Delayed travel-time tomography

Ecole thématique CNRS-CGG-UNSA SEISCOPE – 11-15 septembre 2006

UMR Géosciences Azur – CNRS – IRD – UPMC - UNSA

Jean Virieux

Professeur UNSA

Acknowledgments

•Matthieu Delost (Géosciences Azur/ FSH post-doc) Wavelet model description

•Jean-Xavier Dessa (Géosciences Azur) Travel-time tomography and Full acoustic inversion

•Stéphanie Gautier (Géosciences Azur now at UTMG) Delayed Fresnel tomography

•Céline Gélis (Géosciences Azur now at Amadeus) Full elastic inversion

•Diana Latorre (Géosciences Azur now at INGV) Resolution analysis

•Vadim Monteillier (LGIT/DASE post-doc) Double-difference tomography

•Stéphane Operto (Géosciences Azur/ CNRS CR) full researcher

•Céline Ravaut (Géosciences Azur now at Dublin) Full acoustic inversion

•Tiziana Vanorio (Géosciences Azur now at Stanford) Petro-physics interpretation

With collaboration of Pr. G. Nolet (Princeton University), Jean-Luc Got (LGIT, Chambéry)

Translucid Earth 1Source

Receiver

Same shape !

T(x)

Too diffracting medium : wavefront coherence lost !

Wavefront preserved

Wavefront : T(x)=T0

Travel-time T(x)

Amplitude a(x)

S(t)

)()()(),(

))(()(),(xTieSxaxu

xTtSxatxu

Eikonal equation 2

)(

1)(

)(

1)(

xcxT

xcL

T

T

Lxc x

Two simple interpretation of wavefront evolution

Orthogonal trajectories are rays

T+T

T=cte

Velocity c(x)

L

Grad(T) orthogonal to wavefront

Direction ? : abs or square )(

1))((

22

xcxTx

Transport Equation 3

0).2(

0)()(0

...0

..0

..

2

22

211

'1

21222

22

1112

12222

2

2222

21112

1

222

2112

1

TATAA

TAdivdTAdiv

dSnTAdSnTAdSnTA

dSnTAdSnTA

dSnTAdSnTA

TdSATdSAd

cccc

Tracing neighboring rays defines a ray tube : variation of amplitude depends on section variation

0)()()().(2 2 xTxAxTxA

Ray tracing equations 4T

1 tds

xdt

)()(// xTxcds

xdT

ds

xd

Ray

ds

xd )(sx

ds

Td

.)(. Txctds

d )(. TTcds

Td

)1

(2

)(2 2

2

c

cT

c

ds

Td

)1

(cds

Td

))(

1()

)(

1(

xcds

xd

xcds

d

)(xTp

)(sxq

T=cte

s curvilinear abscisse

evolution of x

Evolution of is given by butT

therefore

evolution of T

Curvature equation known as the ray equation

We often note the slowness vector and the position

System of ray equations 5

cds

pd

pcds

xd

1

ccd

pd

pd

xd

11

cc

dT

pd

pcdT

xd

1

2

ddTc

dscdT

cdsd

2

with

which ODE to select for numerical solving ? Either T or sampling

Many analytical solutions (gradient of velocity; gradient of slowness square)

Time integration of ray equations 6

Initial conditions EASY

1D sampling of 2D/3D medium : FAST

source

receiver

Runge-Kutta second-order integration

Predictor-Corrector integration stiffness

source

receiver Boundary conditions VERY DIFFICULT

?

?

Shooting p ?

Bending x ?

Continuing c ?

AND FROM TIME TO TIME IT FAILS !

But we need 2 points ray tracing because we have a source and a receiver to be connected !

Save p conditions if possible !

Fast marching method (FMM) 7

22

222

)(1

1)()(

x

T

cz

T

cz

T

x

T

z

T

x

T

(tracking interface/wavefront motion : curve and surface evolution)

2D case

Let us assume T is known at a level z=cte

z=cte

z

z + dz

Compute along z=cte by a finite difference

approximation

z

T

Deduce and extend T estimation at depth z+dz

x

T

1D plane wave propagation

Strategy available in 2D and 3D BUT only for the minimum time in each node in the spatial domain (x,y,z).

Possible extension in the phase domain ?

From Sethian, 1999

Back-raytracing strategy 8

Once traveltime T is computed over the grid for one source, we may backtrace using the gradient of T from any point of the medium towards the source (should be applied from each receiver)

The surface {MIN TIME} is convex as time increases from the source : one solution !

A VERY GOOD TOOL

FOR FATT TOMOGRAPHY

Time over the grid Ray

Smooth medium : simple case

Ray tracing by wavefronts 9Evolution over time :

folding of the wavefront is allowed

Dynamic sampling :

undersampling of ray fans

oversampling of ray fans

Keep sampling of the medium by rays « uniform »

Heavy task 2D & 3D !

Example of wavefront evolution in Marmousi model

Hamilton Formulation 10

pp

qq

0

0 q

p

Information around the ray

Ray

2

1

2

1

cd

pd

pd

qd

)1

(2

1),(

22

cppqH

Hd

pd

Hd

qd

x

p

mechanics : ray tracing is a balistic problem

sympletic structure (FUN!)

Meaning of the neighbooring zone – Fresnel zone for example but also anything you wish

Paraxial Ray theory 11

yyd

d

)( y

Estimation of ray tube : amplitude

Estimation of taking-off angles : shooting strategy

…

does not depend on : LINEAR PROBLEM (SIMPLE) !

qpqHppqHd

qd

ppqqHd

qqd

qppp

pp

),(),(

),()(

0000

000

0000

0

p

q

HH

HH

p

q

d

d

pppq

qpqq

00

00

Seismic attributes 12

Travel time evolution with the grid step : blue for FMM and black for recomputed time

One ray Log scale in time

Grid step

S

R

A ray

2 PT ray tracing non-linear problem solved, any attribute could be computed along this line :

-Time (for tomography)

-Amplitude (through paraxial ODE integration fast)

-Polarisation, anisotropy and so on

Moreover, we may bend the ray for a more accurate ray tracing less dependent of the grid step (FMM)

Keep values of p at source and receiver !

Time error over the grid (0)

NOT THE SAME COLOR SCALE

Delayed Travel-time tomography 13

station

source

station

source

station

source

dlzyxudlzyxudlzyxustationsourcet ),,(),,(),,(),( 0

0

0

0

0

0

0

0

0

),,(),(

),,(),(),(

),,(),,(),(

0

0

station

source

station

source

station

source

station

source

dlzyxustationsourcet

dlzyxustationsourcetstationsourcet

dlzyxudlzyxustationsourcet

Consider small perturbations u(x) of the slowness field u(x)

station

source

dlzyxustationsourcet ),,(),(Finding the slowness u(x) from t(s,r) is a difficult problem: only techniques for one variable !

This a LINEAR PROBLEM t(s,r)=G(u)

Discrete Model Space 14

cube

kjikji huzyxu ,,,,),,(

m

m

m

nn

m

n

n

cubekji

cube rayon

kjikji

rayon cubekjikji

u

u

u

u

u

t

u

t

u

t

u

t

t

t

t

t

uu

trst

dlhudlhurst

1

2

1

1

1

1

1

1

2

1

,,

,,,,,,,,

...

),(

),(00

Slowness perturbation description

uGt 0

Matrice of sensitivity or of partial derivatives

Discretisation of the medium fats the ray

Sensitivity matrice is a sparse matrice

LINEAR INVERSE PROBLEM 15

mGduGt

dGmtGu

00

10

10

Updating slowness perturbation values from time residuals

Formally one can write

with the forward problem

Existence, Uniqueness, Stability, RobustnessDiscretisation

Identifiability

of the model

Small errors propagates

Outliers effects

NON-UNIQUENESS & NON-STABILITY : ILL-POSED PROBLEM

REGULARISATION : ILL-POSED -> WELL-POSED

LEAST SQUARES METHOD 16

dGGGm

dGmGG

m

mE

mGdmGdmE

ttest

tt

t

0

1

00

000

00

0)(

)()()(

L2 norm

locates the minimum of E

normal equations

if exists 1

00

GG t

Least-squares estimation

Operator on data will derive a new model : this is called

the generalized inverse

tt GGG 0

1

00

gG0

G0 is a N by M matrice

is a M by M matrice 1

00

GG t

Under-determination M > N

Over-determination N > MMixed-determination – seismic tomography

Maximum Likelihood method 17

One assume a gaussian distribution of data

Joint distribution could be written

)()(

2

1exp)( 0

10 mGdCmGddp d

Where G0m is the data mean and Cd is the data covariance matrice: this method is very similar to the least squares method

)()()()()()( 01

0100 mGdCmGdmEmGdmGdmE dt

)()()( 01

02 mGdWmGdmE d

Even without knowing the matrice Cd, we may consider data weight Wd through

SVD analysis for stability and uniqueness 18

SVD decomposition :

U : (N x N) orthogonal Ut=U-1

V : (M x M) orthogonal Vt=V-1

: (N x M) diagonal matrice Null space for i=0

tVUG 0

UtU=I and VtV=I (not the inverse !)

][

][

0

0

UUU

VVV

p

p

tpp

p VVUUG 000 00

0

Vp and V0 determine the uniqueness while Up and U0 determine the existence of the solution

tppp

tppp

UVG

VUG11

0

0

Up and Vp have now inverses !

Solution, model & data resolution 19

RmmVVmVUUVmGGdGm tp

tppp

tpppest 1

01

01

0 )(The solution is

where Model resolution matrice : if V0=0 then R=VVt=I tppVVR

NddUUmGd tppestest 0

dUUN tppwhere Data resolution matrice : if U0=0 then N=UUt=I

importance matriceGoodness of resolution

SPREAD(R)=

SPREAD(N)=

Spreading functions

2

2

IN

IR

PRIOR INFORMATION 20Hard bounds

Prior model

is the damping parameter controlling the importance of the model mp

Gaussian distribution

Model smoothness

Penalty approach

add additional relations between model parameters (new lines)

)()()()()( 005 pmt

pdt mmWmmmGdWmGdmE

With Wd data weighting and Wm model weighting

tmd

tg

pmt

pdt

GCGCGG

mmCmmmGdCmGdmE

0

110

100

10

104 )()()()()(

)()()()()( 003 pt

pt mmmmmGdmGdmE

BmA i Seismic velocity should be positive

UNCERTAINTY ESTIMATION 21

Least squares solution

Model covariance : uncertainty in the data

curvature of the error function

Sampling the error function around the estimated model often this has to be done numerically

dGdGGGm gttest 00

1

00

1

2

22

1

002

2

0000

2

1cov

cov

covcov

estmm

dest

tdest

dd

gtd

ggtgest

m

Em

GGm

IC

GCGGdGm

Uncorrelated data

A posteriori model covariance matrice 22True a posteriori distribution

Tangent gaussian distribution

S diagonal matrice eigenvalues

U orthogonal matrice eigenvectors

Error ellipsoidal could be estimated

WARNING : formal estimation related to the gaussian distribution hypothesis

tmd

t USUCGCG 10

10

If one can decompose this matrice

Another error function 23pmpmg mmCmmCmE ()(

2

1)( 2/12/1

))(())(( 2/12/1 mgdCmgdC dd

Scalar product on D x M

mC

mgC

mC

dC

mC

mgC

mC

dCmE

m

d

pm

d

m

d

pm

dg 2/1

2/1

2/1

2/1

2/1

2/1

2/1

2/1 )()(

2

1)(

We must minimize 2

2/1

2/1

2/1

2/1 )(

2

1

mC

mgC

mC

dC

m

d

pm

d

which is related to the possible following factorisation

2/10

2/1

2/10

2/11

01

0

m

d

t

m

dmd

C

GC

C

GCCGCG

t

m

d VUC

GC

2/10

2/1

SVD decomposition if possible : please note that this is a sparse matrice good for tomography

Flow chart 24

true ray tracing

data residual

sensitivity matrice

model update

new modelmmm

dGm

mgG

ddd

mgd

m

d

synobs

syn

obs

10

0

)(

collected data

starting modelloop

Calculate for formal uncertainty estimation

small model variation or small errors exit

2

2

mE

Steepest descent methods 25)()( 1 kk mEmE

kk

kk

kkkk

k

kkk

DmE

mEmEd

dE

Ed

mEd

mEmEtmE

)(

)()(

)(

)())((

2

12

1

0

Gradient method

Conjugate gradient

Newton

Quasi-Newton

Gauss-Newton is Quasi-Newton for L2 norm

quadratic approximation of E

Tomographic descent 262

2/1

2/1

2/1

2/1 )(

2

1

mC

mgC

mC

dC

m

d

pm

dMinimisation of this vector

2/1

2/1

m

kdk

C

GCAIf one computes

then

)(

))((

02/1

2/1

km

kdtkk

tk

mmC

dmgCAmAA

Gaussian error distribution of data and of a posteriori model

Easy implementation once Gk has been computed

Extension using Sech transformation (reducing outliers effects while keeping L2 norm simplicity)

LSQR method 27The LSQR method is a conjugate gradient method developped by Paige & Saunders

Good numerical behaviour for ill-conditioned matrices

When compared to an SVD exact solution, LSQR gives main components of the solution while SVD requires the entire set of eigenvectors

Fast convergence and minimal norm solution (zero components in the null space if any)

THE Cm-1/2 MATRICE 28

)exp(2

ji

ij

xxc

Shape independent of

Values depend on

SATURATION

The matrice Cm has a band diagonal shape

- is the standard error (same for all nodes)

is the correlation length

n=nx.ny.nz=104 Cm=USUt (Lanzos decomposition)

tm UUSC 2/12/1

Analysis of coefficients 29

Values independent of n (n>5000)

Values are only related to and

2/12/1 ~0 m

nm

n CC

Typical sizes 200x200x50

deduced from 20x20x5 (few minutes)

Strategy of libraries of Cm-1/2 for

variousand =

Other coefficients could be deduced

R: Cm-1/2 sparse matrice

An example 30

=0.8

v=100 km/s

x=100 km

t=100 s

=0.1

Ray imprints

Same numerical grid for all simulations (either 100x100 or 400x400)

Same results at the limit of numerical precision related to the estimation of the sensitivity matrice

Illustration of selection {,v} 31

= 5 km and v= 3 km/s

Error function analysis will give us optimal values of a priori standard error and correlation length (2D analysis)

v influence

influence

Perspectives 32

• A priori model covariance : location dependent

wavelet decomposition allows local analysis

(work of Matthieu Delost-Geoazur)• Fresnel tomography : introduction of the first Fresnel

zone influence in the forward problem

(work of Stéphanie Gautier-Montpellier)

Conclusion FATT 33

• Selection of an enough fine grid• Selection of the a priori model information• Selection of an initial model• FMM and BRT for 2PT-RT• Time and derivatives estimation• LSQR inversion• Update the model• Uncertainty analysis (Lanzos or numerical)

THANK YOU !

Many figures have come from people I have mentionned in the second slide : again, many thanks to them. Mistakes or misuses of their work is under my responsability.