Reflection & transmission coefficients

Physics prerequisites for acquisition

Learning objectives

State how the P-wave reflection coefficient $R_{\mathrm{PP}}(\theta)$ at a flat interface depends on the incidence angle and on the contrasts across it
Use the two-term Shuey approximation $R_{\mathrm{PP}}(\theta) \approx A + B\sin^2\theta$ , and say where it fails
Apply Snell’s law, $\sin\theta_1/V_{\mathrm{P1}} = \sin\theta_2/V_{\mathrm{P2}}$ , to find the transmitted angle and the critical angle $\theta_{\mathrm c}$
Relate a spread’s far offset and the target depth to the largest angle recorded, $\theta_{\max} \approx \arctan(x_{\max}/2z)$ , and find the critical offset

When a P wave meets a flat interface, part of its energy reflects, part transmits and part converts to S. The reflected P amplitude, the coefficient $R\_{\\mathrm{PP}}$ , depends on the contrasts in P velocity, S velocity and density across the interface, and on the incidence angle $\\theta$ . In acquisition the angle is what you control: with straight rays, a receiver at offset $x$ sees a flat target at depth $z$ at $\\theta \\approx \\arctan(x/2z)$ , so the far offset of the spread decides the largest angle you record.

The exact coefficient and its two-term approximation

The Zoeppritz equations give the exact plane-wave coefficients: four boundary conditions (continuity of displacement and traction) fix the reflected and transmitted P and S amplitudes of an incident P wave. For small contrasts and moderate angles they reduce to Shuey’s two-term form, written here in the Aki and Richards terms:

R\_{\\mathrm{PP}}(\\theta) \\approx A + B\\sin^2\\theta

The intercept $A = \\tfrac12(\\Delta V\_{\\mathrm P}/V\_{\\mathrm P} + \\Delta\\rho/\\rho)$ approximates the normal-incidence coefficient: it is the linearised impedance contrast, so for a strong contrast it differs slightly from the exact zero-offset value (0.283 against 0.280 for the limestone, −0.168 against −0.167 for the gas sand). The gradient $B = \\Delta V\_{\\mathrm P}/(2V\_{\\mathrm P}) - 2(V\_{\\mathrm S}/V\_{\\mathrm P})^2(\\Delta\\rho/\\rho + 2\\Delta V\_{\\mathrm S}/V\_{\\mathrm S})$ says how fast the reflection changes with angle, and it depends on the shear contrast, not only on $V\_{\\mathrm P}$ and density. Each $\\Delta$ is the lower rock minus the upper, and each velocity and density is the average of the two. In the figure, start with the short spread over the gas sand and lengthen it: watch what (c) records, and where in (b) the two-term line leaves the exact curve.

Over the gas sand (Ostrander’s 1984 model) the default spread of 1500 m reaches only 16.7° at 2500 m, and the reflection moves from −0.167 to −0.188 across it: too little to pin down the gradient. Stretch the spread to 5000 m (45°) and the far trace reads −0.319, the brightening with offset that marks the gas. The two-term line is still within 0.02 there. It is not always that good: over the hard sand it reads −0.117 at 42.8° where the exact value is −0.023 (exercise 3), because the exact curve turns up toward the critical angle and a straight line in $\\sin^2\\theta$ cannot follow it.

Snell’s law, the critical angle and the critical offset

The transmitted P ray bends by Snell’s law, $\\sin\\theta\_1/V\_{\\mathrm{P1}} = \\sin\\theta\_2/V\_{\\mathrm{P2}}$ . When the lower rock is faster there is a critical angle, $\\sin\\theta\_{\\mathrm c} = V\_{\\mathrm{P1}}/V\_{\\mathrm{P2}}$ , past which the transmitted P is evanescent: it decays away from the interface and carries no energy down. Between elastic rocks the reflection past $\\theta\_{\\mathrm c}$ is strong but not total, because converted S waves still carry energy away, and $R\_{\\mathrm{PP}}$ becomes complex, so the reflected wavelet changes phase. For shale on limestone $\\theta\_{\\mathrm c}$ = 38.5°, and over a target at 2000 m that angle is reached at the critical offset $x\_{\\mathrm c} = 2z\\tan\\theta\_{\\mathrm c}$ = 3180 m (exercise 2).

For acquisition this cuts both ways. The critical offset is where head waves begin, the refracted arrivals that refraction statics are built on, so the early arrivals of every shot gather (direct wave, refractions, reflections) are organised around it. For amplitude work, traces within about 5° of $\\theta\_{\\mathrm c}$ or past it cannot be fitted with $A + B\\sin^2\\theta$ and are usually muted. Design the spread so the angles the target needs, commonly 30° or more, arrive below that limit: 30° alone needs a far offset of about $1.15\\,z$ (exercise 4).

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.
Ostrander, W. J. (1984). Plane-wave reflection coefficients for gas sands at nonnormal angles of incidence. Geophysics, 49(10), 1637-1648.
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.