1. SEISMIC SOURCES

2. Solution for the displacement potential Properties of the disp potential 1.An outgoing D ’ Alembert solution in the spherical coordinate. 2.Functional shape is the same as the force-time history. Displacement field Near-field termFar-field term

3. 8.3 Elastostatics Purpose – to determine the static displacement u in an isotropic, infinite, homogeneous elastic medium due to a point source at point O. O F

4. 8.3 Elastostatics Three-dimensional delta function (in spherical polar coordinate): With the use of Gauss’ theorem:

5. 8.3.1 Static displacement field due to a single force Elastic equation without time-dependent displacement term (u tt =0) The point force is balanced by the stresses/strain Consider a point force of magnitude F at the origin

6. 8.3.1 Static displacement field due to a single force The equation of equilibrium becomes: Note that the representations of force and displacement field are similar! The solution form (Helmholtz’s theorem):

7. 8.3.1 Static displacement field due to a single force Which can be satisfied by having:

8. If we put A p =A p a and A s =A s a, we obtain two Poisson’s equations: 8.3.1 Static displacement field due to a single force The solutions (of potentials) are

9. 8.3.1 Static displacement field due to a single force Plugging in the potentials A p and A s and expressing the vector operations with indicial notation, and yield the ith component of displacement for a unit force (F=1) in the jth direction, u i j :

10. 8.3.1 Static displacement field due to a single force Somigliana tensor Symmetry : u i j = u j i

11. 8.3.1 Static displacement field due to a single force Thanks to the symmetry, there are only 6 independent permutations:

12. 8.3.1 Static displacement field due to a single force Find the displacements given in polar coordinates due to a point force applied in the x 1 direction (Figure 8.12), using the Jacobian coordinate transformation matrix (8.19)

13. (typo in textbook!) In the x 1 x 3 plane, φ=0: 8.3.1 Static displacement field due to a single force The (x 1 x 3 plane) static radial deformation and shear deformation due to a single force (Fig 8.12)

14. 8.3.2 Static displacement field due to a force couple Figure 8.13 A force couple acting at O’ parallel to the x 1 x 2 plane. The displacement at P is the sum of the displacements from two individual forces O’ P(x 1,x 2,x 3 ) x1 x2 x3

15. The displacement due to the force couple is given by: 8.3.2 Static displacement field due to a force couple

16. Let dξ 2 →0 and F →∞ so that Fdξ 2 → M, a finite moment. Let Force couple acting at the origin (o’=(0,0,0)) Offset in the x2 direction. Moment = M With (8.18) and (8.23)  the static displacement field u i 8.3.2 Static displacement field due to a force couple x1 x2 x3

17. In the same way, we may derive u 2 and u 3 for the single couple 8.3.2 Static displacement field due to a force couple

18. x1 x2 x3 Similarly, for single couple oriented along the x2 axis with offset arm along the x1 direction In general j : direction of force k : direction of offset arm

19. 8.3.3 Static displacement field due to a double couple x1 x2 x3 A double couple in the x 1 x 2 plane Principles of superposition: The displacement is the sum of the displacements from two individual couples (+/- of M)

20. 8.3.3 Static displacement field due to a double couple Convert to spherical polar coordinates with (8.19) On the x 1 x 2 plane, θ=π/2, u θ =0, and Static deformations decay rapidly.

21. 8.3.3 Static displacement field due to a double couple On the x 1 x 2 plane, θ=π/2, u θ =0, and Figure 8.14 Azimuthal pattern of u r and u θ on the x 1 x 2 plane

22. The displacement field due to a shear dislocation can be given by the displacement field due to a distribution of equivalent double couples that are placed in a medium without any dislocation. Since static deformations decay rapidly with distance from the source, ground deformations are usually near the fault  a point-source approximation is never valid  finite fault  numerically discretized distribution of double couples. 8.3.3 Static displacement field due to a double couple

23. Going beyond the simple faulting model in geodetic modeling Incorporating viscoelastic effects of the deeper crust. Adding layering and elastic parameter heterogeneity in the Earth model. Variable slip function or changing fault mechanism. Curved fault plane

24. 8.4 Elastodynamics Elastodynamic equations: Consider a time-dependent body force: Following the same basic procedure as used in elastostatic problem.

25. 8.4 Elastodynamics Again, we seek a solution of the form: Compare it to (8.14)

26. 8.4 Elastodynamics Putting A p =A p a. A s =A s a, we obtain twos scalar equations:

27. 8.4 Elastodynamics: The solution to an inhomogeneous wave equation The solution is: (Buy it, for now) An inhomogeneous wave equation: Where g is a “point” source both in space and time: (Box 2.5)

28. 8.4 Elastodynamics Standard type of D’Alembert-type solution: For a point force at x=(ξ 1, ξ 2,ξ 3 ) applied at t=τ. For a time-dependent point force f(t) applied at x=(ξ 1, ξ 2,ξ 3 )

29. 8.4 Elastodynamics If the source is extended through a volume V, as well as in time

30. 8.4 Elastodynamics  The solutions to (8.33)

31. So far, so good ? Sorry, it’s getting messy …

32. How to deal with this integration ? 8.4 Elastodynamics Integrating over V via the system of concentric spherical shells …

33. 8.4 Elastodynamics

36. Typo in textbook!

37. 8.4 Elastodynamics Stokes solution (for point force in the j direction, located at the origin): Near-field term Far-field term

38. 8.4 Elastodynamics Properties of the far-field P-wave 1.It attenuates as 1/r 2.Arrival time=r/αwith velocity α 3.waveform is proportional to the applied force at the retarded time. 4.The displacement is parallel to the direction from the source (u p ×γ=0) (longitudinal wave) 5.|u p | is proportional to γ j

39. Properties of the far-field P-wave 4. The displacement is parallel to the direction from the source (u p ×γ=0) (longitudinal wave) 5. |u p | is proportional to γ j

40. 8.4 Elastodynamics Properties of the far-field S-wave 1.It attenuates as 1/r 2.Arrival time=r/β with velocityβ 3.waveform is proportional to the applied force at the retarded time. 4.The displacement is perpendicular to the direction from the source (u s ． γ=0) (transverse wave)

41. The displacement field for single couples and double couples can be obtained by differentiating the single-force results w.r.t appropriate coordinates. (The same as we did for the static fields.) Only far-field displacements are discussed from now on. 8.4 Elastodynamics

42. 8.4 Elastodynamics – The displacement field due to a single couple F F2F2

43. x1x1 x2x2 p In the x1-x2 plane Direction cosines for F x1x1 x2x2 p In the x1-x2 plane Direction cosines for F 2

44. x1x1 x2x2 p In the x1-x2 plane

45. For Temporal differentiation of the source time history 8.4 Elastodynamics – The displacement field due to a single couple Expand this term around (t-r 2 /α) in Taylor series.

46. 8.4 Elastodynamics – The displacement field due to a single couple Temporal differentiation of the source time history

47. Near-field terms (Decays as 1/r 2) 8.4 Elastodynamics – The displacement field due to a single couple Far-field displacement (Decays as 1/r ) The far-field displacement is sensitive to particle velocity at the source rather than to particle displacements. (rev: Eq 8.3)

48. Introduce the moment representation: M 8.4 Elastodynamics – The displacement field due to a single couple

49. 8.4 Elastodynamics – The displacement field due to a single couple/double couple Solution for the single couple is x1x1 Δx 2 x2x2 x1x1 Δx 1 x2x2 Symmetry in force direction and offset direction  Solution to a double couple x1 x2 x3 A double couple in the x 1 x 2 plane

50. Elastodynamic Green function G ij : elastodynamic Green function – The displacement field from the simplest source – namely, the unidirectional unit impulse, which is localized precisely in both space and time Notaion – Gij : ith response to impulse force acting on jth direction.

51. 8.4 Elastodynamics – general form of the far-field displacement for a couple Using the notation of Green function, the general form of the far-field displacement field (P and S) for a couple in the pq plane is given by P S M pq : seismic moment tensor (9 couples/dipoles)

52. 8.4 Elastodynamics – Radiation pattern of the far-field displacement for double couple Convert to spherical coordinate system (for p=1, q=3) Time-dependent moment function The far-field displacements are proportional to the moment rate function.

53. 8.4 Elastodynamics – Radiation pattern of the far-field P displacement for double couple x3x3 x1x1 Θ=180° Θ=90° Θ=0° There are two nodal lines (fault plane and auxiliary plane) x1x1 -x 3

54. 8.4 Elastodynamics – Radiation pattern of the far-field S displacement for double couple x1x1 -x 3 x3x3 x1x1 Θ=180° Θ=90° Θ=0° Θ=90° T P T P P: pressure axis T: Tension axis There are 6 nodal points. (There is no nodal line.)

55. 8.4 Elastodynamics – Radiation pattern of the far-field S displacement for double couple

56. 8.4 Elastodynamics – Example of point-source Comparison of observed and synthetic ground motions for June 13, 1980 eruption of Mt.St. Helens (A vertical point force at the source). The comparison can be used to estimate the strength of the eruption.(Kanamori & Given, 1983)

57. 8.4 Elastodynamics – Example of point-source Observed and interpretations of the source mechanism for the 1975 Kalapana, Hawaii event.(Eissler and Kanamori, 1987) Double coupleObserved Single force

58. 8.4 Elastodynamics – the nature of moment rate function – step/delta function t M(t) ● t Step function δ function

59. 8.4 Elastodynamics – the nature of moment rate function – ramp/boxcar function t M(t) ● t Ramp function Boxcar function τ In the case of boxcar function, the area under the boxcar is equal to M 0

60. 8.5 The Seismic Moment Tensor P342 The DC solution given by (8.61) has the corresponding moment tensor: Where M 0 is the scalar factor.

61. 8.5 The Seismic Moment Tensor – M ij in terms of fault parameters From Seth Stein & Michael Wysession “An introduction to seismology, earthquakes, and Earth structure.

62. 8.5 The Seismic Moment Tensor – M ij in terms of fault parameters φfφf δ λ D : slip vector V : fault normal v D Fig8.20 The geographic coordinate system (ray coord.) Double couple M in the geographic frame N φsφs E

63. 8.5 The Seismic Moment Tensor – M ij in terms of fault parameters Express D and v in terms of fault parameters (φ f, δ, λ) Double couple

64. In the same way: 8.5 The Seismic Moment Tensor – double couple Mij in terms of fault parameters

65. 8.5 The Seismic Moment Tensor – moment weighted Green ’ s function It’s possible to construct the P or S motion for a moment tensor by summing the moment weighted Green’s function The basis for many synthetic seismogram programs and waveform inversions.

66. 8.5 The Seismic Moment Tensor – rotation of moment tensor

67. 8.5 The Seismic Moment Tensor – decomposition of Moment tensor In general, a seismic moment tensor need not corresponding to a pure double couple, but the symmetric tensor can still be diagonalized into three orthogonal dipoles. Moment tensors for faulting events are often determined with the constraint tr(M)=0 Isotropic component  a volume change, when tr(M) ≠0

68. 8.5 The Seismic Moment Tensor – decomposition of Moment tensor An isotropic part and three double couples.

69. 8.5 The Seismic Moment Tensor – decomposition of Moment tensor An isotropic part and three CLVDs CLVD: componsated linear vector dipoles

70. 8.5 The Seismic Moment Tensor – decomposition of Moment tensor An isotropic part, a major double couple and a minor double couple.

71. 8.5 The Seismic Moment Tensor – decomposition of Moment tensor Where ε is a measure of the size of the CLVD component relative to the double couple. For a pure double couple, ε=0.

72. Box 8.4 A non-double-couple source Significant non-double components found using waves with different frequencies Comparison of observed P and predictions

73. Focal sphere – Beach Ball

74. Focal sphere – relation between fault planes and stress axes

75. Example of the determination of a complex rupture for the 1976 Guatemala earthquake.

76. x1x1 x2x2 baseball eyeball CLVD

78. The END

79. Focal sphere – Beach Ball From Seth Stein & Michael Wysession “An introduction to seismology, earthquakes, and Earth structure.

80. Focal sphere – Beach Ball From Seth Stein & Michael Wysession “An introduction to seismology, earthquakes, and Earth structure.

81. Focal sphere – Beach Ball

83. 8.4 Elastodynamics p.334 For

84. 8.4 Elastodynamics p.335

85. 8.4 Elastodynamics p.335-336

86. 8.4 Elastodynamics p.340

87. 8.4 Elastodynamics p.339-340

88. 8.4 Elastodynamics p.340-341

89. 8.4 Elastodynamics p.341

90. 8.6 Determination of Faulting Orientation P347

92. 8.6 Determination of Faulting Orientation P348

94. 8.6 Determination of Faulting Orientation P349 1. 2. 3. 4.

95. 8.6 Determination of Faulting Orientation P350

96. 8.6 Determination of Faulting Orientation P351

97. 8.6 Determination of Faulting Orientation P350-351

98. 8.6 Determination of Faulting Orientation P352

100. 8.6 Determination of Faulting Orientation P353

102. Finite fault ～ Discretized distributions of double couples (Figure 8.15) Vertical strike slipFault parallel motions Fault perpendicular motionsVertical motions (Chinnery 1961) + + - -