Seismic Reflections II

Beyond the Flat-Layer Model

ESS 314 Geophysics · University of Washington

Week 3, Lecture 9 · April 22, 2026

Marine Denolle

By the end of this lecture…

[LO-9.1] Derive the dipping-layer travel-time equation; compute up-dip and down-dip apparent velocities; recover true velocity and dip
[LO-9.2] Identify multiple types; predict long-path multiple TWTT and NMO velocity; explain why stacking cannot remove it
[LO-9.3] State the diffraction equation; describe what migration accomplishes
[LO-9.4] Apply Shuey approximation $R(\theta) \approx R(0)+G\sin^2\theta$; classify AVO Classes I–IV
[LO-9.5] Evaluate DL denoising claims; identify two failure modes

Why the Flat-Layer Model Fails: Cascadia

alt text: Accretionary wedge schematic cross-section showing dipping thrust packages with labelled decorations for three non-idealities: orange label for dipping reflectors, red dashed ray path for a surface multiple, and green triangles with fan arrows for a fault-tip diffraction.

Each non-ideality requires a distinct correction: DMO for ①, SRME for ②, migration for ③.

Five Assumptions That Fail

In Lecture 8, the CMP stacking pipeline assumed:

  1. Reflectors are horizontal — no linear term in t2(x)t^2(x)
  2. Only primary reflections — every event is a single bounce
  3. Continuous interfaces — no point scatterers
  4. Noise-free wavefield — no ground roll or surface waves
  5. Only travel times matter — amplitude constant with offset

This lecture relaxes each assumption in turn:

Why it matters → What breaks → The math → How to fix it

① Dipping Reflectors: Geometry

For perpendicular depth hh, dip δ\delta, velocity V1V_1:

td(x)=1V1x2+4hxsin⁡δ+4h2(down-dip)t_d(x) = \frac{1}{V_1}\sqrt{x^2 + 4hx\sin\delta + 4h^2} \quad \text{(down-dip)}

tu(x)=1V1x2−4hxsin⁡δ+4h2(up-dip)t_u(x) = \frac{1}{V_1}\sqrt{x^2 - 4hx\sin\delta + 4h^2} \quad \text{(up-dip)}

Both have t(0)=t0=2h/V1t(0) = t_0 = 2h/V_1 — same zero-offset time.

Key: a linear term $\pm\,(2t_0\sin\delta/V_1)\cdot x$ appears in $t^2$ — the mathematical signature of dip

① Dipping Layer: Asymmetric Curves

alt text: Three-panel figure showing (A) dipping reflector geometry with source and up-dip/down-dip receivers; (B) t(x) curves where down-dip (orange) arrives later and up-dip (blue) earlier than flat (grey dashed); (C) t-squared x-squared plot with curved non-linear trends for dipping cases

Down-dip: MORE moveout (slower apparent VV). Up-dip: LESS moveout (faster apparent VV). All share the same t0t_0.

① CMP Reflection-Point Smear

alt text: Two-panel figure; left shows flat reflector with all CMP gather reflection points coinciding at one location; right shows dipping reflector with reflection points scattered up-dip as offset increases

For dipping reflectors, stacking without DMO correction blurs the subsurface image. DMO repositions reflection points before NMO stacking.

① NMO Velocity and Dip Recovery

Taylor expansion of td(x)t_d(x) at small xx gives:

VNMO,dip=V1cos⁡δ(>V1 for any δ>0)V_\mathrm{NMO,dip} = \frac{V_1}{\cos\delta} \quad (> V_1 \text{ for any } \delta > 0)

Recover V1V_1 and δ\delta from two-survey apparent velocities VdV_d, VuV_u:

V1=2VdVuVd+Vusin⁡δ=Vu−VdVu+VdV_1 = \frac{2V_d V_u}{V_d + V_u} \qquad \sin\delta = \frac{V_u - V_d}{V_u + V_d}

② Multiple Reflections

alt text: Three-panel figure showing (A) ray paths for primary P, long-path multiple M, peg-leg PL, interbed IB; (B) synthetic CMP gather with four hyperbolic events; (C) t-squared x-squared plot where primary and long-path multiple have the same slope annotated as same V_rms

② The Multiple Suppression Problem

Long-path surface multiple TWTT:

tmult2(x)=(2t0)2+x2Vrms2t_\mathrm{mult}^2(x) = (2t_0)^2 + \frac{x^2}{V_\mathrm{rms}^2}

Same NMO velocity as the primary → NMO correction flattens BOTH simultaneously. Stacking cannot suppress the multiple.

Suppression methods:

  • SRME (surface-related multiple elimination): autocorrelation-based prediction and subtraction
  • DL in τ\tau-pp domain: CNN trained to separate primaries from multiples by slope

③ Diffractions: Huygens Principle

Any sharp edge (fault tip, channel boundary, unconformity) acts as a secondary point source of spherical waves.

tdiff(x)=2V1(x−xs)2+zs2t_\mathrm{diff}(x) = \frac{2}{V_1}\sqrt{(x-x_s)^2 + z_s^2}

Key properties vs primary reflections:

  • Uniform amplitude across all offsets (isotropic emission)
  • Energy from a single point, not a planar interface
  • Migration collapses it to the point (xs,zs)(x_s, z_s)

③ Diffractions in the Seismic Section

alt text: Three-panel figure showing (A) depth model with flat reflector at 600 m and fault-tip scatterer at 1000 m; (B) travel-time curves showing flat primary and broader diffraction hyperbola; (C) synthetic seismic section with both events visible

Bowtie patterns (synclines) and diffraction tails (fault tips) are unmigrated artefacts. Migration (Lecture 10) collapses them.

④ Shot Gather Noise and f–k Filtering

Coherent noise in raw shot gathers:

  • Ground roll: V≈300V \approx 300 m/s, f≈5f \approx 5–20 Hz — high amplitude
  • Direct wave: V≈V1V \approx V_1, linear, easily muted
  • Air blast: V≈340V \approx 340 m/s

f–k filter: reject all ∣k∣>f/Vcutoff|k| > f / V_\mathrm{cutoff}, preserving V>VcutoffV > V_\mathrm{cutoff}

Velocity cone: $|k| \leq f / V_\mathrm{cutoff}$ in $f$–$k$ space defines the slope threshold separating slow noise from faster reflections.

④ f–k Ground Roll Suppression

alt text: Three-panel figure showing (A) raw shot gather with annotations for ground roll, direct wave and reflection; (B) f-k spectrum with velocity fan lines at 300, 600, 2000 m/s; (C) filtered gather with ground roll removed

Ground roll occupies the slow fan (high ∣k∣|k| per Hz). Rejecting it preserves reflections (V>600V > 600 m/s).

⑤ AVO: Zoeppritz + Shuey

At oblique incidence θi\theta_i, energy partitions into reflected P, S, transmitted P, S (Zoeppritz equations). Shuey (1985) linearisation:

R(θi)≈R(0)⏟intercept+G⏟gradientsin⁡2θiR(\theta_i) \approx \underbrace{R(0)}_{\text{intercept}} + \underbrace{G}_{\text{gradient}} \sin^2\theta_i

  • R(0)=(Z2−Z1)/(Z2+Z1)R(0) = (Z_2 - Z_1)/(Z_2 + Z_1) — normal-incidence reflectivity
  • GG = AVO gradient, sensitive to Δ(VP/VS)\Delta(V_P/V_S) — fluid content
  • Gas substitution lowers VPV_P, leaves VSV_S unchanged → large ∣G∣|G|

⑤ AVO Classes I–IV

alt text: Two-panel figure showing (A) R(theta) vs theta curves for Classes I through IV and IIp with different slopes and (B) R(0)-G crossplot with background trend, scatter clusters for each class, and quadrant annotations

Gas sands (Class III): negative R(0)R(0) and GG — amplitude brightens with offset. The R(0)R(0)–GG crossplot separates gas from brine-saturated sands.

DL Denoising: U-Net Architecture

alt text: Three-panel figure showing (A) U-Net schematic with encoder green blocks bottleneck orange decoder blue and red skip connections; (B) noisy synthetic CMP gather; (C) denoised gather with improved SNR

Skip connections preserve fine spatial detail. Supervised training requires paired noisy/clean data — unavailable for field data; self-supervised methods train on the noisy data alone.

DL Failure Modes — Critical Evaluation

  1. Domain shift: network trained on synthetic gathers fails on field data where noise is non-stationary and geologically correlated
  2. Physics inconsistency: denoised output may violate AVO, polarity, or reciprocity — creating spurious bright spots
  3. Interpretability gap: cannot determine whether amplitude anomaly is a true DHI or a network artifact

False bright spots from DL denoising have been documented in published case studies.

Worked Example: Dipping Layer

h=800h = 800 m, V1=2000V_1 = 2000 m/s, δ=10°\delta = 10°

Quantity Formula Result
t0t_0 2h/V12h/V_1 0.80 s
VNMOV_\mathrm{NMO} V1/cos⁡δV_1/\cos\delta 2031 m/s
td(2400 m)t_d(2400\,\text{m}) Exact dip eq. 1.553 s
tu(2400 m)t_u(2400\,\text{m}) Exact dip eq. 1.322 s

Check: sin⁡δ=(2421−1704)/(2421+1704)=0.174≈sin⁡10°\sin\delta = (2421-1704)/(2421+1704) = 0.174 \approx \sin10° ✓

Concept Check

  1. A flat-layer NMO correction is applied to a dipping reflector (δ=12°\delta = 12°). Is the corrected gather over- or under-corrected? Quantify the velocity error.

  2. A long-path multiple arrives at t0=1.4t_0 = 1.4 s with Vrms=2400V_\mathrm{rms} = 2400 m/s. What is the parent primary TWTT? Compute the reflector depth.

  3. What distinguishes a diffraction hyperbola from a primary reflection? Name two geological features in the Cascadia wedge that commonly produce diffractions.

  4. A sand–shale interface has R(0)=−0.08R(0) = -0.08 and G=−0.10G = -0.10. What AVO class is this? Is the sand likely gas- or brine-saturated?