geophysical inverse problems with a focus on seismic tomography
DESCRIPTION
Geophysical Inverse Problems with a focus on seismic tomography. CIDER2012- KITP- Santa Barbara. Seismic travel time tomography. Principles of travel time tomography. 1) In the background, “reference” model: Travel time T along a ray g:. v 0 (s) velocity at point s on the ray - PowerPoint PPT PresentationTRANSCRIPT
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Geophysical Inverse Problems
with a focus on seismic tomography
CIDER2012- KITP- Santa Barbara
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Seismic travel time tomography
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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:
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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
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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
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- 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
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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
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• 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
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• 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
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Weighted damped least squares
• More generally, the solution has the form:
][][
:,][][
11211
12
mGdWGGWGWmm
lyequivalentormGdWGWGWGmm
eT
mT
mest
eT
meTest
For more rigorous and complete treatment (incl. non-linear):See Tarantola (1985) Inverse problem theoryTarantola and Valette (1982)
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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
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2
mm
2
dGm
• As you increase the damping parameter , 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 so that model is not overly biased?
• Leads to idea of trade-off analysis.
η
“L curve”
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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.
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Masters, CIDER 2010
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80%
Checkerboard test
R contains theoretical assumptionson wave propagation, parametrizationAnd assumes the problem is linearAfter Masters, CIDER 2010
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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
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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
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Van der Hilst et al., 1998
Slabs…… ...and plumes
Montelli et al., 2004
P velocity tomography
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Vasco and Johnson,1998
P TravelTimeTomography:
RayDensitymaps
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Karason andvan der Hilst,2000
Checkerboard tests
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Honshu
410660
±1.5 %
15
13
05
06
07
08
09
11
12
14
15
13
northern Bonin
±1.5 %410660
1000
Fukao andObayashi2011
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±1.5%
Tonga
Kermadec
06
07
08
09
10
11
12
13
14
15
±1.5%
410660
1000
Fukao andObayashi2011
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PRI-S05Montelli et al., 2005
EPR
South Pacific superswell
Tonga
Fukao andObayashi,2011
6601000
400
S40RTSRitsema et al., 2011
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Rayleigh waveovertones
By including overtones, we can see into the transition zone and the top of the lower mantle.
after Ritsema et al, 2004
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Models from different data subsets
120 km
600 km
1600 km
2800 km
After Ritsema et al., 2004
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Sdiff ScS2
The travel time dataset in this model includes:
Multiple ScS: ScSn
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Coverage of S and P
After Masters, CIDER 2010
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P S
Surface wavesSS
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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
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PAVA
NACT
SS Sdiff
Li and Romanowicz , 1995
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PAVA NACT
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2800 km depthfrom Kustowski, 2006
Waveforms only, T>32 s!20,000 wavepacketsNACT
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To et al, 2005
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Indian Ocean Paths - Sdiffracted
Corner frequencies: 2sec, 5sec, 18 sec To et al, 2005
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To et al., EPSL, 2005
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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
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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%
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French et al, 2012, in prep.
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Courtesy of Scott French
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SEMum2
S40RTSRitsema et al., 2011
French et al., 2012
EPR
South Pacific superswellTonga
Samoa
Easter IslandMacdonald
Fukao andObayashi, 2011
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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