Zoeppritz Equations Calculator: Exact P-Wave Reflection Coefficient vs Shuey

When a P-wave hits an interface between two rocks at an angle, it does not simply bounce. It splits four ways: reflected P, reflected S, transmitted P, and transmitted S, with the energy split among them governed by continuity of displacement and stress across the boundary. Karl Zoeppritz worked out the exact equations in 1919, and more than a century later they remain the ground truth that every AVO approximation is measured against.

Most day-to-day AVO work runs on approximations, and for good reason: Shuey’s two-term form turns messy algebra into an intercept and gradient you can map. But approximations have a service ceiling. This calculator solves the exact P-P reflection coefficient from the full equations (in the standard Aki and Richards formulation) and plots it against the Shuey two-term curve from 0 to 60 degrees, so you can see precisely where the shortcut starts to bend away from the truth for your specific rock pair.

What the inputs mean

Each medium needs three numbers: Vp, Vs, and density. From the incidence angle, Snell’s law fixes one ray parameter that all four scattered waves share, which sets the four angles in the geometry. The coefficient arithmetic then reduces, after a century of algebraic housekeeping by Aki and Richards, to a handful of intermediate terms combined into a single ratio. At vertical incidence everything collapses to the familiar impedance contrast (Z2 – Z1)/(Z2 + Z1), which is a built-in sanity check this calculator passes by construction.

Worked example

• Shale over brine sand: Vp 3000 to 3500 m/s, Vs 1500 to 1900 m/s, density 2.30 to 2.40 g/cc
• Normal incidence R0 = 0.098, a solid positive reflector
• Exact Rpp at 20 degrees: 0.076; at 30 degrees: 0.056
• Shuey two-term at 30 degrees gives 0.047, already about 0.009 low, and the gap keeps widening toward the 59 degree critical angle

When you actually need exact

Below about 25 to 30 degrees of incidence, for modest property contrasts, Shuey and Zoeppritz are practically indistinguishable and the approximation wins on interpretability. The exact solution earns its keep in three situations: far-offset or wide-angle work where 40-plus degree angles carry real information, strong contrasts like a soft seabed or a carbonate cap where the small-contrast assumption behind the approximations breaks down, and anywhere near the critical angle, where transmitted P grazes the interface, amplitudes swell, and the coefficient goes complex. This solver flags the post-critical zone rather than pretending a real number exists there, which is exactly the honesty you want from a reference calculation.

A modeling habit worth keeping: run the exact curve once for every key interface in a prospect before trusting intercept-gradient attributes derived from approximations. If the two curves separate inside your usable angle range, the AVO attributes inherit that error silently, and it is far cheaper to learn that from a calculator than from a mis-ranked prospect.

Related Calculators and Reference

The two-term fit this compares against lives in the AVO intercept-gradient calculator, the normal-incidence limit is the reflection coefficient calculator, and converting offsets to the angles this needs is the job of the offset to angle converter. The SEG Wiki page on the Zoeppritz equations gives the full matrix derivation.