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.
M11 + M22 + M33 = 0
That means independent elements can be represented by 5 independent
parameters:
M11+M22
M11=[(M11+M22)+(M11-M22)]/2
M11-M22
M22=[(M11+M22)-(M11-M22)]/2
M12
M33=-(M11+M22)
M13
M23
This adds the double-couple constraint with no net volume change.
Formulation of source inverse problem
Resulting displacement
u from a force couple
ui (x, t)  Gij (x, t;x 0 , t 0 ) f j (x 0 , t 0 ) 
ˆ k d, t 0 ) f j (x 0 , t 0 )
Gij (x, t;x 0  x
1
Taylor expand second term:
Gij (x,t;x 0  xˆ k d,t 0 ) f j (x 0 ,t 0 )  Gij (x,t;x 0 ,t 0 ) f j (x 0 ,t 0 ) 
Gij (x,t;x 0 ,t 0 )
f j (x 0 ,t 0 ) * d
x k



Gij (x,t;x 0 ,t 0 )
ui (x,t) 
f j (x 0 ,t)d
x k
Gij (x,t;x 0 ,t 0 )
ui (x,t) 
M jk (x 0 ,t0)
x k
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
layer1
layer2
layer3

Need to solve: Mij
basis functions
1 0 0 Use reflectivity


M  0 0 0


0
0
0


0 1 0

 Use reflectivity
M  0 0 0


0
0
0


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
layer1
layer2
layer3
Step 2: add 0.01*Qb(0) to Qb(0) (1%), calculate
synthetic seismogram.
Step 3: calculate difference seismogram by
differencing step2 seismogram and step1
seismogram for each time point.
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
S
 Qb  D  S
Qb
D – S is the point by point subtraction of Observed Seismogram with synthetic
5
seismogram computed by the starting Qb(0) model. Solve for Qb, add to Qb(0).5
Seismic Velocity Inversion with Reflectivity
Similar deal to Q inversion
S
 V  D  S
V
This is actually a waveform inversion approach where both timing and amplitudes
matter.
D
S

D-S
Final prediction
Notable steps:
V_final
depth
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 dS 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
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:
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:
time5km/s
= distance/velocity
5 10 m /s
T _ ultra 

5mm

3
5 103 m
 106 s  1 micron
5km
T _ seismic 
1 sec
5km/s
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…
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
sin( 1) sin( 2 )

v1
v1

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 boneMarrow 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!
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 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)
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
25
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
Fast
Why? More melt (hot, slow) along
the ridge than perpendicular to ridge.
Slow
26
1D 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)
VSH2=L/r
N cor
VSV2=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-2L), 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 1D 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