GE 162 Introduction to Seismology
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California Institute of Technology β Seismological Laboratory GE 162 Introduction to Seismology Lecture notes Jean Paul (Pablo) Ampuero β [email protected] Winter 2013 - 2016
Transcript of GE 162 Introduction to Seismology
California Institute of Technology β Seismological Laboratory
GE 162 Introduction to Seismology Lecture notes
Jean Paul (Pablo) Ampuero β [email protected] Winter 2013 - 2016
GE 162 Introduction to Seismology Winter 2013 - 2016
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Contents 1 Overview and 1D wave equation .......................................................................................................... 5
1.1 Overview, etc ................................................................................................................................ 5
1.2 Longitudinal waves in a rod: derivation of the wave equation .................................................... 5
1.2.1 Description of the problem and kinematics.......................................................................... 5
1.2.2 Dynamics ............................................................................................................................... 6
1.2.3 Rheology ............................................................................................................................... 6
1.2.4 The 1D wave equation .......................................................................................................... 6
2 1D wave equation: solution and main properties ................................................................................ 8
2.1 General solution ............................................................................................................................ 8
2.2 Reflection at one end .................................................................................................................... 8
2.3 Fourier transform .......................................................................................................................... 9
2.4 Harmonic waves ............................................................................................................................ 9
3 Energy, reflection/transmission, normal modes ............................................................................... 11
3.1 Impedance .................................................................................................................................. 11
3.2 Energy considerations for harmonic waves ................................................................................ 11
3.3 Reflection and transmission at a material interface ................................................................... 11
3.4 Normal modes of a finite elastic rod ........................................................................................... 12
3.5 Duality between modes and propagating waves........................................................................ 13
4 Greenβs function. Waves in heterogeneous media. ........................................................................... 14
4.1 Linear invariant systems, Greenβs functions, convolution .......................................................... 14
4.2 Greenβs function for the 1D wave equation ............................................................................... 14
4.3 Waves in heterogeneous medium (WKBJ approximation) ......................................................... 15
5 The 3D elastic wave equation ............................................................................................................. 16
5.1 Strain ........................................................................................................................................... 16
5.2 Stress ........................................................................................................................................... 16
5.3 Momentum equation .................................................................................................................. 16
5.4 Elasticity ...................................................................................................................................... 16
5.5 The seismic wave equation ......................................................................................................... 17
5.6 Itβs a perturbative equation ........................................................................................................ 17
5.7 P and S waves examples ............................................................................................................. 17
5.8 General decomposition into P and S waves ................................................................................ 18
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5.9 Polarization of body waves ......................................................................................................... 18
5.10 Usual characteristics of body waves ........................................................................................... 19
6 Body waves ......................................................................................................................................... 20
6.1 Spherical waves, far-field, near-field .......................................................................................... 20
6.2 Ray theory: eikonal equation, ray tracing ................................................................................... 20
6.3 Rays in depth-dependent media, Snellβs law, refraction ............................................................ 21
7 More on body waves .......................................................................................................................... 22
7.1 Layer over half-space: head waves ............................................................................................. 22
7.2 A steep transition zone ............................................................................................................... 23
7.3 A low velocity zone ..................................................................................................................... 23
7.4 Wave amplitude along a ray ....................................................................................................... 25
7.5 Ray parameter in spherically symmetric Earth ........................................................................... 25
7.6 Body waves in the Earth .............................................................................................................. 25
8 Surface waves I: Love waves ............................................................................................................... 30
8.1 Separation between SH and P-SV waves .................................................................................... 30
8.2 SH reflection and transmission coefficients at a material interface ........................................... 30
8.3 Love waves .................................................................................................................................. 32
8.4 Dispersion relation ...................................................................................................................... 32
8.5 Phase and group velocities ......................................................................................................... 34
8.6 Airy phase .................................................................................................................................... 35
9 Surface waves II: Rayleigh waves ........................................................................................................ 36
9.1 Rayleigh waves ............................................................................................................................ 36
9.2 Surface waves in a heterogeneous Earth .................................................................................... 38
9.3 Implications for tsunami waves .................................................................................................. 39
10 Normal modes of the Earth ............................................................................................................ 40
11 Attenuation and scattering ............................................................................................................. 47
11.1 Attenuation of normal modes .................................................................................................... 47
11.2 A damped oscillator .................................................................................................................... 48
11.3 A propagating wave .................................................................................................................... 48
12 Scattering ........................................................................................................................................ 50
13 Seismic sources. .............................................................................................................................. 55
13.1 Kinematic vs dynamic description of earthquake sources ......................................................... 55
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13.2 Stress glut and equivalent body force ........................................................................................ 55
13.3 Equivalent body force representation of fault slip ..................................................................... 55
13.4 Moment tensor ........................................................................................................................... 56
13.5 Seismic moment and moment magnitude ................................................................................. 58
13.6 Representation theorem ............................................................................................................. 58
14 Seismic sources: moment tensor .................................................................................................... 59
14.1 Greenβs function .......................................................................................................................... 59
14.2 Moment tensor wavefield .......................................................................................................... 59
14.3 Far field and radiation pattern of a double couple source ......................................................... 60
14.4 Surface waves ............................................................................................................................. 65
15 Finite sources .................................................................................................................................. 66
15.1 Kinematic source parameters of a finite fault rupture ............................................................... 66
15.2 Far-field, apparent source time function .................................................................................... 66
15.3 ASTF in the Fraunhofer approximation ....................................................................................... 66
15.4 Haskell pulse model, directivity .................................................................................................. 67
16 Scaling laws ..................................................................................................................................... 69
16.1 Circular crack model.................................................................................................................... 69
16.2 Stress drop, corner frequency, self-similarity ............................................................................. 69
16.3 Energy considerations and moment magnitude scale ................................................................ 70
16.4 Stress drop for Haskell model and break of self-similarity ......................................................... 71
17 Source inversion, near-fault ground motions and isochrone theory.............................................. 72
17.1 Fundamental limitation of far-field source imaging ................................................................... 72
17.2 Source inversion .......................................................................................................................... 72
17.3 Isochrone theory ......................................................................................................................... 73
18 Source inversion and source imaging ............................................................................................. 74
18.1 Source inversion problem ........................................................................................................... 74
18.2 Ill-conditioning of the source inversion problem ........................................................................ 74
18.3 Stacking ....................................................................................................................................... 76
18.4 Array seismology ......................................................................................................................... 76
18.5 Array response ............................................................................................................................ 79
18.6 Coherency stacking ..................................................................................................................... 79
19 Earthquake dynamics I: Fracture mechanics perspective ............................................................... 80
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20 Earthquake dynamics II: Fault friction perspective ......................................................................... 80
21 Inverse problems, part 1 ................................................................................................................. 81
21.1 Earthquake location .................................................................................................................... 81
21.2 Iterative solution ......................................................................................................................... 81
21.3 Solution of inverse problems. ..................................................................................................... 81
21.4 Weighted over-determined problem .......................................................................................... 81
21.5 Uncertainties: model covariance ................................................................................................ 81
21.6 Double difference location ......................................................................................................... 81
22 Inverse problems, part 2 ................................................................................................................. 82
22.1 Travel time tomography, ill-posed problems .............................................................................. 82
22.2 SVD, minimum-norm solution .................................................................................................... 82
22.3 Resolution matrix, model covariance matrix .............................................................................. 82
22.4 Truncated SVD............................................................................................................................. 82
22.5 Regularization ............................................................................................................................. 82
22.6 Bayesian approach ...................................................................................................................... 82
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1 Overview and 1D wave equation
1.1 Overview, etc β’ Have you ever felt an earthquake? β’ About me: Pablo Ampuero, office SM 359, email [email protected], research topics β’ About you: fill entry survey β’ Seismology = study of ground motions induced by seismic waves β’ Show a few seismograms and point to interesting features and their broader significance β’ Applications from traditional to recent: Earth structure (e.g. seismic tomography), search for
natural resources (exploration seismology), earthquake physics, monitoring nuclear explosions, monitoring volcanic activity, assess natural or anthropogenic hazards, mitigate hazards (early warning), helioseismology, landslides, tsunamis, icequakes, sediment transport in rivers, hydrology and brittle deformation of glaciers, β¦
β’ History and the Seismolab: Gutenberg-Richter frequency-magnitude distribution. Richterβs magnitude scale. Kanamoriβs moment magnitude scale. Andersonβs PREM. From Trinet to SCSN, EEW, CSN. Analog to digital. Today: large N, big data, noise, computational seismology.
β’ These lectures are in two parts: basic seismology theory, and basic earthquake source theory.
β’ These lecture notes are a support for class, but not a replacement for it (they are terse and sometimes incomplete). Derivations in blue are not done in detail in class, you should review them at home.
β’ Books. Some are available as ebooks through our library. In brackets are the shorthand for references in this document. for Part 1:
i. Stein and Wysession (S&W) ii. Shearer (S)
iii. Lay and Wallace (LW) iv. More technical: Aki and Richards (AR)
for Part 2 i. Madariaga et al (M)
ii. Scholz (SZ) iii. More technical: Aki and Richards (AR), Freund (F)
β’ What is the most recent significant earthquake you learned about? sign up for USGS email Earthquake Notification Service follow Twitter earthquake reports via @CaltechQuake, @USGSBigQuakes, @USGSted learn about USGS-NEIC and IRIS online products
1.2 Longitudinal waves in a rod: derivation of the wave equation See also S&W 2.2.1
1.2.1 Description of the problem and kinematics [experiment: waves along a rope or slinky β or the projector cable]
Consider a solid rod with cross section S much smaller than length. [Sketch rod and cross section]
Ignore transverse expansion or contraction of the rod. Focus on longitudinal waves: transient deformations parallel to the axis of the rod.
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[Sketch undeformed and deformed rod, annotate] Let π₯π₯ be a location along the rod. At time π‘π‘, its perturbed location is π₯π₯ + π’π’(π₯π₯, π‘π‘). Our goal: determine the evolution of the displacement field π’π’(π₯π₯, π‘π‘) induced by some initial conditions (out of static equilibrium) or by some external forcing.
[discuss initial conditions and forcing]
1.2.2 Dynamics Apply πΉπΉ = ππππ to a small portion of the rod, of length πππ₯π₯.
[Sketch forces on an elementary segment dx. Recall first-order Taylor expansion.] Definition: on a transverse surface located at x, F(x) = force induced by the material located βto the rightβ (xβ>x) on the material located to the left (xβ<x).
πππππ₯π₯ ππ
ππ2π’π’πππ‘π‘2
= πΉπΉ οΏ½π₯π₯ +πππ₯π₯2οΏ½ β πΉπΉ οΏ½π₯π₯ β
πππ₯π₯2οΏ½ = β¦ β πππ₯π₯
πππΉπΉπππ₯π₯
(π₯π₯) (1)
Define stress = force per unit of cross section area: ππ = πΉπΉππ
. Then:
ππππ2π’π’πππ‘π‘2
=πππππππ₯π₯
(2)
1.2.3 Rheology Constitutive relation: How does a material deform in response to an applied stress? Imaginary experiment: compress a rod (static).
[sketch uncompressed, compressed rod. Define notations] [discuss non linear elasticity and inelasticity, e.g. plasticity]
Linear elasticity: πΉπΉ β Ξππ, if Ξππ βͺ ππ. Important assumption: small deformations. [experiment with rubber band]
Actually πΉπΉ β Ξππ/ππ = ππ (strain). [experiment with stack of two rods (larger S)]
Also πΉπΉ β ππ. These properties are summarized by: πΉπΉ = ππππ Ξππ/ππ (3)
where E is a material parameter called Youngβs modulus. ππ = ππΞππ/ππ (4)
Consider an elementary segment of length dx.
[sketch undeformed and deformed rod. Annotate new positions of two ends.] The length of the undeformed rod is ππ = πππ₯π₯. Stretch of the deformed rod: Ξππ = π’π’ οΏ½π₯π₯ + ππππ
2οΏ½ β π’π’ οΏ½π₯π₯ β ππππ
2οΏ½ β πππ₯π₯ πππ’π’/πππ₯π₯.
Dividing by dx: Ξππ/ππ = πππ’π’/πππ₯π₯ (5)
Plugging it into eq. (4): ππ = ππ πππ’π’ /πππ₯π₯ (6)
1.2.4 The 1D wave equation Combining eqs. (2) and (6):
πππππ₯π₯
οΏ½πππππ’π’πππ₯π₯οΏ½ = ππ
ππ2π’π’ πππ‘π‘2
(7)
Assuming E is constant
ππππ2π’π’πππ₯π₯2
= ππ ππ2π’π’ πππ‘π‘2
(8)
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[dimensional analysis: determine the units of ππ/ππ] Define a quantity with units of speed: ππ = οΏ½ππ/ππ = wave speed (celerity). We get the 1D wave equation:
ππ2π’π’πππ₯π₯2
=1ππ2
ππ2π’π’ πππ‘π‘2
(9)
For transverse motions we get the same wave equation (9) but with shear wave speed ππ = οΏ½ππ/ππ, where ππ is the shear modulus. The shear stress is ππ = ππ πππ’π’ /πππ₯π₯. More on this in a later lecture.
[orders of magnitude of c]
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2 1D wave equation: solution and main properties
2.1 General solution The solution comprises waves propagating in both directions at speed ππ, DβAlembertβs solution:
π’π’(π₯π₯, π‘π‘) = ππ(π₯π₯ β πππ‘π‘) + ππ(π₯π₯ + πππ‘π‘) (10)
[Sketch waves snapshots, seismograms, characteristic lines in space-time (x,t) plane] Proof: Define new variables ππ = π₯π₯ β πππ‘π‘ and ππ = π₯π₯ + πππ‘π‘. Apply chain rule:
πππππ₯π₯
=ππππππ
πππππππ₯π₯
+ππππππ
πππππππ₯π₯
=ππππππ
+ππππππ
πππππ‘π‘
= β― = βππππππππ
+ ππππππππ
Apply it again, then plug it into eq (9). After some algebra: ππ2π’π’
ππππππππ= 0
(11)
Integrating this equation with respect to ππ: πππ’π’/ππππ = ππβ² (ππ)
where ππβ² is an arbitrary function. Integrating this with respect to ππ: π’π’(ππ, ππ) = ππ(ππ) + ππ(ππ) (12)
where ππ is an arbitrary function. β‘ ππ and ππ are constrained by the initial conditions. Example: compress the rod, then suddenly release it. The initial conditions are π’π’(π₯π₯, 0) = π’π’0(π₯π₯) and οΏ½ΜοΏ½π’(π₯π₯, 0) = 0. Considering these initial conditions and (10) we get two equations: ππ(π₯π₯) + ππ(π₯π₯) = π’π’0(π₯π₯) and βππππβ²(π₯π₯) + ππππβ²(π₯π₯) = 0. From these we derive: ππ = (π’π’0 + πΆπΆ)/2 and ππ = (π’π’0 β πΆπΆ)/2, where πΆπΆ is a constant. Finally,
π’π’(π₯π₯, π‘π‘) =12π’π’0(π₯π₯ β πππ‘π‘) +
12π’π’0(π₯π₯ + πππ‘π‘)
(13)
Note that πΆπΆ is undetermined but cancels out in the final solution. [Draw this solution]
2.2 Reflection at one end Apply mirror image trick to satisfy boundary conditions at the end of the rod (at π₯π₯ = 0).
[Sketches explaining the mirror image trick] Case 1, fixed displacement (Dirichlet b.c.) π’π’(0, π‘π‘) = 0 achieved by an image wave with opposite amplitude:
π’π’(π₯π₯, π‘π‘) = ππ(π₯π₯ β πππ‘π‘) β ππ(βπ₯π₯ β πππ‘π‘) (14)
Case 2, free stress (Neumann b.c.) π’π’β²(0, π‘π‘) = 0 achieved by an image with same amplitude (note the cancelation of slopes):
π’π’(π₯π₯, π‘π‘) = ππ(π₯π₯ β πππ‘π‘) + ππ(βπ₯π₯ β πππ‘π‘) (15)
In case 2 there is amplification at the boundary: π’π’(0, π‘π‘) = 2ππ
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2.3 Fourier transform Any βwell behavedβ function π’π’(π‘π‘) can be decomposed as a linear superposition (weighted sum) of oscillatory functions ππβππππππ with angular frequency ππ:
π’π’(π‘π‘) =
12ππ
οΏ½ π’π’οΏ½(ππ)ππβππππππππππ+β
ββ
(16)
[Sketch linear superposition. Note diversity of conventions about factor 1/2ππ and frequency] The frequency-dependent weights π’π’οΏ½(ππ) are called the spectral coefficients or Fourier coefficients and are defined by the so-called Fourier transform of π’π’(π‘π‘) :
π’π’οΏ½(ππ) = οΏ½ π’π’(π‘π‘)πππππππππππ‘π‘
+β
ββ
(17)
The function π’π’οΏ½(ππ) is also called the spectrum of π’π’(π‘π‘). Some useful Fourier transform pairs: π’π’(π‘π‘) β· π’π’οΏ½(ππ) (18)
οΏ½ΜοΏ½π’(π‘π‘) β·βππππ π’π’οΏ½(ππ) (19)
οΏ½ΜοΏ½π’(π‘π‘) β·βππ2 π’π’οΏ½(ππ) (20)
2.4 Harmonic waves Taking the Fourier transform of equation (9):
βππ2π’π’οΏ½(π₯π₯,ππ) = ππ2
ππ2π’π’οΏ½πππ₯π₯2
(21)
Solution: π’π’οΏ½(π₯π₯,ππ) = π΄π΄(ππ) ππππππππ + π΅π΅(ππ) ππβππππππ (22)
where A and B are complex valued functions of frequency, to be determined by boundary conditions, and ππ is the wavenumber defined by
ππ = ππ/ππ (23)
Taking the inverse Fourier transform of (22) shows that π’π’(π₯π₯, π‘π‘) is a superposition of harmonic waves of the following form:
π΄π΄ ππππ(ππππβππππ) + π΅π΅ ππβππ(ππππ+ππππ) (24)
[Sketch harmonic wave (real part) at fixed t, then at fixed x]
Some definitions: Frequency: ππ = ππ/2ππ Period (temporal): ππ = 2ππ/ππ = 1/ππ Wavelength (spatial period): ππ = 2ππ/ππ Duality space-time:
ππ = ππ ππ (25)
Short periods = high frequencies = short wavelengths. Long periods = low frequencies = long wavelengths.
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The space-time duality encapsulated in equation (25) has profound implications for how seismologists infer Earth structure and earthquake processes. Long period waves are associated to long wavelengths and are not sensitive to small scale features, which limits the resolution of seismic tomography. Short period waves are associated to short wavelengths and potentially contain detailed information about small scale earthquake rupture processes, but they are also severely affected by the poorly known fine scale structure along the wave path.
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3 Energy, reflection/transmission, normal modes
3.1 Impedance How is stress related to velocity? (dynamic vs kinematic quantities) Shear stress in elastic medium:
ππ = ππ πππ’π’ /πππ₯π₯ Consider a wave π’π’(π₯π₯, π‘π‘) = ππ(π₯π₯ β πππ‘π‘). We have πππ’π’ /πππ₯π₯ = ππβ² and πππ’π’/πππ‘π‘ = βππππβ². Hence
πππ’π’ /πππ₯π₯ = β1/ππ πππ’π’/πππ‘π‘ It implies that a local measurement of velocity allows an estimate of strain. For a shear wave we find that shear stress is proportional to velocity:
ππ = ππ πππ’π’ /πππ₯π₯ = βππ/ππ πππ’π’/πππ‘π‘ This relation defines the impedance ππ/ππ of the material.
3.2 Energy considerations for harmonic waves For a harmonic wave of the form π’π’(π₯π₯, π‘π‘) = π΄π΄ cos(πππ₯π₯ β πππ‘π‘) where π΄π΄ is a real number: Kinetic energy density (per unit volume) πππΎπΎ = πποΏ½ΜοΏ½π’2/2 Potential energy density ππππ = ππ π’π’β²/2 Making use of ππ = πππππ’π’/πππ₯π₯, πππ’π’/πππ₯π₯ = β1/ππ πππ’π’/πππ‘π‘ and ππ2 = ππ/ππ, we can show that
ππππ = πππΎπΎ = ππππ2π΄π΄2sin2(πππ₯π₯ β πππ‘π‘) Integrating over one wavelength we get the average energies, 1
ππ β« β¦πππ₯π₯ππ+ππππ :
πππΎπΎ = ππππ = ππππ2π΄π΄2/4 Total average energy density = ππ = πΈπΈ
ππππππ= πππΎπΎ + ππππ = ππππ2π΄π΄2/2
Energy flux (per unit of cross-section surface, per unit of time) = οΏ½ΜοΏ½π = πΈπΈππππππ
= ππ ππππππππ
= ππ ππ = ππππ ππ2π΄π΄2/2 Note that ππππ = ππ/ππ = impedance. For a harmonic wave of the form π’π’(π₯π₯, π‘π‘) = π΄π΄ππππ(ππππβππππ) where A is a complex number:
οΏ½ΜοΏ½π =12ππππ
ππ2|π΄π΄|2 (26)
where |π΄π΄| is the modulus of π΄π΄.
3.3 Reflection and transmission at a material interface Consider two semi-infinite media in welded contact at π₯π₯ = 0. Medium 1 is in π₯π₯ β€ 0 and has wave speed ππ1 and shear modulus ππ1, medium 2 is in π₯π₯ β₯ 0 and has wave speed ππ2 and shear modulus ππ2. Consider a harmonic shear wave incident from medium 1, of the form ππππ(ππ1ππβππππ), with ππ1 = ππ/ππ1.
[Sketch] In medium 1 we have the incident and reflected waves:
π’π’1(π₯π₯, π‘π‘) = ππππ(ππ1ππβππππ) + π π ππβππ(ππ1ππ+ππππ) (27)
In medium 2 we have the transmitted wave, with ππ2 = ππ/ππ2: π’π’2(π₯π₯, π‘π‘) = ππ ππππ(ππ2ππβππππ) (28)
Boundary conditions at π₯π₯ = 0: Continuity of displacement: π’π’1(0, π‘π‘) = π’π’2(0, π‘π‘) Continuity of shear stress: ππ1πππ’π’1/πππ₯π₯(0, π‘π‘) = ππ2πππ’π’2/πππ₯π₯(0, π‘π‘) Combining the boundary conditions with eqs (27)-(28) yields:
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1 + π π = ππ (29)
1 β π π = πΌπΌ ππ (30)
where πΌπΌ = ππ2ππ2
/ ππ1ππ1
is the impedance ratio at the material interface. Solving this system of 2 equations
with 2 unknowns: ππ = 2/(1 + πΌπΌ) (31)
π π = (1 β πΌπΌ)/(1 + πΌπΌ) (32)
Verifications: If πΌπΌ = 1 (no material contrast), then ππ = 1 and π π = 0. If πΌπΌ = β (Dirichlet b.c.), then ππ = 0 and π π = β1. If πΌπΌ = 0 (Neumann b.c.), then π π = 1, but ππ = 2 instead of ππ = 0! Energy flux is conserved: οΏ½ΜοΏ½πππππππππππππππππ β οΏ½ΜοΏ½ππ π = οΏ½ΜοΏ½πππ (1 β π π 2 = πΌπΌππ2, after multiplying (29) and (30))
3.4 Normal modes of a finite elastic rod Consider a rod of finite length πΏπΏ.
[Discuss the vibrations of a guitar string. Sketch a standing wave.] Consider solutions of the 1D wave equation in the form of a standing wave (we seek a βseparable solutionβ to the PDE):
π’π’(π₯π₯, π‘π‘) = ππ(π₯π₯)ππ(π‘π‘) (33)
Plugging (33) into (9) we get ππ οΏ½ΜοΏ½π = ππ2ππ" ππ and οΏ½ΜοΏ½π
ππ(π‘π‘) = ππ2
ππ"
ππ(π₯π₯)
(34)
The l.h.s. is a function of π‘π‘ only and the r.h.s. is a function of π₯π₯ only. Both are necessarily equal to a constant. We are free to choose a name for the constant, letβs call it βππ2. Equation (34) leads to two separate ODEs, one for ππ(π‘π‘) and one for ππ(π₯π₯):
οΏ½ΜοΏ½π = βππ2ππ (35)
ππ" = βππ2/ππ2 ππ = βππ2ππ (36)
Note that we have introduced the wavenumber ππ = ππ/ππ. Their solutions are ππ(π‘π‘) β sin(πππ‘π‘ + ππ) (37)
ππ(π₯π₯) β sin(πππ₯π₯ + ππ) (38)
where ππ and ππ are constants called phase shifts. Combining them, we get a standing wave solution: π’π’(π₯π₯, π‘π‘) = π΄π΄ sin(πππ‘π‘ + ππ) sin(πππ₯π₯ + ππ) (39)
Assume fixed displacements as boundary conditions on both ends of the rod: π’π’(0, π‘π‘) = π’π’(πΏπΏ, π‘π‘) = 0. Applying these to (35) we get: ππ = 0 and sin(πππΏπΏ) = 0. This is satisfied by a discrete set of admissible wavenumbers:
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ππππ = (ππ + 1) ππ/πΏπΏ (40)
with ππ β β. The general solution is a discrete superposition of standing waves, each with a different wavenumber ππππ and associated frequency ππππ = ππ ππππ :
π’π’(π₯π₯, π‘π‘) = οΏ½π΄π΄ππ sin(πππππ₯π₯) sin(πππππ‘π‘ + ππππ)
β
0
(41)
Each term of this sum is called a mode. The amplitude π΄π΄ππ and phase ππππ are determined by initial conditions. The ππ-th mode has ππ zero-crossings.
[Draw modes] The mode with ππ = 0 is the fundamental mode. The fundamental frequency is ππ0 = ππ0/2ππ = ππ/2πΏπΏ. The others are higher modes or overtones and correspond to higher frequencies and shorter wavelengths.
3.5 Duality between modes and propagating waves Using the trigonometric relation
2 sin ππ sin ππ = cos(ππ β ππ) β cos(ππ + ππ), a mode can be re-written as
π΄π΄ππ2
[cos(ππππ(π₯π₯ β πππ‘π‘) β ππππ) β cos(ππππ(π₯π₯ + πππ‘π‘) + ππππ)] (42)
This is actually a superposition of two propagating waves.
[Mirror image trick, periodic functions, Fourier series.]
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4 Greenβs function. Waves in heterogeneous media.
4.1 Linear invariant systems, Greenβs functions, convolution So far we considered the wave equation (9) without forcing term: waves induced by initial conditions. We are now interested in waves induced by a source ππ(π₯π₯, π‘π‘).
ππ
ππ2π’π’ πππ‘π‘2
β ππππ2π’π’πππ₯π₯2
= ππ(π₯π₯, π‘π‘) (43)
A linear invariant system is defined by the following properties: Input Output ππ(π‘π‘) π’π’(π‘π‘) Multiply ππ ππ(π‘π‘) ππ π’π’(π‘π‘) Add ππ1(π‘π‘) + ππ2(π‘π‘) π’π’1(π‘π‘) + π’π’2(π‘π‘) Delay ππ(π‘π‘ β π‘π‘β²) π’π’(π‘π‘ β π‘π‘β²) Over the time scales of seismic wave propagation the Earth is a linear invariant system: it is linearly elastic and its material properties (density and elastic moduli) are fixed. We define the Greenβs function (aka impulse response, transfer function) as the motion induced by an impulse force πΏπΏ(π‘π‘). The motion produced by an arbitrary force ππ(π‘π‘) can be obtained from the Greenβs function by an integral operation known as convolution: Input Output πΏπΏ(π‘π‘) πΊπΊ(π‘π‘) Impulse response, Greenβs function ππ πΏπΏ(π‘π‘) + ππ πΏπΏ(π‘π‘ β π‘π‘β²) ππ πΊπΊ(π‘π‘) + ππ πΊπΊ(π‘π‘ β π‘π‘β²) Linear combination of impulses β« ππ(π‘π‘β²)πΏπΏ(π‘π‘ β π‘π‘β²)πππ‘π‘β² = ππ(π‘π‘) β« ππ(π‘π‘β²)πΊπΊ(π‘π‘ β π‘π‘β²)πππ‘π‘β² Continuum superposition of impulses = [ππ β πΏπΏ](π‘π‘) = [ππ β πΊπΊ](π‘π‘) = Convolution Important property of the Fourier transform: convolution in time domain is equivalent to multiplication in frequency domain:
ππ β πποΏ½ (Ο) = ππ(Ο) Γ πποΏ½(Ο) (44)
Convolution is easier to compute in spectral domain. Chain of linear invariant systems: π’π’(ππ) = πΊπΊ1(ππ) Γ πΊπΊ2(ππ) Γ β¦ Γ ππ(ππ)
[Ex: source, path, site, instrument or building]
4.2 Greenβs function for the 1D wave equation The solution of the wave equation with an impulsive point source ππ(π₯π₯, π‘π‘) = πΏπΏ(π₯π₯)πΏπΏ(π‘π‘) (per unit force, per second) is called the Greenβs function πΊπΊ(π₯π₯, π‘π‘). It is given by (displacement per unit force, per unit time):
πΊπΊ(π₯π₯, π‘π‘) =
ππ/ππ2
π»π» οΏ½π‘π‘ β|π₯π₯|πποΏ½
(45)
where π»π» is the Heaviside step function. [Sketch it]
Proof: The solution must comprise two symmetric waves propagating away from π₯π₯ = 0, hence it must be of the form πΊπΊ(π₯π₯, π‘π‘) = πΉπΉ(π‘π‘ β |π₯π₯|/ππ). Plugging this into the wave equation, and noting that π»π»β² = πΏπΏ, ππ|ππ|ππππ
= sign(π₯π₯) = 2π»π»(π₯π₯) β 1 and ππsign(π₯π₯)/πππ₯π₯ = 2πΏπΏ(π₯π₯), we get πΉπΉβ² = ππ2πππΏπΏ, hence πΉπΉ = ππ
2πππ»π».
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4.3 Waves in heterogeneous medium (WKBJ approximation) We have previously considered harmonic waves in homogeneous media:
π’π’(π₯π₯, π‘π‘) = π΄π΄ππππ(ππππβππππ) = π΄π΄ππππππ(ππ/ππβππ) (46)
Note that π₯π₯/ππ is the wave travel time over distance π₯π₯. Let us generalize this concept, at least approximately, to smoothly heterogeneous media. We consider spatially variable material properties ππ(π₯π₯) and ππ(π₯π₯). Inspired by harmonic waves, we adopt the following ansatz:
π’π’(π₯π₯, π‘π‘) = π΄π΄(π₯π₯)ππππππ(ππ(ππ)βππ) (47)
where the amplitude π΄π΄(π₯π₯) (a real number) and travel time ππ(π₯π₯) are smoothly varying functions of π₯π₯. Plugging this into the wave equation (7)
πππππ₯π₯
οΏ½ππ(π₯π₯)πππ’π’πππ₯π₯οΏ½ = ππ(π₯π₯)
ππ2π’π’ πππ‘π‘2
(48)
we get after some algebra: (πππ΄π΄β²)β² β ππ2ππππβ²2π΄π΄ + ππππ(2ππππβ²π΄π΄β² + (ππππβ²)β²π΄π΄) = βππππ2π΄π΄ (49)
Separating the real and imaginary parts we get two equations:
ππβ²2 β1c2
=(πππ΄π΄β²)β²
πππ΄π΄ππ2 (50)
2π΄π΄β²
A+
(ππππβ²)β²
ππππβ²= 0
(51)
In the high frequency limit, the r.h.s. of equation (50) can be neglected. This approximation is valid when the typical length scale of material heterogeneities, ππ/|ππβ²|, is much longer than the wavelength. We obtain then the so-called eikonal equation:
ππβ²2 β1c2
= 0 (52)
From which we derive the travel time:
ππ(π₯π₯) = οΏ½πππ₯π₯β²ππ(π₯π₯β²)
ππ
0
(53)
Integrating equation (51), making use of ππβ² = 1/ππ and denoting by ππ(π₯π₯) = ππ(π₯π₯)/ππ(π₯π₯) the local impedance, we get the wave amplitude:
π΄π΄(π₯π₯) = π΄π΄(0)οΏ½ππ(0)/ππ(π₯π₯)
(54)
Note that this is a statement of conservation of energy along the wave path (see section 3.1): ππ(π₯π₯)π΄π΄(π₯π₯)2 = πππππππππ‘π‘πππππ‘π‘.
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5 The 3D elastic wave equation Notation: bold quantities are column vectors, with 3 components in 3D. See also chapters 2 and 3 of Peter Shearerβs book. See Tromp and Dahlenβs book for gravitational, Earthβs rotation and pre-stress effects (not treated here).
5.1 Strain Displacement field: ππ(ππ). Deformation, related to the displacement gradient (a tensor): ππ(ππ + π π ππ) β ππ(ππ) + ππππ:π π ππ. Where ππ = (ππππ ,πππ¦π¦ , πππ§π§) is the gradient differential operator. The displacement gradient ππππ = ππ πππ»π» is a tensor, (βπ’π’)ππππ = πππππ’π’ππ Strain tensor = symmetric part of the displacement gradient: ππ = 1
2οΏ½ ππππ + πππππ»π»οΏ½, ππππππ = 1
2(πππππ’π’ππ + πππππ’π’ππ).
Assumption: small perturbations relative to a static configuration, ππ βͺ 1 [earthquake strain ~ slip/(rupture length) ~ m / 10 km ~0.01%]
5.2 Stress Traction ππ(ππ,ππ) is the force per unit of surface area acting on an oriented surface with normal ππ centered on ππ.
[sketch] The dependence on ππ is encapsulated in the stress tensor: ππ(ππ,ππ) = ππ(ππ) β ππ, π‘π‘ππ = ππππππππππ The i-th column of ππ is the traction on a surface whose normal is the unit vector along the i-th dimension. Owing to conservation of angular momentum, ππ is a symmetric tensor: ππππππ = ππππππ
5.3 Momentum equation Apply πΉπΉ = ππππ to a volume, and
οΏ½πποΏ½ΜοΏ½π ππππ = οΏ½ππ ππππ = οΏ½ππ β ππ ππππ (55)
Applying Gaussβ theorem (aka divergence theorem; applied here to the tensor field ππ(ππ)) we transform the surface integral into a volume integral:
οΏ½πποΏ½ΜοΏ½π ππππ = οΏ½ππ β ππ ππππ (56)
where ππ β ππ is the divergence of ππ, a vector with components (ππ β ππ)ππ = βjΟij This is valid for any volume, hence
πποΏ½ΜοΏ½π β ππ β ππ = 0 (57)
πποΏ½ΜοΏ½π’i β βjΟij = 0 (58)
5.4 Elasticity We also need a constitutive equation relating stress to strain. Assumption: linear elastic material. Not valid near the source, but we assume that non-linearity occurs on length scales smaller than the wavelengths we will investigate. Hookeβs law:
ππ = ππ: ππ (59)
ππππππ = ππππππππππππππππ (60)
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where ππ is the elastic tensor. Its components ππππππππππ are material properties (elastic moduli). Assumption: isotropic elasticity
ππππππ = πππππππππΏπΏππππ + 2ππππππππ (61)
where ππ and ππ are the Lame coefficients.
5.5 The seismic wave equation Combining Hookeβs law with eq (57):
πποΏ½ΜοΏ½π β ππ β (ππ: ππ) = 0 (62)
For isotropic elasticity:
πποΏ½ΜοΏ½π’ππ = (ππ + ππ)
ππ2π’π’πππππ₯π₯πππππ₯π₯ππ
+ ππππ2π’π’πππππ₯π₯ππ2
(63)
which can also be written as πποΏ½ΜοΏ½π = (ππ + ππ)ππ(ππ β ππ) + ππππππππ (64)
where ππππππ = (ππ β ππ)ππ is the Laplacian of ππ.
5.6 Itβs a perturbative equation So far ππ and ππ denote the total stress and displacement, respectively. Consider now that before seismic waves are generated by a certain source, the Earth is in static equilibrium with stress and displacement (ππ0,ππ0) satisfying eq (57) with zero acceleration.
βππ β ππ0 = 0 (65)
We assume that the subsequent transient motion comprises small perturbations (πΏπΏππ, πΏπΏππ) relative to the initial static configuration (ππ0,ππ0). Subtracting (65) from (57) we find that the perturbations (πΏπΏππ, πΏπΏππ) =(ππ,ππ) β (ππ0,ππππ) also satisfy
πππΏπΏοΏ½ΜοΏ½π β ππ β πΏπΏππ = 0 (66)
The material might have a non-linear behavior, ππ = πΉπΉ(ππ). If the perturbations are small enough, we can linearize the constitutive relation near the initial configuration: πΏπΏππ = πΉπΉ(ππ0 + πΏπΏππ) β πΉπΉ(ππ0) β βπΉπΉ: πΏπΏππ. This has the same form as Hookeβs law but for the perturbations, πΏπΏππ = ππ: πΏπΏππ, if we define the effective linear elastic tensor as ππ = βπΉπΉ. Hence the governing equation (62) also applies to the perturbations πΏπΏππ and πΏπΏππ. In the remainder we will be concerned only with the perturbations, but to simplify the notations we will drop the πΏπΏ.
5.7 P and S waves examples Longitudinal waves (P waves): assume ππ(ππ, π‘π‘) = π’π’1(π₯π₯, π‘π‘)πποΏ½ (no dependence on π¦π¦ or π§π§) where πποΏ½ is the unit vector in the ππ direction. Plugging it into the wave equation we get:
ππππ2π’π’1πππ‘π‘2
= (ππ + 2ππ) ππ2π’π’1 πππ₯π₯2
(67)
This is a wave equation like eq (9), with P wave speed
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ππππ = οΏ½
ππ + 2ππππ
(68)
Transverse waves (S waves): assume ππ(ππ, π‘π‘) = π’π’3(π₯π₯, π‘π‘)πποΏ½
ππππ2π’π’3πππ‘π‘2
= ππ ππ2π’π’3 πππ₯π₯2
(69)
S wave speed:
ππππ = οΏ½ππ/ππ
(70)
Rocks usually have ππ β ππ (a Poisson solid), hence ππππ β β3ππππ. Typically wave speeds ~ several km/s, but can be ~ 100 m/s on the shallowest layers.
5.8 General decomposition into P and S waves Letβs show that the 3D wavefield is a superposition of two types of body waves. Using the following vector identity
ππ2ππ = ππ(ππ β ππ) β ππ Γ ππ Γ ππ (71)
where ππ Γ ππ is the curl of ππ, eq (64) can be written as οΏ½ΜοΏ½π = ππππ2 ππ(ππ β ππ) β ππππ2 ππ Γ ππ Γ ππ (72)
The Helmholtz theorem states that any sufficiently smooth, rapidly decaying 3D vector field ππ can be decomposed as the sum of a curl-free vector field and a divergence-free vector field. This decomposition can also be expressed as the sum of the gradient of a scalar potential ππ and the curl of a divergence-free vector potential ππ (i.e. ππ β ππ = 0):
ππ = ππππ + ππ Γ ππ (73)
The curl of the gradient of any 3D scalar field is always the zero vector, hence the vector field ππππ is curl-free (ππ Γ ππππ = ππ). The divergence of the curl of any 3D scalar field is always the zero, hence the vector field ππ Γ ππ is divergence-free (ππ β (ππ Γ ππ) = 0). Combining (73) and the divergence and curl of (72), respectively, we arrive at two partial differential equations (actually not quite, see AR for a more rigorous derivation):
οΏ½ΜοΏ½π = ππππ2β2ππ (74)
οΏ½ΜοΏ½π = ππππ2 β2ππ (75)
These are two 3D wave equations for P wave and S wave potentials, respectively.
5.9 Polarization of body waves A particular solution of the 3D wave equation for the scalar potential is a plane wave:
ππ(ππ, π‘π‘) = π΄π΄ππππ(ππβ ππβππππ) (76)
where ππ is the wave vector, which indicates the direction of propagation. [Sketch]
The phase velocity of the plane wave is ππ/|ππ|. The wave equation is satisfied if
ππ/|ππ| = ππππ (77)
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The P wave displacement is [show it] ππππ = βππ = ππππ (78)
It is parallel to the direction of wave propagation ππ. Similarly, assuming that the vector potential ππ is a plane wave one can show that the S wave displacement is
πππΊπΊ = ππ Γ ππ = ππ Γ ππ (79)
This is perpendicular to the wave propagation direction ππ.
5.10 Usual characteristics of body waves P vs S:
β’ speed: S is slower. Typically, ππππ/ππππ = β3 and epicentral distance (km) β 8 Γ P-S travel time (s) β’ polarization (see above). Refraction at shallow depth P vertical, S horizontal β’ amplitude: S is stronger β’ frequency content: S is often lower frequency (attenuation)
[You can guess the distance to an earthquake if you feel P and S waves.
Principle of early warning systems: P information carrier, S damage carrier.]
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6 Body waves
6.1 Spherical waves, far-field, near-field We now consider the wavefield induced by an explosive (isotropic) point-source with source-time-function ππ(π‘π‘). Considering the isotropy of the problem, we seek radially symmetric solutions that derive from a scalar potential ππ(ππ, π‘π‘) satisfying
β2ππ β1ππππ2οΏ½ΜοΏ½π = β4πππΏπΏ(ππ)ππ(π‘π‘) (80)
It can be shown that the solution is ππ(ππ, π‘π‘) = β
1ππππ οΏ½π‘π‘ β
πππππποΏ½ (81)
The displacement field, ππ = βππ, is radially symmetric. Its radial component is π’π’ππ(ππ, π‘π‘) = 1
ππ2ππ οΏ½π‘π‘ β ππ
πππποΏ½ β 1
ππππππππΜ οΏ½π‘π‘ β ππ
πππποΏ½ (82)
The first term describes the near-field term and has the following properties: β’ It decays as 1/ππ2 β’ A persistent source (i.e. ππ β 0 when π‘π‘ β β) leaves a static residual displacement
The second term describes the far-field motion and has the following properties: β’ Transient, it vanishes after the passage of the wave (ππΜ is non-zero only over a finite interval) β’ 1/r decay, consistent with conservation of total energy ππ β 4ππππ2πποΏ½ΜοΏ½π’2 (energy density times
surface area of the spherical wavefront) The far-field term dominates over the near-field term when ππ β« ππππ/ππ = ππ/2ππ, i.e. at high-frequencies / long distances. (Take the Fourier transform of π’π’ππ, then compare the two terms).
6.2 Ray theory: eikonal equation, ray tracing In smoothly heterogeneous media, approximate solutions of the form
ππ(ππ, π‘π‘) = π΄π΄(ππ)ππππππ(ππ(ππ)βππ) (83)
can be found at high frequencies corresponding to wavelengths much shorter than the characteristic length scales of heterogeneity of the material properties (see also LW p.72). The travel time ππ(ππ) satisfies the 3D eikonal equation:
|ππππ|2 = 1/ππ(ππ)2 (84)
Solving this non-linear differential equation with a given origin point (ππ(ππ0) = 0) gives the spatial distribution of travel times. The contours of ππ(ππ) are the wave fronts. The normal to these contours is parallel to ππππ. The paths connecting these normals define rays.
[sketch wave fronts and rays] Denoting the local ray direction by the slowness vector ππ = ππππ and defining a curvilinear coordinate ππ along a ray, the ray tracing problem is formulated as: given an origin point, ππ(ππ = 0), and an initial ray direction (take-off vector), ππ(ππ = 0), compute the ray path ππ(ππ) and slowness ππ(ππ) by solving:
ππππππππ
= ππ(ππ) ππ (85)
ππππππππ
= πποΏ½1
ππ(ππ)οΏ½ (86)
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6.3 Rays in depth-dependent media, Snellβs law, refraction Consider the special case of a medium with depth-dependent material properties: ππ(ππ) = ππ(π§π§). From the last equation we get ππππππ/ππππ = πππππ¦π¦/ππππ = 0: the horizontal components of the slowness vector are constant and rays remain confined on a vertical plane. The constant horizontal slowness (aka
apparent slowness) defines the ray parameter ππ = οΏ½ππππ2 + πππ¦π¦2.
Because |ππ| = 1/ππ, we have ππ = sinππ /ππ and πππ§π§ = cosππ /ππ, where ππ is the angle between the ray and the vertical axis. The constancy of the ray parameter gives Snellβs law at the interface between two materials:
ππ = sinππ1/ππ1 = sinππ2 /ππ2 (87)
Continuity of the wave front along an interface also implies Snellβs law. It can also be shown for plane waves (not restricted to high-frequency ray theory). It is also implied by Fermatβs principle: ray path is optimal, it has shortest travel time.
[Analogy: getting from A to B across a river, by running and swimming.] If wave speed increases as a function of depth, Snellβs law implies that upward rays get refracted (bent) towards the vertical. Downward rays refract towards horizontal and eventually turn up. Refraction potentially brings information back to the surface.
[Draw a downward refracted ray. Add more rays]
Wave gets reflected at the surface, so this can repeat multiple times. Rays with more vertical take-off (lower p) resurface further away. Prograde branch. Its ray parameter is ππ = sin(ππ0) /ππ(0). At the turning point sin(ππ) = 1. From Snellβs law, the depth of the turning point is such that ππ = 1/ππ(π§π§). [Q: what is the angle of a ray emitted by a shallow source that penetrates down to depth where S wave
speed is 3 km/s, if shallow speed is 300 m/s?]
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7 More on body waves
7.1 Layer over half-space: head waves Assume shallow layer is slower. Body waves: direct, reflected and transmitted.
[Draw rays. L&W fig 3.9] Angle of transmitted wave becomes horizontal (sin(ππ) = 1) when incidence reaches a critical angle ππππ such that
sin(ππππ) =πππ π πππ π π π πππππππ π ππ
(88)
The critically refracted wave is called a head wave or refracted wave. [Plot travel times. L&W fig 3.10. Order: direct, reflected, head]
Key features of the travel time curves are related to Earth structure. Asymptotes give velocities of both layers. Head wave appears beyond a critical distance. It has a cross-over distance with the direct wave (should I take the highway or not? Depends on how far I am going). Both depend on depth of interface and velocity contrast. Seismic refraction imaging. Ray parameter p = slope dT/dX of travel time curve T(X). It can be estimated by a small-aperture array of seismometers. Application: discriminate deep from shallow sources.
Earth example: Moho discontinuity crust/mantle. Moho depth = a few 10 km in continents, shallower in oceans. Direct wave Pg. Reflected wave PmP. Head wave Pn. Crossover at 150 km continental, 30 km oceanic crust. Discovered by Mohorovicic (1909). Plus: S waves, P-SV converted phases, multiple reflections. Pn waveforms are complicated. Note that SH cannot convert into P.
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7.2 A steep transition zone [Shearerβs Fig bottom p72. Fig 4.5]
Figure by P. Richards http://www.ldeo.columbia.edu/~richards/ARhtml/add_to_Sec9.4.html
SW Fig 3.4-6
Triplication. Prograde and retrograde branches. Complicated waveforms. Earth examples: Upper mantle 440 and 610 discontinuities (triplications at 15 and 24 deg). Compare to layer over half-space: long vs short wavelength views of the same structure. Frequency-dependency of the wavefield. At end of these branches: caustics, energy focusing (multiple rays converge on the same point).
[Discuss caustics at bottom of a pool]
7.3 A low velocity zone [Shearerβs fig 4.9]
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Fig by P. Richards http://www.ldeo.columbia.edu/~richards/ARhtml/add_to_Sec9.4.html
SW Fig 3.4-7
Shadow zone. Ex: vp drops at the core-mantle boundary then keeps increasing, shadow zone from 103 to 140 deg. Trapped waves (guided waves) if source is inside the LVZ.
[Shearerβs fig 4.10]
Ex: SOFAR (Sound Fixing and Ranging) channel in oceans. T waves. Less than 12 deg from horizontal. Earthquakes may induce them by diffraction at bathymetry.
[Determine the geometrical spreading of a guided wave from energy argument] Guided waves have slow geometrical spreading β 1/βππ. Actually decay a bit faster, 1/ππππ with 1
2< ππ <
1, because of energy leakage. Nevertheless, they persist over longer distances than other body waves. Other ex: fault zone guided waves.
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7.4 Wave amplitude along a ray Energy travels along the ray. Consider a group of neighboring rays, a ray beam of cross-section dS.
[Sketch a ray beam] Energy density flux of plane wave = 1
2πππ΄π΄2ππ2, where ππ = ππππ = impedance.
Energy flux through a cross-section of a ray beam is conserved along the ray path and β πππ΄π΄2 ππππ Hence
π΄π΄1π΄π΄2
= οΏ½ππ2ππππ2ππ1ππππ1
(89)
Ex: spherical wave = ππ(ππππππ/2)2 , hence π΄π΄ β 1/ππ.
οΏ½ ππππππππ2
βΌ spreading factor.
Wave focusing shrinks the spreading factor, hence amplifies wave motion (more waves arriving together). [Shearer section 6.2]
ππ β 1/ οΏ½πππππππποΏ½
Discuss caustics again.
7.5 Ray parameter in spherically symmetric Earth In a spherically symmetric Earth we define a modified ray parameter such that it is conserved along a great circle path (along the intersection of the vertical plane containing the ray and the Earth surface):
ππβ² =
ππ sin(ππ)ππ
(90)
In a layer with constant speed, trigonometric arguments show that [Sketch Shearerβs fig 4.12] ππβ² = πππππππππ‘π‘πππππ‘π‘
At a material interface, this can be derived from Snellβs law. In terms of great-circle distance Ξ in radians, ππππ = ππππΞ, ππβ² = ππππ = ππππππ/ππππ
ππβ² = ππππ/ππΞ (91)
7.6 Body waves in the Earth Figures from S&W section 3.5 1D Earth models (PREM, IASP91, etc). Main concepts developed in the 40s. Provide reference model for studies of lateral heterogeneities, and inform about physical, chemical, thermal and mineralogical state of the Earthβs materials. Seismological data: arrival time of several phases (families of ray paths).
[Examples of what seismologists do with deep seismic phases]
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+ Dββ = region with reduced gradient above the CMB (2700-2900 km)
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8 Surface waves I: Love waves
8.1 Separation between SH and P-SV waves
In depth-dependent media rays remain confined in the vertical plane of their take-off vector (see 6.3). SV waves = shear motion within the plane of the ray SH waves = shear motion normal to the plane of the ray In a depth dependent medium SH waves are decoupled from P-SV waves, i.e. the equations governing these two systems of waves are independent.
8.2 SH reflection and transmission coefficients at a material interface
Show that SH displacement field of the form π’π’(π₯π₯, π§π§, π‘π‘)πποΏ½ satisfies a scalar wave equation with S-wave speed. Consider two materials in contact along a planar interface, with speeds π½π½1 and π½π½2, and an incident plane wave arriving to the interface from medium 1. Formulate the problem:
β’ Write the plane wave displacement field in both materials, composed of incident, reflected and transmitted waves of amplitude 1, R and T, respectively (z points down):
π’π’1(π₯π₯, π§π§, π‘π‘) = ππππππ(ππππ+ππ1π§π§βππ) + π π ππππππ(ππππβππ1π§π§βππ) π’π’2(π₯π₯, π§π§, π‘π‘) = ππ ππππππ(ππππβππ2π§π§βππ)
where ππ = πππ₯π₯ππ
is the ray parameter, ππ1 = πππ§π§1
ππ= οΏ½
1π½π½12β ππ2 and ππ2 = πππ§π§2
ππ= οΏ½
1π½π½22β ππ2. If ππ < 1/π½π½ππ,
the incidence/reflection/transmission angles are well defined (real) and ππππ = cos ππππ. β’ Write the boundary conditions: continuity of displacement and shear stress at the interface.
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β’ Obtain a system of two linear equations with two unknowns (R and T). β’ Solve it (defining impedances ππ1 = ππ1/π½π½1 and ππ2 = ππ2/π½π½2):
π π = ππ1ππ1βππ2Ξ·2ππ1Ξ·1+ππ2Ξ·2
and ππ = 2 ππ1ππ1ππ1Ξ·1+ππ2Ξ·2
If ππ < 1/π½π½ππ we can write it as:
π π = ππ1 cos ππ1βππ2 cos ππ2ππ1 cos ππ1+ππ2 cos ππ2
and ππ = 2 ππ1 cos ππ1ππ1 cos ππ1+ππ2 cos ππ2
If π½π½1 < π½π½2 there is a critical angle ππππ defined by
sin ππππ = π½π½1/π½π½2 such that any wave with incidence angle wider than ππππ (post-critical) emerges parallel to the interface (ππ2 = ππ/2). Post-critical reflections have |π π | = 1 (total reflection) and incur a phase shift. The refracted wave in the post-critical range decays exponentially with distance to the interface: this
kind of wave is called inhomogeneous or evanescent wave: if ππ > 1π½π½2
, the quantity ππ2 = οΏ½1π½π½22β ππ2 is
imaginary and the displacement in the bottom half-space is π’π’2(π₯π₯, π§π§, π‘π‘) = ππ ππππππ(ππππβππ)ππβππ πΌπΌπΌπΌ(ππ2)π§π§ . The exponential decay has a characteristic depth β 1
πΌπΌπΌπΌ(ππ2)ππ. If ππ β« 1/π½π½2 the penetration depth is ~ 1
ππππβ ππππ
In the special case of a free surface (ππ2 = 0) we get total reflection (π π = 1) at all incident angles.
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8.3 Love waves Consider a soft layer of thickness h over a half space. Post-critical waves are trapped by total reflection at the surface and at the material interface.
Constructive interference between these trapped waves leads to surface waves known as Love waves, a system of post-critical waves that propagate horizontally and whose amplitude below the soft layer decays exponentially with depth (they are evanescent). The penetration depth of a surface wave is βΌ ππ (horizontal wavelength).
The amplitude of surface waves decays as 1/βππππ where ππ is horizontal propagation distance. Proof: energy is proportional to wave amplitude squared and is conserved over a ring of surface β ππππ (as opposed to a sphere of surface β ππ2 in the case of body waves).
8.4 Dispersion relation The longer wavelengths / lower frequencies penetrate deeper, hence probe faster speeds of the medium: hints that the Love wave speed depends on frequency, a phenomenon called dispersion. Derivation of dispersion relation:
β’ Write displacement fields as plane waves in each of the two media π’π’1(π₯π₯, π§π§, π‘π‘) = π΄π΄ππππππ(ππππ+ππ1π§π§βππ) + π΅π΅ππππππ(ππππβππ1π§π§βππ)
π’π’2(π₯π₯, π§π§, π‘π‘) = πΆπΆ ππππππ(ππππβππ2π§π§βππ) β’ Apply boundary conditions (3): zero stress at the surface (at z=0) and continuity of displacement
and shear stress at the interface (at z=h) β’ Obtain a homogeneous (zero right-hand-side) system of 3 linear equations with 3 unknowns (A,
B and C).
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β’ A homogeneous system has non-trivial (non-zero) solutions only if its determinant is zero. This condition yields the following equation relating horizontal speed ππ = ππ/ππ to frequency, known as a dispersion relation:
tanοΏ½βπππ½π½1
οΏ½1 β οΏ½π½π½1πποΏ½2
οΏ½ =ππ2ππ1
οΏ½οΏ½π½π½2ππ οΏ½2β 1
οΏ½1 β οΏ½π½π½1ππ οΏ½2
For any frequency ππ, this equation can be solved to get a speed ππ(ππ). Because of the trigonometric (periodic) function on the l.h.s., solutions are not always unique; in some frequency ranges we get a set of speeds ππππ(ππ).
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Fundamental mode. (ππ = 0) Penetration depth is shorter at high frequency, probes shallower part of the crust, lower velocity. Two extremes: ππ βΌ π½π½1 at high frequency, ππ βΌ π½π½2 at low frequency. General appearance of Love wave seismogram: different frequencies arrive at different times. Overtones. (ππ > 0) In some frequency ranges the dispersion relation has multiple solutions: the n-th higher modes (overtone) appears at frequencies higher than cutoff frequency
ππππ =ππππ
βοΏ½1π½π½12
β 1π½π½22
8.5 Phase and group velocities Consider the superposition of two harmonic waves with slightly different parameters, (ππ1,ππ1) = (ππ0 +πΏπΏππ,ππ0 + πΏπΏππ) and (ππ1,ππ1) = (ππ0 β πΏπΏππ,ππ0 β πΏπΏππ):
cos(ππ1π₯π₯ β ππ1π‘π‘) + cos(ππ2π₯π₯ β ππ2π‘π‘) We can rewrite it as
2 cos(ππ0π₯π₯ β ππ0π‘π‘) cos(πΏπΏπππ₯π₯ β πΏπΏπππ‘π‘) Product of two terms: a fast (high-frequency) carrier modulated by a slow (low frequency) envelope.
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Phase velocity = ππ(ππ) = πππππ₯π₯
= propagation speed of the carrier
Group velocity = ππ(ππ) = πΏπΏπππΏπΏππ
= propagation speed of the envelope Compare: in homogeneous media, ππ/ππ = ππ. Phase velocity is constant, does not depend on frequency: no dispersion. Group velocity is also = ππ.
8.6 Airy phase
ππ =ππππππππ
=1
ππ οΏ½ππππ οΏ½ππππ
=ππ
1 βππππππππππππ
=ππ
1 + ππππππππππππ
where ππ = 2ππππ
= period.
Because ππππππππ
> 0, the group velocity U may have a minimum.
Surface waves with a range of frequencies close to the minimum U have very similar U, hence they arrive close together and add up to create a high amplitude phase known as Airy phase (see example in next lecture).
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9 Surface waves II: Rayleigh waves
9.1 Rayleigh waves See L&W 4.1 Displacement wavefield: ππ = ππππ + ππ Γ ππ In 2D (π₯π₯, π§π§), the SV potential is off-plane, its curl is in-plane: ππ = (0,πππ¦π¦, 0). P-SV potentials in the form of harmonic waves (z points down):
ππ = π΄π΄ ππβπποΏ½ππππβπππ₯π₯ππβπππ§π§πππ§π§οΏ½ = π΄π΄ ππππππ(ππππ+πππππ§π§βππ) πππ¦π¦ = π΅π΅ ππππππ(ππππβπππ π π§π§βππ)
where ππ = πππ₯π₯ππ
, ππππ = πππ§π§ππ
ππ= οΏ½
1ππππ2 β ππ2 and ππππ = πππ§π§ππ
ππ= οΏ½
1ππππ2 β ππ2.
Evanescent waves if ππ > 1ππππ
> 1ππππ
, leads to surface wave confined near the surface ~ππβππ πΌπΌπΌπΌ(ππ)π§π§ Free surface boundary condition, at π§π§ = 0:
πππ§π§π§π§ = πποΏ½π’π’ππ,ππ + π’π’π§π§,π§π§οΏ½ + 2πππ’π’π§π§,π§π§ = 0 πππππ§π§ = πποΏ½π’π’π§π§,ππ + π’π’ππ,π§π§οΏ½ = 0
This leads to a linear system of two equations and two unknowns (A and B). Non-trivial solutions exist only if its determinant is zero. That condition leads to
οΏ½2 βππ2
ππππ2οΏ½2
β 4οΏ½1 βππ2
ππππ2 οΏ½1 β
ππ2
ππππ2= 0
where ππ = 1/ππ is the apparent velocity. For a given value of the ratio ππππ/ππππ, this equation has a single solution ππ/ππππ < 1, the Rayleigh wave speed. For a Poisson solid (ππ = 1/4), we find ππ β 0.92 ππππ.
[LW fig 4.6]
Ground motion (at the surface) is elliptical and retrograde. Ellipticity is depth-dependent. Penetration depth ~ horizontal wavelength ~ 1/frequency. Sensitivity to material properties and sources depends on depth. Hypocenter depth acts as a filter: shallower earthquakes excite more efficiently the higher frequencies.
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In a homogeneous half-space, Rayleigh wave speed is constant, there is no dispersion. But in a medium with depth-dependent wave speed, surface waves of lower frequency probe deeper materials, and hence Rayleigh waves are dispersive.
Surface wave dispersion in PREM.
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Stacked long-period vertical seismograms (positive black, negative white). Note R1, R2 (left), R3, R4 (right). Dispersion. Phase vs group velocity.
9.2 Surface waves in a heterogeneous Earth Love wave ansatz: π’π’ = β Γ ππ where ππ = (0,0,ππ(π§π§)ππ(π₯π₯,π¦π¦,ππ)) and ππ(π§π§) is the Love wave eigenfunction at frequency ππ derived from the local 1D velocity model. Plug it into the seismic wave equation, leads to
ππ2πππππ₯π₯2
+ππ2πππππ¦π¦2
+ππ2
ππ2(ππ, π₯π₯,π¦π¦) ππ = 0
Where ππ(ππ, π₯π₯,π¦π¦) is the local Love wave phase velocity at location (π₯π₯,π¦π¦) and frequency ππ. Applying WKBJ approximation leads to ray theory along the surface. Great circle path. Focusing. Laterally βtrappedβ surface waves in basins. Body to surface wave conversion at sediment edges.
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9.3 Implications for tsunami waves
The dispersion at short periods was explained in a homework. The derivation assumed a rigid seafloor. The dispersion at long periods is only observed for mega-earthquakes. It is due to the coupling between wave height - water pressure changes β and elastic deformation of the crust (deformable seafloor). See Tsai et al (2013).
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10 Normal modes of the Earth
Modes in an acoustic sphere [see derivation in LW p 156-158]:
β’ Write scalar wave equation in spherical coordinates. β’ Assume a separable solution π’π’(ππ,ππ,ππ, π‘π‘) = π π (ππ)Ξ(ππ)Ξ¦(ππ)ππ(π‘π‘) an plug it in wave equation β’ Find 4 separate differential equations β’ Solutions are related to well-known special functions β’ The surface dependence Ξ(ππ)Ξ¦(ππ) is given by spherical harmonics
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o ππ = angular order number, aka spherical harmonic degree. Positive integer. Total number of nodal lines (zero crossing lines)
o ππ = azimuthal order number. 2ππ + 1 integer values: βππ β€ ππ β€ ππ. Number of nodal lines through the pole = |ππ|.
β’ The radial dependence π π (ππ) is given by spherical Bessel functions
ππππ(ππ) = π₯π₯ππ οΏ½β 1πππππππποΏ½ππ sin ππ
ππ where π₯π₯ = ππππ
ππ
β’ The time dependence ππ(π‘π‘) is sinusoidal, with frequency πππππΌπΌ
β’ Applying boundary conditions at the surface and center of the sphere leads to a family of admissible frequencies, or eigenfrequencies ππππ
πΌπΌ for each admissible pairs ππ, ππ , where ππ = radial order number
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Mode decomposition in the elastic Earth.
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Properties of eigenfrequencies ππππππ :
β’ They do not depend on azimuthal order ππ β’ Singlets of same ππ are grouped in multiplets. β’ Degeneracy: in a perfectly radial, isotropic, non-rotating Earth, all singlets in a multiplet have the
same eigenfrequency. In reality there is mode splitting
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Duality modes β waves:
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11 Attenuation and scattering
11.1 Attenuation of normal modes Attenuation: the amplitude of modal vibrations decays exponentially with time due to anelastic dissipation processes, including shear heating at grain boundaries, dislocation sliding of crystal defects.
Q is usually β« 1.
In one cycle, energy (amplitude squared) decays by a factor πΈπΈοΏ½ππ+2ππππ οΏ½
πΈπΈ(ππ) = exp οΏ½β 2πππποΏ½ β 1 β 2ππ
ππ .
Energy loss per cycle: ΞπΈπΈπΈπΈ
= β2ππππ
.
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11.2 A damped oscillator Mass-spring-dashpot system, single degree of freedom, subject to an impulse force:
πποΏ½ΜοΏ½π₯ + πποΏ½ΜοΏ½π₯ + πΎπΎπ₯π₯ = πΏπΏ(π‘π‘) Displacement:
π₯π₯(π‘π‘) = π΄π΄ exp(βππππ0π‘π‘) sin (οΏ½1 β ππ2ππ0π‘π‘) where ππ0 = οΏ½πΎπΎ/ππ and ππ = ππ/ππππ0 Here ππ = 1/2ππ.
In a standard linear solid, Q is not constant but depends on frequency (see LW p 111-112):
11.3 A propagating wave Consider a plane wave, π’π’(π₯π₯, π‘π‘) = π΄π΄(π₯π₯) exp οΏ½βππππ οΏ½π‘π‘ β ππ
πποΏ½οΏ½, in an attenuating medium. In a frame that
tracks the wave front (βriding the waveβ), wave amplitude decays with increasing travel time ππ:
π΄π΄(ππ) = π΄π΄0 exp οΏ½βππππ2ππ
οΏ½
or, equivalently, with increasing propagation distance π₯π₯ = ππππ:
π΄π΄(π₯π₯) = π΄π΄0 exp οΏ½βπππ₯π₯2ππππ
οΏ½
Hence,
π’π’(π₯π₯, π‘π‘) = π΄π΄0exp οΏ½βπππ₯π₯2ππππ
οΏ½ exp οΏ½βππππ οΏ½π‘π‘ βπ₯π₯πποΏ½οΏ½ = π΄π΄0 exp οΏ½βππππ οΏ½π‘π‘ β
π₯π₯ππβ²οΏ½οΏ½
where 1ππβ²
=1ππ
+ππ
2ππππ
ππβ² =ππ
1 + ππ2ππ
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Along a ray in a smoothly heterogeneous medium:
π’π’(π₯π₯, π‘π‘) = π΄π΄0exp οΏ½βπππ‘π‘β
2οΏ½ expοΏ½βππππ(π‘π‘ β ππ(π₯π₯))οΏ½
where ππ(π₯π₯) = β« ππππ
ππ(ππ)πππππ¦π¦ = travel time,
and π‘π‘β = β« ππππ
ππ(ππ)ππ(ππ)πππππ¦π¦ =characteristic attenuation time
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π΄π΄(π₯π₯) = π΄π΄0 exp οΏ½βπππ₯π₯2ππππ
οΏ½
In regions with stronger attenuation (lower Q) strong ground motion has shorter reach. At fixed Q, higher frequencies are damped more severely. This reduces the signal-to-noise ratio at high frequencies and makes it challenging to extract information about small scales of Earthβs structure or earthquake sources.
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12 Scattering Movie of wave front interacting with a low velocity anomaly: http://web.utah.edu/thorne/movies/Movie_Low_Velocity_Anomaly_720p.wmv Distortion and healing of the wave front. Multipathing: rays that follow different paths but arrive at the same time at a station
Scattering: sharp features act as point sources, but they donβt generate energy, they only redistribute it. Movie of scattered wavefield in a randomly heterogeneous medium: http://web.utah.edu/thorne/movies/Scattering_720p.wmv
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Seismograms recorded at different epicentral distances have similar coda envelopes:
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Observed energy (waveform envelope squared) decays as
ππ~ exp οΏ½βπππ‘π‘πππποΏ½ /π‘π‘ππ
where ππππ = coda attenuation quality factor.
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Single scattering. The ellipsoids in the previous figure are isochrones: surfaces grouping positions of scattering points whose scattered waves reach the station at the same time. Waves from scattering sources in a given isochrone have same arrival time but different amplitude. For an elementary scattering volume containing a collection of scattering points, define ππ0 = scattering coefficient = scattered energy per unit of volume, averaged over all directions. It combines information about the βstrengthβ of each scatterer (size, material contrast, etc) and the density of scatterers. A plane wave leaks out energy by scattering. Its energy decays as exp (βππ0π₯π₯). Defining scattering attenuation as πππ π ππ = ππ/ππ0ππ, the energy decay is βΌ exp (βπππ₯π₯/πππππ π ππ). If A=source, B=scattering point and C=station, the energy scattered by point B is:
ππ β ππ01πππ΄π΄π΄π΄2
ππ01πππ΄π΄π΅π΅2
The total contribution integrated over one isochrone is
ππ β ππ0 οΏ½1πππ΄π΄π΄π΄2
ππ01πππ΄π΄π΅π΅2
πππππππ π π π ππβπππ π ππππ(ππ)
The amplitude of coherent waves adds up. The amplitude squared (energy) of incoherent waves add up. Travel time π‘π‘ = (πππ΄π΄π΄π΄ + πππ΄π΄π΅π΅)/ππ. It can be shown that sufficiently long after the first arrival, when π‘π‘ β« ππ/ππ,
ππ βππ0π‘π‘2
Long after the passage of the main wave front, the scattered energy is independent on distance to the source A. In practice, a phenomenological attenuation factor (βcoda Qβ) needs to be included, representing energy loss of the main wave front due to scattering and intrinsic attenuation:
ππ βππ0π‘π‘2
expβπππ‘π‘ππππ
where 1ππππ
= 1ππππ
+ 1πππ π
.
Multiple scattering.
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At longer times, multiple scattering dominates over single scattering. A strong scattering process can be described by diffusion with diffusivity π π = ππ/3ππ0. The total energy in the medium is given by the classical solution of the 3D diffusion equation:
πππππ π ππ~ππ0
(4πππ π π‘π‘)3/2 exp οΏ½βππ2
4π π π‘π‘οΏ½
An alternative approximation: energy uniformly distributed behind the direct wave front. Energy gets redistributed by scattering. The direct phase (first-arrival) loses energy to scattering, its energy decays with travel time as exp (βππ π‘π‘/πππ π ππ). Energy partitioning between direct and scattered field, and overall conservation:
πππππ π ππ~E0 exp οΏ½βπππ‘π‘πππ π ππ
οΏ½ +4ππ3
(πππ‘π‘)3πππ π ππ = ππ0
Leads to
πππ π ππ~3ππ04ππ
1 β ππβ
πππππππ π ππ
(πππ‘π‘)3
In practice, both equations need to be corrected by an intrinsic attenuation factor exp (βππ π‘π‘/ππππ) Applications Interferometry. Ex: in optical fiber. In the crust.
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13 Seismic sources.
13.1 Kinematic vs dynamic description of earthquake sources Standard earthquake model: sudden slip along a pre-existing fault surface Slip = displacement discontinuity (offset) across the fault Faults: nature vs models. Slip on thin fault vs inelastic deformation inside a thick fault zone. Kinematic source model = describes what happened on the fault: space-time distribution of slip velocity Dynamic source model = describes why it happened: forces governing unstable slip (e.g. fault friction) The next few lectures will deal with kinematic sources.
13.2 Stress glut and equivalent body force See Dahlen & Tromp section 5.2 Reminder: seismic wave equation (momentum equation & constitutive βlawβ β elasticity)
πποΏ½ΜοΏ½π’ = β β ππ + ππ ππ = ππ: ππ
where the source ππ is a density of body forces (distributed in the volume). The deformation in the earthquake source region is inelastic, a departure from our usual model of elastic media. Our goal here is to derive an equivalent body-force representation of an earthquake source, so that we can still use the elastic seismic wave equation to evaluate ground motions. Momentum equation:
πποΏ½ΜοΏ½π’ = β β πππππππ‘π‘ππ = β β πππΌπΌπ π ππππππ + β β οΏ½πππππππ‘π‘ππ β πππΌπΌπ π πππππποΏ½ = β β πππΌπΌπ π ππππππ β β β ππβ
where πππΌπΌπ π ππππππ is the stress given by the idealized Hookeβs law, and the mismatch between model and true stress is called stress glut:
ππβ = πππΌπΌπ π ππππππ β πππππππ‘π‘ππ We define a density of body forces as
ππ = ββ β ππβ Then
πποΏ½ΜοΏ½π’ = β β πππΌπΌπ π ππππππ + ππ Example: plastic deformation. As an example of departure from Hookeβs law, consider an elasto-plastic material. Strain is partitioned into elastic and plastic components, ππ = πππππππππ π ππππππ + πππππππππ π ππππππ, the true stress is related by Hookeβs law to the elastic strain, πππππππ‘π‘ππ = ππ: πππππππππ π ππππππ, and the model stress is what you get by trying to apply Hookeβs law to the total strain instead, πππΌπΌπ π ππππππ = ππ: ππ. Then,
ππβ = ππ: πππππππππ π ππππππ Plastic deformation is a source of seismic waves. For an isotropic elastic medium:
ππππππβ = πππππππππππππππ π πππππππΏπΏππππ + 2ππππππππ
πππππππ π ππππππ Example: explosion source, ππππππ
πππππππ π ππππππ = 13ΞπππππΏπΏππππ and we get an isotropic stress glut:
ππππππβ = οΏ½ππ + 23πποΏ½ Ξππ
πππΏπΏππππ.
13.3 Equivalent body force representation of fault slip Consider a vertical strike-slip fault. The trace of the fault at the surface is parallel to πποΏ½ (perpendicular to πποΏ½). Slip D is defined as the displacement offset across the fault. Slip is horizontal, parallel to πποΏ½, i.e. near the fault the inelastic displacement is: ππ βΌ π·π·π»π»(π¦π¦) πποΏ½.
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The gradient tensor of the inelastic displacement, (ππππ)ππππ = πππ’π’ππ/πππ₯π₯ππ, has only one non-zero component: πππ‘π‘π₯π₯πππ¦π¦
= π·π·πΏπΏ(π¦π¦). The only non-zero components of the inelastic strain tensor, ππ = 12
(ππππ + πππππ»π»), are:
πππππ¦π¦ = πππ¦π¦ππ = 12πππ‘π‘π₯π₯πππ¦π¦
= 12
π·π·πΏπΏ(π¦π¦).
The only non-zero components of the associated stress glut tensor are: πππππ¦π¦β = πππ¦π¦ππβ = πππ·π·πΏπΏ(π¦π¦). If the source is localized at a point (very small fault): πππππ¦π¦β = πππ¦π¦ππβ = πππ·π·πΏπΏ(π¦π¦)πΏπΏ(π₯π₯)πΏπΏ(π§π§) = πππ·π·πΏπΏ(ππ). Equivalent body force:
ππ = βππ β ππβ = βοΏ½πππππππ¦π¦β
πππ¦π¦,πππππ¦π¦ππβ
πππ₯π₯, 0οΏ½
ππ = πππ·π· οΏ½πππΏπΏπΏπΏπ¦π¦
πποΏ½ +πππΏπΏπΏπΏπ₯π₯
πποΏ½ οΏ½
Itβs a double-couple source, the sum of two force couples: πππΏπΏπΏπΏπ¦π¦πποΏ½ = a pair of forces parallel to πποΏ½, same amplitude but opposite sign,
separated by an arm parallel to πποΏ½. πππΏπΏπΏπΏπππποΏ½ = a conjugate force couple, forces parallel to πποΏ½, arm parallel to πποΏ½.
It has zero net force and zero net torque. Two conjugate planes: fundamental ambiguity. For a small source, it cannot be distinguished from the radiated wavefield which of the two possible planes is the causative fault. A measure of the source amplitude is the seismic moment density (per unit of fault surface): ππ = πππ·π·. Definition: the product ππππ of two vectors ππ and ππ is a tensor whose components are (ππππ)ππππ = ππππππππ. We can write ππππ βΌ π·π·πΏπΏ(π¦π¦) πποΏ½πποΏ½, where πποΏ½πποΏ½ is a tensor whose only non-zero component is (πποΏ½πποΏ½)12 = 1. Inelastic strain ππ = 1
2π·π·πΏπΏ(π¦π¦)(πποΏ½πποΏ½ + πποΏ½πποΏ½).
More generally, define normal vector n and slip direction unit vector e. Potency density tensor:
ππ =12π·π·(ππππ + ππππ)
ππππππ =12π·π·(ππππππππ + ππππππππ)
Moment density tensor: ππ = πππ·π· ππ β ππ πΌπΌππ + πππ·π·(ππππ + ππππ)
ππππππ = πππ·π·πππππππππΏπΏππππ + πππ·π·(ππππππππ + ππππππππ) The first term is zero if the fault does not open (no offset normal to the fault: ππ β ππ = 0). Extended source. Define seismic moment ππ0 = β« β« πππ·π· = πππππ·π·οΏ½ and potency ππ0 = β« β« π·π· = πππ·π·οΏ½, where ππ is the rupture surface and π·π·οΏ½ is the average slip.
13.4 Moment tensor ππππππ = β« β«ππππππ
Itβs a symmetric tensor. Not restricted to double-couple sources, it can represent sources other than faulting.
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Decomposition into isotropic and deviatoric parts:
ππ = πππΌπΌπππΌπΌ + πππ·π·πΈπΈππ where πππΌπΌπππΌπΌ = 1
3π‘π‘ππππππππ(ππ) πΌπΌππ.
The moment tensor of a double-couple source is purely deviatoric (zero trace) and has zero determinant. The decomposition of the deviatoric part into double-couple and non-double couple parts is not unique. One conventional decomposition, πππ·π·πΈπΈππ = πππ·π·π΅π΅ + πππ΅π΅πΆπΆπππ·π·, is defined such that the DC component is the largest possible. CLVD stands for βcompensated linear vector dipoleβ. It can appear if the source has a slip component normal to the fault plane (e.g. an inflating magma dyke), if two faults of different orientation slip simultaneously, or if the shear modulus drops close to the fault due to rock damage (increase in micro-crack density) induced by large dynamic stresses near the rupture front (e.g. Ben-Zion and Ampuero, 2009). If we diagonalize the seismic moment tensor as ππ = diag(ππ1,ππ2,ππ3), then
πππΌπΌπππΌπΌ =13
(ππ1 + ππ2 + ππ3) diag(1,1,1)
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πππ·π·π΅π΅ =12
(ππ1 βππ3) diag(1,0,β1)
πππ΅π΅πΆπΆπππ·π· = οΏ½ππ2 β13
(ππ1 + ππ2 + ππ3)οΏ½ diag οΏ½β12
, 1,β12οΏ½
13.5 Seismic moment and moment magnitude Seismic moment is a scalar that quantifies the βamplitudeβ of a moment tensor (its norm, in N.m):
ππ0 =1β2
οΏ½ππππππ2 οΏ½
12
Moment magnitude is a logarithmic scale related to the seismic moment:
πππ π =23
log10(ππ0) β 6
13.6 Representation theorem Greenβs function πΊπΊππππ(ππ, π‘π‘; ππ) is the n-th component of displacement at location ππ and time t induced by a point force at location ππ set at π‘π‘ = 0 in the p-th direction, i.e. ππ = πΏπΏ(π‘π‘)πΏπΏ(ππ β ππ)πποΏ½ππ Displacement induced by a point-force with arbitrary orientation and source time function ππ(π‘π‘):
π’π’ππ(ππ, π‘π‘) = πΊπΊππππ β ππππ Displacement induced by an extended distribution of point-forces ππ(ππ, π‘π‘):
π’π’ππ(ππ, π‘π‘) = οΏ½πΊπΊππππ(ππ, π‘π‘; ππ) β ππππ(ππ, π‘π‘)ππ3ππ
For a moment tensor source, ππππ = βππππππ,ππ and
π’π’ππ(ππ, π‘π‘) = βοΏ½πΊπΊππππ β ππππππ,ππππ3ππ
Integrating by parts:
π’π’ππ(ππ, π‘π‘) = οΏ½πΊπΊππππ β ππππππππππππ2ππ + οΏ½πΊπΊππππ,ππ β ππππππππ3ππ
The surface integral (first term on r.h.s.) vanishes if the integration surface is taken sufficiently far from the source region, where ππππππ vanishes. We obtain the representation theorem, a relation between wave field and moment tensor source:
π’π’ππ(ππ, π‘π‘) = οΏ½πΊπΊππππ,ππ β ππππππππ3ππ
Practical significance: if we know the Greenβs function, we can compute the whole wave field induced by an arbitrary moment tensor source by convolution with the gradients of the Greenβs function, πΊπΊππππ,ππ(ππ, π‘π‘; ππ).
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14 Seismic sources: moment tensor
14.1 Greenβs function Sketch of derivation: LamΓ© potentials of the wave field: = βΟ + β Γ ππ , with β β ππ = 0. Decompose a point-force source ππ in Helmholtz potentials: ππ = βΞ¦ + β Γ Ξ¨ with β β Ξ¨ = 0. The potentials satisfy wave equations with source terms:
οΏ½ΜοΏ½π = ππππ2β2ππ + Ξ¦/ππ and οΏ½ΜοΏ½π = ππππ2β2ππ + Ξ¨/ππ The solutions are
ππ(ππ, π‘π‘) = 14ππππππ
2βΞ¦οΏ½ππ,ππβ ππ
πππποΏ½
ππππ3ππ and ππ(ππ, π‘π‘) = 1
4ππππππ2β
Ξ¨οΏ½ππ,ππβ πππππποΏ½
ππππ3ππ
For any vector field πΎπΎ we have β2ππ = β(β β ππ) β β Γ (β Γ ππ). Hence we can derive the force potentials from a single vector potential ππ that satisfies Poissonβs equation, β2πΎπΎ = ππ, if Ξ¦ = β β W and Ξ¨ = ββ Γ W. The solution of Poissonβs equation is ππ = β 1
4ππβππππππ3ππ. For a point source: πΎπΎ = βππ(ππ)
4ππππ.
Knowing πΎπΎ, we can now evaluate Ξ¦ and Ξ¨, then ππ and ππ, and finally π’π’ (the Greenβs function). For a force with source time function ππ(π‘π‘):
π’π’ππ(ππ, π‘π‘) = 14ππππ
οΏ½3πΎπΎπππΎπΎππ β πΏπΏπππποΏ½1ππ3
β« πππΉπΉππ(π‘π‘ β ππ)ππππππ/ππ_ππ ππ/ππ_ππ (near field)
+ 14ππππππππ
2 πΎπΎπππΎπΎππ1ππ
πΉπΉππ οΏ½π‘π‘ βπππππποΏ½ (far-field P wave)
+ 14ππππππππ
2 οΏ½πΎπΎπππΎπΎππ β πΏπΏπππποΏ½1ππ
πΉπΉππ οΏ½π‘π‘ βπππππποΏ½ (far-field S wave)
where πΎπΎππ = π₯π₯ππ/ππ are direction cosines.
14.2 Moment tensor wavefield The representation theorem for a moment tensor source (last lecture):
π’π’ππ(ππ, π‘π‘) = οΏ½πΊπΊππππ,ππ β ππππππππ3ππ
involves the derivatives of the Greenβs function. So, taking appropriate derivatives of the result in the previous section:
π’π’ππ(ππ, π‘π‘) = π π ππ(πΈπΈ)4ππππ
1ππ4
β« ππππππππ(π‘π‘ β ππ)ππππππππππππππππ
(near field)
+ π π πΌπΌππ(πΈπΈ)4ππππππππ
21ππ2
ππππππ οΏ½π‘π‘ βπππππποΏ½ + π π πΌπΌππ(πΈπΈ)
4ππππππππ21ππ2
ππππππ οΏ½π‘π‘ βπππππποΏ½ (intermediate field, P and S)
+ π π πΉπΉππ(πΈπΈ)4ππππππππ
31r
οΏ½ΜοΏ½πππππ οΏ½π‘π‘ βπππππποΏ½ + π π πΉπΉππ(πΈπΈ)
4ππππππππ31r
οΏ½ΜοΏ½πππππ οΏ½π‘π‘ βπππππποΏ½ (far field, P and S)
Discuss properties of near-field, intermediate-field, far-field terms.
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14.3 Far field and radiation pattern of a double couple source In a spherical coordinate system (ππ,ππ,ππ) related to the fault orientation, the far-field P wave displacement field is:
π’π’ππ(π₯π₯, π‘π‘) βΌ sin2ΞΈ cosΟ π«π«οΏ½4ππππππππ
31r
οΏ½ΜοΏ½π0 οΏ½π‘π‘ βπππππποΏ½
+cos2ΞΈ cosΟ πποΏ½ + cosΞΈ sinΟ πποΏ½
4ππππππππ31r
οΏ½ΜοΏ½π0 οΏ½π‘π‘ βπππππποΏ½
P and S radiation patterns:
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Based on the sign of far-field P waves (up or down) we can infer the focal mechanism of an earthquake, which constrains the orientation of faulting (barring the fundamental ambiguity between the two conjugate fault planes)
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Focal mechanisms are a useful information for seismo-tectonic studies, for instance to infer the orientation of strain accumulation in the crust:
The far-field displacement is proportional to the seismic moment rate ππ0Μ (π‘π‘). This provides information about the temporal evolution of the rupture. Once we have determined the focal mechanism and the distance to the source, the time-integral of the far-field displacement gives an estimate of the seismic moment ππ0.
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Complications due to depth phases:
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14.4 Surface waves
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15 Finite sources
15.1 Kinematic source parameters of a finite fault rupture Far from the source and for wavelengths longer than the rupture size, we can consider an earthquake as a point source characterized by its
β’ seismic moment ππ0 (or moment magnitude πππ π ), β’ moment tensor ππππππ β’ source time function οΏ½ΜοΏ½π0(π‘π‘).
Close to the fault we need to describe in more detail the space-time distribution of the source: [sketch]
β’ Slip. β’ Slip velocity. β’ Rupture front. Rupture speed. β’ Rupture duration. β’ Healing front. Rise time (local slip duration).
15.2 Far-field, apparent source time function Far-field displacement induced by an extended source with slip rate οΏ½ΜοΏ½π·(ππ, π‘π‘):
ππ(ππ, π‘π‘) = οΏ½π π ππ(ππ,ππ)
4ππππππππ31ππ
πποΏ½ΜοΏ½π·Ξ£
οΏ½ππ, π‘π‘ βπππππποΏ½ πποΏ½ ππ2ππ
If the source area is small compared to the distance ππ between the fault and the receiver, we can approximate the Greenβs function by its value at the center of the rupture area (subscript 0):
ππ(ππ, π‘π‘) βπ π ππ(ππ0,ππ0)
4ππππππππ3 πποΏ½0ππ0
πποΏ½ οΏ½ΜοΏ½π· οΏ½ππ, π‘π‘ βπππππποΏ½ ππ2ππ
Ξ£
The integral term defines the apparent source time function (ASTF):
Ξ©P(ππ, π‘π‘) = οΏ½ οΏ½ΜοΏ½π· οΏ½ππ, π‘π‘ βπππππποΏ½ ππ2ππ
Ξ£
We can similarly define an ASTF for S waves, Ξ©S(ππ, π‘π‘). The ASTF is not only a property of the source, it depends also on the location of the observer relative to the source and on the type of wave considered (P or S). The time integral of the ASTF
οΏ½Ξ©P or S(ππ, π‘π‘) πππ‘π‘ = οΏ½ π·π·(ππ) ππ2ππΞ£
= ππ0
is the seismic potency, which does not depend on the location of the observer nor on wave type.
15.3 ASTF in the Fraunhofer approximation Let ππ0 be the position of the receiver relative to a reference point on the fault, ππ its position relative to an arbitrary point on the fault, and ππ an arbitrary position on the fault (relative to the reference point). Far from the source, ππ βͺ ππ0 and
ππ = βππ0 β ππβ = (ππ02 β 2ππ β ππππ + ππ2)1/2 β ππ0 β ππ β πποΏ½0 +ππ2 β (ππ β πποΏ½)2
2ππ0+ ππ οΏ½
ππππ0οΏ½3
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The Fraunhofer approximation amounts to keep only the zero and first order terms: ππ β ππ0 β ππ β πποΏ½0
Define the time relative to the wave arrival from the reference point, ππ = π‘π‘ β ππ0/ππ. The simplified expression of ASTF, in the Fraunhofer approximation, is:
Ξ©(πποΏ½ππ, ππ) = οΏ½ οΏ½ΜοΏ½π· οΏ½ππ, ππ +ππ β πποΏ½0ππ
οΏ½ ππ2ππΞ£
The ASTF depends on the direction πποΏ½ππ from which we observe the source. Range of validity of the Fraunhofer approximation. Valid if the rupture size, πΏπΏ = max(ππ), is small compared to distance and wavelength:
πΏπΏ βͺ οΏ½ππππ02
The range of validity depends on frequency (through ππ). It is less restrictive than the validity condition of the point-source approximation (πΏπΏ βͺ ππ). Proof: Denote ππβ the Fraunhofer approximation of the distance ππ, and ππ = ππ β ππβ the residual of this approximation. The Fourier transform of οΏ½ΜοΏ½π·(ππ, π‘π‘ β ππ/ππ) is οΏ½ΜοΏ½π·(ππ,ππ) exp(ππππππ/ππ). We can justify exp(ππππππ/ππ) β exp(ππππππβ/ππ) if ππππ/ππ βͺ ππ/2. This leads to the quarter-wavelength rule: ππ βͺ ππ/4.
Applying it to the residual ππ β ππ2β(ππβ πποΏ½)2
2ππ0 (the third order term in the Taylor expansion of ππ) we get,
conservatively, πΏπΏ βͺ οΏ½ππππ0/2. 2D example: letβs consider a problem with lower dimensionality, a linear (1D) fault embedded in a 2D elastic medium. Let ππ0 be the take-off angle of the seismic ray leaving the reference point, relative to the fault line direction.
Ξ©(ππ0, ππ) = οΏ½ οΏ½ΜοΏ½π· οΏ½ππ, ππ +ππ cosππ0
πποΏ½ ππππ
Ξ£
[Sketch οΏ½ΜοΏ½π·(ππ, π‘π‘). Construct graphically the ASTF for ππ0 = 0π π , 90π π , 180π π .
At ππ0 = 90π π we recover the STF. Comparing the other angles, introduce the directivity effect: the duration of the ASTF depends on ππ0.
Shorter duration yields larger amplitude because the time-integral of ASTF is the seismic potency, regardless of ππ0.]
15.4 Haskell pulse model, directivity Haskell model Consider a pulse propagating with constant slip π·π·, rise time π‘π‘πππππ π and rupture speed ππππ on a rectangular fault of width ππ and length πΏπΏ. The slip rate function is assumed invariant. οΏ½ΜοΏ½π·(π₯π₯, π‘π‘) = π·π· οΏ½ΜοΏ½π οΏ½π‘π‘ β ππ
π£π£πποΏ½ if π₯π₯ β [0, πΏπΏ] and π§π§ β [0,ππ]
= 0 elsewhere Far-field spectrum of a Haskell source Its ASTF is
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Ξ©(πποΏ½ππ, ππ) = π·π·οΏ½ οΏ½ΜοΏ½π οΏ½ππ β ππ/ππππ +ππ β πποΏ½0ππ
οΏ½ ππππ ππππΞ£
If ππ is small,
Ξ©(πποΏ½ππ, ππ) = π·π·πποΏ½ οΏ½ΜοΏ½π οΏ½ππ β ππ/ππππ +ππ cos ππ0
πποΏ½ ππππ
πΆπΆ
0
Its Fourier transform is
Ξ©(πποΏ½ππ,ππ) = βππππ ππ(ππ)π·π·πππΏπΏοΏ½ expοΏ½ππππππ οΏ½1ππππβ
cos ππ0ππ
οΏ½οΏ½πππππΆπΆ
0= βππππ ππ(ππ)π·π·πππΏπΏ
sinππππ
ππππππ
where ππ = πππΆπΆ2οΏ½ 1π£π£ππβ cosππ0
πποΏ½.
If ππ(π‘π‘) is a boxcar function with duration π‘π‘πππππ π , then ππ(ππ) = (1 β πππππππππππππ π )/ππ2π‘π‘πππππ π , and
|Ξ©(πποΏ½ππ,ππ)| = πππΏπΏπ·π· οΏ½sinππππ
sinπππ‘π‘πππππ π πππ‘π‘πππππ π
οΏ½
[Plot π π ππππ ππ
ππ, indicate zero-crossings.
Log-log plot spectrum of ASTF.] Spectral shape and two corners frequencies The ASTF spectrum exhibits three distinct behaviors, from low to high-frequencies: flat, 1/Ο and 1/Ο2. These regimes are separated by corner frequencies
ππ1(ππ0) = π£π£πππΆπΆ
οΏ½1 β π£π£ππππ
cosππ0οΏ½ and ππ2 = 1/π‘π‘πππππ π Directivity effect The directivity effect (azimuth-dependence) appears in the lower corner frequency, ππ1.
[Plot ππ1 as a function of ππ0]
Far-field waveform shape Displacement seismogram is a trapezoid. Velocity is made of two bumps and nothing in between: far-field radiation occurs during initiation and arrest (abrupt changes of rupture speed) but not during steady-state rupture propagation.
[Sketch: far-field displacement and velocity waveform. Indicate time scales] Directivity more intuitively: 1/ππ1 is the duration of the ASTF.
[Sketch: relation between ASTF duration
and the arrival times of waves radiated by the two ends of a rupture]
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16 Scaling laws
16.1 Circular crack model Circular rupture, constant rupture speed, final radius R. Rupture front, healing front, duration:
ππ β 2π π /ππππ. Only one characteristic time-scale, event duration. Spectrum has only one corner frequency, ππππ, that separates flat spectrum at low-f and 1/ππ2 at high-f:
Ξ©(ππ) βππ0
1 + οΏ½πππππποΏ½2
with
ππππ =πππππππ π
β 1/ππ
where ππ is a factor of order 1 that depends mildly on rupture speed (ππ = 0.44 for ππππ = 0.9ππππ).
16.2 Stress drop, corner frequency, self-similarity
Ξππ =7ππ16
πππ·π·π π
Considering also ππ0 = πππ·π·ππππ2, we get
Ξππ =7
16 οΏ½ππππππππππ
οΏ½3
ππ0
This shows how to estimate stress drop from far-field observations (assuming ππππ). Corner frequencies carry information about rupture duration. Attenuation distorts the spectral shape and makes it hard to determine corner frequencies, especially for small events (trade-off between Q and fc).
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Self-similarity: if Ξππ and ππππ do not depend on earthquake size:
ππ0 β π π 3 ππ β π π
π’π’(π‘π‘) β οΏ½ΜοΏ½π0 βΌ ππ0/ππ β π π 2 β ππ02/3
οΏ½ΜοΏ½π’(π‘π‘) β οΏ½ΜοΏ½π0 βΌ ππ0/ππ2 β π π β ππ01/3
16.3 Energy considerations and moment magnitude scale Energy radiated to the far-field:
ππ β β« οΏ½ΜοΏ½π’2πππ‘π‘ β οΏ½ΜοΏ½π’2 ππ β ππ0 log10 ππ = log10 ππ0 + β―
Moment magnitude:
πππ π =23
log10(ππ0) β 6.0
Then:
logππ =32πππ π + β―
One magnitude unit = 30 times more energy radiated, 30 times larger moment, 10 times larger ground displacement, 3 times larger ground velocity (ignoring attenuation) Itβs also useful to have in mind relations between magnitude and earthquake size:
πππ π = 2 log10 π π + β― One magnitude unit = 3 times larger rupture size, 3 times longer rupture duration. In terms of rupture surface area A:
πππ π = log10 π΄π΄ + β― One magnitude unit = 10 times larger rupture area.
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16.4 Stress drop for Haskell model and break of self-similarity Once rupture growth saturates the depth of the seismogenic zone, it has no choice but to become an elongated rupture pulse, like in Haskellβs model (rectangular rupture, width W and length L). Its moment is
ππ0 = πππ·π·πππΏπΏ
The elastic stiffness is controlled by the shortest rupture length. If πΏπΏ β« ππ:
Ξππ βΌπππππ·π·
Hence, ππ0 βΌ Ξππππ2πΏπΏ
Its rupture duration is controlled by the longest rupture length, ππ = πΆπΆπ£π£ππ
. Hence, the corner frequency
(ππππ βΌ 1/ππ) now scales as ππππ βΌ Ξππππ2ππππ Γ ππ0
β1
Saturation of the seismogenic depth breaks self-similarity. A change in aspect ratio can also happen at smaller scales, due to fault heterogeneities.
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17 Source inversion, near-fault ground motions and isochrone theory
17.1 Fundamental limitation of far-field source imaging The Apparent Source Time Function in the Fraunhofer approximation is
Ξ©(πποΏ½ππ, ππ) = οΏ½ οΏ½ΜοΏ½π· οΏ½ππ, ππ +ππ β πποΏ½0ππ
οΏ½ ππ2ππΞ£
Its temporal Fourier transform (ππ β ππ) is
Ξ©(πποΏ½ππ,ππ) = οΏ½ οΏ½ΜοΏ½π·(ππ,ππ) exp οΏ½βiΟππ β πποΏ½0ππ
οΏ½ ππ2ππΞ£
The spatial Fourier transform of a function ππ(ππ) defined on the fault surface (ππ β Ξ£) is
ππ(ππ) = οΏ½ ππ(ππ) exp(βi ππ β ππ)ππ2ππΞ£
where ππ is a wavenumber vector along the fault. Hence, the ASTF is related to the spatial Fourier transform of slip rate:
Ξ©(πποΏ½ππ,ππ) = οΏ½ΜοΏ½π· οΏ½ππ =πππΈπΈοΏ½0ππ
,πποΏ½
where πΈπΈοΏ½0 = πποΏ½0 β (πποΏ½0 β ππ)ππ is the projection of πποΏ½0 on the fault surface Ξ£. If we were able to measure οΏ½ΜοΏ½π·(ππ,ππ) for all ππ and ππ we could readily infer οΏ½ΜοΏ½π·(ππ, π‘π‘) by inverse Fourier transform. However, |πΈπΈοΏ½0| < 1 and the ASTF only samples on-fault wavenumber vectors such that |ππ/ππ| > ππ, i.e. with along-fault phase velocity larger than wave speed. Hence far-field source imaging is limited to along-fault wavelengths ππ > ππ/ππ. Indeed, all perturbations with |ππ/ππ| < ππ are associated to evanescent waves with exponential decay in the fault-normal direction, which do not make it to far field distances. To extract finer information about source processes, near-field ground motion recordings are needed.
17.2 Source inversion See Ide (2007). Goal: given seismograms recorded at N seismic stations during an earthquake, infer the spatio-temporal distribution of slip rate on the fault. Representation theorem:
π’π’ππ(ππ, π‘π‘) = οΏ½πΊπΊππππ,ππ β ππππππππ3ππ
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17.3 Isochrone theory See Bernard and Madariaga (1984) and Spudich and Frazer (1984).
Ground-motion simulation using isochrone theory for a hypothetical Mw 6.7 earthquake. Top: Slip
distribution (colors), rupture time (white contours) and hypocenter (red star). Constant rupture speed is assumed. Bottom: Isochrone quantities and computed seismograms at three sites. Top row: S-wave arrival
time contours. Second row: isochrones (black) and isochrone integrand (colors). Third row: isochrones and isochrone velocity (colors). Bottom row: Fault-normal velocity seismograms dominated by large S
wave pulses (in cm/s; peak velocity indicated at the end of each trace). From Mai (2007).
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18 Source inversion and source imaging
18.1 Source inversion problem For given data d and design matrix G (Greenβs functions), find the model m that minimizes the cost
function Ο2 = οΏ½|ππ β πΊπΊππ|οΏ½2 . The formal solution is ππ = (πΊπΊπππΊπΊ)β1πΊπΊππππ = πΊπΊ#ππ. Proof: set ππΟ2
πππΌπΌ= 0.
18.2 Ill-conditioning of the source inversion problem The Singular Value Decomposition (SVD) of matrix G is
πΊπΊ = ππππππππ where U[N,M] and V[M,M] are orthonormal matrices (ππππππ = πΌπΌ, ππππππ = πΌπΌ), and ππ[M,M] is a diagonal matrix consisting of positive singular values ππππ, ππ = 1, . . . ,ππ, sorted in descending order. The columns ππ(ππ) of matrix V are called right-singular vectors. They are also eigenvectors of matrix πΊπΊπππΊπΊ[M,M], forming an orthonormal basis system in the model space, and ππππ2 are its eigenvalues. Columns ππ(ππ) of matrix U are called left-singular vectors. They are projections of basis vectors ππ(ππ) into the data space,
ππ(ππ) = πΊπΊππ(ππ) /ππππ i.e. normalized seismograms related to the individual singular vectors. The generalized solution of the inverse problem, ππ = πΊπΊ#ππ can be expressed as a linear combination of basis vectors ππ(ππ):
ππ = β πποΏ½ππ ππ(ππ)ππππ=1 where πποΏ½ππ = ππ(ππ) β ππ/ππππ
Similarly, the data vector can be expressed as ππ = β οΏ½ΜοΏ½πππ ππ(ππ)
ππππ=1 where οΏ½ΜοΏ½πππ = ππ(ππ) β ππ
The spectral components of data and model are thus related by πποΏ½ππ = οΏ½ΜοΏ½πππ/ππππ
Therefore, the smaller is the singular value ππππ, the less sensitive is the data component to a given change of the corresponding model component; in other words, the singular value bears information about the sensitivity of the data to the particular basis function in the model space. Also, if ππππ is small, errors in the data or in the G matrix get amplified. Hence the components of the data associated to small eigenvalues are hardly recoverable by the inverse problem, they define an effective null space.
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18.3 Stacking Stacking (adding up) seismograms enhances the signal-to-noise ratio. Central limit theorem: stacking N seismograms reduces the noise by βππ.
18.4 Array seismology Plane wave impinging on a linear array and 2D array:
Parameters: back-azimuth and horizontal slowness (related to wave speed and incidence angle i):
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Seismogram recorded by a reference station in the array (signal + noise):
π’π’1(π‘π‘) = ππ(π‘π‘) + ππ1(π‘π‘)
Seismogram recorded by the i-th station of the array, at position ππππ relative to the reference station: π’π’ππ(π‘π‘) = ππ(π‘π‘ β ππππ β π’π’βπ π ππ) + ππππ(π‘π‘)
We are assuming that the stations are close enough (small ππππ) so that the signal shapes are similar, i.e. ππππ(π‘π‘) = ππ1(π‘π‘). In practice this requires high coherency (similarity) of the wavefield across the array.
Classical beamforming: The delay-and-sum beam is defined as
ππ(π‘π‘, ππ) =1πποΏ½π’π’ππ(π‘π‘ + ππππ β ππ)ππ
ππ=1
For the signal model assumed: ππ(π‘π‘, ππ = π’π’βπ π ππ) = ππ(π‘π‘) + πποΏ½(π‘π‘)
where πποΏ½(π‘π‘) is the average noise, whose amplitude has been reduced by βππ. If ππ β π’π’βπ π ππ then the signals do not stack up coherently and ππ(π‘π‘, ππ) is small.
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18.5 Array response
18.6 Coherency stacking β¦
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19 Earthquake dynamics I: Fracture mechanics perspective [Hand-written notes and slides to be cleaned up] Stress intensity factor Energy release rate Crack tip equation of motion Radiated energy
20 Earthquake dynamics II: Fault friction perspective [Hand-written notes and slides to be cleaned up] Friction (laboratory, physical mechanisms, usual constitutive relations) Slip-weakening and process zone size Earthquake nucleation Fluid and thermal effects in fault weakening Cracks versus pulses Supershear rupture
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21 Inverse problems, part 1
21.1 Earthquake location
21.2 Iterative solution
21.3 Solution of inverse problems.
21.4 Weighted over-determined problem
21.5 Uncertainties: model covariance
21.6 Double difference location
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22 Inverse problems, part 2
22.1 Travel time tomography, ill-posed problems
22.2 SVD, minimum-norm solution
22.3 Resolution matrix, model covariance matrix
22.4 Truncated SVD
22.5 Regularization
22.6 Bayesian approach
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