Earth's Interior: Seismic Waves & Inversion
Earth's Interior: Seismic Waves & Inversion
Seismology probes Earth's interior using elastic waves generated by earthquakes. By measuring travel times of P-waves (compressional) and S-waves (shear), geophysicists construct a detailed model of Earth's layered structure.
Definition
P-wave velocity \(v_P = \sqrt{(K + 4G/3)/\rho}\) depends on bulk modulus \(K\), shear modulus \(G\), and density \(\rho\). S-waves have \(v_S = \sqrt{G/\rho}\) and cannot propagate in liquids (\(G=0\)).
Key Result
Snell's law for seismic waves: \(\sin\theta_i/v_i = \sin\theta_r/v_r = p\) (ray parameter). Discontinuities in \(v\) create reflected and refracted phases; total internal reflection creates shadow zones.
Example 1
At 2890 km depth (core-mantle boundary), P-wave velocity jumps from 13.7 to 8.0 km/s. The resulting S-wave shadow zone (103°–142° epicentral distance) revealed that the outer core is liquid.
Example 2
Seismic tomography inverts arrival-time residuals using iterative algorithms to image 3D velocity anomalies. Subducting slabs appear as fast (cold) anomalies; hotspot plumes appear as slow (hot) anomalies beneath Hawaii and Iceland.
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Practice
- Why do S-waves not penetrate the outer core?
- Explain how the epicenter of an earthquake is located using three seismograph stations.
- What is the Mohorovičić discontinuity?
- How does seismic tomography differ from medical CT scanning?
Show Answer Key
1. S-waves are shear waves requiring a rigid medium. The outer core is liquid (molten iron-nickel), which cannot support shear stress, so S-waves cannot propagate through it. This creates the S-wave shadow zone (103°–180°).
2. Each station measures the P-S time difference, giving the distance to the epicenter (using travel-time curves). Three distances define three circles on Earth's surface; their intersection (trilateration) gives the epicenter location.
3. The Moho is the seismic discontinuity between Earth's crust and mantle, where P-wave velocity jumps from ~6.5 to ~8 km/s. It lies at ~35 km depth under continents and ~7 km under oceans. Discovered by Andrija Mohorovičić in 1909.
4. Both use waves (seismic vs. X-ray) and reconstruct internal structure from boundary measurements via inverse problems. Key difference: seismic tomography uses curved ray paths (refraction in a heterogeneous medium) and irregular source/receiver geometry, while CT uses straight parallel rays with controlled geometry. Seismic inversion is more ill-posed.