1. II.1 Theoretical Seismology 2: 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
This talk is supported by information in the training handout from page 40.

2. Magnitude for Local Tsunami
(Example)
JMA Magnitude (Tsuboi, 1954)
M=log 10A log10 Δ －0.83
A : Half of maximum total amplitude [μm]
Δ : Epicentral distance [km]
Example of a seismogram.
First wave = P wave, second S wave. P = primary wave, S secondary wave. After P and S are the surface waves

3. Faulting Seismic waves
Wave propagate in all directions from the fault. This shows the double couple fault mechanism. The causes changes in polarity of the P wave (first motion is either up or down).
Seismic waves

4. Travel Time and Distance
As the S wave is slower than the P wave, the difference in arrival time increases with distance from the fault. This is used for locating the epicentre.

5. Homogeneous Earth
If the Earth was constant velocity seismology would be easy! But velocity changes in the Earth

6. Ray Paths in a Layered Medium
a1 > a2 q1 q2 a1
Faster q1
Ｓlower q2
Slower
Faster a２
The ray path is deflected at velocity changes – the change in direction is calculated using Snell’s Law
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
Moho is one of the key boundaries in the Earth. Marks the velocity increase at the base of the crust.

8. Ray Paths in a Layered Medium
Distance
Time a1
The first arrival at the station is related to the velocity structure of the Earth and distance from the earthquake. The deeper path may be longer, but the velocity increase means that after a certain distance energy taking the longer path actually arrives first. a2 a3

9. Structure in the Earth Crust-Mantle Core-Mantle 440 km 660 km
The velocity in the mantle increases fairly constantly with depth, so rather than sharp changes in direction we see constant curving of the ray paths through the Earth.
The outer core is liquid and so there are no S waves in the outer core and P waves are slower in the outer core than the mantle

10. The ray paths though the mantle are reasonably simple

11. The core makes the ray paths more complicated
The core makes the ray paths more complicated. The P wave slows and is deflected deeper into the earth

12. Forward Branch Backward Branch
Then rays setting off at a steeper angle actually hit the surface nearer to the source – called the backward branch

13. Forward Branch Backward Branch Shadow Zone
Steeper angle leads to another forward branch, but there remains a area with no P arrivals = Shadow

14. ・ 1912 Gutenberg observed shadow zone 105o to 143o
PcP
・ 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
Shadow
Zone
PcP
Shadow
Zone P
Forward Branch
Backward Branch
Forward
Branch

15. PcP
PcP name from: P = P wave, c = outer core reflection
Core Reflections

16. Why are observed seismograms so complicated ?

17. Structure: Free Surface
Earth is a not homogenous whole-space
Free surface causes many complications
- surface waves　
- reflections (pP, sP, sS)
depth phase　
In the phases shown on this slide the lower case “p” or “s’ indicates that the energy bounced of the Earth surface near to the earthquake. These phases are important for estimating the depth of the earthquake.

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

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

21. 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)

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

23. - Period Wavelength 0.01 to 50 sec 50 m to 500 km 10 to 350 sec
Body waves
（P・S）
0.01 to 50 sec
50 m to 500 km
Surface waves
10 to 350 sec
30 to 1000 km
Free Oscillations
350 to 3600 sec　　(6 min to 1 hour)
1000 to km
Static Displacements -
The frequency of the waves varies and therefore the response of the seismometer is important – it controls what data is observed. Modern broadband instruments will record all of these frequencies (except the static displacements).

24. Static Displacements
Example of static displacements - This waterfall was generated by the 1999 earthquake
Bei-Fung Bridge near Fung-Yan city, 1999 Chi-Chi, Taiwan earthquake

25. Static displacements Co-seismic deformation of 2003 Tokachi-oki
Earthquake (M8.0)
Static displacements can be measured geodetically

26. Free Oscillations l=1 m=1
Houseman

27. 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

28. Thank you for your attention

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

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

32. 1-D Wave Equation Solution T = wave period w = angular frequency
LW 3.2.1

33. 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

34. 3-D Wave Equation with Source
spatial 2nd derivative
Near-field Terms (Static Displacements)
Solution
Far-field Terms (P, S Waves)

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

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

37. Surface Waves Love Rayleigh Period (sec) Shearer, Fig. 8.1
Group　Velocity (km/sec)
Love
Rayleigh
Period (sec) S
Shearer, Fig. 8.1

38. Generation of Tsunami from Near-field Term

39. Normal Modes Free Oscillations of the Earth 1960 Chile Earthquake
(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

40. Free Oscillations l=1 m=2
Houseman

41. Free Oscillations l=1 m=3
Houseman

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

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