MODERN GLOBAL SEISMOLOGY BODY WAVES AND RAY THEORY-2

2 3.1 The Eikonal Equation and Ray Geometry Direction cosine of associated with the ray is (dx 1 /ds,dx 2 /ds,dx 3 /ds) and must satisfy:

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4 Connection between W and dx/ds ∵ ▽ W is the normal to the wavefront surface ▽W▽W Unit normal dx/ds

5 Compare (3.17) with eikonal equation Define a -1 =n=c 0 /c(x) (index of refraction) Normal equations

6 How does the normal to the wavefront change along the path of the ray ?

7 Ray-path equation Ray-path equation describe how the normal to the wavefront change along the path of the ray – it depends on the spatial change of velocity structure. Two initial conditions control the behavior of (3.20) 1.The direction in which the ray leaves the reference point, 2.The position of the reference point s 0.

8 A simple case for ray-path equation: c=c(x 3 ), i.e., the velocity only changes with depth.

9 Consider only x 1 x 3 plane At a given point, the direction cosine of the ray:

10 p – ray parameter, or horizontal slowness. p varies from 0 to 1/c. Angle i – angle of incidence – the inclination of a ray measured from the vertical at any given depth. For a prescribed reference point and takeoff angle, a ray will have a constant ray parameter, p, for the entire path. This is also known as Snell’s law, which can be derived from Fermat’s principle.

11 Stretch your legs

12 Now, let’s explore the story behind the second equation of (3.22) ?

13 The curvature of the ray is proportional to the velocity gradient (dc/dx 3 ) The initial angle and the velocity structure determine the distance at which ray will emerge at the surface.

14 Can be used to predict where and when a ray will arrive

15 Integrate over depth Travel time

16 Relationship between T and X

17 Travel time equation travel time depends on X and Z and is separable ! vertical travel time and horizontal travel time p : horizontal slowness η: vertical slowness dT/dX = p (the change in travel time with horizontal distance is equal to ray parameter)

18 Amplitude of a seismic arrival Total energy on the hemispherical wavefront = K Energy per unit area = K/2 πr 2 Energy in the ring bounded by two takeoff angles i 0 and i 0 +di 0.

19 dX cos i 0 (suume source at surface) Energy density at the surface

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21 Energy density dT/dX = p  Energy density is proportional to the change in ray parameter with distance.

22 Stretch your legs

Travel Times in a Layered Earth

24 Apparent velocity in the horizontal direction p: horizontal slowness

25 α 2 > α 1  critical refraction critical angel Critical reflection  head wave

26 Direct ReflectedHead wave Source on the surface Travel time for direct/reflected/head waves in a flat layer

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29 Head wave in layered structure Turning wave in a continuous structure Travel time equations icic

30 Direct Reflected Head wave Source on the surface Determining the travel time curves

31 Direct Reflected Head wave Xc : crossover distance

32 At the crossover distance, the travel times of the direct arrival and head wave are equal: Fig 3.11 P n =head wave; P g =direct arrival

33 From the seismic observations, draw the travel time curves From the slopes of direct arrival and headwave  α 1 and α 2 The crossover distance  thickness

34 In 1909, Andrija Mohorovicic analysed the records of an earthquake in Croatia. Close to the epicenter he found one single arrival (Pg) the direct wave from the focus. Beyond 200km there were two arrivals: a new arrival (Pn) overtaking Pg. Mohorovicic identified that Pg and Pn travelled at a speed of ~5.4 and ~7.9 km/s respectively. He also calculated that the jump in velocity in that part of the world occured at a depth of 54km.  Andrija Mohorovicic had just discovered the the existence of the mantle underneath the continental crust.  Moho discontinuity Andrija Mohorovicic

35 P m P: moho reflected P n : moho head wave P g : direct wave

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37 (1936) I. Lehmann ( )I. Lehmann ( ) ( ) B. Gutenberg ( )B. Gutenberg ( ) (1909) A. Mohorovicic ( )A. Mohorovicic ( )

38 Travel-time equation in 2-layered structure A to B 1st layer

39 Travel-time equation in 2-layered structure n-layers

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41 Figure 3.13 Travel-time curve for a finely layered Earth. The first arrivel is comprised of short segments of the head-wave cruves for each layer, over the limited distance range between crossover points.

42 1/α 3 1/α 1 1/α 4 α1α1 α2α2 α3α3 α4α4 α 2 <α 1 Pseudothickness = th 1 +th 2 (η 2 /η 1 ) Low velocity layer

43 Blind zone P*:Conrad head wave When T2<T1 …

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Travel-Time Curves in a Continuous Medium n-layers Continuous medium

46 Ray paths Travel time P vs X triplication shadow

Travel Times in a Spherical Earth Figure 3.19

48 Anything new for the p in spherical Earth ? The ray parameter p is precisely the slope of the travel- time curve, as it was for the flat-Earth case except that distance is now measured in angular degrees.

49 A homogeneous sphere v=v 0 The travel time curve is not a straight line even thought the velocity is constant

50 Travel-time equation in a Spherical Earth With (3.64) and (3.65), we may remove ds

51 r 0 = the Earth radius r t = the deepest point of penetration With (3.64) and (3.65), we may remove dt

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Wave Amplitude, Energy, and Geometric Spreading Fig 3.6 (flat Earth) Homogeneous Spherical Earth v=v 0

57 v=v 0 For a homogeneous Earth, T=2r 0 sin(Δ/2)/v 0 Energy decays as 1/R 2 (Amplitude decays as 1/R)

58 At B and C, dp/dX ~ ∞  E ~ ∞ (Caustic)  Ray theory breaks down here (WHY ?)

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60 Stretch your legs

61 Total work done to a mass suspended in a spring Potential energy in a small volume under stress SH plane wave propagating in the x1 direction, with all motion in x2 direction Energy of seismic signals Average strain energy during a complete wavelength

62 The amplitude is modified during propagation by: 1.Geometric spreading 2.Reflection and refraction at a boundary 3.Attenuation

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Partitioning of Seismic Energy at a Boundary Why we need SV ? P P P’ SV ’ SV

66 Ray geometry is governed by Snell’s law (Mode conversion, Generalized Snell’s law) All of them have the same ray parameter, p P P P’ SV ’ SV

67  The ray geometry of the wave interaction is described by Snell’s law and ray theory  How about the amplitude partitioning ?

68 solid Welded interface Continuity across the interface Stress Displacement Boundary conditions on a welded interface The stresses and displacements must be “transmitted” across the interface. P waves alone can not satisfy B.C. solid

69 solid Continuity across the interface Normal traction Normal displacement fluid Boundary conditions on a solid-fluid interface Only normal stresses and normal displacements can be “transmitted” across the interface. The tangential displacements are not continuous and the tangential tractions must vanish.

70 Boundary conditions on a free surface All tractions must be zero, and no explicit restriction is placed on the displacements.

71 Potential of the incident plane P wave P P P’ SV ’ SV x1x1 x3x3

72 Similarly, for reflected/refracted waves P P P’ SV ’ SV x1x1 x3x3

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74 A simple case: fluid-fluid P P P

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80 P Free sureface

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82 Displacement transmission and reflection coefficients for vertical incidence

83 Non-vertical incidence (α 2 <α 1 ) P P P i1i1 i2i2

84 Non-vertical incidence (α 2 <α 1 )

85 Beyond the intramission angle, reflection coefficient is negative and decrease to a value of -1 at grazing incidence. (i=90)

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87 If α 2 >α 1, head waves are produced at the critical angle. when i>i c  Post-critical reflection

88 post-critical reflection

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91 Track the behavior of a particular wavefront via constant phase argument An frequency-dependent apparent time  Dispersion : wavespeed is frequency dependent (lower frequency signals arrive earlier)  Wavefront is “spread out”

92 The change in the reflected pulse shape as a function of incidence angle given in degrees at the left of the pulse.

93 Fast  slow media Slow  fast media

94 P  SV : mode conversions Sp and Ps are generated at a sediment-bedrock interface.  Depth of interface and vp/vs ratio in the crust.

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98 Post-critical reflection of a SH wave: Decays exponentially away from the interface in the media 2 SH

99  Evanescent waves or Inhomogeneous waves  surface waves

100 Attenuation The Earth is not perfectly elastic! Propagating waves attenuate with time due to various energy-loss mechanisms. Successive conversion of potential energy (particle position) to kinetic energy (particle velocity) as a wave propagates is not perfectly reversible, and other work is done: Internal friction, such as movements along mineral dislocations shear heating at grain boundaries Phenomenological description - seismic attenuation

101 Figure 3.31 Phenomenological model for seismic attenuation. Natural frequency

102 Introducing attenuation

103 Harmonic oscillation that decays exponentially with time Quality factor Q

104 Q = the fractional loss of energy per cycle of oscillation.

105 Figure 3.32: Effects of attenuation on a seismic pulse. Geometrical spreading + attenuation  amplitude decreases Attenuation  pulse broadening. Why ?

106 Large Q  weak attenuation (cold, high seismic velocity) Small Q  strong attenuation. (warm, low seismic velocity) Q α >Q β Intrinsic attenuation occurs almost entirely in shear, associated with lateral movements of lattice effects and grain boundaries. (Q α ～ 9/4 Q β ) Energy loss through non-elastic processes is usually measured by intrinsic attenuation and parameterized with Q.

107 Frequency dependency of Q.

108 Standard linear solid (Visco-elastic solid) Dashpot (viscous element)

109 The response to harmonic waves depends on the product of the frequency and the relaxation time. For wave periods that are very short or very long compared to the relaxation time, there is little attenuation.

110 Absorption peak of a standard linear solid

111 Relaxation spectrum for a polycrystalline material showing attenuation peaks at different frequencies due to different microscopic mechanisms.

112 Schematic model to explain the observation that Q is roughly constant over a wide range of frequencies. The superposition of absorption peaks for different compositions at different temperatures and pressures yields a flat absorption band.

113 Dispersion in a standard linear solid

114 Dispersion due to attenuation

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118 Scattering/coda waves APSE, the Apollo Passive Seismic Experiment. ( ) Figure 3.B5.1 Three-comp seismograms recording the impact of and Apollo lander on the moon. Seismograms ring for more than 1h.

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120 Advantage of using τ(p) ? (τ(p) is a single value function)

121 Ray pathsTravel timep vs Xτ vs p triplication Shadow

122 Figure 3.26