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・ Rays Snell ’ s Law Structure of the Earth ・ Seismic Waves

Theoretical Seismology: Wave Propagation. ・ Rays Snell ’ s Law Structure of the Earth ・ Seismic Waves Near-Field Terms (Static Displacements) Far-Field Terms (P, S, Surface waves) ・ Normal modes Free oscillations of the Earth. Magnitude for Local Tsunami.

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・ Rays Snell ’ s Law Structure of the Earth ・ Seismic Waves

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  1. Theoretical Seismology: Wave Propagation ・ Rays Snell’s Law Structure of the Earth ・ Seismic Waves Near-Field Terms (Static Displacements) Far-Field Terms (P, S, Surface waves) ・ Normal modes Free oscillations of the Earth

  2. Magnitude for Local Tsunami (Example) JMA Magnitude (Tsuboi, 1954) M=log 10A + 1.73 log10 Δ －0.83 A : Half of maximum total amplitude [μm] Δ : Epicentral distance [km]

  3. Faulting Seismic waves

  4. Travel Time and Distance

  5. Homogeneous Earth

  6. a1 q1 q2 a２ a1 > a2 Ray Paths in a Layered Medium Faster a1 q1 Ｓlower Slower q2 Faster a２ a1 < a2

  7. Moho Andrija Mohorovicic (1857-1936) Found seismic discontinuity at 30 km depth in the Kupa Valley (Croatia). Mohorovicic discontinuity or ‘Moho’ Boundary between crust and mantle

  8. Time 1/a3 1/a2 1/a1 Distance Ray Paths in a Layered Medium a1 a2 a3

  9. Structure in the Earth Crust-Mantle Core-Mantle 440 km 660 km

  10. Forward Branch Backward Branch

  11. Forward Branch Shadow Zone Forward Branch Backward Branch

  12. PcP Shadow Zone ・ 1912 Gutenberg observed shadow zone 105o to 143o ・ 1939 Jeffreys fixed depth of core at 2898 km (using PcP) Backward Branch Forward Branch PKP Forward Branch PcP Shadow Zone P Forward Branch Forward Branch Backward Branch

  13. PcP Core Reflections

  14. Why are observed seismograms so complicated ?

  15. Structure: Free Surface Earth is a not homogenous whole-space Free surface causes many complications - surface waves - reflections (pP, sP, sS) depth phase

  16. Surface Wave and Maximum Amplitude Observed in Japan. Δ=57(deg) Max Amp., 40 min after occurrence. (Ms, 20 deg ≦ Δ ≦ 160 deg)

  17. Seismogram of a distant earthquake Fig.16 ( LR: Rayleigh wave, LQ: Love wave )

  18. January 26, 2001 Gujarat, India Earthquake (Mw7.7) vertical Rayleigh Waves radial transverse Love Waves Recorded in Japan at a distance of 57o (6300 km)

  19. Seismic Waves Aspects of Waves not Explained by Ray Theory ・ Different types of waves (P, S) ・ Surface Waves ・ Static Displacements ・ Frequency content

  20. Period Wavelength

  21. Static Displacements Bei-Fung Bridge near Fung-Yan city, 1999 Chi-Chi, Taiwan earthquake

  22. Static displacements Co-seismic deformation of 2003 Tokachi-oki Earthquake (M8.0)

  23. Free Oscillations l=1 m=1 Houseman http://earth.leeds.ac.uk/~greg/?Sphar/index.html

  24. Summary Rays Earth structure causes complicated ray paths through the Earth (P, PKP, PcP) Wave theory explains ・ P and S waves ・ Static displacements ・ Surface waves Normal Modes The Earth rings like a bell at long periods

  25. Thank you for your attention

  26. Snell’s Law Fermat’s Principle Rays q1 Air Water q2 sin q1 / sin q2 = n21

  27. Wave Equation 1-D wave equation c = propagation speed Slinky: constant velocity wave propagation, no mass transfer, different from circulation eq.

  28. 1-D Wave Equation Solution T = wave period w = angular frequency LW 3.2.1

  29. Wave Period and Wavelength Velocity 6 km/s Space x wavelength 300 km wavelength Time t period 50 s frequency = 1/period= 0.02 hz period Velocity = Wavelength / Period

  30. 3-D Wave Equation with Source source spatial 2nd derivative Near-field Terms (Static Displacements) Solution Far-field Terms (P, S Waves)

  31. r/a r/b r/a r/b Near-field terms • ・ Static displacements • ・ Only significant close to the fault • ・ Source of tsunamis t　→

  32. Far-field Terms • ・ Propagating Waves • ・ No net displacement • ・ P waves • ・ S waves

  33. Surface Waves GroupVelocity (km/sec) Love Rayleigh Period (sec) S Shearer, Fig. 8.1

  34. Generation of Tsunami from Near-field Term

  35. Normal Modes (Stein and Gellar 1978) Free Oscillations of the Earth 1960 Chile Earthquake (Daishinji, Fukui Prefecture) Useful for studies of ・ Interior of the Earth ・ Largest earthquakes

  36. Free Oscillations l=1 m=2 Houseman http://earth.leeds.ac.uk/~greg/?Sphar/index.html

  37. Free Oscillations l=1 m=3 Houseman http://earth.leeds.ac.uk/~greg/?Sphar/index.html

  38. Toroidal and Spheroidal Modes Toroidal Spheroidal Dahlen and Tromp Fig. 8.5, 8.17

  39. Natural Vibrations of the Earth Shearer Ch.8.6 Lay and Wallace, Ch. 4.6

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