2.1 Seismic waves and their properties

Seismic waves are the primary tool geophysicists use to probe Earth's interior. Because we can't directly sample most of what lies beneath the surface, we rely on how these waves travel, reflect, and attenuate to reconstruct the planet's layered structure.

Two broad categories exist: body waves that propagate through Earth's interior, and surface waves that travel along or near the surface. Their distinct behaviors in different materials form the basis of nearly everything we know about Earth's deep structure.

Seismic Wave Types and Characteristics

Body Waves

P-waves (primary or compressional waves) are the fastest seismic waves and arrive first on a seismogram. They propagate through both solids and liquids.

• Particle motion is parallel to the direction of propagation, producing alternating zones of compression and rarefaction.
• Velocities range from ~1.5 km/s in unconsolidated sediments to over 13 km/s in the inner core.

S-waves (secondary or shear waves) are slower and travel only through solid materials.

• Particle motion is perpendicular to the propagation direction, producing a shearing deformation.
• S-wave velocity is typically about 60% of the P-wave velocity in the same solid material.
• S-waves cannot propagate through liquids because liquids have zero shear modulus. This is why S-waves vanish at the outer core boundary, which is one of the strongest pieces of evidence that the outer core is liquid.

Surface Waves

Surface waves are generated when body waves interact with Earth's free surface. They carry most of the destructive energy during earthquakes.

Rayleigh waves produce elliptical particle motion in the vertical plane (both vertical and horizontal components), creating a ground-rolling effect.

• They are the slowest seismic waves and typically cause the most structural damage.
• Their velocity is slightly less than the S-wave velocity in the same material.

Love waves produce horizontal particle motion perpendicular to the propagation direction (side-to-side shearing).

• Faster than Rayleigh waves, but slower than body waves.
• Confined to shallow depths and require a low-velocity layer overlying a higher-velocity layer to propagate (a waveguide effect). Without this velocity contrast, Love waves don't exist.

Factors Influencing Seismic Wave Propagation

Elastic Properties and Density

Seismic wave velocity depends on the balance between a material's stiffness (elastic moduli) and its density. Stiffer materials transmit waves faster; denser materials slow them down.

The governing equations are:

• P-wave velocity: $V_p = \sqrt{\frac{K + \frac{4}{3}\mu}{\rho}}$
• S-wave velocity: $V_s = \sqrt{\frac{\mu}{\rho}}$

where $K$ is the bulk modulus (resistance to uniform compression), $\mu$ is the shear modulus (resistance to shearing), and $\rho$ is density.

Notice that $V_s$ depends only on $\mu$. When $\mu = 0$ (as in a liquid), $V_s = 0$, confirming that S-waves can't travel through fluids.

Temperature and pressure both change with depth, and they compete:

• Increasing temperature reduces elastic moduli by weakening atomic bonds and promoting thermal expansion. This tends to decrease velocity.
• Increasing pressure increases elastic moduli and density by compressing the material and closing pore space. This tends to increase velocity.

In most of Earth's interior, the pressure effect wins, so velocity generally increases with depth.

Anisotropy and Heterogeneity

Anisotropy means the elastic properties of a material vary with direction. Seismic waves traveling in different directions through the same rock will have different velocities.

• Common in layered sedimentary sequences and foliated metamorphic rocks, where mineral grains or layers create a preferred orientation.
• One observable consequence is shear wave splitting (birefringence): an S-wave entering an anisotropic region splits into two orthogonally polarized components that travel at different speeds.

Heterogeneities (inclusions, fractures, compositional changes) scatter and attenuate seismic energy.

• Scattering redistributes wave energy in multiple directions and generates coda waves, the long tail of scattered arrivals that follow the main phases on a seismogram.
• Attenuation from scattering preferentially removes high-frequency energy, because shorter wavelengths interact more strongly with small-scale heterogeneities.

Seismic Wave Velocity and Attenuation

Velocity Variations in Earth Materials

Seismic velocities span a wide range depending on material type and depth. Representative values:

Velocity generally increases with depth, but not smoothly. Sharp jumps occur at major compositional or phase boundaries:

• Crustal velocity structure: gradual increase with depth, punctuated by the Moho at the base of the crust.
• Mantle velocity structure: gradual increase with depth, with notable discontinuities at ~410 km and ~660 km corresponding to mineral phase transitions (e.g., olivine to wadsleyite, ringwoodite to bridgmanite + ferropericlase).

Attenuation Mechanisms and Quality Factor

As seismic waves propagate, they lose energy. The quality factor $Q$ quantifies this: higher $Q$ means less energy loss per oscillation cycle, and more efficient propagation.

$Q(\omega) = \frac{\omega}{2\alpha(\omega)V}$

where $\omega$ is angular frequency, $\alpha$ is the attenuation coefficient, and $V$ is wave velocity.

Three main mechanisms contribute to amplitude decay:

1. Intrinsic absorption converts elastic energy to heat through anelastic processes like grain boundary sliding and dislocation motion. It increases with temperature and is strongest in partially molten or fluid-rich regions (the asthenosphere, magma chambers).

2. Scattering redistributes energy due to heterogeneities. Its strength depends on the size and velocity contrast of the heterogeneities relative to the seismic wavelength. Highly fractured fault zones and volcanic regions show pronounced scattering.

3. Geometrical spreading reduces energy density as the wavefront expands away from the source. This occurs even in a perfectly elastic medium:

   • Body wave amplitude decays as $1/r$
   • Surface wave amplitude decays as $1/\sqrt{r}$ where $r$ is distance from the source. The slower decay of surface waves is why they dominate seismograms at large epicentral distances.

Earth's Internal Structure from Seismic Waves

Seismic Wave Arrival Times and Amplitudes

The time difference between P-wave and S-wave arrivals (S-P time) at a station increases with distance from the earthquake source, because the two wave types travel at different speeds and the gap between them grows over longer paths.

• With S-P times from at least three stations, you can triangulate the epicenter using travel-time curves.
• Abrupt changes in the S-P time relationship at certain distances signal major internal discontinuities (e.g., the core-mantle boundary).

Amplitude information complements travel times. Lower-than-expected amplitudes along a particular path suggest the waves crossed a region of high attenuation, potentially indicating partial melt or elevated fluid content.

Seismic Discontinuities and Velocity Structure

At a discontinuity, seismic waves reflect and convert between P and S types. The major discontinuities that define Earth's layered structure:

• Mohorovičić discontinuity (Moho): crust-mantle boundary. P-wave velocity jumps from ~6.0–7.5 km/s to ~7.5–8.5 km/s. Depth varies from ~5–10 km under oceans to ~30–70 km under continents.
• Core-mantle boundary (CMB) at ~2,891 km depth: P-wave velocity drops sharply from ~13.0 km/s to ~8.0 km/s, and S-waves disappear entirely, indicating the outer core is liquid.
• Inner core boundary (ICB) at ~5,150 km depth: P-wave velocity increases from ~10.5 km/s to ~11.0 km/s, and S-waves reappear, indicating the inner core is solid.

Because velocity generally increases with depth, seismic rays curve back toward the surface according to Snell's law:

$\frac{\sin\theta_1}{V_1} = \frac{\sin\theta_2}{V_2}$

where $\theta$ is the angle of incidence or refraction and $V$ is wave velocity. This curving creates shadow zones, angular ranges where direct P or S arrivals are absent. The P-wave shadow zone between ~104° and ~140° from the epicenter, and the complete absence of direct S-waves beyond ~104°, were the key observations that revealed the liquid outer core.

Seismic Tomography and 3D Earth Structure

Seismic tomography extends the 1D radial velocity model into three dimensions. The basic idea: if a wave arrives earlier than predicted by the 1D model, it traveled through faster-than-average material; if it arrives late, it crossed slower-than-average material.

Three main approaches:

1. Travel-time tomography: inverts residuals (observed minus predicted travel times) for 3D velocity perturbations. The most widely used method.
2. Attenuation tomography: inverts amplitude decay patterns for 3D variations in $Q$.
3. Waveform tomography: fits the full waveform (shape and amplitude), yielding higher-resolution models but at greater computational cost.

Tomographic images have revealed major lateral structures in the mantle:

• Mantle plumes: narrow, low-velocity, high-attenuation columns rising from near the CMB to the surface, associated with hotspot volcanism (e.g., Hawaii, Iceland).
• Subducting slabs: high-velocity, low-attenuation sheets extending from trenches down into the lower mantle, representing cold oceanic lithosphere sinking at convergent boundaries (e.g., beneath the Pacific Ring of Fire).
• Large low-shear-velocity provinces (LLSVPs): two broad, low-velocity regions in the lowermost mantle beneath Africa and the Pacific. Their origin remains debated, with hypotheses ranging from purely thermal anomalies to compositionally distinct material, possibly primordial.