Zoeppritz Equation Calculator

Calculator Inputs

Vp1 (upper medium)

P-wave velocity, m/s.

Vs1 (upper medium)

S-wave velocity, m/s.

ρ1 (upper medium)

Density, consistent units.

Vp2 (lower medium)

P-wave velocity, m/s.

Vs2 (lower medium)

S-wave velocity, m/s.

ρ2 (lower medium)

Density, consistent units.

Incident P angle θ1

Degrees from the interface normal.

Compute Coefficients Reset

Example Data Table

Formula Used

Zoeppritz coefficients come from enforcing displacement and traction continuity at an interface between two isotropic elastic half-spaces.

• p = sin(θ1) / Vp1 (ray parameter, constant across the interface)
• sin(θ2) = p·Vp2, sin(φ1) = p·Vs1, sin(φ2) = p·Vs2 (Snell’s law)
• Matrix system: M · [R_P, R_S, T_P, T_S]^T = b, solved by Gaussian elimination

This implementation uses the common exact 4×4 matrix form for a P-wave incident from above, producing displacement amplitude coefficients for reflected and transmitted P and SV waves.

How to Use This Calculator

1. Enter P-wave velocity, S-wave velocity, and density for the upper and lower media.
2. Provide the incident P-wave angle θ1 in degrees from the normal.
3. Press Compute Coefficients to view angles and Zoeppritz outputs.
4. Use CSV or PDF to export a clean calculation record.
5. If you approach critical angles, interpret results cautiously.

Professional Article

1) Why the Zoeppritz solution is used

Seismic interfaces rarely behave like simple acoustic boundaries. The Zoeppritz equations describe how an incident P wave partitions energy into reflected P, reflected SV, transmitted P, and transmitted SV modes across an elastic contact. This mode conversion is the physical basis of amplitude‑versus‑offset (AVO) behavior, elastic impedance inversion, and lithology or fluid discrimination in reflection seismology.

2) Parameters that control amplitudes

Six rock properties drive the response: P velocity, S velocity, and density for each medium. Together they set acoustic impedance Z=ρVp and shear impedance μ=ρVs². Typical near‑surface sedimentary ranges are Vp 1500–4500 m/s, Vs 500–2500 m/s, and density 1.8–2.7. Stiffer carbonates often exceed Vp 5500 m/s.

3) Ray parameter and transmitted angles

The calculator starts with the ray parameter p=sin(θ1)/Vp1. Snell’s law then gives transmitted and converted angles using sin(angle)=p·velocity. These angles control traction and displacement continuity terms, which is why small angle changes can noticeably affect coefficients, especially where contrasts in Vs are large.

4) PP reflection trends with angle

At normal incidence, the PP reflection coefficient is approximated by R0=(Z2−Z1)/(Z2+Z1). With increasing θ1, elastic effects add curvature and gradient. A common quality check is to compare the exact PP result with a two‑term Shuey estimate for moderate angles and weak contrasts. Large deviations suggest strong contrasts or angle limits.

5) PS conversion as a diagnostic

Converted SV reflections often peak at mid angles because shear coupling depends on both Vs and density. In practice, stronger PS energy can indicate appreciable shear contrast, fractured zones, or changes in rigidity. The reflected SV coefficient provides a direct measure of conversion potential for multicomponent surveys.

6) Critical angles and evanescent behavior

When |p·V|>1, the corresponding angle becomes complex and the wave turns evanescent. This file uses a real‑angle clamp to keep outputs stable and clearly flags critical handling. For rigorous work near critical incidence, a complex‑valued formulation is recommended because phase and decay length become important.

7) Practical consistency checks

Use realistic velocities where Vs<Vp and densities that match lithology. Watch for impossible combinations (for example, very low Vs with extremely high Vp). Compare the sign of R0 with impedance contrast and confirm that angle outputs increase smoothly with θ1 under typical layering.

8) Interpretation workflow with exported reports

A repeatable workflow is to test multiple angles and property scenarios, then export CSV or PDF for documentation. Start with expected end‑member rocks, evaluate PP and PS sensitivity to fluids or porosity, and record the most diagnostic angle ranges. These steps support transparent AVO screening and rock‑physics calibration.

FAQs

1) What do the coefficients represent?

They are displacement amplitude ratios for each reflected or transmitted mode relative to the incident P wave, computed from exact elastic boundary conditions.

2) Which angle definition is used?

θ1 is measured from the interface normal. A larger θ1 means a more grazing incidence along the boundary.

3) Do density units matter?

Use any consistent density unit in both layers. Only ratios appear in the equations, but consistency is essential for meaningful impedances and coefficients.

4) Why can results change rapidly with angle?

Angles control both Snell’s law geometry and the traction terms. Near critical conditions, small angle shifts can cause large amplitude changes and phase effects.

5) What happens beyond a critical angle here?

The code clamps the sine term to keep real angles and adds a note. For accurate near‑critical modeling, use a complex Zoeppritz implementation.

6) How should I interpret negative PP reflection?

A negative value often indicates the lower medium has lower acoustic impedance than the upper medium, such as a gas‑charged sand beneath a wet sand.

7) Why include a Shuey approximation output?

It provides a fast reasonableness check for moderate angles and weak contrasts, helping you spot cases where full elastic behavior dominates.