (when t0,n−t0,n−1t_{0,n} - t_{0,n-1} is small).

Velocity Analysis: Semblance Panel

Semblance S(V,τ)S(V,\tau): coherence of the NMO-corrected CMP gather at trial velocity VV and time τ\tau.

S(V,τ)=[∑jdj(τ+Δtj)]2N ∑j[dj(τ+Δtj)]2∈[0,1]S(V, \tau) = \frac{\left[\sum_j d_j(\tau + \Delta t_j)\right]^2}{N\,\sum_j \left[d_j(\tau + \Delta t_j)\right]^2} \in [0, 1]

Reading the semblance panel:

  • Pick maxima tracing a velocity function V(t0)V(t_0) from shallow to deep
  • Velocity should increase with depth for a normal gradient profile
  • Multiples appear at lower velocity than primaries at the same t0t_0

CMP Stack: The Full Pipeline

After semblance velocity picking, NMO correction, and mute:

s(t)=1Nfold∑j=1NfolddjNMO(t)s(t) = \frac{1}{N_\mathrm{fold}} \sum_{j=1}^{N_\mathrm{fold}} d_j^\mathrm{NMO}(t)

Each CMP produces one stacked trace. Assembled side by side → the 2D stacked section (the cross-section we opened the lecture with).

Post-stack: deconvolution → migration (Lecture 10) → interpretation

Worked Example: Two-Layer NMO + Dix

V1=1800V_1 = 1800 m/s, h1=900h_1 = 900 m; V2=2600V_2 = 2600 m/s, h2=700h_2 = 700 m

Reflector t0t_0 (s) VrmsV_\mathrm{rms} (m/s)
1 1.000 1800
2 1.538 2048

Dix recovery: V2=(20482×1.538−18002×1.000)/0.538=2600V_2 = \sqrt{(2048^2 \times 1.538 - 1800^2 \times 1.000)/0.538} = 2600 m/s ✓

SOTA: DL Velocity Analysis

Traditional velocity picking: manual, time-consuming, subjective for millions of CMPs in 3D surveys.

CNN semblance pickers: input = semblance image; output = V(t0)V(t_0) curve. Match expert picks within 1–2% RMS.

Bayesian uncertainty: output p(VNMO∣t0)p(V_\mathrm{NMO} \mid t_0) — widest uncertainty at large TWTTs and near-zero fold zones.

Physics-constrained inversion: embed Dix equation as a hard constraint → interval velocities guaranteed consistent with observed VrmsV_\mathrm{rms}.

Concept Check

  1. Two layers: V1=2000V_1 = 2000 m/s at t0=0.80t_0 = 0.80 s; Vrms,2=2300V_\mathrm{rms,2} = 2300 m/s at t0=1.40t_0 = 1.40 s. Compute V2V_2 with Dix. Compute the depth to reflector 2.

  2. NMO is applied to a CMP gather using a velocity that is 5% too low. Are the hyperbolas over- or under-corrected? What does the gather look like after correction?

  3. A semblance panel shows a peak at (V=2000 m/s,  t0=1.0 s)(V = 2000 \text{ m/s},\; t_0 = 1.0 \text{ s}) and another at (V=1500 m/s,  t0=2.0 s)(V = 1500 \text{ m/s},\; t_0 = 2.0 \text{ s}). What is the second event most likely to be?

  4. A 48-fold stack has SNRsingle=0.5\mathrm{SNR}_\mathrm{single} = 0.5. What is SNRstack\mathrm{SNR}_\mathrm{stack}? Is the reflector visible?