Moment Tensor Inversion Symmetry of the Moment Tensor

Moment Tensor Inversion Symmetry of the Moment Tensor: Mij = Mji Now assume the earthquake is a pure double couple, that means there is no net volume change, which means trace must be 0. M 11 + M 22 + M 33 = 0 That means independent elements can be represented by 5 independent parameters: M 11+M 22 M 11=[(M 11+M 22)+(M 11 -M 22)]/2 M 11 -M 22=[(M 11+M 22)-(M 11 -M 22)]/2 M 12 M 33=-(M 11+M 22) M 13 M 23 This adds the double-couple constraint with no net volume change. Formulation of source inverse problem Resulting displacement u from a force couple 1

Taylor expand second term: Bingo! AX=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 2

Simple Scheme of Moment Tensor Inversion using Reflectivity Known Quantities: H, Vp, Vs, r, Qa, Qb layer 1 Need to solve: Mij basis functions Use reflectivity layer 2 layer 3 Use reflectivity 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)! 3

Earthquake Source Time Func 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 function Time 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. 4

Q inversion using Reflectivity Known Quantities: H, Vp, Vs, r, Qa, Qb(0) Wish to solve Qb(final) Step 1: Assume Qb(0), use reflectivity layer 1 layer 2 layer 3 Step 2: add 0. 01*Qb(0) to Qb(0) (1%), calculate synthetic seismogram. Step 3: calculate difference seismogram by differencing step 2 seismogram and step 1 seismogram for each time point. Step 4: divide the difference seismogram by 0. 01, this gives a numerical derivative of d. S/d. Qb (d. S is the differential seismogram) Step 5: Based on linear equation D – S is the point by point subtraction of Observed Seismogram with synthetic 5 seismogram computed by the starting Qb(0) model. Solve for DQb, add to Qb(0). 5

Seismic Velocity Inversion with Reflectivity Similar deal to Q inversion This is actually a waveform inversion approach where both timing and amplitudes matter. D S D-S Final prediction depth Notable steps: V_final V_0 velocity (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 d. S for that layer. (2) Put all layers derivatives in A matrix

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. Ultrasound Bone Imaging This is a great example of critical thinking based body waves, surface waves, and seismic sources. Bone Cross-section Marrow Disclaimer: NO HUMAN BONES INVOLVED here.

DR. Watson 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 Thanks to Lauren Stieglitz

Educated Guesses: 1. 2. 3. 4. There is probably P wave (first arrival) There maybe S, but we have no clue where it is hiding. There may be “guided” waves or “tubular” waves, but where? ? ? We can describe these waves using normal modes. Guided Waves Mode Description Similar to SOFAR channels (or T waves) in the ocean, or Wispering Galery (France, or Wispering Wall of China) High Velocity Low Velocity High Velocity Critically reflected Body waves entrapped between two high velocity layers, behaving like a surface wave. 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 1 D 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: Reflectivity: Freq: 1 MHz Distance 5 mm Freq: 1 Hz Distance 5 km Bone Vel=5 km/s Seismic Vel=5 km/s Now, basic relationship we have known for eons: 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 1 D 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…

Effect (or the lack of) Bone Beneath Marrow 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!

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!

Separation of S and Rayleigh 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!

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

Travel Time Moveouts The first phase group is entirely the result of P-to-S converted energy… Reflected PP coincides with P (conjoined twins) S P S head P PS S+R

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 2 D and 3 D effects!

A few notes about seismic anisotropy 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 3 D velocity perturbations relative to a 1 D reference model, say PREM (1982, 1984 and onwards). (4) Study of anisotropy (Mid 1980 s): Love-vs-Rayleigh wave analysis, shear wave splitting for S and Sc. S waves (polarization anisotropy), Pn wave speed anisotropy (azimuthal), anisotropic inversions for radial anisotropy (polarization + directional) 22

Seismic anisotropy: (1) presence of anisotropic fabric or minerals: ----- Lattice Preferred Orientation (LPO) Flow direction is parallel to direction of fast shear or compression velocity ----- Anisotropic effect induced at the refraction surface (behavior of head waves) (2) shape preferred orientation (SPO) Homogeneous silt layers fast Slow direction l < stack thickness h Fluid filled dikes fast Slow direction 23

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

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East Pacific Rise: the Fastest spreading ridge in the world (18 cm/year). Waves going through a ocean ridge will “see” different speeds depending on how it goes across it. Parallel to Ridge: Slow Perpendicular to Ridge: Fast Why? More melt (hot, slow) along the ridge than perpendicular to ridge. Slow 26

1 D wave velocities (PREM) comparing with tomographic result at a given region (VSH >VSV, reflecting horizontality globally). 27

Conventional wisdom: Flow direction is parallel to direction of fast shear or compression velocity Radial anisotropy: Simplest anisotropy with 5 independent elastic parameters 28

S waves: 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 N cor VSV 2=N/r This means waves will travel in different speeds depending on the polarization, or particle motion. F is more complex and depends on velocities at intermediate incidence angles, usually quantified by h =N/(A-2 L), where h varies between 0 and 1. Simple questions: 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. Which velocity is higher, VSV or VSH in general? Answer: near surface, usually VSH. 29

Gu et al. , 2004 Love and Rayleigh Wave Anisotropy: Black trace: Love waves on the transverse comp. Red trace: Synthetic Love waves using a earth Model 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. 30

Data constraints: 1. Body Waves under Radial Anisotropy Gu et al. 2005, EPSL (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: 2. Radial sensitivity of Mode eigen-frequency to SV and SH speed of the earth structure. Gu et al. , 2004 (EPSL) For toridal modes, mode is sensitive to both, but for sphoridal modes, mostly sensitive to SV. 32

Shear Wave Tomography (perturbations to reference 1 D model velocities at given depths) Gu et al. , 2004 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) 33

Global shear anisotropic inversions Ekstrom & Dziewonski, 1998 Interpretation by Gung & Romanowicz (04) 34

Azimuthal Anisotropy: changes in shear Velocity inside horizontal layers. Shear Wave Splitting 35

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. 36

Data fit: Shaded indicate “cross-convolution functions” & nonshaded plots are particle motions of Radial vs. Transverse before & after introducing anisotropy 37

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 38 not too clear in the Alberta Basin where angles are anomalous!

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) 39