Reflection, refraction, and Snell’s law

Prerequisites

Learning objectives

State what happens to energy when a seismic wave meets a layer boundary
Apply Snell’s law to compute the angle of a refracted ray
Explain the critical angle, and why total reflection is a fluid result that rock does not quite reach
Connect these ideas to what a reflection seismogram records

The whole business of reflection seismology depends on one physical fact: when a wave reaches a boundary between two materials with different velocity or density, part of its energy bounces back (the reflection) and part continues forward, bent (the transmission, which we also call refraction). What we record at the surface is a long sequence of those echoes, one echo for each boundary the wave met on its way down.

The three things that happen at every boundary

Some energy reflects back: the reflected P wave leaves at the same angle as the incidence angle (measured from the normal to the boundary). This is what the reflection seismogram records.
Some energy transmits through at a new angle, computed from the velocity contrast. This is the wave that continues downward and may reflect off a deeper boundary.
How the energy splits depends on the contrast in acoustic impedance (density × velocity) across the boundary. A big contrast makes a bright reflection, a tiny one an invisible reflection. We will make this quantitative in Section 1.1.

For now, focus on the second point, the angle of the transmitted ray. This is governed by Snell’s law, one of the oldest and most reliable statements in physics:

\dfrac{\sin \theta_1}{V_1} = \dfrac{\sin \theta_2}{V_2}

where $\theta_1$ is the incidence angle (measured from the vertical normal to the boundary), $\theta_2$ is the transmitted angle, and $V_1$ , $V_2$ are the wave velocities in the upper and lower media. Rearranging gives the useful working form: $\sin \theta_2 = \sin \theta_1 \cdot \dfrac{V_2}{V_1}$ .

Which way does the ray bend?

If $V_2 > V_1$ (faster lower layer), the transmitted ray bends away from the normal, $\theta_2 > \theta_1$ . The ray spreads out into the faster rock.
If $V_2 < V_1$ (slower lower layer), the transmitted ray bends toward the normal, $\theta_2 < \theta_1$ . The ray steepens.
If $V_2 = V_1$ , $\theta_2 = \theta_1$ : no bending. The boundary is invisible only if the density is also the same; equal velocities with different densities still reflect, because reflection follows impedance.

Figure 0.2 opens with $V_1$ = 2200 m/s over $V_2$ = 3500 m/s and $\theta_1$ = 25°: the transmitted P ray leaves at 42.2°, bent away from the normal. Drag $\theta_1$ up and watch (b): the transmitted curve climbs faster than the reflected one and reaches 90° at 38.9°. In (a) the transmitted ray swings toward the boundary as you go (78.4° out at 38°) and is gone at 39°: past 38.9° no P wave enters the lower rock. That angle is the critical angle. Exactly at it the transmitted ray runs along the boundary; exercise 1 sets 2000 m/s over 4000 m/s, where the critical angle is exactly 30°, and there (a) draws the ray lying on the boundary.

The critical angle

The critical angle $\theta_{\mathrm c}$ is the incidence angle at which the transmitted ray would travel parallel to the boundary ( $\theta_2 = 90^\circ$ ). Setting $\sin \theta_2 = 1$ in Snell’s law:

\sin \theta_{\mathrm c} = V_1 / V_2

Between two fluids, which carry no shear, all the energy then reflects: total reflection. Rock is different, because it carries shear waves too. With the figure’s default rocks at 55°, 49% of the energy reflects as P and the other 51% leaves as converted S waves (exercise 3 compares fluid and rock at 50°: 100% against 26%). At exactly the critical angle the transmitted wave runs along the boundary at $V_2$ and sheds a head wave back upward, the arrival refraction surveys use. A critical angle requires $V_2 > V_1$ (the lower layer must be faster), which is why we do not always have one.

Why does any of this matter for interpretation? Two reasons. First, when we record reflection seismic we usually care about the near-vertical echoes, small incidence angles where rays barely bend. That is why stacked seismic data tries to approximate "normal-incidence" reflectivity. Second, as we push to wider angles (the far offsets in our acquisition), the bending matters more and reflection strength changes with angle, as plate (c) shows. That is AVO (amplitude versus offset), a central topic in Part 5 and the reason angles matter even in post-stack work.

One last piece: when a P wave meets a boundary between rocks at any angle other than straight on, it does not produce just one reflected and one transmitted wave. It produces four: reflected P and S, transmitted P and S. This is mode conversion, drawn as the dashed rays in Figure 0.2. Straight down nothing converts; the converted S grows as the angle opens (at the default 25° it takes 2.5% of the energy on reflection and 5.0% on transmission). It is recorded mainly on the horizontal components of multicomponent surveys, and on P data it shows up as reflection strength lost at wide angles.

References

Sheriff, R. E., & Geldart, L. P. (1995). Exploration Seismology (2nd ed.). Cambridge University Press.
Aki, K., & Richards, P. G. (2002). Quantitative Seismology (2nd ed.). University Science Books.
Yilmaz, Ö. (2001). Seismic Data Analysis (2 vols.). Society of Exploration Geophysicists.
Sheriff, R. E. (2002). Encyclopedic Dictionary of Applied Geophysics. Society of Exploration Geophysicists.