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Symmetry of the Moment Tensor: Mij = MjiNow assume the earthquake is a pure double couple, that means there is no net volume change, which means trace must be 0.

M11 + M22 + M33 = 0

That means independent elements can be represented by 5 independent parameters:

M11+M22

M11-M22

M12

M23M13

M11=[(M11+M22)+(M11-M22)]/2

M22=[(M11+M22)-(M11-M22)]/2M33=-(M11+M22)

This adds the double-couple constraint with no net volume change.

Formulation of source inverse problem

Resulting displacement u from a force couple

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ui(x, t) =Gij (x, t;x0, t0) f j (x0, t0) −

Gij (x, t;x0 − ˆ x kd, t0) f j (x0, t0)

Moment Tensor Inversion

2

Taylor expand second term:

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Gij (x, t;x0 − ˆ x kd, t0) f j (x0, t0) =Gij (x, t;x0, t0) f j (x0, t0) −

∂Gij (x, t;x0, t0)

∂xkf j (x0, t0) * d

€

ui (x,t) =∂Gij (x,t;x 0,t0 )

∂xkf j (x 0 ,t)d

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ui (x,t) =∂Gij (x,t;x 0,t0 )

∂xkM jk (x 0 ,t0)

Bingo! A X = D

Choices of inversion parameters:

(1) Can invert for 5 moment tensor elements(2) Heck, you can invert for 6, thereby removing the

double-couple condition

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Simple Scheme of Moment Tensor Inversion using Reflectivity

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M =

1 0 0

0 0 0

0 0 0

⎛

⎝

⎜ ⎜ ⎜

⎞

⎠

⎟ ⎟ ⎟

Known Quantities:H, Vp, Vs, r, Qa, Qb

layer1

layer2

layer3

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M =

0 1 0

0 0 0

0 0 0

⎛

⎝

⎜ ⎜ ⎜

⎞

⎠

⎟ ⎟ ⎟

Use reflectivity

Use reflectivity

basis functions

Do this for each element you need to solve, then you have 5 or 6 traces (or basis functions), they should be multiplied by the corresponding weights (or moment tensors). Your problem is to solve for the weights. This is Weights * basis = data. The data is the observed seismogram (a long vector)!

Need to solve: Mij

4

Time domain convolution:

For earthquakes, large ones can have a rupture process that last tens of seconds. To model it, people convolve a point source with a time functionTime Domain Inversion Recipe: Step 1: Generate synthetic seismogram assuming source pattern and structure are known. Assume source duration is 1 sec.Step 2: Shift the seismogram by 1 sec, name 1 seismogram 2. Step 3: Repeat Step 2 by say, 200 sec, if source duration is no longer than that.Step 4: These shifted seismograms are now the basis function, solve for the weight to each synthetic to mimic the observed.

Earthquake Source Time Func

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Q inversion using ReflectivityKnown Quantities:H, Vp, Vs, r, Qa, Qb(0)

layer1

layer2

layer3

Step 1: Assume Qb(0), use reflectivity

Wish to solve Qb(final)

Step 3: calculate difference seismogram by differencing step2 seismogram and step1 seismogram for each time point.

Step 2: add 0.01*Qb(0) to Qb(0) (1%), calculate synthetic seismogram.

Step 4: divide the difference seismogram by 0.01, this gives a numerical derivative of dS/dQb (dS is the differential seismogram)

Step 5: Based on linear equation

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∂S∂Qb

∗ΔQb = D− S

D – S is the point by point subtraction of Observed Seismogram with synthetic seismogram computed by the starting Qb(0) model. Solve for DQb, add to Qb(0).

Seismic Velocity Inversion with Reflectivity

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∂S∂V

∗ΔV = D− SSimilar deal to Q inversion

This is actually a waveform inversion approach where both timing and amplitudes matter.

DS

D-SFinal prediction

velocity

depthNotable steps:(1) Numerically compute derivatives above as a time series for each layer (i.e., perturb the velocity of a specific layer by a tiny bit, the outcome seismogram is dS for that layer. (2) Put all layers derivatives in A matrix

V_0

V_final

Ultrasound Bone Imaging

Bone Cross-section

Disclaimer: NO HUMAN BONES INVOLVED here.

Marrow

Putting all things together: A key objective of the course is to make you think critically and creatively based on available concepts/tools. Chances are, you will never need to write a large code such as reflectivity method, but you are very likely to utilize a tool similar to that (web is a beautiful thing) to perform moment tensor inversion, waveform matching or inversions.

This is a great exampleof critical thinking based body waves, surface waves, and seismic sources.

Thanks to Lauren Stieglitz

Bone Ultrasound Record

Aha, I have a theory---

The first arrival must be P wave, then… S arrival, wait, but it is so complex… And… Ah, Surface wave. Group 1 is P+S and Group 2 is surface wave… What, time does not work? And amplitudes are wrong??? Where are the guided waves then…?? Oh, Holmes, I am hopelessly confused…

P

??

?? ??

DR. Watson

Educated Guesses:

1. There is probably P wave (first arrival)2. There maybe S, but we have no clue where it is hiding.3. There may be “guided” waves or “tubular” waves, but where???4. We can describe these waves using normal modes.

Similar to SOFAR channels (or T waves) in the ocean, or Wispering Galery (France,or Wispering Wall of China)

Guided Waves

High Velocity

High Velocity

Low Velocity

Critically reflected Body waves entrapped between two high velocity layers, behaving like a surface wave.

Mode Description

Describe the sound waves as standing waves caused by vibrations at material’s natural frequencies. (Mode-Ray Duality)Caveat: not physically intuitive, no time-domain wiggles!!

Lets start with simple 1D assumption… and seismic source

Sherlock Holmes

First realization: Ultrasound is P wave, therefore the P wave radiation pattern should be perfectly suited. In this case, the maximum dilatation is going horizontal, exactly the direction of the transmitted ultrasound wave after the wedge

Scaling Assumptions (why would reflectivity code work?):Ultrasound: Freq: 1 MHz Distance 5 mm Bone Vel=5 km/sReflectivity: Freq: 1 Hz Distance 5 km Seismic Vel=5 km/s

Now, basic relationship we have known for eons:

€

time = distance/velocity

€

T_ultra=5mm

5km/s=

5×10−3m

5×103m/s

=10−6s=1 micron

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T _ seismic =5km

5km /s=1 sec

What does this mean? It means that the seismic problem is perfectly suited for this problem, provided that one just simply have to interpret the time axis as microns (microseconds) as opposed to sec, then all is good!

Waveform Fitting Using Best 1D Model

The culprits causing these inflicted wounds (pulses) on the victim (ultrasound records) are:

P wave in top CB

P-S conversions in top CB

S wave in top CB

Surface wave in top CB

Marrow P and S multiples

Innocent: cortical bone on the bottom

But Holmes, how do you know!!?? Tell me, what are these two main phase groups then?

Watson, they are marrow related phases since CB on top (Model 1, black) does not produce groups 2, 3, 4…

Why is the cortical bone beneath the marrow INNOCENT?

Because these two models produce identical waveforms. That narrows down our search for the true culprits!

Effect (or the lack of) Bone Beneath Marrow

Ok, say in the first group, what is the cause of all that complexity?

Model 1 (black) does not allow reflections and conversions. The results: No more complexities!

So where are the famous guided waves?

Model 1 (black) uses a slower bone speed to widen the difference between S and Rayleigh waves. Look, they are separate!

Separation of S and Rayleigh

Mystery of the First Phase Group

As you have guessed, they are converted and reflected P, S waves in the top cortical bone segment.

Travel Time Table

€

sin(θ1)

v1

=sin(θ2)

v1

Travel Time MoveoutsP

S+R

S PheadSPS

The first phase group is entirely the result of P-to-S converted energy… Reflected PP coincides with P (conjoined twins)

Mystery of the Second, Third Phase Groups (refracted/transmitted waves and receiver functions)

Transmitted and Converted Waves at the Cortical bone-Marrow Boundary (CMB)

Simple Conclusion:

The Cortical bone – Marrow-cortical Bone (CMB) system is, intrinsically, no different from modeling of Porous media and seismic imaging of the Core – Mantle Boundary (CMB). Simple ray theory can explain most of the variations, despite complexities related to 2D and 3D effects!

Chronology of Seismic Structure Analysis:

(1) ravel time observations, building travel time curves (Jeffreys & Bullen tables), obtaining major divisions inside

the earth, core, mantle… Also, obtaining subsurface structure (roughly) from reflectors.

(2) Study of 1-D earth structure (inversion)PREM (Preliminary Reference Earth Model, by Dziewonski & Anderson, the “Half Million Dollar Paper!” (1981)

(3) Study of 3D velocity perturbations relative to a 1D reference model, say PREM (1982, 1984 and onwards).

(4) Study of anisotropy (Mid 1980s): Love-vs-Rayleigh wave analysis, shear wave splitting for S and ScS waves (polarization anisotropy), Pn wave speed anisotropy (azimuthal), anisotropic inversions for radial anisotropy (polarization + directional)

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A few notes about seismic anisotropy

Seismic anisotropy: (1) presence of anisotropic fabric or minerals: ----- Lattice Preferred Orientation (LPO)

fast

Slow direction

Homogeneous silt layers

l < stack thickness h

(2) shape preferred orientation (SPO)

Flow direction is parallel to direction of fast shear or compression velocity

Fluid filled dikes

fast Slow direction

----- Anisotropic effect induced at the refraction surface (behavior of head waves)

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Pn analysis for P (head waves) traveling along Moho 24

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Waves going through a ocean ridge will “see” different speeds depending on how it goes across it.

Parallel to Ridge: SlowPerpendicular to Ridge: Fast

East Pacific Rise: the Fastest spreading ridge in the world (18 cm/year).

Fast

SlowWhy? More melt (hot, slow) alongthe ridge than perpendicular to ridge. 26

1D wave velocities (PREM) comparing with tomographic result at a given region (VSH >VSV, reflecting horizontality globally).

27

Conventional wisdom:

Flow direction is parallelto direction of fast shearor compression velocity

Radial anisotropy:

Simplest anisotropy with 5independent elastic parameters

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L corresponds to m for Z direction (think of the first two indices as normal in X or Y pointing to Z, the last two indices represents response to strain along X or Y with directional derivative in Z) VSH

2=L/r

S waves:

N cor responds to C1212 (m) in the Y directionVSV

2=N/r

This means waves will travel in different speeds depending on the polarization, or particle motion.

Why choose transverse isotropy with a vertical symmetry axis?Answer: things settle down in layers (or so called the principle of horizontality). It is all based on common knowledge.

F is more complex and depends on velocities at intermediate incidence angles, usually quantified by h =N/(A-2L), where h varies between 0 and 1.

Simple questions:

Which velocity is higher, VSV or VSH in general?Answer: near surface, usually VSH.

29

Love and Rayleigh Wave Anisotropy:

Black trace: Love waves on the transverse comp.Red trace: Synthetic Love waves using a earthModel obtained from the Rayleigh wave of the same source-receiver pair.

Indication: How particles move will result in different arrival time, if material speed is different in different directions.

Gu et al., 2004

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(1) Travel time of a SH-Polarized wave is sensitive to both Vsh and Vsv(2) Sensitivity to Vsh and Vsv depends both on polarization and direction of

the ray path(3) SV-polarized waves are only sensitive to vertical wave speed (Vsv).

Gu et al, 2004

Data constraints: 1. Body Waves under Radial Anisotropy

Gu et al. 2005, EPSL

Gu et al., 2004 (EPSL)

Data Constraints: 2. Radial sensitivity of Mode eigen-frequency to SV and SH speed of the earth structure.

For toridal modes, mode is sensitive to both, but for sphoridal modes, mostly sensitive to SV.

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200 km

300 km

400 km

You see that for the most part, the “tomographic” images show consistent patterns between SV and SH. Especially at 200 km, continents are blue (fast, colder, older) and oceans are red (slow, hotter, younger)

Gu et al., 2004

Shear Wave Tomography (perturbations to reference 1D model velocities at given depths)

33

34

Global shear anisotropic inversionsEkstrom & Dziewonski, 1998

Interpretation by Gung & Romanowicz (04)

35

Azimuthal Anisotropy: changes in shear Velocity inside horizontal layers. Shear Wave Splitting

36

Particle motions before and after splitting measurements

Plotted above are comparisons of radial and transverse components. The right panels show no anisotropy, which result in a 45 deg straight line. The left shows strong anisotropy prior to time shift (anisotropy) and rotation of the records to null axis.

37

Data fit: Shaded indicate “cross-convolution functions” & non-shaded plots are particle motions of Radial vs. Transverse before & after introducing anisotropy

38

Splitting time & fast direction + velocity

Gu et al., 2011

Splitting angles (fast direction indicated by arrows) show northeast-southwest directions near the foothills of the Rockies, consistent with plate motion of north America. The splitting times of 1+ sec indicate significant horizontal anisotropy. However, the story is not too clear in the Alberta Basin where angles are anomalous!

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Interpreted Flow in and around Alberta

Maybe the result of moving continental root (below 100 km) that channels the flow like boat does to water when moving? (Gu et al., 2012, in preparation)