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