geophysics · avo
Zoeppritz PP Reflection Coefficient
M·x = b (4×4 elastic boundary-condition system); Rpp = x₁
click formula to derive ↑
Inputs
ft/s
ft/s
g/cc
ft/s
ft/s
g/cc
°
Description
Solves the exact Zoeppritz (1919) boundary-condition equations for the P-P reflection coefficient as a function of incidence angle, without the small-contrast linearization used by Aki-Richards or Shuey. Serves as the definitive benchmark for QC'ing those faster approximations at any specific well or interface.
Variables
| Symbol | Unit | Description |
|---|---|---|
| Rpp(θ) | — | PP Reflection Coefficient |
| Vp₁ | ft/s | P-wave velocity of the layer above the interface, from sonic log or check shot over the upper interval. |
| Vs₁ | ft/s | S-wave velocity of the layer above the interface, from a dipole/multipole sonic log over the upper interval. |
| ρ₁ | g/cc | Bulk density of the layer above the interface. |
| Vp₂ | ft/s | P-wave velocity of the layer below the interface, from sonic log or check shot over the lower interval. |
| Vs₂ | ft/s | S-wave velocity of the layer below the interface, from a dipole/multipole sonic log over the lower interval. |
| ρ₂ | g/cc | Bulk density of the layer below the interface. |
| θ₁ | ° | P-wave angle of incidence measured from the interface normal, e.g. the angle corresponding to a target offset/angle-stack. |
Assumptions
- A single, sharp, welded (no-slip) interface between two homogeneous, isotropic elastic half-spaces
- All six elastic/density parameters and the incidence angle are independently known, not fitted
- Pre-critical incidence — both the transmitted P wave and transmitted/reflected S waves remain real (propagating), not evanescent
Limitations
- Complex-valued (post-critical) reflection beyond the P-wave or S-wave critical angle is not computed — flagged explicitly rather than silently wrong
- Assumes one sharp interface only — does not model thin-bed tuning or interbed multiples
- Highly sensitive to Vs, often the least-certain input (dipole sonic quality, or Vp/Vs-derived estimates) — propagate that uncertainty into any AVO interpretation drawn from this result
Use Cases
- → Approximation QC benchmark: Compute the exact Rpp(θ) at a specific well/interface to check how far Aki-Richards or Shuey diverge before trusting them for far-angle inversion.
- → Synthetic angle-gather modeling: Forward-model a full angle range for well-tie and AVO-attribute calibration ahead of seismic inversion.
- → Critical-angle / wide-angle screening: Identify the incidence angle beyond which post-critical effects (total internal reflection, head waves) become relevant for survey design.
Related Calculations
Region Notes
Permian Basin
Wolfcamp shale-on-carbonate contacts often have Vp2>Vp1 by 20-40%, putting the critical angle around 45-55° — well within angle-stack ranges used in some Permian AVO workflows, so post-critical flagging matters here in practice, not just in theory.
Gulf of Mexico
Class III unconsolidated gas sands (Vp2<Vp1) never reach a P-wave critical angle, so the full 0-70° range in this calculator remains in the real, pre-critical regime — a useful QC contrast against Permian-style hard-below-soft interfaces.
Global
Always cross-check a well's exact Rpp(θ) against Aki-Richards/Shuey at the same angle before committing to a linearized inversion — large disagreement signals either a strong elastic contrast or an angle beyond the linearization's comfort zone.
References
Primary source
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