GE 162 Introduction to Seismology

California Institute of Technology – Seismological Laboratory GE 162 Introduction to Seismology

Lecture notes

Jean Paul (Pablo) Ampuero – [email protected] Winter 2013 - 2016

GE 162 Introduction to Seismology Winter 2013 - 2016

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 Green’s function. Waves in heterogeneous media...... 14 4.1 Linear invariant systems, Green’s functions, convolution ...... 14 4.2 Green’s 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 It’s a perturbative equation ...... 17 5.7 P and S waves examples ...... 17 5.8 General decomposition into P and S waves ...... 18

GE 162 Introduction to Seismology Winter 2013 - 2016

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, Snell’s 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

GE 162 Introduction to Seismology Winter 2013 - 2016

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 Green’s 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

GE 162 Introduction to Seismology Winter 2013 - 2016

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

GE 162 Introduction to Seismology Winter 2013 - 2016

1 Overview and 1D wave equation

1.1 Overview, etc • Have you ever felt an earthquake? • About me: Pablo Ampuero, office SM 359, email [email protected], research topics • About you: fill entry survey • Seismology = study of ground motions induced by seismic waves • Show a few seismograms and point to interesting features and their broader significance • 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, … • History and the Seismolab: Gutenberg-Richter frequency-magnitude distribution. Richter’s magnitude scale. Kanamori’s moment magnitude scale. Anderson’s PREM. From Trinet to SCSN, EEW, CSN. Analog to digital. Today: large N, big data, noise, computational seismology. • These lectures are in two parts: . basic seismology theory, and . basic earthquake source theory. • 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. • 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) • 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 – 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.

GE 162 Introduction to Seismology Winter 2013 - 2016

[Sketch undeformed and deformed rod, annotate] Let be a location along the rod. At time , its perturbed location is + ( , ). Our goal: determine the evolution of the displacement field ( , ) induced by some initial conditions (out𝑥𝑥 of static equilibrium) or by some external𝑡𝑡 forcing. 𝑥𝑥 𝑢𝑢 𝑥𝑥 𝑡𝑡 [discuss initial conditions and𝑢𝑢 𝑥𝑥forcing]𝑡𝑡

1.2.2 Dynamics Apply = to a small portion of the rod, of length . [Sketch forces on an elementary segment dx. Recall first-order Taylor expansion.] Definition:𝐹𝐹 𝑚𝑚𝑚𝑚on a transverse surface located at x, F(x) = force𝑑𝑑𝑑𝑑 induced by the material located “to the right” (x’>x) on the material located to the left (x’linear elasticity and inelasticity, e.g. plasticity] Linear elasticity: , if . Important assumption: small deformations. [experiment with rubber band] Actually / 𝐹𝐹=∝ Δ (strain𝑙𝑙 Δ𝑙𝑙).≪ 𝑙𝑙 [experiment with stack of two rods (larger S)] Also 𝐹𝐹 .∝ TheseΔ𝑙𝑙 𝑙𝑙 properties𝜖𝜖 are summarized by: = / (3) where𝐹𝐹 E∝ is𝑆𝑆 a material parameter called Young’s modulus. 𝐹𝐹 =𝑆𝑆𝑆𝑆 Δ𝑙𝑙/ 𝑙𝑙 (4)

Consider an elementary segment of length dx. 𝜎𝜎 𝐸𝐸Δ𝑙𝑙 𝑙𝑙 [sketch undeformed and deformed rod. Annotate new positions of two ends.] The length of the undeformed rod is = . Stretch of the deformed rod: = + / . 𝑙𝑙 𝑑𝑑𝑑𝑑 Dividing by dx: 𝑑𝑑𝑑𝑑 𝑑𝑑𝑑𝑑 Δ𝑙𝑙 𝑢𝑢 �𝑥𝑥 2 � − 𝑢𝑢 �𝑥𝑥 − 2 � ≈ 𝑑𝑑𝑑𝑑 𝜕𝜕𝜕𝜕 𝜕𝜕𝜕𝜕 / = / (5) Plugging it into eq. (4): Δ𝑙𝑙=𝑙𝑙 𝜕𝜕𝜕𝜕 /𝜕𝜕𝜕𝜕 (6)

1.2.4 The 1D wave equation 𝜎𝜎 𝐸𝐸 𝜕𝜕𝜕𝜕 𝜕𝜕𝜕𝜕 Combining eqs. (2) and (6): (7) = 2 𝜕𝜕 𝜕𝜕𝜕𝜕 𝜕𝜕 𝑢𝑢 Assuming E is constant �𝐸𝐸 � 𝜌𝜌 2 𝜕𝜕𝜕𝜕 𝜕𝜕𝜕𝜕 𝜕𝜕𝑡𝑡 (8) = 2 2 𝜕𝜕 𝑢𝑢 𝜕𝜕 𝑢𝑢 𝐸𝐸 2 𝜌𝜌 2 𝜕𝜕𝑥𝑥 𝜕𝜕𝑡𝑡 6

GE 162 Introduction to Seismology Winter 2013 - 2016

[dimensional analysis: determine the units of / ] Define a quantity with units of speed: = / = wave speed (celerity). We get the 1D wave 𝐸𝐸 𝜌𝜌 equation: 𝑐𝑐 �𝐸𝐸 𝜌𝜌 1 (9) = 2 2 𝜕𝜕 𝑢𝑢 𝜕𝜕 𝑢𝑢 2 2 2 𝜕𝜕𝑥𝑥 𝑐𝑐 𝜕𝜕𝑡𝑡 For transverse motions we get the same wave equation (9) but with shear wave speed = / , where is the shear modulus. The shear stress is = / . More on this in a later lecture. [orders of magnitude of c] 𝑐𝑐 �𝜇𝜇 𝜌𝜌 𝜇𝜇 𝜎𝜎 𝜇𝜇 𝜕𝜕𝜕𝜕 𝜕𝜕𝜕𝜕

GE 162 Introduction to Seismology Winter 2013 - 2016

2 1D wave equation: solution and main properties

2.1 General solution The solution comprises waves propagating in both directions at speed , D’Alembert’s solution: (10) ( , ) = ( ) + ( + ) 𝑐𝑐 [Sketch waves snapshots,𝑢𝑢 seismograms,𝑥𝑥 𝑡𝑡 𝑓𝑓 𝑥𝑥 −characteristic𝑐𝑐𝑐𝑐 𝑔𝑔 𝑥𝑥 lines𝑐𝑐𝑐𝑐 in space-time (x,t) plane]

Proof: Define new variables = and = + . Apply chain rule:

𝜉𝜉 𝑥𝑥 − 𝑐𝑐𝑐𝑐 = 𝜂𝜂 +𝑥𝑥 𝑐𝑐𝑐𝑐 = + 𝜕𝜕 𝜕𝜕 𝜕𝜕𝜕𝜕 𝜕𝜕 𝜕𝜕𝜕𝜕 𝜕𝜕 𝜕𝜕 𝜕𝜕𝜕𝜕 𝜕𝜕=𝜕𝜕 𝜕𝜕𝜕𝜕 = 𝜕𝜕𝜕𝜕 𝜕𝜕𝜕𝜕 + 𝜕𝜕𝜕𝜕 𝜕𝜕𝜕𝜕 𝜕𝜕 𝜕𝜕 𝜕𝜕 Apply it again, then plug it into eq (9). After some⋯ algebra−𝑐𝑐 : 𝑐𝑐 𝜕𝜕𝜕𝜕 𝜕𝜕𝜕𝜕 𝜕𝜕𝜕𝜕 (11) = 0 2 Integrating this equation with respect to : 𝜕𝜕 𝑢𝑢 /𝜕𝜕𝜕𝜕𝜕𝜕𝜕𝜕= ( ) where is an arbitrary function. Integrating𝜉𝜉 this with respect to : ′ 𝜕𝜕𝜕𝜕 𝜕𝜕𝜕𝜕 𝑔𝑔′ 𝜂𝜂 (12) 𝑔𝑔 ( , ) = ( ) + ( ) 𝜂𝜂 where is an arbitrary function. □ 𝑢𝑢 𝜉𝜉 𝜂𝜂 𝑓𝑓 𝜉𝜉 𝑔𝑔 𝜂𝜂

and 𝑓𝑓 are constrained by the initial conditions. Example: compress the rod, then suddenly release it. The initial conditions are ( , 0) = ( ) and 𝑓𝑓( , 0)𝑔𝑔= 0. Considering these initial conditions and (10) we get two equations: ( ) + ( ) = ( ) 0 and ( ) + ( ) = 0. From these we derive: = ( + )/2 and = (𝑢𝑢 𝑥𝑥 )/2𝑢𝑢, where𝑥𝑥 is a 0 constant.𝑢𝑢̇ 𝑥𝑥 ′ Finally, ′ 𝑓𝑓 𝑥𝑥 𝑔𝑔 𝑥𝑥 𝑢𝑢 𝑥𝑥 0 0 −𝑐𝑐𝑓𝑓 𝑥𝑥 𝑐𝑐𝑔𝑔 𝑥𝑥 1 𝑓𝑓 𝑢𝑢1 𝐶𝐶 𝑔𝑔 𝑢𝑢 − 𝐶𝐶 𝐶𝐶 (13) ( , ) = ( ) + ( + ) 2 2

Note that is undetermined but 𝑢𝑢cancels𝑥𝑥 𝑡𝑡 out 𝑢𝑢in0 the𝑥𝑥 − final𝑐𝑐𝑐𝑐 solution𝑢𝑢0 𝑥𝑥. 𝑐𝑐𝑐𝑐 [Draw this solution] 𝐶𝐶 2.2 Reflection at one end Apply mirror image trick to satisfy boundary conditions at the end of the rod (at = 0). [Sketches explaining the mirror image trick] Case 1, fixed displacement (Dirichlet b.c.) (0, ) = 0 achieved by an image wave𝑥𝑥 with opposite amplitude: 𝑢𝑢 𝑡𝑡 (14) ( , ) = ( ) ( ) ( ) Case 2, free stress (Neumann b.c.) 𝑢𝑢 𝑥𝑥0,𝑡𝑡 =𝑓𝑓0 𝑥𝑥achieved− 𝑐𝑐𝑐𝑐 − by𝑓𝑓 an−𝑥𝑥 image− 𝑐𝑐𝑐𝑐 with same amplitude (note the cancelation of slopes): 𝑢𝑢′ 𝑡𝑡 (15) ( , ) = ( ) + ( ) ( ) In case 2 there is amplification at the𝑢𝑢 𝑥𝑥boundary𝑡𝑡 𝑓𝑓 :𝑥𝑥 −0𝑐𝑐,𝑐𝑐 = 𝑓𝑓 2 − 𝑥𝑥 − 𝑐𝑐𝑐𝑐 𝑢𝑢 𝑡𝑡 𝑓𝑓

GE 162 Introduction to Seismology Winter 2013 - 2016

2.3 Fourier transform Any “well behaved” function ( ) can be decomposed as a linear superposition (weighted sum) of oscillatory functions with angular frequency : 𝑢𝑢 𝑡𝑡 −𝑖𝑖𝑖𝑖𝑖𝑖 1 (16) 𝑒𝑒 ( ) = +𝜔𝜔∞ ( ) 2 −𝑖𝑖𝑖𝑖𝑖𝑖 [Sketch linear superposition. Note𝑢𝑢 𝑡𝑡 diversity �of conventions𝑢𝑢� 𝜔𝜔 𝑒𝑒 𝑑𝑑 about𝑑𝑑 factor 1/2 and frequency] 𝜋𝜋 −∞ The frequency-dependent weights ( ) are called the spectral coefficients or Fourier coefficients and are defined by the so-called Fourier transform of ( ) : 𝜋𝜋 𝑢𝑢� 𝜔𝜔 (17) ( ) = 𝑢𝑢+∞𝑡𝑡 ( ) 𝑖𝑖𝑖𝑖𝑖𝑖 The function ( ) is also called the spectrum𝑢𝑢� 𝜔𝜔 of �( )𝑢𝑢. Some𝑡𝑡 𝑒𝑒 u𝑑𝑑seful𝑑𝑑 Fourier transform pairs: −∞ ( ) ( ) (18) 𝑢𝑢� 𝜔𝜔 𝑢𝑢 𝑡𝑡 (𝑢𝑢) 𝑡𝑡 ⟷ 𝑢𝑢� 𝜔𝜔( ) (19)

𝑢𝑢̇(𝑡𝑡) ⟷ −𝑖𝑖𝑖𝑖 𝑢𝑢� (𝜔𝜔) (20) 2 𝑢𝑢̈ 𝑡𝑡 ⟷ −𝜔𝜔 𝑢𝑢� 𝜔𝜔 2.4 Harmonic waves Taking the Fourier transform of equation (9): (21) ( , ) = 2 Solution: 2 2 𝜕𝜕 𝑢𝑢� −𝜔𝜔 𝑢𝑢� 𝑥𝑥 𝜔𝜔 𝑐𝑐 2 𝜕𝜕𝑥𝑥 (22) ( , ) = ( ) + ( ) 𝑖𝑖𝑖𝑖𝑖𝑖 −𝑖𝑖𝑖𝑖𝑖𝑖 where A and B are complex valued𝑢𝑢 �functions𝑥𝑥 𝜔𝜔 of𝐴𝐴 frequency𝜔𝜔 𝑒𝑒 , 𝐵𝐵to 𝜔𝜔be determined𝑒𝑒 by boundary conditions, and is the wavenumber defined by (23) 𝑘𝑘 = / ( ) Taking the inverse Fourier transform of (22) shows𝑘𝑘 that𝜔𝜔 𝑐𝑐 , is a superposition of harmonic waves of the following form: 𝑢𝑢 𝑥𝑥 𝑡𝑡 (24) ( ) + ( ) 𝑖𝑖 𝑘𝑘𝑘𝑘−𝜔𝜔𝜔𝜔 −𝑖𝑖 𝑘𝑘𝑘𝑘+𝜔𝜔𝜔𝜔 𝐴𝐴 𝑒𝑒 𝐵𝐵 𝑒𝑒 [Sketch harmonic wave (real part) at fixed t, then at fixed x] Some definitions: Frequency: = /2 Period (temporal): = 2 / = 1/ Wavelength (spatial period): 𝑓𝑓 = 2𝜔𝜔 / 𝜋𝜋 𝑇𝑇 𝜋𝜋 𝜔𝜔 𝑓𝑓 Duality space-time: 𝜆𝜆 𝜋𝜋 𝑘𝑘 (25) =

Short periods = high frequencies𝜆𝜆 𝑐𝑐 𝑇𝑇 = short wavelengths. Long periods = low frequencies = long wavelengths.

GE 162 Introduction to Seismology Winter 2013 - 2016

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.

GE 162 Introduction to Seismology Winter 2013 - 2016

3 Energy, reflection/transmission, normal modes

3.1 Impedance How is stress related to velocity? (dynamic vs kinematic quantities) Shear stress in elastic medium: = / Consider a wave ( , ) = ( ). We have / = and / = . Hence /𝜎𝜎 =𝜇𝜇 𝜕𝜕1𝜕𝜕/ 𝜕𝜕 𝜕𝜕 / It implies that a local𝑢𝑢 𝑥𝑥 measurement𝑡𝑡 𝑓𝑓 𝑥𝑥 − 𝑐𝑐𝑐𝑐 of velocity allows𝜕𝜕𝜕𝜕 𝜕𝜕 𝜕𝜕an estimate𝑓𝑓′ 𝜕𝜕 of𝜕𝜕 strain.𝜕𝜕𝜕𝜕 − For𝑐𝑐𝑐𝑐 ′a shear wave we find that shear stress is proportional to velocity:𝜕𝜕𝜕𝜕 𝜕𝜕𝜕𝜕 − 𝑐𝑐 𝜕𝜕𝜕𝜕 𝜕𝜕𝜕𝜕 = / = / / This relation defines the impedance / of the material. 𝜎𝜎 𝜇𝜇 𝜕𝜕𝜕𝜕 𝜕𝜕𝜕𝜕 −𝜇𝜇 𝑐𝑐 𝜕𝜕𝜕𝜕 𝜕𝜕𝜕𝜕 3.2 Energy considerations 𝝁𝝁for𝒄𝒄 harmonic waves For a harmonic wave of the form ( , ) = cos( ) where is a real number: Kinetic energy density (per unit volume) = /2 Potential energy density 𝑢𝑢 𝑥𝑥=𝑡𝑡 /𝐴𝐴2 𝑘𝑘𝑘𝑘 − 𝜔𝜔2𝜔𝜔 𝐴𝐴 𝐾𝐾 Making use of = / , / = 1′/ /𝑒𝑒 and𝜌𝜌𝑢𝑢 ̇ = / , we can show that 𝑃𝑃 𝑒𝑒 𝜎𝜎=𝑢𝑢 = sin 2( ) 𝜎𝜎 𝜇𝜇𝜇𝜇𝜇𝜇 𝜕𝜕𝜕𝜕 𝜕𝜕𝜕𝜕 𝜕𝜕𝜕𝜕 − 𝑐𝑐 𝜕𝜕𝜕𝜕 𝜕𝜕𝜕𝜕 𝑐𝑐 𝜇𝜇 𝜌𝜌 Integrating over one wavelength we get the average2 energies,2 2 … : 𝑒𝑒𝑃𝑃 𝑒𝑒𝐾𝐾 𝜌𝜌𝜔𝜔 𝐴𝐴 𝑘𝑘𝑘𝑘 − 𝜔𝜔𝜔𝜔 = = /41 𝑥𝑥+𝜆𝜆 𝜆𝜆 ∫𝑥𝑥 𝑑𝑑𝑑𝑑 2 2 Total average energy density = = = 𝐾𝐾 + 𝑃𝑃 = /2 𝐸𝐸 𝜀𝜀 𝜀𝜀 𝜌𝜌𝜔𝜔 𝐴𝐴 2 2 = = = = /2 Energy flux (per unit of cross-section𝜀𝜀 𝑆𝑆 𝑆𝑆𝑆𝑆surface,𝜀𝜀𝐾𝐾 per𝜀𝜀𝑃𝑃 unit𝜌𝜌 of𝜔𝜔 time)𝐴𝐴 = Note that = / = impedance. 𝐸𝐸 𝑑𝑑𝑑𝑑 2 2 𝐸𝐸̇ 𝑆𝑆𝑆𝑆𝑆𝑆 𝜀𝜀 𝑑𝑑𝑑𝑑 𝑐𝑐 𝜀𝜀 𝜌𝜌𝜌𝜌 𝜔𝜔 𝐴𝐴 For a harmonic wave of the form ( , ) = ( ) where A is a complex number: 𝜌𝜌𝜌𝜌 𝜇𝜇 𝑐𝑐 𝑖𝑖 𝑘𝑘𝑘𝑘1 −𝜔𝜔𝜔𝜔 (26) = | | 𝑢𝑢 𝑥𝑥 𝑡𝑡 𝐴𝐴𝑒𝑒 2 | | 𝜇𝜇 2 2 where is the modulus of . 𝐸𝐸̇ 𝜔𝜔 𝐴𝐴 𝑐𝑐 3.3 Reflection𝐴𝐴 and transmission𝐴𝐴 at a material interface Consider two semi-infinite media in welded contact at = 0. Medium 1 is in 0 and has wave speed and shear modulus , medium 2 is in 0 and has wave speed and shear modulus . Consider a harmonic shear wave incident from medium 1, of the𝑥𝑥 form ( ), with 𝑥𝑥 ≤= / . 1 1 2 2 𝑐𝑐 𝜇𝜇 𝑥𝑥 ≥ [Sketch] 𝑖𝑖 𝑘𝑘1𝑥𝑥−𝜔𝜔𝜔𝜔𝑐𝑐 𝜇𝜇 1 1 In medium 1 we have the incident and reflected waves: 𝑒𝑒 𝑘𝑘 𝜔𝜔 𝑐𝑐 (27) ( , ) = ( ) + ( ) 𝑖𝑖 𝑘𝑘1𝑥𝑥−𝜔𝜔𝜔𝜔= / −𝑖𝑖 𝑘𝑘1𝑥𝑥+𝜔𝜔𝜔𝜔 In medium 2 we have the transmitted𝑢𝑢1 𝑥𝑥 wave,𝑡𝑡 𝑒𝑒with 𝑅𝑅 𝑒𝑒: ( ) (28) ( , ) =𝑘𝑘2 𝜔𝜔 𝑐𝑐2 = 0 𝑖𝑖 𝑘𝑘2𝑥𝑥−𝜔𝜔𝜔𝜔 Boundary conditions at : 𝑢𝑢2 𝑥𝑥 𝑡𝑡 𝑇𝑇 𝑒𝑒 Continuity of displacement: (0, ) = (0, ) Continuity of shear𝑥𝑥 stress: / (0, ) = / (0, ) 1 2 𝑢𝑢 𝑡𝑡 𝑢𝑢 𝑡𝑡 1 1 2 2 Combining the boundary conditions with𝜇𝜇 eqs𝜕𝜕𝑢𝑢 (27)𝜕𝜕𝜕𝜕-(28)𝑡𝑡 yield𝜇𝜇s: 𝜕𝜕𝜕𝜕 𝜕𝜕𝜕𝜕 𝑡𝑡

GE 162 Introduction to Seismology Winter 2013 - 2016

(29) 1 + = (30) 1 𝑅𝑅= 𝑇𝑇 where = / is the impedance ratio at the −material𝑅𝑅 𝛼𝛼 interface.𝑇𝑇 Solving this system of 2 equations 𝜇𝜇2 𝜇𝜇1 with 2 unknowns: 𝛼𝛼 𝑐𝑐2 𝑐𝑐1 (31) = 2/(1 + ) (32) =𝑇𝑇(1 )/(1𝛼𝛼+ )

𝑅𝑅 − 𝛼𝛼 𝛼𝛼 Verifications:

If = 1 (no material contrast), then = 1 and = 0. If = (Dirichlet b.c.), then = 0 and = 1. If 𝛼𝛼 = 0 (Neumann b.c.), then = 1,𝑇𝑇 but = 2𝑅𝑅 instead of = 0! Energy𝛼𝛼 ∞ flux is conserved: 𝑇𝑇 =𝑅𝑅 − (1 = , after multiplying (29) and (30)) 𝛼𝛼 𝑅𝑅 𝑇𝑇 2 𝑇𝑇2 3.4 Normal modes of a finite𝐸𝐸̇𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖 elastic− 𝐸𝐸̇𝑅𝑅 rod𝐸𝐸̇𝑇𝑇 − 𝑅𝑅 𝛼𝛼𝑇𝑇 Consider a rod of finite length . [Discuss the vibrations of a guitar string. Sketch a standing wave.] Consider solutions of the 1D wave𝐿𝐿 equation in the form of a standing wave (we seek a “separable solution” to the PDE): (33) ( , ) = ( ) ( ) " Plugging (33) into (9) we get = 𝑢𝑢 and𝑥𝑥 𝑡𝑡 𝑋𝑋 𝑥𝑥 𝑇𝑇 𝑡𝑡 2 " (34) 𝑋𝑋 𝑇𝑇̈ 𝑐𝑐 𝑋𝑋 𝑇𝑇 ( ) = ( ) The l.h.s. is a function of only and the r.h.s.𝑇𝑇 ̈ is a function2 𝑋𝑋 of only. Both are necessarily equal to a 𝑡𝑡 𝑐𝑐 𝑥𝑥 constant. We are free to choose a name for𝑇𝑇 the constant,𝑋𝑋 let’s call it . Equation (34) leads to two separate ODEs, one for 𝑡𝑡( ) and one for ( ): 𝑥𝑥 2 −𝜔𝜔 (35) 𝑇𝑇 𝑡𝑡 𝑋𝑋 𝑥𝑥 = 2 (36) " = 𝑇𝑇̈ /−𝜔𝜔 𝑇𝑇= 2 = 2 / 2 Note that we have introduced the wavenumber𝑋𝑋 −𝜔𝜔 𝑐𝑐 𝑋𝑋 . Their−𝑘𝑘 𝑋𝑋solutions are (37) ( ) 𝑘𝑘sin𝜔𝜔( 𝑐𝑐+ ) (38) 𝑇𝑇(𝑡𝑡 ) ∝ sin(𝜔𝜔𝜔𝜔 + 𝜓𝜓)

where and are constants called phase𝑋𝑋 shifts𝑥𝑥 ∝. Combining𝑘𝑘𝑘𝑘 𝜙𝜙 them, we get a standing wave solution: (39) 𝜓𝜓 𝜙𝜙 ( , ) = sin( + ) sin( + ) 𝑢𝑢 𝑥𝑥 𝑡𝑡 𝐴𝐴 𝜔𝜔𝜔𝜔 𝜓𝜓 𝑘𝑘𝑘𝑘 𝜙𝜙 Assume fixed displacements as boundary conditions on both ends of the rod: (0, ) = ( , ) = 0. Applying these to (35) we get: = 0 and sin( ) = 0. This is satisfied by a discrete set of admissible wavenumbers: 𝑢𝑢 𝑡𝑡 𝑢𝑢 𝐿𝐿 𝑡𝑡 𝜙𝜙 𝑘𝑘𝑘𝑘

GE 162 Introduction to Seismology Winter 2013 - 2016

(40) = ( + 1) / with . The general solution is a discrete𝑘𝑘𝑛𝑛 superposition𝑛𝑛 𝜋𝜋 of𝐿𝐿 standing waves, each with a different wavenumber and associated frequency = : 𝑛𝑛 ∈ ℕ (41) 𝑛𝑛 𝑛𝑛 𝑛𝑛 𝑘𝑘 ( , ) = ∞𝜔𝜔 sin𝑐𝑐 𝑘𝑘( ) sin( + )

Each term of this sum is called a𝑢𝑢 mode𝑥𝑥 𝑡𝑡 . The� amplitude𝐴𝐴𝑛𝑛 𝑘𝑘𝑛𝑛 𝑥𝑥 and phase𝜔𝜔𝑛𝑛𝑡𝑡 𝜙𝜙𝑛𝑛 are determined by initial 0 conditions. The -th mode has zero-crossings. 𝑛𝑛 𝑛𝑛 [Draw modes]𝐴𝐴 𝜙𝜙 The mode with 𝑛𝑛 = 0 is the fundamental𝑛𝑛 mode. The fundamental frequency is = /2 = /2 . The others are higher modes or overtones and correspond to higher frequencies and shorter 0 0 wavelengths. 𝑛𝑛 𝑓𝑓 𝜔𝜔 𝜋𝜋 𝑐𝑐 𝐿𝐿

3.5 Duality between modes and propagating waves Using the trigonometric relation 2 sin sin = cos( ) cos( + ), a mode can be re-written as 𝑎𝑎 𝑏𝑏 𝑎𝑎 − 𝑏𝑏 − 𝑎𝑎 𝑏𝑏 (42) [cos( ( ) ) cos( ( + ) + )] 2 𝐴𝐴𝑛𝑛 This is actually a superposition of two𝑘𝑘𝑛𝑛 propagating𝑥𝑥 − 𝑐𝑐𝑐𝑐 − 𝜙𝜙 𝑛𝑛waves.− 𝑘𝑘𝑛𝑛 𝑥𝑥 𝑐𝑐𝑐𝑐 𝜙𝜙𝑛𝑛

[Mirror image trick, periodic functions, Fourier series.]

GE 162 Introduction to Seismology Winter 2013 - 2016

4 Green’s function. Waves in heterogeneous media.

4.1 Linear invariant systems, Green’s 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 ( , ). (43) = ( , ) 2 2𝑓𝑓 𝑥𝑥 𝑡𝑡 𝜕𝜕 𝑢𝑢 𝜕𝜕 𝑢𝑢 𝜌𝜌 2 − 𝜇𝜇 2 𝑓𝑓 𝑥𝑥 𝑡𝑡 A linear invariant system is defined by the𝜕𝜕 following𝑡𝑡 𝜕𝜕 𝑥𝑥properties: Input  Output ( )  ( ) Multiply ( )  ( ) Add 𝑓𝑓 (𝑡𝑡 ) + ( )  𝑢𝑢 (𝑡𝑡 ) + ( ) Delay 𝑎𝑎(𝑓𝑓 𝑡𝑡 )  𝑎𝑎(𝑢𝑢 𝑡𝑡 ) 1 2 1 2 𝑓𝑓 𝑡𝑡 𝑓𝑓 𝑡𝑡 𝑢𝑢 𝑡𝑡 𝑢𝑢 𝑡𝑡 Over the time scales𝑓𝑓 𝑡𝑡 − of𝑡𝑡 seismic′ wave propagation𝑢𝑢 𝑡𝑡 − 𝑡𝑡′ the Earth is a linear invariant system: it is linearly elastic and its material properties (density and elastic moduli) are fixed. We define the Green’s function (aka impulse response, transfer function) as the motion induced by an impulse force ( ). The motion produced by an arbitrary force ( ) can be obtained from the Green’s function by an integral operation known as convolution: 𝛿𝛿 𝑡𝑡 𝑓𝑓 𝑡𝑡 Input  Output ( )  ( ) Impulse response, Green’s function ( ) + ( )  ( ) + ( ) Linear combination of impulses ( ) ( )𝛿𝛿 𝑡𝑡 = ( )′  𝐺𝐺 𝑡𝑡( ) ( ) ′ Continuum superposition of impulses ′ 𝑎𝑎 𝛿𝛿 𝑡𝑡 ′ =𝑏𝑏 [𝛿𝛿 𝑡𝑡 −]𝑡𝑡( ) 𝑎𝑎 𝐺𝐺 𝑡𝑡′ =𝑏𝑏[𝐺𝐺 𝑡𝑡′−](𝑡𝑡 ) = Convolution ∫ 𝑓𝑓 𝑡𝑡 𝛿𝛿 𝑡𝑡 − 𝑡𝑡 𝑑𝑑𝑑𝑑′ 𝑓𝑓 𝑡𝑡 ∫ 𝑓𝑓 𝑡𝑡 𝐺𝐺 𝑡𝑡 − 𝑡𝑡 𝑑𝑑𝑑𝑑′ Important property of𝑓𝑓 ∗ the𝛿𝛿 Fourier𝑡𝑡 transform: convolution𝑓𝑓 ∗ 𝐺𝐺 in𝑡𝑡 time domain is equivalent to multiplication in frequency domain: (44) ( ) = ( ) × ( )

Convolution is easier to compute in spectral𝑓𝑓�∗ 𝑔𝑔 domain.ω 𝑓𝑓̃ ω 𝑔𝑔� ω Chain of linear invariant systems: ( ) = ( ) × ( ) × … × ( ) [Ex: source, path, site, instrument or building] 𝑢𝑢 𝜔𝜔 𝐺𝐺1 𝜔𝜔 𝐺𝐺2 𝜔𝜔 𝑓𝑓 𝜔𝜔 4.2 Green’s function for the 1D wave equation The solution of the wave equation with an impulsive point source ( , ) = ( ) ( ) (per unit force, per second) is called the Green’s function ( , ). It is given by (displacement per unit force, per unit time): 𝑓𝑓 𝑥𝑥 𝑡𝑡 𝛿𝛿 𝑥𝑥 𝛿𝛿 𝑡𝑡 𝐺𝐺 𝑥𝑥 𝑡𝑡 / | | (45) ( , ) = 2 𝑐𝑐 𝜇𝜇 𝑥𝑥 where is the Heaviside step function.𝐺𝐺 𝑥𝑥 𝑡𝑡 𝐻𝐻 �𝑡𝑡 − � [Sketch it] 𝑐𝑐 Proof: The𝐻𝐻 solution must comprise two symmetric waves propagating away from = 0, hence it must be of the form ( , ) = ( | |/ ). Plugging this into the wave equation, and noting that = , | | 𝑥𝑥 = sign( ) = 2 ( ) 1 and sign( )/ = 2 ( ), we get = , hence = . ′ 𝐺𝐺 𝑥𝑥 𝑡𝑡 𝐹𝐹 𝑡𝑡 − 𝑥𝑥 𝑐𝑐 𝐻𝐻 𝛿𝛿 𝜕𝜕 𝑥𝑥 ′ 𝑐𝑐 𝑐𝑐 𝜕𝜕𝜕𝜕 𝑥𝑥 𝐻𝐻 𝑥𝑥 − 𝜕𝜕 𝑥𝑥 𝜕𝜕𝜕𝜕 𝛿𝛿 𝑥𝑥 𝐹𝐹 2𝜇𝜇 𝛿𝛿 𝐹𝐹 2𝜇𝜇 𝐻𝐻 14

GE 162 Introduction to Seismology Winter 2013 - 2016

4.3 Waves in heterogeneous medium (WKBJ approximation) We have previously considered harmonic waves in homogeneous media: (46) ( , ) = ( ) = ( / ) / 𝑖𝑖 𝑘𝑘𝑘𝑘−𝜔𝜔𝜔𝜔 𝑖𝑖𝑖𝑖 𝑥𝑥 𝑐𝑐−𝑡𝑡 Note that is the wave travel time𝑢𝑢 𝑥𝑥 over𝑡𝑡 distance𝐴𝐴𝑒𝑒 . Let u𝐴𝐴s𝑒𝑒 generalize this concept, at least approximately, to smoothly heterogeneous media. We consider spatially variable material properties ( ) and 𝑥𝑥( 𝑐𝑐). Inspired by harmonic waves, we adopt𝑥𝑥 the following ansatz: ( ( ) ) (47) 𝜌𝜌 𝑥𝑥 𝜇𝜇 𝑥𝑥 ( , ) = ( ) ( ) 𝑖𝑖𝑖𝑖 𝑇𝑇 (𝑥𝑥 −)𝑡𝑡 where the amplitude (a real number)𝑢𝑢 𝑥𝑥 and𝑡𝑡 travel𝐴𝐴 𝑥𝑥 time𝑒𝑒 are smoothly varying functions of . Plugging this into the wave equation (7) 𝐴𝐴 𝑥𝑥 𝑇𝑇 𝑥𝑥 𝑥𝑥(48) ( ) = ( ) 2 𝜕𝜕 𝜕𝜕𝜕𝜕 𝜕𝜕 𝑢𝑢 we get after some algebra: �𝜇𝜇 𝑥𝑥 � 𝜌𝜌 𝑥𝑥 2 𝜕𝜕𝜕𝜕 𝜕𝜕𝜕𝜕 𝜕𝜕𝑡𝑡 (49) ( ) + (2 + ( ) ) = ′ ′ 2 ′2 ′ ′ ′ ′ 2 Separating the real and imaginary𝜇𝜇𝐴𝐴 − 𝜔𝜔parts𝜇𝜇𝜇𝜇 we𝐴𝐴 get𝑖𝑖 𝑖𝑖two𝜇𝜇 equations𝑇𝑇 𝐴𝐴 𝜇𝜇: 𝑇𝑇 𝐴𝐴 −𝜌𝜌𝜔𝜔 𝐴𝐴 1 ( ) (50) = c ′ ′ ′2 𝜇𝜇𝐴𝐴 2 (2 ) 2 (51) 𝑇𝑇 −+ = 0 A ′ ′ 𝜇𝜇′𝜇𝜇𝜔𝜔 𝐴𝐴 𝜇𝜇𝑇𝑇 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, 𝜇𝜇𝑇𝑇/| |, is much longer than the wavelength. We obtain then the so-called eikonal equation: ′ 𝜇𝜇1 𝜇𝜇 (52) = 0 c ′2 From which we derive the travel time: 𝑇𝑇 − 2 (53) ( ) = 𝑥𝑥 ( ) 𝑑𝑑𝑑𝑑′ Integrating equation (51), making use of = 1/ and denoting′ by ( ) = ( )/ ( ) the local 𝑇𝑇 𝑥𝑥 �0 impedance, we get the wave amplitude: ′ 𝑐𝑐 𝑥𝑥 𝑇𝑇 𝑐𝑐 𝑍𝑍 𝑥𝑥 𝜇𝜇 𝑥𝑥 𝑐𝑐 𝑥𝑥 (54) ( ) = (0) (0)/ ( ) Note that this is a statement of conservation of energy along the wave path (see section 3.1): 𝐴𝐴 𝑥𝑥 𝐴𝐴 �𝑍𝑍 𝑍𝑍 𝑥𝑥 ( ) ( ) = . 2 𝑍𝑍 𝑥𝑥 𝐴𝐴 𝑥𝑥 𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐

GE 162 Introduction to Seismology Winter 2013 - 2016

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 Shearer’s book. See Tromp and Dahlen’s book for gravitational, Earth’s rotation and pre-stress effects (not treated here).

5.1 Strain Displacement field: ( ). Deformation, related to the displacement gradient (a tensor): ( + ) ( ) + : . Where = ( , ,𝒖𝒖 )𝒙𝒙 is the gradient differential operator. The displacement gradient = is a tensor, ( ) = 𝒖𝒖 𝒙𝒙 𝒅𝒅𝒅𝒅 ≈ 𝒖𝒖 𝒙𝒙 𝛁𝛁𝒖𝒖 𝒅𝒅𝒅𝒅 𝛁𝛁 𝜕𝜕𝑥𝑥 𝜕𝜕𝑦𝑦 𝜕𝜕𝑧𝑧 Strain tensor = symmetric part of the 𝑻𝑻displacement gradient: = + , = ( + ). 𝛁𝛁𝒖𝒖 𝛁𝛁 𝒖𝒖 ∇𝑢𝑢 𝑖𝑖𝑖𝑖 𝜕𝜕𝑗𝑗𝑢𝑢𝑖𝑖 Assumption: small perturbations relative to a static configuration,1 1 𝑻𝑻 1 2 𝑖𝑖𝑖𝑖 2 𝑖𝑖 𝑗𝑗 𝑗𝑗 𝑖𝑖 [earthquake strain ~ slip/(rupture length)𝜀𝜀 ~ m �/ 𝛁𝛁10𝒖𝒖 km𝛁𝛁 ~0.𝒖𝒖 0�1%]𝜀𝜀 𝜕𝜕 𝑢𝑢 𝜕𝜕 𝑢𝑢 𝜀𝜀 ≪ 5.2 Stress Traction ( , ) is the force per unit of surface area acting on an oriented surface with normal centered on . 𝒕𝒕 𝒙𝒙 𝒏𝒏 [sketch] 𝒏𝒏 The dependence𝒙𝒙 on is encapsulated in the stress tensor: ( , ) = ( ) , = The i-th column of is the traction on a surface whose normal is the unit vector along the i-th 𝑖𝑖 𝑖𝑖𝑖𝑖 𝑗𝑗 dimension. 𝒏𝒏 𝒕𝒕 𝒙𝒙 𝒏𝒏 𝜎𝜎 𝒙𝒙 ∙ 𝒏𝒏 𝑡𝑡 𝜎𝜎 𝑛𝑛 Owing to conservation𝜎𝜎 of angular momentum, is a symmetric tensor: =

𝑖𝑖𝑖𝑖 𝑗𝑗𝑗𝑗 5.3 Momentum equation 𝜎𝜎 𝜎𝜎 𝜎𝜎 Apply = to a volume, and (55) 𝐹𝐹 𝑀𝑀𝑀𝑀 = = Applying Gauss’ theorem (aka divergence� 𝜌𝜌𝒖𝒖̈ theorem;𝑑𝑑𝑑𝑑 � 𝒕𝒕applied𝑑𝑑𝑑𝑑 � here𝜎𝜎 ∙to𝒏𝒏 the𝑑𝑑𝑑𝑑 tensor field ( )) we transform the surface integral into a volume integral: 𝜎𝜎 𝒙𝒙 (56) = ( ) where is the divergence of , a vector� 𝜌𝜌 with𝒖𝒖̈ 𝑑𝑑𝑑𝑑 components� 𝛁𝛁 ⋅ 𝜎𝜎 𝑑𝑑𝑑𝑑 = This is valid for any volume, hence 𝑖𝑖 j ij 𝛁𝛁 ⋅ 𝜎𝜎 𝜎𝜎 𝛁𝛁 ⋅ 𝜎𝜎 ∂ σ (57) = 0

𝜌𝜌𝒖𝒖̈ − 𝛁𝛁 ⋅ 𝜎𝜎 = 0 (58)

𝜌𝜌𝑢𝑢̈ i − ∂jσij 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. Hooke’s law: (59) = : (60) 𝜎𝜎= 𝑐𝑐 𝜀𝜀

𝜎𝜎𝑖𝑖𝑖𝑖 𝑐𝑐𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝜀𝜀𝑘𝑘𝑘𝑘 16

GE 162 Introduction to Seismology Winter 2013 - 2016 where is the elastic tensor. Its components are material properties (elastic moduli). Assumption: isotropic elasticity 𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖 𝑐𝑐 𝑐𝑐 (61) = + 2 where and are the Lame coefficients. 𝜎𝜎𝑖𝑖𝑖𝑖 𝜆𝜆𝜀𝜀𝑘𝑘𝑘𝑘𝛿𝛿𝑖𝑖𝑖𝑖 𝜇𝜇𝜀𝜀𝑖𝑖𝑖𝑖

𝜆𝜆 𝜇𝜇 5.5 The seismic wave equation Combining Hooke’s law with eq (57): (62) ( : ) = 0

For isotropic elasticity: 𝜌𝜌𝒖𝒖̈ − 𝛁𝛁 ⋅ 𝑐𝑐 𝜀𝜀

(63) = ( + ) + 2 2 𝜕𝜕 𝑢𝑢𝑖𝑖 𝜕𝜕 𝑢𝑢𝑖𝑖 which can also be written as 𝑖𝑖 2 𝜌𝜌𝑢𝑢̈ 𝜆𝜆 𝜇𝜇 𝑖𝑖 𝑗𝑗 𝜇𝜇 𝑗𝑗 𝜕𝜕𝑥𝑥 𝜕𝜕𝑥𝑥 𝜕𝜕𝑥𝑥 (64) = ( + ) ( ) + 𝟐𝟐 where = ( ) is the Laplacian𝜌𝜌𝒖𝒖 ̈ of 𝜆𝜆. 𝜇𝜇 𝛁𝛁 𝛁𝛁 ⋅ 𝒖𝒖 𝜇𝜇𝛁𝛁 𝒖𝒖 𝟐𝟐 𝛁𝛁 𝒖𝒖 𝛁𝛁 ⋅ 𝛁𝛁 𝒖𝒖 𝒖𝒖 5.6 It’s a perturbative equation So far and 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 ( , 𝜎𝜎) satisfying𝒖𝒖 eq (57) with zero acceleration. (65) 𝜎𝜎0 𝒖𝒖0 = 0 ( ) We assume that the subsequent transient motion−𝛁𝛁 ⋅comprises𝜎𝜎0 small perturbations , relative to the initial static configuration ( , ). Subtracting (65) from (57) we find that the perturbations ( , ) = ( , ) ( , ) also satisfy 𝛿𝛿𝛿𝛿 𝛿𝛿𝒖𝒖 0 0 𝜎𝜎 𝒖𝒖 𝛿𝛿𝛿𝛿 𝛿𝛿𝒖𝒖(66) 𝜎𝜎 𝒖𝒖 − 𝜎𝜎0 𝒖𝒖𝟎𝟎 = 0 𝜌𝜌𝜌𝜌𝒖𝒖̈ − 𝛁𝛁 ⋅ 𝛿𝛿𝛿𝛿 The material might have a non-linear behavior, = ( ). If the perturbations are small enough, we can linearize the constitutive relation near the initial configuration: = ( + ) ( ) : . This has the same form as Hooke’s law but for the perturbations,𝜎𝜎 𝐹𝐹 𝜀𝜀 = : , if we define the effective linear 0 0 elastic tensor as = . Hence the governing equation (62) also𝛿𝛿 𝛿𝛿applies𝐹𝐹 𝜀𝜀 to the𝛿𝛿𝛿𝛿 perturbations− 𝐹𝐹 𝜀𝜀 ≈ ∇ 𝐹𝐹 𝛿𝛿and𝛿𝛿 . In the remainder we will be concerned only with the perturbations,𝛿𝛿𝛿𝛿 𝑐𝑐 but𝛿𝛿𝛿𝛿 to simplify the notations we will drop the . 𝑐𝑐 ∇𝐹𝐹 𝛿𝛿𝒖𝒖 𝛿𝛿𝛿𝛿

5.7 P and𝛿𝛿 S waves examples Longitudinal waves (P waves): assume ( , ) = ( , ) (no dependence on or ) where is the unit vector in the direction. Plugging it into the wave equation we get: 1 𝒖𝒖 𝒙𝒙 𝑡𝑡 𝑢𝑢 𝑥𝑥 𝑡𝑡 𝒙𝒙� 𝑦𝑦 𝑧𝑧 𝒙𝒙� (67) = ( + 2 ) 𝒙𝒙 2 2 1 1 This is a wave equation like eq (9), with P wave𝜕𝜕 𝑢𝑢 speed 𝜕𝜕 𝑢𝑢 𝜌𝜌 2 𝜆𝜆 𝜇𝜇 2 𝜕𝜕𝑡𝑡 𝜕𝜕𝑥𝑥

GE 162 Introduction to Seismology Winter 2013 - 2016

+ 2 (68) = 𝜆𝜆 𝜇𝜇 Transverse waves (S waves): assume ( , ) =𝑐𝑐𝑃𝑃 ( �, ) 𝜌𝜌 (69) 3 = 𝒖𝒖 𝒙𝒙 𝑡𝑡 𝑢𝑢2 𝑥𝑥 𝑡𝑡 𝒛𝒛� 2 3 3 S wave speed: 𝜕𝜕 𝑢𝑢 𝜕𝜕 𝑢𝑢 𝜌𝜌 2 𝜇𝜇 2 𝜕𝜕𝑡𝑡 𝜕𝜕𝑥𝑥 (70) = /

𝑐𝑐𝑆𝑆 �𝜇𝜇 𝜌𝜌 Rocks usually have (a Poisson solid), hence 3 . Typically wave speeds ~ several km/s, but can be ~ 100 m/s on the shallowest layers. 𝜆𝜆 ≈ 𝜇𝜇 𝑐𝑐𝑃𝑃 ≈ √ 𝑐𝑐𝑆𝑆 5.8 General decomposition into P and S waves Let’s show that the 3D wavefield is a superposition of two types of body waves. Using the following vector identity (71) = ( ) × × × 2 where is the curl of , eq (64) can𝛁𝛁 𝒖𝒖 be written𝛁𝛁 𝛁𝛁 ⋅ 𝒖𝒖 as − 𝛁𝛁 𝛁𝛁 𝒖𝒖 (72) 𝛁𝛁 𝒖𝒖 𝒖𝒖 = ( ) × × 2 2 The Helmholtz theorem states that any𝒖𝒖̈ sufficiently𝑐𝑐𝑃𝑃 𝛁𝛁 𝛁𝛁 ⋅ 𝒖𝒖 smooth,− 𝑐𝑐𝑆𝑆 𝛁𝛁 rapidly𝛁𝛁 𝒖𝒖 decaying 3D vector field 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 and the curl of a divergence𝒖𝒖 -free vector potential (i.e. = 0): 𝜙𝜙 (73) 𝝍𝝍 𝛁𝛁 ⋅ 𝝍𝝍 = + × The curl of the gradient of any 3D scalar field𝒖𝒖 is always𝛁𝛁𝜙𝜙 𝛁𝛁the zero𝝍𝝍 vector, hence the vector field is curl- free ( × = ). The divergence of the curl of any 3D scalar field is always the zero, hence the vector field × is divergence-free ( ( × ) = 0). Combining (73) and the divergence and curl 𝛁𝛁of𝜙𝜙 (72), respectively,𝛁𝛁 𝛁𝛁𝜙𝜙 we 𝟎𝟎arrive at two partial differential equations (actually not quite, see AR for a more rigorous𝛁𝛁 derivation)𝝍𝝍 : 𝛁𝛁 ⋅ 𝛁𝛁 𝝍𝝍 (74) = 2 2 (75) 𝜙𝜙̈ = 𝑐𝑐𝑃𝑃 ∇ 𝜙𝜙 2 2 These are two 3D wave equations for P wave and𝝍𝝍̈ S𝑐𝑐 wave𝑆𝑆 ∇ 𝝍𝝍 potentials, respectively. 5.9 Polarization of body waves A particular solution of the 3D wave equation for the scalar potential is a plane wave: (76) ( , ) = ( ) 𝑖𝑖 𝒌𝒌⋅𝒙𝒙−𝜔𝜔𝜔𝜔 where is the wave vector, which indicates𝜙𝜙 𝒙𝒙the𝑡𝑡 direction𝐴𝐴𝑒𝑒 of propagation. [Sketch] The phase𝒌𝒌 velocity of the plane wave is /| |. The wave equation is satisfied if 𝜔𝜔 𝒌𝒌 (77) /| | =

𝜔𝜔 𝒌𝒌 𝑐𝑐𝑃𝑃 18

GE 162 Introduction to Seismology Winter 2013 - 2016

The P wave displacement is [show it] (78) = =

It is parallel to the direction of wave propagation𝒖𝒖𝑃𝑃 ∇. 𝜙𝜙 𝒌𝒌𝜙𝜙

Similarly, assuming that the vector potential is a𝒌𝒌 plane wave one can show that the S wave displacement is 𝝍𝝍 (79) = × = ×

This is perpendicular to the wave propagation𝒖𝒖𝑺𝑺 direction𝛁𝛁 𝝍𝝍 𝒌𝒌. 𝝍𝝍 5.10 Usual characteristics of body waves 𝒌𝒌 P vs S: • speed: S is slower. Typically, / = 3 and epicentral distance (km) 8 × P-S travel time (s) • polarization (see above). Refraction at shallow depth  P vertical, S horizontal 𝑃𝑃 𝑆𝑆 • amplitude: S is stronger 𝑐𝑐 𝑐𝑐 √ ≈ • 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.]

GE 162 Introduction to Seismology Winter 2013 - 2016

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 ( ). Considering the isotropy of the problem, we seek radially symmetric solutions that derive from a scalar potential ( , ) satisfying 𝑓𝑓 𝑡𝑡 1 (80) 𝜙𝜙 𝑟𝑟 𝑡𝑡 = 4 ( ) ( ) 2 It can be shown that the solution is 2 ̈ ∇ 𝜙𝜙 − 𝑃𝑃 𝜙𝜙 − 𝜋𝜋𝜋𝜋 𝑟𝑟 𝑓𝑓 𝑡𝑡 𝑐𝑐 1 (81) ( , ) = 𝑟𝑟 The displacement field, = , is radially𝜙𝜙 𝑟𝑟symmetric.𝑡𝑡 − 𝑓𝑓Its� 𝑡𝑡radial− �component is 𝑟𝑟 𝑐𝑐𝑃𝑃 ( , ) = (82) 𝒖𝒖 ∇𝜙𝜙 1 𝑟𝑟 1 𝑟𝑟 2 The first term describes the near𝑢𝑢-field𝑟𝑟 𝑟𝑟 𝑡𝑡term𝑟𝑟 and𝑓𝑓 � has𝑡𝑡 − the𝑐𝑐𝑃𝑃� following− 𝑟𝑟𝑐𝑐𝑃𝑃 𝑓𝑓̇ �𝑡𝑡 properties:− 𝑐𝑐𝑃𝑃� • It decays as 1/ • A persistent source2 (i.e. 0 when ) leaves a static residual displacement The second term describes𝑟𝑟 the far-field motion and has the following properties: • Transient, it vanishes after𝑓𝑓 ≠ the passage𝑡𝑡 → of∞ the wave ( is non-zero only over a finite interval) • 1/r decay, consistent with conservation of total energy 4 (energy density times ̇ surface area of the spherical wavefront) 𝑓𝑓 2 2 The far-field term dominates over the near-field term when 𝐸𝐸 ∝/ 𝜋𝜋𝑟𝑟= 𝜌𝜌𝑢𝑢/̇2 , i.e. at high-frequencies / long distances. (Take the Fourier transform of , then compare the two terms). 𝑟𝑟 ≫ 𝑐𝑐𝑃𝑃 𝜔𝜔 𝜆𝜆 𝜋𝜋 6.2 Ray theory: eikonal equation, 𝑢𝑢ray𝑟𝑟 tracing In smoothly heterogeneous media, approximate solutions of the form (83) ( , ) = ( ) ( ( ) ) 𝑖𝑖𝑖𝑖 𝑇𝑇 𝑥𝑥 −𝑡𝑡 can be found at high frequencies corresponding𝜙𝜙 𝒙𝒙 𝑡𝑡 to𝐴𝐴 wavelengths𝒙𝒙 𝑒𝑒 much shorter than the characteristic length scales of heterogeneity of the material properties (see also LW p.72). The travel time ( ) satisfies the 3D eikonal equation: 𝑇𝑇 𝒙𝒙 (84) | | = 1/ ( ) 2 2 ( ) = 0 Solving this non-linear differential equation with𝛁𝛁𝑇𝑇 a given𝑐𝑐 origin𝒙𝒙 point ( ) gives the spatial distribution of travel times. The contours of ( ) are the wave fronts. The normal to these contours is 0 parallel to . The paths connecting these normals define rays. 𝑇𝑇 𝒙𝒙 [sketch𝑇𝑇 wave𝒙𝒙 fronts and rays] Denoting the𝛁𝛁𝑇𝑇 local ray direction by the slowness vector = and defining a curvilinear coordinate along a ray, the ray tracing problem is formulated as: given an origin point, ( = 0), and an initial ray direction (take-off vector), ( = 0), compute the ray path𝒔𝒔 𝛁𝛁(𝑇𝑇) and slowness ( ) by solving: 𝜉𝜉 𝒙𝒙 𝜉𝜉 (85) 𝒔𝒔 𝜉𝜉 = ( ) 𝒙𝒙 𝜉𝜉 𝒔𝒔 𝜉𝜉 𝑑𝑑𝒙𝒙 1 (86) = 𝑐𝑐 𝒙𝒙 𝒔𝒔 𝑑𝑑𝑑𝑑 ( ) 𝑑𝑑𝒔𝒔 𝛁𝛁 � � 𝑑𝑑𝑑𝑑 𝑐𝑐 𝒙𝒙

GE 162 Introduction to Seismology Winter 2013 - 2016

6.3 Rays in depth-dependent media, Snell’s law, refraction

Consider the special case of a medium with depth-dependent material properties: ( ) = ( ). From the last equation we get / = / = 0: the horizontal components of the slowness vector are constant and rays remain confined on a vertical plane. The constant horizontal𝑐𝑐 𝒙𝒙 slowness𝑐𝑐 𝑧𝑧 (aka 𝑑𝑑𝑠𝑠𝑥𝑥 𝑑𝑑𝑑𝑑 𝑑𝑑𝑠𝑠𝑦𝑦 𝑑𝑑𝑑𝑑 apparent slowness) defines the ray parameter = + . | | 2 2 Because = 1/ , we have = sin / and 𝑝𝑝= cos�𝑠𝑠𝑥𝑥 / 𝑠𝑠, 𝑦𝑦where is the angle between the ray and the vertical axis. The constancy of the ray parameter gives Snell’s law at the interface between two 𝑧𝑧 materials:𝒔𝒔 𝑐𝑐 𝑝𝑝 𝜃𝜃 𝑐𝑐 𝑠𝑠 𝜃𝜃 𝑐𝑐 𝜃𝜃 (87) = sin / = sin /

Continuity of the wave front along an interface𝑝𝑝 𝜃𝜃also1 𝑐𝑐 1implies 𝜃𝜃Snell’s2 𝑐𝑐2 law. It can also be shown for plane waves (not restricted to high-frequency ray theory). It is also implied by Fermat’s 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, Snell’s 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 = sin( ) / (0). At the turning point sin( ) = 1. From Snell’s law, the depth of the turning point is such that = 1/ ( ). 0 [Q: what is the angle𝑝𝑝 of a ray𝜃𝜃 emitted𝑐𝑐 by a shallow source that penetrate𝜃𝜃 s down to depth where S wave 𝑝𝑝speed is𝑐𝑐 3𝑧𝑧 km/s, if shallow speed is 300 m/s?]

GE 162 Introduction to Seismology Winter 2013 - 2016

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( ) = 1) when incidence reaches a critical angle such that 𝜃𝜃 𝜃𝜃𝑐𝑐 sin( ) = (88) 𝑐𝑐𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠 The critically refracted wave is called a head wave𝑐𝑐 or refracted wave. 𝜃𝜃 𝑓𝑓𝑓𝑓𝑓𝑓𝑓𝑓 [Plot travel times. L&W fig 3.10. Order:𝑐𝑐 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.

GE 162 Introduction to Seismology Winter 2013 - 2016

7.2 A steep transition zone [Shearer’s 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 [Shearer’s fig 4.9]

GE 162 Introduction to Seismology Winter 2013 - 2016

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.

[Shearer’s 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 1/ . Actually decay a bit faster, 1/ with < < 1, because of energy leakage. Nevertheless, they persist over longer distances than other𝑛𝑛 body1 waves. ∝ 𝑟𝑟 𝑟𝑟 2 𝑛𝑛 Other ex: fault zone guided waves. √

GE 162 Introduction to Seismology Winter 2013 - 2016

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 = , where = = impedance. Energy flux through a cross-section of1 a ray2 beam2 is conserved along the ray path and 2 𝑍𝑍𝐴𝐴 𝜔𝜔 𝑍𝑍 𝜌𝜌𝜌𝜌 Hence 2 ∝ 𝑍𝑍𝐴𝐴 𝑑𝑑𝑑𝑑 (89) = 𝐴𝐴1 𝑍𝑍2𝑑𝑑𝑆𝑆2 Ex: spherical wave = ( /2) , hence 1/ . � 𝐴𝐴2 𝑍𝑍1𝑑𝑑𝑆𝑆1 2 𝜋𝜋 𝑟𝑟𝑟𝑟𝑟𝑟 𝐴𝐴 ∝ spreading𝑟𝑟 factor. 𝑑𝑑𝑑𝑑 2 Wave focusing shrinks the spreading factor,�𝑑𝑑𝜃𝜃 hence∼ amplifies wave motion (more waves arriving together). [Shearer section 6.2] 1/ 𝑑𝑑𝑑𝑑 Discuss caustics again. 𝐸𝐸 ∝ � � 𝑑𝑑𝑑𝑑 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): sin( ) (90) = In a layer with constant speed, trigonometric arguments𝑟𝑟 𝜃𝜃show that [Sketch Shearer’s fig 4.12] 𝑝𝑝′ = 𝑐𝑐 ′ At a material interface, this can be derived from𝑝𝑝 Snell’s𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐 𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐law. In terms of great-circle distance in radians, = , = = / ′ (91) Δ 𝑑𝑑𝑑𝑑 = 𝑟𝑟𝑟𝑟Δ/ 𝑝𝑝 𝑟𝑟𝑟𝑟 𝑟𝑟𝑟𝑟𝑟𝑟 𝑑𝑑𝑑𝑑 ′ 𝑝𝑝 𝑑𝑑𝑑𝑑 𝑑𝑑Δ 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 Earth’s materials. Seismological data: arrival time of several phases (families of ray paths).

[Examples of what seismologists do with deep seismic phases]

GE 162 Introduction to Seismology Winter 2013 - 2016

+ D’’ = region with reduced gradient above the CMB (2700-2900 km)

GE 162 Introduction to Seismology Winter 2013 - 2016

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 ( , , ) satisfies a scalar wave equation with S-wave speed. Consider two materials in contact along a planar𝑢𝑢 𝑥𝑥 interface,𝑧𝑧 𝑡𝑡 𝒚𝒚� with speeds and , and an incident plane wave arriving to the interface from medium 1. 1 2 Formulate the problem: 𝛽𝛽 𝛽𝛽 • 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): ( , , ) = ( ) + ( ) ( ) ( , 𝑖𝑖𝑖𝑖, )𝑝𝑝𝑝𝑝=+𝜂𝜂1 𝑧𝑧−𝑡𝑡 𝑖𝑖𝑖𝑖 𝑝𝑝𝑝𝑝 −𝜂𝜂1𝑧𝑧−𝑡𝑡 1 𝑢𝑢 𝑥𝑥 𝑧𝑧 𝑡𝑡 𝑒𝑒 𝑖𝑖𝑖𝑖 𝑝𝑝𝑝𝑝𝑅𝑅−𝜂𝜂𝑒𝑒2𝑧𝑧−𝑡𝑡 where = is the ray parameter2 , = = and = = . If < 1/ , 𝑢𝑢 𝑥𝑥 𝑧𝑧 𝑡𝑡 1𝑇𝑇 𝑒𝑒 2 𝑘𝑘𝑥𝑥 𝑘𝑘𝑧𝑧 1 𝑘𝑘𝑧𝑧 1 2 2 2 2 the incidence/reflection/transmission𝑝𝑝 𝜔𝜔 𝜂𝜂1 angles𝜔𝜔 are�𝛽𝛽 well1 − 𝑝𝑝defined𝜂𝜂 (real)2 𝜔𝜔 and �𝛽𝛽2=−cos𝑝𝑝 . 𝑝𝑝 𝛽𝛽𝑘𝑘 • Write the boundary conditions: continuity of displacement and shear stress at the interface. 𝜂𝜂𝑘𝑘 𝑗𝑗𝑘𝑘 30

GE 162 Introduction to Seismology Winter 2013 - 2016

• Obtain a system of two linear equations with two unknowns (R and T). • Solve it (defining impedances = / and = / ):

= 𝑍𝑍1 𝜇𝜇1 𝛽𝛽 1 𝑍𝑍2and𝜇𝜇 2 𝛽𝛽2 = 𝑍𝑍1𝜂𝜂1−𝑍𝑍2η2 2 𝑍𝑍1𝜂𝜂1 𝑍𝑍1η1+𝑍𝑍2η2 𝑍𝑍1η1+𝑍𝑍2η2 If < 1/ we can write𝑅𝑅 it as: 𝑇𝑇

𝑘𝑘 𝑝𝑝 𝛽𝛽 = and = 𝑍𝑍1 cos 𝑗𝑗1−𝑍𝑍2 cos 𝑗𝑗2 2 𝑍𝑍1 cos 𝑗𝑗1 𝑍𝑍1 cos 𝑗𝑗1+𝑍𝑍2 cos 𝑗𝑗2 𝑍𝑍1 cos 𝑗𝑗1+𝑍𝑍2 cos 𝑗𝑗2 𝑅𝑅 𝑇𝑇

If < there is a critical angle defined by sin = / 1 2 𝑐𝑐 such𝛽𝛽 that𝛽𝛽 any wave with incidence𝑗𝑗 angle wider than (post-critical) emerges parallel to the interface 𝑐𝑐 1 2 ( = /2). 𝑗𝑗 𝛽𝛽 𝛽𝛽 𝑐𝑐 Post-critical reflections have | | = 1 (total reflection𝑗𝑗) and incur a phase shift. 2 The𝑗𝑗 refracted𝜋𝜋 wave in the post-critical range decays exponentially with distance to the interface: this 𝑅𝑅 kind of wave is called inhomogeneous or evanescent wave: if > , the quantity = is 1 1 ( ) 2( ) 2 imaginary and the displacement in the bottom half-space is 𝑝𝑝( , 𝛽𝛽2, ) = 𝜂𝜂2 �𝛽𝛽2 − 𝑝𝑝 . The 1/ 𝑖𝑖𝑖𝑖 𝑝𝑝𝑝𝑝−𝑡𝑡 −𝜔𝜔 𝐼𝐼𝐼𝐼 𝜂𝜂~2 𝑧𝑧 exponential decay has a characteristic depth ( ) . If 2 the penetration depth is 1 𝑢𝑢 𝑥𝑥 𝑧𝑧 𝑡𝑡 𝑇𝑇 𝑒𝑒 𝑒𝑒 1

∝ 𝐼𝐼𝐼𝐼 𝜂𝜂2 𝜔𝜔 𝑝𝑝 ≫ 𝛽𝛽2 𝑝𝑝𝑝𝑝 ∝ 𝜆𝜆𝑥𝑥 In the special case of a free surface ( = 0) we get total reflection ( = 1) at all incident angles.

𝑍𝑍2 𝑅𝑅

GE 162 Introduction to Seismology Winter 2013 - 2016

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 (horizontal wavelength).

∼ 𝜆𝜆

The amplitude of surface waves decays as 1/ where is horizontal propagation distance. Proof: energy is proportional to wave amplitude squared and is conserved over a ring of surface (as opposed to a sphere of surface in the case√𝜆𝜆 𝜆𝜆of body waves).𝑟𝑟 2 ∝ 𝜆𝜆𝜆𝜆 8.4 Dispersion relation∝ 𝑟𝑟 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: • Write displacement fields as plane waves in each of the two media ( , , ) = ( ) + ( ) ( ) ( , , ) =𝑖𝑖𝑖𝑖 𝑝𝑝 𝑝𝑝+𝜂𝜂1𝑧𝑧−𝑡𝑡 𝑖𝑖𝑖𝑖 𝑝𝑝𝑝𝑝−𝜂𝜂1𝑧𝑧−𝑡𝑡 1 • Apply boundary conditions𝑢𝑢 (3):𝑥𝑥 𝑧𝑧 zero𝑡𝑡 stress𝐴𝐴𝑒𝑒 at the𝑖𝑖𝑖𝑖 surface𝑝𝑝𝑝𝑝−𝜂𝜂2𝑧𝑧−𝐵𝐵 (at𝑡𝑡𝑒𝑒 z=0) and continuity of displacement 2 and shear stress at the interface𝑢𝑢 (at𝑥𝑥 𝑧𝑧z=h)𝑡𝑡 𝐶𝐶 𝑒𝑒 • Obtain a homogeneous (zero right-hand-side) system of 3 linear equations with 3 unknowns (A, B and C).

GE 162 Introduction to Seismology Winter 2013 - 2016

• A homogeneous system has non-trivial (non-zero) solutions only if its determinant is zero. This condition yields the following equation relating horizontal speed = / to frequency, known as a dispersion relation: 𝒄𝒄 𝟏𝟏 𝒑𝒑 1 2 tan 1 = 𝛽𝛽2 2 �� − ℎ𝜔𝜔 𝛽𝛽1 𝑍𝑍2 𝑐𝑐 � � − � � � 1 𝛽𝛽1 𝑐𝑐 𝑍𝑍1 2 𝛽𝛽1 � − � � 𝑐𝑐 For any frequency , this equation can be solved to get a speed ( ). Because of the trigonometric (periodic) function on the l.h.s., solutions are not always unique; in some frequency ranges we𝜔𝜔 get a set of speeds ( ). 𝑐𝑐 𝜔𝜔

𝑐𝑐𝑛𝑛 𝜔𝜔

GE 162 Introduction to Seismology Winter 2013 - 2016

Fundamental mode. ( = 0) Penetration depth is shorter at high frequency, probes shallower part of the crust, lower velocity. Two extremes: 𝑛𝑛 at high frequency, at low frequency. General appearance of Love wave seismogram: different frequencies arrive at different times. 1 2 𝑐𝑐 ∼ 𝛽𝛽 𝑐𝑐 ∼ 𝛽𝛽 Overtones. ( > 0) In some frequency ranges the dispersion relation has multiple solutions: the n-th higher modes (overtone) appears𝑛𝑛 at frequencies higher than cutoff frequency = 1𝑛𝑛𝑛𝑛 1 𝜔𝜔𝑛𝑛 ℎ� 2 − 2 𝛽𝛽1 𝛽𝛽2 8.5 Phase and group velocities Consider the superposition of two harmonic waves with slightly different parameters, ( , ) = ( + , + ) and ( , ) = ( , ): 1 1 0 cos( ) + cos( ) 𝑘𝑘 𝜔𝜔 𝑘𝑘 0 1 1 0 0 𝛿𝛿We𝛿𝛿 𝜔𝜔can rewrite𝛿𝛿𝛿𝛿 it as𝑘𝑘 𝜔𝜔 𝑘𝑘 − 𝛿𝛿𝛿𝛿 𝜔𝜔 − 𝛿𝛿𝛿𝛿 1 1 2 2 2 cos𝑘𝑘( 𝑥𝑥 − 𝜔𝜔 𝑡𝑡 ) cos( 𝑘𝑘 𝑥𝑥 − 𝜔𝜔 𝑡𝑡) Product of two terms: a fast (high-frequency) carrier modulated by a slow (low frequency) envelope. 0 0 𝑘𝑘 𝑥𝑥 − 𝜔𝜔 𝑡𝑡 𝛿𝛿𝛿𝛿𝛿𝛿 − 𝛿𝛿𝛿𝛿𝛿𝛿

GE 162 Introduction to Seismology Winter 2013 - 2016

Phase velocity = ( ) = = propagation speed of the carrier 𝜔𝜔 Group velocity = 𝑐𝑐 (𝜔𝜔 ) =𝑘𝑘𝑥𝑥 = propagation speed of the envelope 𝛿𝛿𝛿𝛿 𝑈𝑈 𝜔𝜔 𝛿𝛿𝛿𝛿 Compare: in homogeneous media, / = . Phase velocity is constant, does not depend on frequency: no dispersion. Group velocity is also = . 𝜔𝜔 𝑘𝑘 𝑐𝑐 8.6 Airy phase 𝑐𝑐 1 = = = = 𝑑𝑑𝑑𝑑 1 𝑐𝑐 1 +𝑐𝑐 𝑈𝑈 𝜔𝜔 𝜔𝜔 𝑑𝑑𝑑𝑑 𝑇𝑇 𝑑𝑑𝑑𝑑 𝑑𝑑𝑑𝑑 𝑑𝑑 � � − where = = period. 𝑐𝑐 𝑐𝑐 𝑑𝑑𝑑𝑑 𝑐𝑐 𝑑𝑑𝑑𝑑 2𝜋𝜋 𝑑𝑑𝑑𝑑 > 0 Because𝑇𝑇 𝜔𝜔 , the group velocity U may have a minimum. 𝑑𝑑𝑑𝑑 𝑑𝑑𝑑𝑑

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).

GE 162 Introduction to Seismology Winter 2013 - 2016

9 Surface waves II: Rayleigh waves

9.1 Rayleigh waves See L&W 4.1 Displacement wavefield: = + × In 2D ( , ), the SV potential is off-plane, its curl is in-plane: = (0, , 0). P-SV potentials in the form𝒖𝒖 of 𝛁𝛁harmonic𝜙𝜙 𝛁𝛁 waves𝝍𝝍 (z points down): 𝑥𝑥 𝑧𝑧 𝝍𝝍 𝜓𝜓𝑦𝑦 = = ( ) 𝑃𝑃 ( ) −𝑖𝑖�𝜔𝜔𝜔𝜔−=𝑘𝑘𝑥𝑥𝑥𝑥− 𝑘𝑘𝑧𝑧 𝑧𝑧� 𝑖𝑖𝑖𝑖 𝑝𝑝𝑝𝑝+𝜂𝜂𝑃𝑃𝑧𝑧−𝑡𝑡 𝜙𝜙 𝐴𝐴 𝑒𝑒 𝑖𝑖𝑖𝑖 𝑝𝑝𝑝𝑝−𝜂𝜂𝐴𝐴𝑠𝑠𝑧𝑧𝑒𝑒−𝑡𝑡 = = = 𝑦𝑦= = where , 𝑃𝑃 and 𝜓𝜓 𝑆𝑆𝐵𝐵 𝑒𝑒 . 𝑥𝑥 𝑧𝑧 𝑧𝑧 𝑘𝑘 𝑘𝑘 1 2 𝑘𝑘 1 2 2 2 ( ) Evanescent𝑝𝑝 𝜔𝜔waves𝜂𝜂𝑃𝑃 if 𝜔𝜔 > �>𝑐𝑐𝑃𝑃 −,𝑝𝑝 leads to𝜂𝜂 𝑆𝑆surface𝜔𝜔 wave�𝑐𝑐𝑆𝑆 −confined𝑝𝑝 near the surface ~ 1 1 Free surface boundary condition, at = 0: −𝜔𝜔 𝐼𝐼𝐼𝐼 𝜂𝜂 𝑧𝑧 𝑝𝑝 𝑐𝑐𝑆𝑆 𝑐𝑐𝑃𝑃 𝑒𝑒 = , + , + 2 , = 0 𝑧𝑧 = + = 0 𝑧𝑧𝑧𝑧 𝑥𝑥 𝑥𝑥 , 𝑧𝑧 𝑧𝑧 , 𝑧𝑧 𝑧𝑧 This leads to a linear system of two𝜎𝜎 equations𝜆𝜆�𝑢𝑢 and two𝑢𝑢 �unknowns𝜇𝜇𝑢𝑢 (A and B). Non-trivial solutions exist 𝑥𝑥𝑥𝑥 𝑧𝑧 𝑥𝑥 𝑥𝑥 𝑧𝑧 only if its determinant is zero. That condition𝜎𝜎 leads𝜇𝜇�𝑢𝑢 to 𝑢𝑢 �

2 4 1 1 = 0 2 2 2 2 𝑐𝑐 𝑐𝑐 𝑐𝑐 where = 1/ is the apparent velocity.� − For2� a− given� value− 2 of� the− ratio2 / , this equation has a single 𝑐𝑐𝑆𝑆 𝑐𝑐𝑆𝑆 𝑐𝑐𝑃𝑃 solution / < 1, the Rayleigh wave speed. For a Poisson solid ( = 1/4), we find 0.92 . 𝑆𝑆 𝑃𝑃 𝑐𝑐 𝑝𝑝 [LW fig 4.6] 𝑐𝑐 𝑐𝑐 𝑐𝑐 𝑐𝑐𝑆𝑆 𝜈𝜈 𝑐𝑐 ≈ 𝑐𝑐𝑆𝑆

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.

GE 162 Introduction to Seismology Winter 2013 - 2016

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.

GE 162 Introduction to Seismology Winter 2013 - 2016

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: = × where = (0,0, ( ) ( , , )) and ( ) is the Love wave eigenfunction at frequency derived from the local 1D velocity model. Plug it into the seismic wave equation, leads to 𝑢𝑢 ∇ 𝜓𝜓 𝜓𝜓 𝑍𝑍 𝑧𝑧 𝑆𝑆 𝑥𝑥 𝑦𝑦 𝜔𝜔 𝑍𝑍 𝑧𝑧 + + = 0 𝜔𝜔 2 2 ( ,2 , ) 𝜕𝜕 𝑆𝑆 𝜕𝜕 𝑆𝑆 𝜔𝜔 Where ( , , ) is the local Love wave 2phase velocity2 2 at location𝑆𝑆 ( , ) and frequency . Applying WKBJ approximation leads to ray theory𝜕𝜕𝑥𝑥 along𝜕𝜕𝑦𝑦 the surface.𝑐𝑐 𝜔𝜔 𝑥𝑥 𝑦𝑦 Great circle𝑐𝑐 𝜔𝜔 path.𝑥𝑥 𝑦𝑦 𝑥𝑥 𝑦𝑦 𝜔𝜔 Focusing. Laterally “trapped” surface waves in basins. Body to surface wave conversion at sediment edges.

GE 162 Introduction to Seismology Winter 2013 - 2016

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 – and elastic deformation of the crust (deformable seafloor). See Tsai et al (2013).

GE 162 Introduction to Seismology Winter 2013 - 2016

10 Normal modes of the Earth

Modes in an acoustic sphere [see derivation in LW p 156-158]: • Write scalar wave equation in spherical coordinates. • Assume a separable solution ( , , , ) = ( ) ( ) ( ) ( ) an plug it in wave equation • Find 4 separate differential equations • Solutions are related to well-known𝑢𝑢 𝑟𝑟 𝜃𝜃 𝜙𝜙special𝑡𝑡 functions𝑅𝑅 𝑟𝑟 Θ 𝜃𝜃 Φ 𝜙𝜙 𝑇𝑇 𝑡𝑡 • The surface dependence ( ) ( ) is given by spherical harmonics

Θ 𝜃𝜃 Φ 𝜙𝜙

GE 162 Introduction to Seismology Winter 2013 - 2016

o = angular order number, aka spherical harmonic degree. Positive integer. Total number of nodal lines (zero crossing lines) 𝑙𝑙 o = azimuthal order number. 2 + 1 integer values: . Number of nodal lines through the pole = | |. • The radial𝑚𝑚 dependence ( ) is given by spherical𝑙𝑙 Bessel functions−𝑙𝑙 ≤ 𝑚𝑚 ≤ 𝑙𝑙 𝑚𝑚 𝑅𝑅 𝑟𝑟( ) = where = 1 𝑑𝑑 𝑙𝑙 sin 𝑥𝑥 𝜔𝜔𝜔𝜔 • The time dependence ( ) is sinusoidal,𝑙𝑙 with frequency 𝑗𝑗𝑙𝑙 𝑥𝑥 𝑥𝑥 �− 𝑥𝑥 𝑑𝑑𝑑𝑑� 𝑥𝑥 𝑥𝑥 𝑐𝑐 • Applying boundary conditions at the surface and center of 𝑚𝑚the sphere leads to a family of 𝑙𝑙 admissible frequencies,𝑇𝑇 or𝑡𝑡 eigenfrequencies for each𝜔𝜔 admissible pairs , , where = radial order number 𝑚𝑚 𝑙𝑙 𝜔𝜔 𝑙𝑙 𝑚𝑚 𝑛𝑛

GE 162 Introduction to Seismology Winter 2013 - 2016

Mode decomposition in the elastic Earth.

GE 162 Introduction to Seismology Winter 2013 - 2016

Properties of eigenfrequencies : • They do not depend on azimuthal order 𝑛𝑛 𝑙𝑙 • Singlets of same are grouped𝜔𝜔 in multiplets. • Degeneracy: in a perfectly radial, isotropic,𝑚𝑚 non-rotating Earth, all singlets in a multiplet have the same eigenfrequency.𝑚𝑚 In reality there is mode splitting

GE 162 Introduction to Seismology Winter 2013 - 2016

Duality modes – waves:

GE 162 Introduction to Seismology Winter 2013 - 2016

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 1. = exp 1 In one cycle,≫ energy (amplitude squared) decays by a factor ( 2)𝜋𝜋 . 𝐸𝐸�𝑡𝑡+ 𝜔𝜔 � 2𝜋𝜋 2𝜋𝜋 Energy loss per cycle: = . 𝐸𝐸 𝑡𝑡 �− 𝑄𝑄 � ≈ − 𝑄𝑄 Δ𝐸𝐸 2𝜋𝜋 𝐸𝐸 − 𝑄𝑄 47

GE 162 Introduction to Seismology Winter 2013 - 2016

11.2 A damped oscillator

Mass-spring-dashpot system, single degree of freedom, subject to an impulse force: + + = ( ) Displacement: 𝑀𝑀𝑥𝑥̈ 𝜂𝜂𝑥𝑥̇ 𝐾𝐾𝐾𝐾 𝛿𝛿 𝑡𝑡 ( ) = exp( ) sin ( 1 ) where = / and = / 2 0 � 0 Here = 1/𝑥𝑥2 𝑡𝑡. 𝐴𝐴 −𝜖𝜖𝜔𝜔 𝑡𝑡 − 𝜖𝜖 𝜔𝜔 𝑡𝑡 0 0 𝜔𝜔 �𝐾𝐾 𝑀𝑀 𝜖𝜖 𝜂𝜂 𝑀𝑀 𝜔𝜔 𝑄𝑄 𝜖𝜖

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, ( , ) = ( ) exp , in an attenuating medium. In a frame that 𝑥𝑥 tracks the wave front (“riding the wave”), wave amplitude decays with increasing travel time : 𝑢𝑢 𝑥𝑥 𝑡𝑡 𝐴𝐴 𝑥𝑥 �−𝑖𝑖𝑖𝑖 �𝑡𝑡 − 𝑐𝑐�� ( ) = exp 2 𝑇𝑇 𝜔𝜔𝜔𝜔 or, equivalently, with increasing propagation𝐴𝐴 𝑇𝑇 distance𝐴𝐴0 �=− : � 𝑄𝑄 ( ) = exp 𝑥𝑥 𝑐𝑐2𝑐𝑐 𝜔𝜔𝜔𝜔 Hence, 𝐴𝐴 𝑥𝑥 𝐴𝐴0 �− � 𝑐𝑐𝑐𝑐 ( , ) = exp exp = exp 2 𝜔𝜔𝜔𝜔 𝑥𝑥 𝑥𝑥 where 𝑢𝑢 𝑥𝑥 𝑡𝑡 𝐴𝐴0 �− � �−𝑖𝑖𝑖𝑖 �𝑡𝑡 − �� 𝐴𝐴0 �−𝑖𝑖𝑖𝑖 �𝑡𝑡 − ′�� 𝑐𝑐𝑐𝑐 1 1 𝑐𝑐 𝑐𝑐 = + 2 𝑖𝑖 ′ 𝑐𝑐 𝑐𝑐 𝑐𝑐𝑐𝑐 = 1 + ′ 𝑐𝑐2 𝑐𝑐 𝑖𝑖 𝑄𝑄 48

GE 162 Introduction to Seismology Winter 2013 - 2016

Along a ray in a smoothly heterogeneous medium: ( , ) = exp exp ( ( )) 2 ∗ where 𝜔𝜔𝑡𝑡 𝑢𝑢 𝑥𝑥 𝑡𝑡 𝐴𝐴0 �− � �−𝑖𝑖𝑖𝑖 𝑡𝑡 − 𝑇𝑇 𝑥𝑥 � ( ) = = ( ) travel time, 𝑑𝑑𝑑𝑑 and 𝑇𝑇 𝑥𝑥 ∫𝑟𝑟𝑟𝑟𝑟𝑟 𝑐𝑐 𝑥𝑥 = = ( ) ( ) characteristic attenuation time 𝑑𝑑𝑑𝑑 ∗ 𝑡𝑡 ∫𝑟𝑟𝑟𝑟𝑟𝑟 𝑄𝑄 𝑥𝑥 𝑐𝑐 𝑥𝑥

GE 162 Introduction to Seismology Winter 2013 - 2016

( ) = exp 2 𝜔𝜔𝜔𝜔 𝐴𝐴 𝑥𝑥 𝐴𝐴0 �− � In regions with stronger attenuation (lower Q) strong ground𝑐𝑐 motion𝑐𝑐 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 Earth’s structure or earthquake sources.

GE 162 Introduction to Seismology Winter 2013 - 2016

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 don’t generate energy, they only redistribute it. Movie of scattered wavefield in a randomly heterogeneous medium: http://web.utah.edu/thorne/movies/Scattering_720p.wmv

GE 162 Introduction to Seismology Winter 2013 - 2016

Seismograms recorded at different epicentral distances have similar coda envelopes:

GE 162 Introduction to Seismology Winter 2013 - 2016

Observed energy (waveform envelope squared) decays as ~ exp / 𝜔𝜔𝜔𝜔 𝑛𝑛 𝐸𝐸 �− � 𝑡𝑡 where = coda attenuation quality factor. 𝑄𝑄𝑐𝑐

𝑄𝑄𝑐𝑐

GE 162 Introduction to Seismology Winter 2013 - 2016

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 = scattering coefficient = scattered energy per unit of volume, averaged over all directions. It combines information 0 about the “strength” of each scatterer (size, material contrast, etc) and the density of𝑔𝑔 scatterers. A plane wave leaks out energy by scattering. Its energy decays as exp ( ). Defining scattering attenuation as = / , the energy decay is exp ( / ). 0 If A=source, B=scattering𝑠𝑠𝑠𝑠 point and C=station, the energy scattered𝑠𝑠𝑠𝑠 by −point𝑔𝑔 𝑥𝑥 B is: 𝑄𝑄 𝜔𝜔 𝑔𝑔0𝑐𝑐 ∼1 −1𝜔𝜔𝜔𝜔 𝑐𝑐𝑐𝑐

𝐸𝐸 ∝ 𝐸𝐸0 2 𝑔𝑔0 2 The total contribution integrated over one isochrone𝑟𝑟𝐴𝐴𝐴𝐴 is 𝑟𝑟𝐵𝐵𝐵𝐵 1 1

( ) 0 2 0 2 𝐸𝐸 ∝ 𝐸𝐸 � 𝐴𝐴𝐴𝐴 𝑔𝑔 𝐵𝐵𝐵𝐵 𝑑𝑑𝑑𝑑 𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖ℎ𝑟𝑟𝑟𝑟𝑟𝑟𝑟𝑟 𝑡𝑡 𝑟𝑟 𝑟𝑟 The amplitude of coherent waves adds up. The amplitude squared (energy) of incoherent waves add up. Travel time = ( + )/ . It can be shown that sufficiently long after the first arrival, when / , 𝑡𝑡 𝑟𝑟𝐴𝐴𝐴𝐴 𝑟𝑟𝐵𝐵𝐵𝐵 𝑐𝑐 𝑡𝑡 ≫ 𝑟𝑟 𝑐𝑐 0 Long after the passage of the main wave front, the scattered𝐸𝐸 energy is independent on distance to the 𝐸𝐸 ∝ 2 source A. 𝑡𝑡 In practice, a phenomenological attenuation factor (“coda Q”) needs to be included, representing energy loss of the main wave front due to scattering and intrinsic attenuation: exp 𝐸𝐸0 𝜔𝜔𝜔𝜔 = + 2 where . 𝐸𝐸 ∝ − 𝑐𝑐 1 1 1 𝑡𝑡 𝑄𝑄 𝑄𝑄𝑐𝑐 𝑄𝑄𝑖𝑖 𝑄𝑄𝑠𝑠 Multiple scattering.

GE 162 Introduction to Seismology Winter 2013 - 2016

At longer times, multiple scattering dominates over single scattering. A strong scattering process can be described by diffusion with diffusivity = /3 . The total energy in the medium is given by the classical solution of the 3D diffusion equation: 𝜅𝜅 𝑐𝑐 𝑔𝑔0 ~ exp (4 ) / 4 2 𝐸𝐸0 𝑟𝑟 𝐸𝐸𝑡𝑡𝑡𝑡𝑡𝑡 3 2 �− � An alternative approximation: energy uniformly𝜋𝜋𝜋𝜋 distributed𝑡𝑡 behind𝜅𝜅𝜅𝜅 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 ( / ). Energy partitioning between direct and scattered field, and overall conservation: 𝑠𝑠𝑠𝑠 −𝜔𝜔 𝑡𝑡 𝑄𝑄 4 ~E exp + ( ) = 3 𝜔𝜔𝜔𝜔 𝜋𝜋 3 Leads to 𝐸𝐸𝑡𝑡𝑡𝑡𝑡𝑡 0 �− � 𝑐𝑐𝑐𝑐 𝐸𝐸𝑠𝑠𝑠𝑠 𝐸𝐸0 𝑄𝑄𝑠𝑠𝑠𝑠 3 1 𝜔𝜔𝜔𝜔 ~ − 4 ( ) 𝑄𝑄𝑠𝑠𝑠𝑠 𝐸𝐸0 − 𝑒𝑒 𝐸𝐸𝑠𝑠𝑠𝑠 3 In practice, both equations need to be corrected by𝜋𝜋 an intrinsic𝑐𝑐𝑐𝑐 attenuation factor exp ( / ) 𝑖𝑖 Applications −𝜔𝜔 𝑡𝑡 𝑄𝑄 Interferometry. Ex: in optical fiber. In the crust.

GE 162 Introduction to Seismology Winter 2013 - 2016

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 “law” – elasticity) = + = : where the source is a density of body forces𝜌𝜌 𝑢𝑢(distributed̈ ∇ ⋅ 𝜎𝜎 𝑓𝑓in the volume). The deformation in the earthquake source region is inelastic, a departure𝜎𝜎 from𝑐𝑐 𝜀𝜀our usual model of elastic media. Our goal here is to derive an equivalent𝑓𝑓 body-force representation of an earthquake source, so that we can still use the elastic seismic wave equation to evaluate ground motions. Momentum equation: = = + 𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡 = 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑙𝑙 where is the stress𝜌𝜌 given𝑢𝑢̈ ∇ by⋅ 𝜎𝜎the idealized∇ ⋅ 𝜎𝜎 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚Hooke’s ∇law,⋅ � 𝜎𝜎∗and the− 𝜎𝜎mismatc�h between model and true stress𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 is called stress glut: ∇ ⋅ 𝜎𝜎 − ∇ ⋅ 𝜎𝜎 𝜎𝜎 = We define a density of body forces as ∗ 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡 𝜎𝜎 𝜎𝜎= − 𝜎𝜎 Then ∗ =𝑓𝑓 −∇ ⋅ 𝜎𝜎 + 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 Example: plastic deformation. As an example𝜌𝜌𝑢𝑢̈ of∇ departure⋅ 𝜎𝜎 from𝑓𝑓 Hooke’s law, consider an elasto-plastic material. Strain is partitioned into elastic and plastic components, = + , the true stress is related by Hooke’s law to the elastic strain, = : , and the𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒 model stress𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝 is what you get by trying to apply Hooke’s law to the total strain𝑡𝑡𝑡𝑡 instead,𝑡𝑡𝑡𝑡 𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒 =𝜀𝜀 : 𝜀𝜀. Then, 𝜀𝜀 𝜎𝜎 = : 𝑐𝑐 𝜀𝜀 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 Plastic deformation is a source of seismic waves.∗ 𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝜎𝜎 𝑐𝑐 𝜀𝜀 For an isotropic elastic medium: 𝜎𝜎 𝑐𝑐 𝜀𝜀 = + 2 ∗ 𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝 𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝 𝜎𝜎𝑖𝑖𝑖𝑖 𝜆𝜆𝜀𝜀𝑘𝑘𝑘𝑘 𝛿𝛿𝑖𝑖𝑖𝑖 𝜇𝜇𝜀𝜀𝑖𝑖𝑖𝑖 Example: explosion source, = and we get an isotropic stress glut: 𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝 1 Δ𝑉𝑉 = + 𝜀𝜀𝑖𝑖𝑖𝑖 3 𝑉𝑉 𝛿𝛿𝑖𝑖𝑖𝑖 . ∗ 2 Δ𝑉𝑉 𝑖𝑖𝑖𝑖 𝑖𝑖𝑖𝑖 13.3 Equivalent body force representation𝜎𝜎 �𝜆𝜆 3 𝜇𝜇� of𝑉𝑉 𝛿𝛿fault slip Consider a vertical strike-slip fault. The trace of the fault at the surface is parallel to (perpendicular to ). Slip D is defined as the displacement offset across the fault. Slip is horizontal, parallel to , i.e. near the fault the inelastic displacement is: ( ) . 𝒙𝒙� 𝒚𝒚� 𝒙𝒙� 𝒖𝒖 ∼ 𝐷𝐷𝐷𝐷 𝑦𝑦 𝒙𝒙� 55

GE 162 Introduction to Seismology Winter 2013 - 2016

The gradient tensor of the inelastic displacement, ( ) = / , has only one non-zero component: = ( ). The only non-zero components of the inelastic strain tensor, = ( + ), are: 𝛁𝛁𝒖𝒖 𝑖𝑖𝑖𝑖 𝜕𝜕𝑢𝑢𝑖𝑖 𝜕𝜕𝑥𝑥𝑗𝑗 𝜕𝜕𝑢𝑢𝑥𝑥 1 𝑻𝑻 𝜕𝜕𝜕𝜕 = 𝐷𝐷𝐷𝐷 𝑦𝑦= = ( ). 𝜀𝜀 2 𝛁𝛁𝒖𝒖 𝛁𝛁𝒖𝒖 1 𝜕𝜕𝑢𝑢𝑥𝑥 1 The only non-zero components of the associated stress glut tensor are: = = ( ). If the 𝜀𝜀𝑥𝑥𝑥𝑥 𝜀𝜀𝑦𝑦𝑦𝑦 2 𝜕𝜕𝜕𝜕 2 𝐷𝐷𝐷𝐷 𝑦𝑦 source is localized at a point (very small fault): = = ( ) ( )∗ ( ) =∗ ( ). 𝑥𝑥𝑥𝑥 𝑦𝑦𝑦𝑦 Equivalent body force: ∗ ∗ 𝜎𝜎 𝜎𝜎 𝜇𝜇𝜇𝜇𝜇𝜇 𝑦𝑦 𝜎𝜎𝑥𝑥𝑥𝑥 𝜎𝜎𝑦𝑦𝑦𝑦 𝜇𝜇𝜇𝜇𝜇𝜇 𝑦𝑦 𝛿𝛿 𝑥𝑥 𝛿𝛿 𝑧𝑧 𝜇𝜇𝜇𝜇𝜇𝜇 𝒙𝒙 = = , , 0 ∗ ∗ ∗ 𝜕𝜕𝜎𝜎𝑥𝑥𝑥𝑥 𝜕𝜕𝜎𝜎𝑦𝑦𝑦𝑦 𝒇𝒇 −𝛁𝛁 ⋅ 𝜎𝜎 − � � = +𝜕𝜕𝜕𝜕 𝜕𝜕 𝜕𝜕 𝜕𝜕𝜕𝜕 𝜕𝜕𝜕𝜕 It’s a double-couple source,𝒇𝒇 𝜇𝜇the𝜇𝜇 �sum𝒙𝒙� of two force𝒚𝒚� � couples: 𝛿𝛿𝛿𝛿 𝛿𝛿𝛿𝛿 = a pair of forces parallel to , same amplitude but opposite sign, 𝜕𝜕𝜕𝜕 separated by an arm parallel to . 𝛿𝛿𝛿𝛿 𝒙𝒙� 𝒙𝒙� = a conjugate force couple, forces parallel to , arm parallel to . 𝒚𝒚� 𝜕𝜕𝜕𝜕 𝛿𝛿𝛿𝛿 𝒚𝒚� 𝒚𝒚� 𝒙𝒙� 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): = .

Definition: the product of two vectors and is a tensor whose components are ( ) 𝑚𝑚= 𝜇𝜇𝜇𝜇. We can write ( ) , where is a tensor whose only non-zero component is ( ) = 1. 𝒂𝒂𝒂𝒂 𝒂𝒂 𝒃𝒃 𝒂𝒂𝒂𝒂 𝑖𝑖𝑖𝑖 𝑎𝑎𝑖𝑖𝑏𝑏𝑗𝑗 Inelastic strain = ( )( + ). 𝛁𝛁𝒖𝒖 ∼ 𝐷𝐷𝐷𝐷 𝑦𝑦 𝒚𝒚�𝒙𝒙� 𝒚𝒚�𝒙𝒙� 𝒚𝒚�𝒙𝒙� 12 More generally, define1 normal vector n and slip direction unit vector e. Potency density tensor: 𝜀𝜀 2 𝐷𝐷𝐷𝐷 𝑦𝑦 𝒚𝒚�𝒙𝒙� 𝒙𝒙�𝒚𝒚� 1 = ( + ) 2 1 𝑝𝑝= 𝐷𝐷( 𝒏𝒏𝒏𝒏 +𝒆𝒆𝒆𝒆 ) 2 Moment density tensor: 𝑝𝑝𝑖𝑖𝑖𝑖 𝐷𝐷 𝑛𝑛𝑖𝑖𝑒𝑒𝑗𝑗 𝑒𝑒𝑖𝑖𝑛𝑛𝑗𝑗 = + ( + ) = + ( + ) 𝑚𝑚 𝜆𝜆𝜆𝜆 𝒏𝒏 ⋅ 𝒆𝒆 𝐼𝐼𝐼𝐼 𝜇𝜇𝜇𝜇 𝒏𝒏𝒏𝒏 𝒆𝒆𝒆𝒆 𝑖𝑖𝑖𝑖 𝑘𝑘 𝑘𝑘 𝑖𝑖𝑖𝑖 𝑖𝑖 𝑗𝑗 𝑗𝑗 𝑖𝑖 The first term is zero if the fault does𝑚𝑚 not𝜆𝜆 open𝜆𝜆𝑛𝑛 𝑒𝑒 (no𝛿𝛿 offset𝜇𝜇𝜇𝜇 normal𝑒𝑒 𝑛𝑛 to𝑒𝑒 𝑛𝑛the fault: = 0).

Extended source. Define seismic moment = = and potency 𝒏𝒏=⋅ 𝒆𝒆 = , where is the rupture surface and is the average slip. 𝑀𝑀0 ∫ ∫ 𝜇𝜇𝜇𝜇 𝜇𝜇𝜇𝜇𝐷𝐷� 𝑃𝑃0 ∫ ∫ 𝐷𝐷 𝑆𝑆𝐷𝐷� 𝑆𝑆 13.4 Moment tensor𝐷𝐷� = It’s a symmetric tensor. 𝑖𝑖𝑗𝑗 𝑖𝑖𝑖𝑖 Not restricted to double-couple sources, it can𝑀𝑀 represent∫ ∫ 𝑚𝑚 sources other than faulting.

GE 162 Introduction to Seismology Winter 2013 - 2016

Decomposition into isotropic and deviatoric parts: = + where = (𝐼𝐼𝐼𝐼𝐼𝐼) . 𝐷𝐷𝐷𝐷𝐷𝐷 𝑀𝑀 𝑀𝑀 𝑀𝑀 𝐼𝐼𝐼𝐼𝐼𝐼 1 𝑀𝑀 3 𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡 𝑀𝑀 𝐼𝐼𝐼𝐼

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, = + , is defined such that the DC component is the largest possible. 𝐷𝐷𝐷𝐷𝐷𝐷 𝐷𝐷𝐷𝐷 𝐶𝐶𝐶𝐶𝐶𝐶𝐶𝐶 CLVD stands for “compensated linear𝑀𝑀 vector𝑀𝑀 dipole”.𝑀𝑀 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 = diag( , , ), then 1 = ( + + ) diag1 2(1,13,1) 3 𝑀𝑀 𝑚𝑚 𝑚𝑚 𝑚𝑚 𝐼𝐼𝐼𝐼𝐼𝐼 𝑀𝑀 𝑚𝑚1 𝑚𝑚2 𝑚𝑚3 57

GE 162 Introduction to Seismology Winter 2013 - 2016

1 = ( ) diag(1,0, 1) 2 𝐷𝐷𝐷𝐷 1 1 3 1 1 = 𝑀𝑀 ( 𝑚𝑚 +− 𝑚𝑚 + ) diag− , 1, 3 2 2 𝐶𝐶𝐶𝐶𝐶𝐶𝐶𝐶 𝑀𝑀 �𝑚𝑚2 − 𝑚𝑚1 𝑚𝑚2 𝑚𝑚3 � �− − � 13.5 Seismic moment and moment magnitude Seismic moment is a scalar that quantifies the “amplitude” of a moment tensor (its norm, in N.m): 1 = 2 1 2 2 Moment magnitude is a logarithmic scale related𝑀𝑀0 to the�𝑀𝑀 seismic𝑖𝑖𝑖𝑖� moment: 2 √ = log ( ) 6 3 𝑀𝑀𝑤𝑤 10 𝑀𝑀0 − 13.6 Representation theorem Green’s function ( , ; ) is the n-th component of displacement at location and time t induced by a point force at location set at = 0 in the p-th direction, i.e. = ( ) ( ) 𝑛𝑛𝑛𝑛 Displacement induced𝐺𝐺 𝒙𝒙 by𝑡𝑡 a𝝃𝝃 point-force with arbitrary orientation and source time𝒙𝒙 function ( ): 𝒑𝒑 𝝃𝝃 𝑡𝑡 ( , ) = 𝒇𝒇 𝛿𝛿 𝑡𝑡 𝛿𝛿 𝒙𝒙 − 𝝃𝝃 𝒙𝒙� Displacement induced by an extended distribution of point-forces ( , ): 𝒇𝒇 𝑡𝑡 𝑢𝑢𝑛𝑛 𝒙𝒙 𝑡𝑡 𝐺𝐺𝑛𝑛𝑛𝑛 ∗ 𝑓𝑓𝑝𝑝 ( ) ( ) ( , ) = , ; ,𝒇𝒇 𝝃𝝃 𝑡𝑡 3 For a moment tensor source, 𝑢𝑢=𝑛𝑛 𝒙𝒙 𝑡𝑡 , and� 𝐺𝐺𝑛𝑛𝑛𝑛 𝒙𝒙 𝑡𝑡 𝝃𝝃 ∗ 𝑓𝑓𝑝𝑝 𝝃𝝃 𝑡𝑡 𝑑𝑑 𝜉𝜉 𝑓𝑓𝑝𝑝 −𝑚𝑚(𝑝𝑝𝑝𝑝, 𝑘𝑘) = , 3 Integrating by parts: 𝑢𝑢𝑛𝑛 𝒙𝒙 𝑡𝑡 − � 𝐺𝐺𝑛𝑛𝑛𝑛 ∗ 𝑚𝑚𝑝𝑝𝑝𝑝 𝑘𝑘𝑑𝑑 𝜉𝜉 ( , ) = + , 2 3 The surface integral (first𝑢𝑢 term𝑛𝑛 𝒙𝒙 𝑡𝑡on r.h.s.)� 𝐺𝐺 vanishes𝑛𝑛𝑛𝑛 ∗ 𝑚𝑚𝑝𝑝𝑝𝑝 if𝑛𝑛 𝑘𝑘the𝑑𝑑 𝜉𝜉integration� 𝐺𝐺𝑛𝑛𝑛𝑛 surface𝑘𝑘 ∗ 𝑚𝑚𝑝𝑝 𝑝𝑝is𝑑𝑑 taken𝜉𝜉 sufficiently far from the source region, where vanishes. We obtain the representation theorem, a relation between wave field and moment tensor source: 𝑚𝑚𝑝𝑝𝑝𝑝 ( , ) = , 3 Practical significance: if we know the𝑢𝑢 Green’s𝑛𝑛 𝒙𝒙 𝑡𝑡 function,� 𝐺𝐺𝑛𝑛𝑛𝑛 we𝑘𝑘 ∗can𝑚𝑚𝑝𝑝 compute𝑝𝑝𝑑𝑑 𝜉𝜉 the whole wave field induced by an arbitrary moment tensor source by convolution with the gradients of the Green’s function, , ( , ; ).

𝐺𝐺𝑛𝑛𝑛𝑛 𝑘𝑘 𝒙𝒙 𝑡𝑡 𝝃𝝃

GE 162 Introduction to Seismology Winter 2013 - 2016

14 Seismic sources: moment tensor

14.1 Green’s function Sketch of derivation: Lamé potentials of the wave field: = + × , with = 0. Decompose a point-force source in Helmholtz potentials: = + × with = 0. The potentials satisfy wave equations∇ withϕ source∇ 𝜓𝜓 terms:∇ ⋅ 𝜓𝜓 = 𝒇𝒇 + / and 𝒇𝒇 = ∇Φ ∇+ Ψ/ ∇ ⋅ Ψ The solutions are 2 2 2 2 ̈ 𝑃𝑃 ̈ 𝑆𝑆 𝜙𝜙 𝑐𝑐 ∇, 𝜙𝜙 Φ 𝜌𝜌 𝜓𝜓 𝑐𝑐 ∇ 𝜓𝜓 Ψ 𝜌𝜌 , ( , ) = 𝑟𝑟 and ( , ) = 𝑟𝑟 Φ�𝜉𝜉 𝑡𝑡−𝑐𝑐 � Ψ�𝜉𝜉 𝑡𝑡−𝑐𝑐 � 1 𝑃𝑃 3 1 𝑆𝑆 3 2 2 𝜙𝜙 𝒙𝒙 𝑡𝑡 4𝜋𝜋𝑐𝑐𝑃𝑃 ∭ 𝑟𝑟 𝑑𝑑 𝜉𝜉 𝜓𝜓 𝒙𝒙 𝑡𝑡 4𝜋𝜋𝑐𝑐𝑆𝑆 ∭ 𝑟𝑟 𝑑𝑑 𝜉𝜉 For any vector field we have = ( ) × ( × ). Hence we can derive the force potentials from a single vector potential2 that satisfies Poisson’s equation, = , if = W and 𝑾𝑾 ∇ 𝑊𝑊 ∇ ∇ ⋅ 𝑊𝑊 − ∇ ∇ 𝑊𝑊 ( ) = × W. The solution of Poisson’s equation is = . For a point2 source: = . 𝑊𝑊 ∇ 𝑾𝑾 𝒇𝒇 Φ ∇ ⋅ Knowing , we can now evaluate and , then and , and1 finally𝑓𝑓 3 (the Green’s function). 𝒇𝒇 𝑡𝑡 Ψ −∇ 𝑊𝑊 − 4𝜋𝜋 𝑟𝑟 𝑑𝑑 𝜉𝜉 𝑾𝑾 − 4𝜋𝜋𝜋𝜋 For a force with source time function ( ): ∭ 𝑾𝑾 Φ Ψ 𝜙𝜙 𝜓𝜓 𝑢𝑢 ( ) 𝑭𝑭 𝑡𝑡 / _ ( ) , = 3 / _ (near field) 1 1 𝑟𝑟 𝑐𝑐 𝑆𝑆 3 𝑢𝑢𝑖𝑖 𝒙𝒙 𝑡𝑡 4𝜋𝜋𝜋𝜋 � 𝛾𝛾𝑖𝑖𝛾𝛾𝑗𝑗 − 𝛿𝛿𝑖𝑖𝑖𝑖� 𝑟𝑟 ∫𝑟𝑟 𝑐𝑐 𝑃𝑃 𝜏𝜏𝐹𝐹𝑗𝑗 𝑡𝑡 − 𝜏𝜏 𝑑𝑑𝑑𝑑 + (far-field P wave) 1 1 𝑟𝑟 2 4𝜋𝜋𝜋𝜋𝑐𝑐𝑃𝑃 𝛾𝛾𝑖𝑖𝛾𝛾𝑗𝑗 𝑟𝑟 𝐹𝐹𝑗𝑗 �𝑡𝑡 − 𝑐𝑐𝑃𝑃� + (far-field S wave) 1 1 𝑟𝑟 2 4𝜋𝜋𝜋𝜋𝑐𝑐𝑆𝑆 �𝛾𝛾𝑖𝑖𝛾𝛾𝑗𝑗 − 𝛿𝛿𝑖𝑖𝑖𝑖� 𝑟𝑟 𝐹𝐹𝑗𝑗 �𝑡𝑡 − 𝑐𝑐𝑆𝑆� where = / are direction cosines.

𝛾𝛾𝑖𝑖 𝑥𝑥𝑖𝑖 𝑟𝑟 14.2 Moment tensor wavefield The representation theorem for a moment tensor source (last lecture):

( , ) = , 3 involves the derivatives of the Green’s𝑢𝑢𝑛𝑛 function.𝒙𝒙 𝑡𝑡 �So, taking𝐺𝐺𝑛𝑛𝑛𝑛 𝑘𝑘 ∗appropriate𝑚𝑚𝑝𝑝𝑝𝑝𝑑𝑑 𝜉𝜉 derivatives of the result in the previous section: ( ) ( , ) = 𝑟𝑟 ( ) (near field) 𝑅𝑅𝑁𝑁 𝜸𝜸 1 𝑐𝑐𝑆𝑆 𝑛𝑛 4 𝑟𝑟 𝑝𝑝𝑝𝑝 𝑢𝑢 𝒙𝒙 𝑡𝑡 4𝜋𝜋𝜋𝜋 𝑟𝑟 ∫𝑐𝑐𝑃𝑃 𝜏𝜏𝑀𝑀 𝑡𝑡 − 𝜏𝜏 𝑑𝑑𝑑𝑑 ( ) ( ) + + (intermediate field, P and S) 𝑅𝑅𝐼𝐼𝐼𝐼 𝜸𝜸 1 𝑟𝑟 𝑅𝑅𝐼𝐼𝐼𝐼 𝜸𝜸 1 𝑟𝑟 2 2 2 2 𝑃𝑃 𝑝𝑝𝑝𝑝 𝑃𝑃 𝑆𝑆 𝑝𝑝𝑝𝑝 𝑆𝑆 4𝜋𝜋𝜋𝜋𝑐𝑐( )𝑟𝑟 𝑀𝑀 �𝑡𝑡 − 𝑐𝑐 � 4𝜋𝜋𝜋𝜋𝑐𝑐 (𝑟𝑟) 𝑀𝑀 �𝑡𝑡 − 𝑐𝑐 � + + (far field, P and S) 𝑅𝑅𝐹𝐹𝐹𝐹 𝜸𝜸 1 𝑟𝑟 𝑅𝑅𝐹𝐹𝐹𝐹 𝜸𝜸 1 𝑟𝑟 3 3 4𝜋𝜋𝜋𝜋𝑐𝑐𝑃𝑃 r ̇ 𝑝𝑝𝑝𝑝 𝑐𝑐𝑃𝑃 4𝜋𝜋𝜋𝜋𝑐𝑐𝑆𝑆 r ̇ 𝑝𝑝𝑝𝑝 𝑐𝑐𝑆𝑆 Discuss properties of near-field,𝑀𝑀 intermediate�𝑡𝑡 − � -field, far𝑀𝑀-field� 𝑡𝑡terms.− �

GE 162 Introduction to Seismology Winter 2013 - 2016

14.3 Far field and radiation pattern of a double couple source

In a spherical coordinate system ( , , ) related to the fault orientation, the far-field P wave displacement field is: 𝑟𝑟 𝜃𝜃 𝜙𝜙

( , ) sin2θ cosϕ 𝐫𝐫� 1 𝑟𝑟 𝑛𝑛 3 0 𝑢𝑢 𝑥𝑥 𝑡𝑡 ∼ 4𝜋𝜋𝜋𝜋𝑐𝑐𝑃𝑃 r 𝑀𝑀̇ �𝑡𝑡 − 𝑐𝑐𝑃𝑃� cos2 cos + cos sin 1 + 4 r θ ϕ 𝛉𝛉� θ ϕ 𝛉𝛉� 𝑟𝑟 3 ̇ 0 𝑃𝑃 𝑀𝑀 �𝑡𝑡 − 𝑆𝑆� P and S radiation patterns:𝜋𝜋𝜋𝜋 𝑐𝑐 𝑐𝑐

GE 162 Introduction to Seismology Winter 2013 - 2016

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)

GE 162 Introduction to Seismology Winter 2013 - 2016

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 ( ). This provides information about the temporal evolution of the rupture. Once we have determined the focal mechanism and the 0 distance to the source, the time-integral of the far-field displacement gives𝑀𝑀̇ 𝑡𝑡 an estimate of the seismic moment .

0 𝑀𝑀

GE 162 Introduction to Seismology Winter 2013 - 2016

Complications due to depth phases:

GE 162 Introduction to Seismology Winter 2013 - 2016

14.4 Surface waves

GE 162 Introduction to Seismology Winter 2013 - 2016

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 • seismic moment (or moment magnitude ), • moment tensor 0 𝑤𝑤 • 𝑀𝑀 ( ) 𝑀𝑀 source time function𝑖𝑖𝑖𝑖 . 𝑀𝑀 0 Close to the fault we need to 𝑀𝑀describė 𝑡𝑡 in more detail the space-time distribution of the source: [sketch] • Slip. • Slip velocity. • Rupture front. Rupture speed. • Rupture duration. • 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 ( , ): ( , ) 1 ( , ) = , 𝐷𝐷̇ 𝝃𝝃 𝑡𝑡 4 𝑅𝑅𝑃𝑃 𝜃𝜃 𝜙𝜙 𝑟𝑟 2 3 ̇ 𝒖𝒖 𝒙𝒙 𝑡𝑡 � 𝜇𝜇𝐷𝐷 �𝝃𝝃 𝑡𝑡 − 𝑃𝑃� 𝒓𝒓� 𝑑𝑑 𝜉𝜉 If the source area is small compared to theΣ distance𝜋𝜋𝜋𝜋𝑐𝑐𝑃𝑃 𝑟𝑟 between the𝑐𝑐 fault and the receiver, we can approximate the Green’s function by its value at the center of the rupture area (subscript 0): ( , ) 𝑟𝑟 ( , ) , 4 𝑅𝑅𝑃𝑃 𝜃𝜃0 𝜙𝜙0 𝒓𝒓�0 𝑟𝑟 2 3 ̇ 𝒖𝒖 𝒙𝒙 𝑡𝑡 ≈ 0 𝜇𝜇 � 𝐷𝐷 �𝝃𝝃 𝑡𝑡 − 𝑃𝑃� 𝑑𝑑 𝜉𝜉 The integral term defines the apparent source𝜋𝜋𝜋𝜋𝑐𝑐𝑃𝑃 time𝑟𝑟 functΣion (ASTF): 𝑐𝑐

( , ) = , 𝑟𝑟 2 ΩP 𝒙𝒙 𝑡𝑡 � 𝐷𝐷̇ �𝝃𝝃 𝑡𝑡 − � 𝑑𝑑 𝜉𝜉 We can similarly define an ASTF for S waves, (Σ , ). 𝑐𝑐𝑃𝑃 The ASTF is not only a property of the source, it depends also on the location of the observer relative to S the source and on the type of wave consideredΩ (P𝒙𝒙 or𝑡𝑡 S). The time integral of the ASTF

( , ) = ( ) = 2 � ΩP or S 𝒙𝒙 𝑡𝑡 𝑑𝑑𝑑𝑑 � 𝐷𝐷 𝝃𝝃 𝑑𝑑 𝜉𝜉 𝑃𝑃0 is the seismic potency, which does not depend on the locationΣ of the observer nor on wave type.

15.3 ASTF in the Fraunhofer approximation Let be the position of the receiver relative to a reference point on the fault, its position relative to an arbitrary point on the fault, and an arbitrary position on the fault (relative to the reference point). 0 Far 𝒓𝒓from the source, and 𝒓𝒓 𝝃𝝃 ( ) = 𝜉𝜉 ≪ 𝑟𝑟0= ( 2 + ) / + + 2 2 2 3 2 2 1 2 𝜉𝜉 − 𝝃𝝃 ⋅ 𝒓𝒓� 𝜉𝜉 𝑟𝑟 ‖𝒓𝒓0 − 𝝃𝝃‖ 𝑟𝑟0 − 𝝃𝝃 ⋅ 𝒓𝒓𝟎𝟎 𝜉𝜉 ≈ 𝑟𝑟0 − 𝝃𝝃 ⋅ 𝒓𝒓�0 𝑂𝑂 � � 𝑟𝑟0 𝑟𝑟0 66

GE 162 Introduction to Seismology Winter 2013 - 2016

The Fraunhofer approximation amounts to keep only the zero and first order terms:

Define the time relative to the wave arrival from𝑟𝑟 ≈ the𝑟𝑟0 − reference𝝃𝝃 ⋅ 𝒓𝒓�0 point, = / . The simplified expression of ASTF, in the Fraunhofer approximation, is: 𝜏𝜏 𝑡𝑡 − 𝑟𝑟0 𝑐𝑐 ( , ) = , + 𝝃𝝃 ⋅ 𝒓𝒓�0 2 Ω 𝒓𝒓�𝟎𝟎 𝜏𝜏 � 𝐷𝐷̇ �𝝃𝝃 𝜏𝜏 � 𝑑𝑑 𝜉𝜉 The ASTF depends on the direction from whichΣ we observe𝑐𝑐 the source.

𝟎𝟎 Range of validity of the Fraunhofer𝒓𝒓� approximation. Valid if the rupture size, = max( ), is small compared to distance and wavelength: 𝐿𝐿 𝜉𝜉

2 𝜆𝜆𝑟𝑟0 The range of validity depends on frequency (through𝐿𝐿 ≪ � ). It is less restrictive than the validity condition of the point-source approximation ( ). Proof: Denote the Fraunhofer approximation of the𝜆𝜆 distance , and = the residual of this approximation. The∗ Fourier transform of ( , / ) is ( , ) exp( / ). We∗ 𝐿𝐿 can≪ 𝜆𝜆justify exp( / ) 𝑟𝑟exp( / ) if / /2. This leads to the quarter𝑟𝑟 -wavelength𝜖𝜖 𝑟𝑟 − 𝑟𝑟 rule: /4. ( ) 𝐷𝐷̇ 𝝃𝝃 𝑡𝑡 − 𝑟𝑟 𝑐𝑐 𝐷𝐷̇ 𝝃𝝃 𝜔𝜔 𝑖𝑖𝑖𝑖𝑖𝑖 𝑐𝑐 Applying it to the residual∗ (the third order term in the Taylor expansion of ) we get, 𝑖𝑖𝑖𝑖𝑖𝑖 𝑐𝑐 ≈ 𝑖𝑖𝑖𝑖𝑟𝑟 𝑐𝑐 𝜔𝜔2𝜔𝜔 𝑐𝑐 ≪2 𝜋𝜋 𝜖𝜖 ≪ 𝜆𝜆 𝜉𝜉 − 𝝃𝝃⋅𝒓𝒓� conservatively, /𝜖𝜖2≈. 2𝑟𝑟0 𝑟𝑟

0 2D example: let’s𝐿𝐿 ≪consider�𝜆𝜆𝑟𝑟 a problem with lower dimensionality, a linear (1D) fault embedded in a 2D elastic medium. Let be the take-off angle of the seismic ray leaving the reference point, relative to the fault line direction. 0 𝜃𝜃 cos ( , ) = , + 𝜉𝜉 𝜃𝜃0 0 ̇ Ω 𝜃𝜃 𝜏𝜏 �Σ 𝐷𝐷 �𝜉𝜉 𝜏𝜏 � 𝑑𝑑𝑑𝑑 [Sketch ( , ). Construct graphically the ASTF𝑐𝑐 for = 0 , 90 , 180 . At = 90 we recover the STF. 𝑜𝑜 𝑜𝑜 𝑜𝑜 ̇ 0 Comparing the other angles,𝐷𝐷 𝜉𝜉 𝑡𝑡 introduce the directivity𝑜𝑜 effect: the duration𝜃𝜃 of the ASTF depends on . 0 Shorter duration yields larger amplitude𝜃𝜃 because the time-integral of ASTF is the seismic potency, 0 regardless of .] 𝜃𝜃

15.4 Haskell pulse model, directivity 𝜃𝜃0

Haskell model Consider a pulse propagating with constant slip , rise time and rupture speed on a rectangular fault of width and length . The slip rate function is assumed invariant. 𝐷𝐷 𝑡𝑡𝑟𝑟𝑟𝑟𝑟𝑟 𝑣𝑣𝑟𝑟 ( , ) = if [0, ] and [0, ] 𝑊𝑊 𝑥𝑥 𝐿𝐿 = 0 elsewhere 𝐷𝐷̇ 𝑥𝑥 𝑡𝑡 𝐷𝐷 𝑠𝑠̇ �𝑡𝑡 − 𝑣𝑣𝑟𝑟� 𝑥𝑥 ∈ 𝐿𝐿 𝑧𝑧 ∈ 𝑊𝑊

Far-field spectrum of a Haskell source Its ASTF is

GE 162 Introduction to Seismology Winter 2013 - 2016

( , ) = / + 𝝃𝝃 ⋅ 𝒓𝒓�0 If is small, 𝟎𝟎 𝑟𝑟 Ω 𝒓𝒓� 𝜏𝜏 𝐷𝐷 �Σ 𝑠𝑠̇ �𝜏𝜏 − 𝜉𝜉 𝑣𝑣 � 𝑑𝑑𝑑𝑑 𝑑𝑑𝑑𝑑 𝑐𝑐cos ( , ) = / + 𝑊𝑊 𝐿𝐿 𝜉𝜉 𝜃𝜃0 Its Fourier transform is 𝟎𝟎 𝑟𝑟 Ω 𝒓𝒓� 𝜏𝜏 𝐷𝐷𝐷𝐷 �0 𝑠𝑠̇ �𝜏𝜏 − 𝜉𝜉 𝑣𝑣 � 𝑑𝑑𝑑𝑑 1 cos 𝑐𝑐 sin ( , ) = ( ) exp = ( ) 𝐿𝐿 0 𝑖𝑖𝑖𝑖 𝟎𝟎 𝜃𝜃 𝑋𝑋 where Ω=𝒓𝒓� 𝜔𝜔 −𝑖𝑖𝑖𝑖 𝑠𝑠 .𝜔𝜔 𝐷𝐷𝐷𝐷𝐷𝐷 �0 �𝑖𝑖𝑖𝑖𝑖𝑖 � 𝑟𝑟 − �� 𝑑𝑑𝑑𝑑 −𝑖𝑖𝑖𝑖 𝑠𝑠 𝜔𝜔 𝐷𝐷𝐷𝐷𝐷𝐷 𝑒𝑒 𝜔𝜔𝜔𝜔 1 cos 𝜃𝜃0 𝑣𝑣 𝑐𝑐 𝑋𝑋 If ( ) is a boxcar function with duration , then ( ) = (1 )/ , and 𝑋𝑋 2 �𝑣𝑣𝑟𝑟 − 𝑐𝑐 � sin sin 𝑖𝑖𝑖𝑖𝑡𝑡𝑟𝑟𝑟𝑟𝑟𝑟 2 𝑠𝑠 𝑡𝑡 | ( , 𝑡𝑡𝑟𝑟)𝑟𝑟𝑟𝑟| = 𝑠𝑠 𝜔𝜔 − 𝑒𝑒 𝜔𝜔 𝑡𝑡𝑟𝑟𝑟𝑟𝑟𝑟 𝑋𝑋 𝜔𝜔𝑡𝑡𝑟𝑟𝑟𝑟𝑟𝑟 Ω 𝒓𝒓�𝟎𝟎 𝜔𝜔 𝑊𝑊𝑊𝑊𝑊𝑊 � � 𝑋𝑋 𝜔𝜔𝑡𝑡𝑟𝑟𝑟𝑟𝑟𝑟 [Plot , indicate zero-crossings. Log-𝑠𝑠log𝑠𝑠𝑠𝑠 𝑋𝑋 plot spectrum of ASTF.] 𝑋𝑋

Spectral shape and two corners frequencies The ASTF spectrum exhibits three distinct behaviors, from low to high-frequencies: flat, 1/ and 1/ . These regimes are separated by corner frequencies 2 ω ω ( ) = 1 cos and = 1/ 𝑟𝑟 𝑟𝑟 𝑣𝑣 𝑣𝑣 𝜔𝜔1 𝜃𝜃0 𝐿𝐿 � − 𝑐𝑐 𝜃𝜃0� 𝜔𝜔2 𝑡𝑡𝑟𝑟𝑟𝑟𝑟𝑟 Directivity effect The directivity effect (azimuth-dependence) appears in the lower corner frequency, .

1 [Plot as a function of ] 𝜔𝜔

1 0 Far-field waveform shape 𝜔𝜔 𝜃𝜃 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/ is the duration of the ASTF.

1 𝜔𝜔[Sketch: relation between ASTF duration and the arrival times of waves radiated by the two ends of a rupture]

GE 162 Introduction to Seismology Winter 2013 - 2016

16 Scaling laws

16.1 Circular crack model Circular rupture, constant rupture speed, final radius R. Rupture front, healing front, duration: 2 / . Only one characteristic time-scale, event duration. 𝑟𝑟 Spectrum has only one corner frequency, , that𝑇𝑇 ≈ separates𝑅𝑅 𝑣𝑣 flat spectrum at low-f and 1/ at high-f: 2 𝑓𝑓𝑐𝑐 ( ) 𝜔𝜔 1 +𝑃𝑃0 Ω 𝜔𝜔 ≈ 2 𝑓𝑓 with � � 𝑓𝑓𝑐𝑐 = 1/ 𝑟𝑟 where is a factor of order 1 that depends mildly𝑘𝑘 on𝑣𝑣 rupture speed ( = 0.44 for = 0.9 ). 𝑓𝑓𝑐𝑐 ≈ 𝑇𝑇 𝑅𝑅 16.2 Stress𝑘𝑘 drop, corner frequency, self-similarity 𝑘𝑘 𝑣𝑣𝑟𝑟 𝑐𝑐𝑆𝑆 7 = 16 = 𝜋𝜋 𝐷𝐷 Considering also , we get Δ𝜎𝜎 𝜇𝜇 2 7 𝑅𝑅 0 𝑀𝑀 𝜇𝜇𝜇𝜇𝜇𝜇𝑟𝑟 = 3 16 𝑐𝑐 This shows how to estimate stress drop from far-field observations𝑓𝑓 0 (assuming ). Δ𝜎𝜎 � 𝑟𝑟� 𝑀𝑀 Corner frequencies carry information about rupture duration𝑘𝑘𝑣𝑣 . 𝑟𝑟 𝑣𝑣 Attenuation distorts the spectral shape and makes it hard to determine corner frequencies, especially for small events (trade-off between Q and fc).

GE 162 Introduction to Seismology Winter 2013 - 2016

Self-similarity: if and do not depend on earthquake size:

𝑟𝑟 Δ𝜎𝜎 𝑣𝑣 3 0 / ( ) 𝑀𝑀 ∝/ 𝑅𝑅 𝑇𝑇 ∝ 𝑅𝑅 / ( ) / 2 2 3 𝑢𝑢 𝑡𝑡 ∝ 𝑀𝑀̇ 0 ∼ 𝑀𝑀0 𝑇𝑇 ∝ 𝑅𝑅 ∝ 𝑀𝑀0 2 1 3 16.3 Energy considerations𝑢𝑢 ̇ and𝑡𝑡 ∝ moment𝑀𝑀̈ 0 ∼ 𝑀𝑀0 𝑇𝑇 magnitude∝ 𝑅𝑅 ∝ 𝑀𝑀0 scale Energy radiated to the far-field:

log 2= log 2 + 0 𝐸𝐸 ∝ ∫ 𝑢𝑢̇ 𝑑𝑑𝑑𝑑 ∝ 𝑢𝑢̇ 𝑇𝑇 ∝ 𝑀𝑀 10 𝐸𝐸 10 𝑀𝑀0 ⋯

Moment magnitude: 2 = log ( ) 6.0 3 Then: 𝑤𝑤 10 0 𝑀𝑀 3 𝑀𝑀 − log = + 2 One magnitude unit = 30 times more energy radiated, 30 times larger moment, 10 times larger ground 𝐸𝐸 𝑀𝑀𝑤𝑤 ⋯ displacement, 3 times larger ground velocity (ignoring attenuation)

It’s also useful to have in mind relations between magnitude and earthquake size: = 2 log + One magnitude unit = 3 times larger rupture size, 3 times longer rupture duration. 𝑤𝑤 10 In terms of rupture surface area A: 𝑀𝑀 𝑅𝑅 ⋯ = log + One magnitude unit = 10 times larger rupture area. 𝑤𝑤 10 𝑀𝑀 𝐴𝐴 ⋯

GE 162 Introduction to Seismology Winter 2013 - 2016

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 Haskell’s model (rectangular rupture, width W and length L). Its moment is =

The elastic stiffness is controlled by the shortest𝑀𝑀0 rupture𝜇𝜇𝜇𝜇𝜇𝜇𝜇𝜇 length. If :

𝐿𝐿 ≫ 𝑊𝑊 Hence, 𝜇𝜇 Δ𝜏𝜏 ∼ 𝐷𝐷 𝑊𝑊 2 0 Its rupture duration is controlled by the longest𝑀𝑀 rupture∼ Δ𝜏𝜏𝑊𝑊 length,𝐿𝐿 = . Hence, the corner frequency 𝐿𝐿 ( 1/ ) now scales as 𝑇𝑇 𝑣𝑣𝑟𝑟 × 𝑐𝑐 𝑓𝑓 ∼ 𝑇𝑇 2 −1 Saturation of the seismogenic depth breaks𝑓𝑓𝑐𝑐 self∼ Δ-similarity.𝜏𝜏𝑊𝑊 𝑣𝑣𝑟𝑟 𝑀𝑀A 0change in aspect ratio can also happen at smaller scales, due to fault heterogeneities.

GE 162 Introduction to Seismology Winter 2013 - 2016

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

( , ) = , + 𝝃𝝃 ⋅ 𝒓𝒓�0 2 Ω 𝒓𝒓�𝟎𝟎 𝜏𝜏 � 𝐷𝐷̇ �𝝃𝝃 𝜏𝜏 � 𝑑𝑑 𝜉𝜉 Its temporal Fourier transform ( ) is Σ 𝑐𝑐

( 𝜏𝜏 →, 𝜔𝜔) = ( , ) exp i 𝝃𝝃 ⋅ 𝒓𝒓�0 2 Ω 𝒓𝒓�𝟎𝟎 𝜔𝜔 � 𝐷𝐷̇ 𝝃𝝃 𝜔𝜔 �− ω � 𝑑𝑑 𝜉𝜉 The spatial Fourier transform of a function Σ ( ) defined on the fault𝑐𝑐 surface ( ) is

( ) = 𝑓𝑓 𝝃𝝃 ( ) exp( i ) 𝝃𝝃 ∈ Σ 2 where is a wavenumber vector along the fault. Hence, the ASTF is related to the spatial Fourier 𝑓𝑓 𝒌𝒌 �Σ 𝑓𝑓 𝝃𝝃 − 𝛏𝛏 ⋅ 𝒌𝒌 𝑑𝑑 𝜉𝜉 transform of slip rate: 𝒌𝒌 ( , ) = = , 0 = ( ) 𝜔𝜔𝜸𝜸� where is the projectionΩ 𝒓𝒓�𝟎𝟎 𝜔𝜔 of 𝐷𝐷 oṅ � 𝒌𝒌the fault surface𝜔𝜔� . If we were able to measure ( , ) for all and we could readily infer ( , ) by inverse𝑐𝑐 Fourier transform. However, | | < 1 0 0 0 0 and the𝜸𝜸� ASTF𝒓𝒓� only− samples𝒓𝒓� ⋅ 𝒏𝒏 𝒏𝒏 on-fault wavenumber𝒓𝒓� vectors such that | /Σ | > , i.e. with along-fault 0 𝐷𝐷phasė 𝒌𝒌 𝜔𝜔 velocity larger𝒌𝒌 than𝜔𝜔 wave speed. Hence 𝐷𝐷faṙ 𝝃𝝃-field𝑡𝑡 source imaging is limited to along-fault 𝜸𝜸� wavelengths > / . Indeed, all perturbations with | / | < are associated𝜔𝜔 𝑘𝑘 𝑐𝑐 to evanescent waves with exponential decay in the fault-normal direction, which do not make it to far field distances. To extract finer informat𝜆𝜆 𝑐𝑐 𝑓𝑓ion about source processes, near𝜔𝜔-field𝑘𝑘 ground𝑐𝑐 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:

( , ) = , 3 𝑢𝑢𝑖𝑖 𝒙𝒙 𝑡𝑡 � 𝐺𝐺𝑖𝑖𝑖𝑖 𝑘𝑘 ∗ 𝑚𝑚𝑗𝑗𝑗𝑗𝑑𝑑 𝜉𝜉

GE 162 Introduction to Seismology Winter 2013 - 2016

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).

GE 162 Introduction to Seismology Winter 2013 - 2016

18 Source inversion and source imaging

18.1 Source inversion problem For given data d and design matrix G (Green’s functions), find the model m that minimizes the cost = | | = ( ) = # = 0. function . The formal solution is . Proof: set 2 2 2 𝑇𝑇 −1 𝑇𝑇 𝑑𝑑χ 18.2 Illχ-conditioning� 𝑑𝑑 − 𝐺𝐺𝐺𝐺 � of the source inversion𝑚𝑚 𝐺𝐺 problem𝐺𝐺 𝐺𝐺 𝑑𝑑 𝐺𝐺 𝑑𝑑 𝑑𝑑𝑑𝑑 The Singular Value Decomposition (SVD) of matrix G is = where U[N,M] and V[M,M] are orthonormal matrices ( 𝑇𝑇 = , = ), and [M,M] is a diagonal matrix consisting of positive singular values ,𝐺𝐺 = 1𝑈𝑈𝑈𝑈, . .𝑉𝑉.𝑇𝑇, , sorted𝑇𝑇 in descending order. The columns 𝑈𝑈 𝑈𝑈 𝐼𝐼 𝑉𝑉 𝑉𝑉 𝐼𝐼 𝛬𝛬 ( ) of matrix V are called right-singular vectors. They are also eigenvectors of matrix [M,M], 𝜆𝜆𝑖𝑖 𝑖𝑖 𝑀𝑀 forming an orthonormal basis system in the model space, and are its eigenvalues. Columns𝑇𝑇 ( ) of 𝑉𝑉 𝑖𝑖 𝐺𝐺 𝐺𝐺 matrix U are called left-singular vectors. They are projections of 2basis vectors into the data space, 𝑖𝑖 ( ) 𝑖𝑖 = / 𝜆𝜆 𝑈𝑈 ( ) ( ) 𝑖𝑖 i.e. normalized seismograms related to the individual singular vectors. 𝑉𝑉 𝑖𝑖 𝑖𝑖 𝑖𝑖 The generalized solution of the inverse problem,𝑈𝑈 𝐺𝐺=𝑉𝑉 # 𝜆𝜆 can be expressed as a linear combination of basis vectors ( ): = 𝑚𝑚 𝐺𝐺 𝑑𝑑 = / 𝑖𝑖 ( ) where ( ) 𝑉𝑉 𝑀𝑀 𝑖𝑖=1 𝑖𝑖 𝑖𝑖 𝑖𝑖 𝑖𝑖 𝑖𝑖 Similarly, the data vector can𝑚𝑚 be expressed∑ 𝑚𝑚� 𝑉𝑉 as 𝑚𝑚� 𝑈𝑈 ⋅ 𝑑𝑑 𝜆𝜆 = ( ) where = ( ) 𝑁𝑁 𝑖𝑖=1 𝑖𝑖 𝑖𝑖 𝑖𝑖 𝑖𝑖 The spectral components of data𝑑𝑑 and∑ model𝑑𝑑̃ 𝑈𝑈 are thus related by𝑑𝑑̃ 𝑈𝑈 ⋅ 𝑑𝑑 = /

Therefore, the smaller is the singular value , 𝑚𝑚the�𝑖𝑖 less𝑑𝑑̃ 𝑖𝑖sensitive𝜆𝜆𝑖𝑖 is the data component to a given change of the corresponding model component; in other words, the singular value bears information about the 𝑖𝑖 sensitivity of the data to the particular basis𝜆𝜆 function in the model space. Also, if is small, errors in the data or in the G matrix get amplified. Hence the components of the data associated to small eigenvalues 𝑖𝑖 are hardly recoverable by the inverse problem, they define an effective null space.𝜆𝜆

GE 162 Introduction to Seismology Winter 2013 - 2016

18.3 Stacking Stacking (adding up) seismograms enhances the signal-to-noise ratio. Central limit theorem: stacking N seismograms reduces the noise by .

18.4 Array seismology √𝑁𝑁 Plane wave impinging on a linear array and 2D array:

Parameters: back-azimuth and horizontal slowness (related to wave speed and incidence angle i):

GE 162 Introduction to Seismology Winter 2013 - 2016

Seismogram recorded by a reference station in the array (signal + noise): ( ) = ( ) + ( )

Seismogram recorded by the i-th station of𝑢𝑢1 the𝑡𝑡 array,𝑓𝑓 𝑡𝑡 at position𝑛𝑛1 𝑡𝑡 relative to the reference station: ( ) = ( ) + ( ) 𝑟𝑟𝑖𝑖 We are assuming that the stations are𝑢𝑢𝑖𝑖 close𝑡𝑡 enough𝑓𝑓 𝑡𝑡 − 𝑟𝑟 𝑖𝑖(small⋅ 𝑢𝑢ℎ𝑜𝑜𝑜𝑜 ) so𝑛𝑛 𝑖𝑖that𝑡𝑡 the signal shapes are similar, i.e. ( ) = ( ). In practice this requires high coherency (similarity) of the wavefield across the array. 𝑟𝑟𝑖𝑖 Classical𝑓𝑓𝑖𝑖 𝑡𝑡 𝑓𝑓 beamforming:1 𝑡𝑡 The delay-and-sum beam is defined as 1 ( , ) = 𝑁𝑁 ( + )

For the signal model assumed: 𝑏𝑏 𝑡𝑡 𝑣𝑣 � 𝑢𝑢𝑖𝑖 𝑡𝑡 𝑟𝑟𝑖𝑖 ⋅ 𝑣𝑣 𝑁𝑁 𝑖𝑖=1 ( , = ) = ( ) + ( ) where ( ) is the average noise, whose amplitude has been reduced by . ℎ𝑜𝑜𝑜𝑜 If then the signals do not stack𝑏𝑏 𝑡𝑡 up𝑣𝑣 coherently𝑢𝑢 𝑓𝑓and𝑡𝑡 ( 𝑛𝑛�, 𝑡𝑡) is small. 𝑛𝑛� 𝑡𝑡 √𝑁𝑁 𝑣𝑣 ≠ 𝑢𝑢ℎ𝑜𝑜𝑜𝑜 𝑏𝑏 𝑡𝑡 𝑣𝑣

GE 162 Introduction to Seismology Winter 2013 - 2016

18.5 Array response

18.6 Coherency stacking …

GE 162 Introduction to Seismology Winter 2013 - 2016

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

GE 162 Introduction to Seismology Winter 2013 - 2016

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

GE 162 Introduction to Seismology Winter 2013 - 2016

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