Reflection & transmission coefficients

Physics prerequisites for acquisition

Learning objectives

State the angle-dependent reflection coefficient R(θ) for a P-wave at a flat interface
Use the 2-term Shuey approximation R(θ) ≈ A + B sin²θ to estimate near- and far-offset response
Apply Snell's law to predict the transmitted ray angle
Recognise the critical angle beyond which a transmitted P-wave becomes evanescent

When a P-wave hits an interface between two media, part of the energy reflects back and part transmits into the lower medium. How much of each depends on the angle of incidence and on the contrast in velocity and density across the interface.

The Zoeppritz equations, summarised

The exact Zoeppritz equations describe how an incident P-wave partitions into reflected P, reflected S (SV), transmitted P, and transmitted SV at a flat interface. In practice we almost always use approximations. The most popular for survey design and AVO analysis is the 2-term Shuey:

R(θ) \approx A + B\sin^2\theta

where A is the intercept (normal-incidence reflection coefficient) and B is the gradient (how the reflection changes with angle). Both A and B are simple combinations of ΔV/V and Δρ/ρ across the interface.

Snell, critical angles, and evanescence

The transmitted P-ray bends per Snell’s law, sinθ₂/V₂ = sinθ₁/V₁. If V₂ > V₁ there is a critical angle beyond which no P-wave can transmit into the lower medium — all energy reflects back. In the widget, push θ past that angle and the transmitted ray disappears.

For acquisition: critical reflections and refractions are how refraction statics work. The direct wave, refracted head-wave, and reflections form the early-arrival package on every shot gather. Knowing where the critical angle sits lets you predict whether a target even has reflected energy at the offsets you’re proposing.

References

Aki, K., Richards, P. G. (2002). Quantitative Seismology (2nd ed.). University Science Books.
Shuey, R. T. (1985). A simplification of the Zoeppritz equations. Geophysics, 50(4), 609–614.
Sheriff, R. E., Geldart, L. P. (1995). Exploration Seismology (2nd ed.). Cambridge University Press.
Yilmaz, Ö. (2001). Seismic Data Analysis: Processing, Inversion, and Interpretation of Seismic Data (2 vols.). SEG Investigations in Geophysics 10.