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Geophysical Inverse Problems
with a focus on seismic tomography
CIDER2012- KITP- Santa Barbara
Seismic travel time tomography
1) In the background, “reference” model: Travel time T along a ray :g
v0(s) velocity at point s onthe rayu= 1/v is the “slowness”
Principles of travel time tomography
The ray path g is determined by the velocity structure using Snell’s law. Ray theory.
2) Suppose the slowness u is perturbed by an amount du small enoughthat the ray path g is not changed.
The travel time is changed by:
lij is the distance travelled by ray i in block jv0
j is the reference velocity (“starting model”) in block j
Solving the problem: “Given a set of travel time perturbations dTi on an ensemble of rays {i=1…N}, determine the perturbations (dv/v0)j in a 3Dmodel parametrized in blocks (j=1…M}” is solving an inverse problem ofthe form:
d= data vector= travel time pertubations dTm= model vector = perturbations in velocity
G has dimensions M x N
Usually N (number of rays) > M (number of blocks):“over determined system”
We write:
GTG is a square matrix of dimensions MxMIf it is invertible, we can write the solution as:
where (GTG)-1 is the inverse of GTGIn the sense that (GTG)-1(GTG) = I, I= identity matrix
“least squares solution” – equivalent to minimizing ||d-Gm||2
- G contains assumptions/choices:- Theory of wave propagation (ray theory)- Parametrization (i.e. blocks of some size)
In practice, things are more complicated because GTG, in general, is singular:
“””least squares solution”Minimizes ||d-Gm||2
Some Gij are null ( lij=0)-> infinite elements in the inverse matrix
How to choose a solution?
• Special solution that maximizes or minimizes some desireable property through a norm
• For example:– Model with the smallest size (norm): mTm=||
m||2=(m12+m2
2+m32+…mM
2)1/2
– Closest possible solution to a preconceived model <m>: minimize ||m-<m>||2
regularization
• Minimize some combination of the misfit and the solution size:
• Then the solution is the “damped least squares solution”:
mmeem TT 2)(
dGIGGm TT 12ˆ
e=d-Gm
Tikhonov regularization
• We can choose to minimize the model size, – eg ||m||2 =[m]T[m] - “norm damping”
• Generalize to other norms.– Example: minimize roughness, i.e. difference
between adjacent model parameters.– Consider ||Dm||2 instead of ||m||2 and
minimize:
– More generally, minimize:
mWmDmDmDmDm mTTTT
)()( mmWmm mT
<m> reference model
Weighted damped least squares
• More generally, the solution has the form:
][][
:,
][][
11211
12
mGdWGGWGWmm
lyequivalentor
mGdWGWGWGmm
eT
mT
mest
eT
meTest
For more rigorous and complete treatment (incl. non-linear):See Tarantola (1985) Inverse problem theoryTarantola and Valette (1982)
Concept of ‘Generalized Inverse’• Generalized inverse (G-g) is the matrix in the
linear inverse problem that multiplies the data to provide an estimate of the model parameters;
– For Least Squares
– For Damped Least Squares
– Note : Generally G-g ≠G-1
dGm gˆ
TTg GGGG1
TTg GIGGG12
2
mm
2
dGm
• As you increase the damping parameter e, more priority is given to model-norm part of functional.– Increases Prediction Error– Decreases model structure – Model will be biased toward
smooth solution
• How to choose e so that model is not overly biased?
• Leads to idea of trade-off analysis.
η
“L curve”
Model Resolution Matrix• How accurately is the value of an inversion parameter
recovered?• How small of an object can be imaged ?
• Model resolution matrix R:
– R can be thought of as a spatial filter that is applied to the true model to produce the estimated values.
• Often just main diagonal analyzed to determine how spatial resolution changes with position in the image.
• Off-diagonal elements provide the ‘filter functions’ for every parameter.
Masters, CIDER 2010
80%
Checkerboard test
R contains theoretical assumptionson wave propagation, parametrizationAnd assumes the problem is linearAfter Masters, CIDER 2010
Ingredients of an inversion
• Importance of sampling/coverage– mixture of data types
• Parametrization– Physical (Vs, Vp, ρ, anisotropy,
attenuation)– Geometry (local versus global functions,
size of blocks)• Theory of wave propagation
– e.g. for travel times: banana-donut kernels/ray theory
P S
Surface waves
SS
50 mn
P, PPS, SSArrivals well separated on the seismogram, suitable for traveltime measurements
Generally:- Ray theory- Iterative back projection techniques- Parametrization in blocks
Van der Hilst et al., 1998
Slabs……...and plumes
Montelli et al., 2004
P velocity tomography
Vasco and Johnson,1998
P TravelTimeTomography:
RayDensitymaps
Karason andvan der Hilst,2000
Checkerboard tests
Honshu
410660
±1.5 %
15
13
05
06
07
08
09
11
12
14
15
13
northern Bonin
±1.5 %
410660
1000
Fukao andObayashi2011
±1.5%
Tonga
Kermadec
06
07
08
09
10
11
12
13
14
15
±1.5%
410660
1000
Fukao andObayashi2011
PRI-S05
Montelli et al., 2005
EPR
South Pacific superswell
Tonga
Fukao andObayashi,2011
6601000
400
S40RTS
Ritsema et al., 2011
Rayleigh waveovertones
By including overtones, we can see into the transition zone and the top of the lower mantle.
after Ritsema et al, 2004
Models from different data subsets
120 km
600 km
1600 km
2800 km
After Ritsema et al., 2004
Sdiff ScS2
The travel time dataset in this model includes:
Multiple ScS: ScSn
Coverage of S and P
After Masters, CIDER 2010
P S
Surface wavesSS
Full Waveform Tomography Long period (30s-400s) 3- component seismic
waveforms
Subdivided into wavepackets and compared in time domain to synthetics.
u(x,t) = G(m) du = A dm A= ∂u/∂m contains Fréchet derivatives of G
UC B e r k e l e y
PAVA
NACT
SS Sdiff
Li and Romanowicz , 1995
PAVA NACT
2800 km depth
from Kustowski, 2006
Waveforms only, T>32 s!20,000 wavepacketsNACT
To et al, 2005
Indian Ocean Paths - Sdiffracted
Corner frequencies: 2sec, 5sec, 18 sec To et al, 2005
To et al., EPSL, 2005
Full Waveform Tomography using SEM:
UC B e r k e l e y
Replace mode synthetics by numerical syntheticscomputed using the Spectral Element Method (SEM)
Data
Synthetics
SEMum (Lekic and Romanowicz, 2011) S20RTS (Ritsema et al. 2004)
70 km
125 km
180 km
250 km
-12%
+8%
-7%
+9%
-6%
+8%
-5%
+5%
-7%
+6%
-6%
+8%
-4%
+6%
-3.5%
+3%
French et al, 2012, in prep.
Courtesy of Scott French
SEMum2
S40RTS
Ritsema et al., 2011
French et al., 2012
EPR
South Pacific superswellTonga
Samoa
Easter IslandMacdonald
Fukao andObayashi, 2011
Summary: what’s important in global mantle tomography
• Sampling: improved by inclusion of different types of data: surface waves, overtones, body waves, diffracted waves…
• Theory: to constrain better amplitudes of lateral variations as well as smaller scale features (especially in low velocity regions)
• Physical parametrization: effects of anisotropy!!• Geographical parametrization: local/global basis
functions
• Error estimation
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