geophysics · avo
Aki-Richards AVO Approximation
R(θ) ≈ ½Δρ/ρ̄ − 2(V̄s/Vp₁)²(Δρ/ρ̄)sin²θ + ½(ΔVp/V̄p)/cos²θ̄ − 4(V̄s/Vp₁)²(ΔVs/V̄s)sin²θ
click formula to derive ↑
Inputs
ft/s
ft/s
g/cc
ft/s
ft/s
g/cc
°
Description
Computes the Aki-Richards (1980) 4-term linearized approximation to the Zoeppritz P-P reflection coefficient, the general-purpose form most AVO/elastic inversion workflows implement directly. Distinct from Shuey's re-expression in that it works directly in Vp/Vs/density contrasts rather than a Poisson's-ratio-flavored intercept/gradient basis.
Variables
| Symbol | Unit | Description |
|---|---|---|
| R(θ) | — | AVO Reflectivity |
| Vp₁ | ft/s | P-wave velocity of the layer above the interface. |
| Vs₁ | ft/s | S-wave velocity of the layer above the interface. |
| ρ₁ | g/cc | Bulk density of the layer above the interface. |
| Vp₂ | ft/s | P-wave velocity of the layer below the interface. |
| Vs₂ | ft/s | S-wave velocity of the layer below the interface. |
| ρ₂ | g/cc | Bulk density of the layer below the interface. |
| θ₁ | ° | P-wave angle of incidence at the interface. |
Assumptions
- Contrasts in Vp, Vs, and ρ across the interface are small relative to their average values
- A single, sharp, welded interface, matching Zoeppritz's own assumptions
- Pre-critical incidence angle
Limitations
- Accuracy degrades above roughly 30-40° or for large elastic contrasts — cross-check against zoeppritz_pp when in doubt
- Like Zoeppritz, undefined beyond the P-wave critical angle (flagged, not silently wrong)
- A single R(θ) at one angle doesn't capture the full angle trend — see shuey_avo for the intercept/gradient decomposition used in crossplotting
Use Cases
- → Well-to-seismic AVO calibration: Forward-model expected amplitude at a specific offset/angle to calibrate against a real angle-stack trace.
- → Synthetic angle-gather generation: Evaluate at multiple θ to build a synthetic angle-dependent reflectivity trace for inversion QC.
- → Elastic/AVO inversion input: Supply the general-purpose reflectivity term most simultaneous/AVO inversion codes implement directly.
Related Calculations
Region Notes
Permian Basin
Wolfcamp/Bone Spring shale-carbonate contrasts often exceed the ~30-40° comfort zone at typical far-stack angles — spot-check against zoeppritz_pp before trusting far-angle Permian inversions.
Gulf of Mexico
Class III gas-sand contrasts in unconsolidated Miocene-Pliocene sediments are comparatively small in Vp/Vs terms, so Aki-Richards typically tracks the exact Zoeppritz solution well across the full angle range used in practice.
Global
Always confirm the sign/polarity convention (SEG normal vs. reversed) matches your seismic data before comparing computed R(θ) against a real angle-stack amplitude.
References
Primary source
Primary source
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