Wave propagation in solid medium in time By J. Virieux and S. Operto Ecole thématique CNRS-CGG-UNSA SEISCOPE 11-15 septembre 2006 UMR Géosciences Azur.

Wave propagation in solid medium in time

By J. Virieux and S. Operto

Ecole thématique CNRS-CGG-UNSA SEISCOPE

11-15 septembre 2006

UMR Géosciences Azur – CNRS – IRD – UPMC - UNSA

Acknowledgments

•Victor Cruz-Atienza (Géosciences Azur on leave for SDSU) FDTD

• Matthieu Delost (Géosciences Azur on leave) Wavelet tomography

•Céline Gélis (Géosciences Azur now at Amadeous) Full wave elastic imaging

• Bernhard Hustedt (Géosciences Azur now at Shell) Wavelet decomposition of PDE

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

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

Spice group in Europe : http://www.spice-rtn.org

FDTD introduction :

ftp://ftp.seismology.sk/pub/papers/FDM-Intro-SPICE.pdf

By P. Moczo, J. Kristek and L. Halada

Non-Translucid Earth 1

Inside the Earth, discontinuities are present which lead to converted phases, especially in the crust : three characteristic times in seismograms/traces

We need techniques for modelling these waves which can be quite complex

ANATOMY OF GLOBAL-OFFSET DATA

Velocity gradient at interfaces : diving waves

Anatomy of seismic waves phases

From Stéphane Operto

Anatomy of global-offset seismograms:Continuous sampling of apertures from transmission to reflection

Critical incidence – total reflection

Upgoing conic wave

Critical distance

Interface wave

Conic wave

Root wave

Asymptotic « convergence » between direct and super-critical reflected waves

Diving wave

Synthetic seismograms

Head or conic wave

Diving Wave

LA PROPAGATION DES ONDES I

Tenseur de déformation à partir du déplacement

Tenseur de contrainte exprime les forces internes (séismes)

Le PFD en présence de forces

La loi de Hooke avec les coeffs. élastiques

L’équation est dite

l’équation de l’élastodynamiqueUne rhéologie simple pour des milieux LHI

en fonction des coefficients de Lamé !

LA PROPAGATION DES ONDES IIL’équation élastodynamique en milieu linéaire et élastique

en milieu linéaire, élastique. C’est un système de 3 équations du second ordre aux dérivées partielles définissant les composantes ui(x,t). Un système à 9 équations peut aussi être construit à partir des vitesses et des contraintes :

où la fonction mij est non nulle dans les régions sources.

LA PROPAGATION DES ONDES IIIL’équation élastodynamique en milieu linéaire, élastique et isotrope s’écrit

On utilisera fi ou fi - mij suivant les besoins. On parlera de systèmes de forces équivalents.

LA PROPAGATION DES ONDES IVDans des milieux liquides, on préfère travailler avec la pression et la vitesse des particules

où la fonction q(x,t) s’appelle la source volumique en vitesse et est définie par

On en déduit les équations d’onde acoustique

LA PROPAGATION DES ONDES VSi on élimine la vitesse des particules, on obtient l’équation d’onde acoustique scalaire pour la pression p(x,t) :

avec

Si on suppose que la masse volumique est homogène, on a

avec

qui est l’équation d’onde scalaire que l’on retrouve dans différents livres

LA PROPAGATION DES ONDES VISi on élimine la pression, on obtient l’équation d’onde acoustique vectorielle

avec la force en vitesse suivante

Cette équation est un cas particulier de l’équation dérivée de l’équation élastodynamique. En général, on ne l’étudie pas séparément et on ne considère que l’équation d’onde acoustique scalaire.

),(),(),(

)(

1),( 00002

2

2txStxf

t

txP

xctxP p

Considérons l’équation d’onde scalaire

Si les termes sont nuls, alors l’excitation peut se déduire d’un terme excitation en divergence :

pfijm

ii txft

txP

xctxP '002

2

2),(

),(

)(

1),(

),(),(),(),(),(

)(

1),( 00

0000002

2

2txf

z

txf

y

txf

x

txf

t

txP

xctxP zyx

que nous pouvons mettre sous une forme vectorielle

LES FONCTIONS DE GREEN

)()(),;,(

)(

1),;,( 002

002

200 ttxxt

txtxG

xctxtxG

où c(x) est la vitesse et la distribution dirac est notée par

et peut se voir comme une fonction de valeur infinie en zéro.

La réponse impulsionnelle définit la fonction de Green G(x,t;x0,t0) du milieu où la source ponctuelle se trouve en x0 et l’impulsion est donnée en t0,.tandis que l’on calcule la solution au point x et au temps t.

Les solutions en milieu homogène

Solution 1D Solution 2D Solution 3D

Certaines caractéristiques communes mais d’autres très différentes comme la trainée à 2D

The corner-edge as a complex example

ODE versus PDE formulations

GOAL : find ways to transform differential operators into algebraic operators in order to use linear algebra at the end

Ayydt

d

yAydt

d

)(

Dyt

y

yDt

y

)(

O.D.E

Ordinary differential Equations

P.D.E

Partial Differential Equations

Linear

Non-linear

Symmetry between space and time ?

An apparent easy waySpectral methods allow to go directly to this algebraic structure

x

uc

t

u2

22

2

2

x

ucu

2

222

ukcu ˆˆ 222 Dispersion relation has to be verified BUT conditions have to be expressed in this dual space : here is the difficulty !

Pseudo-spectral approach : a remedy for a precise and fast strategy

Go to the dual space only for computing spatial derivatives and goes back to the standard space for equations and conditions

Frequency approach of Pratt : the opposite way around

3D Elasto-dynamic equations

Divide by the density will leave medium properties only on the RHS

The previous PDE form is then retrieved

P-SV equations

Elastic properties

No attenuation

Medium properties vary from point to point

No spatial derivatives of these medium properties

One-dimensional scalar wave

x

uc

t

u2

22

2

2

The wave solution is u(x,t)=F(x+ct)+G(x-ct) whatever are F and G (to be checked)

The wave is defined by pulsation , wavelength , wavenumber k and frequency f and period T. We have the following relations

cc

f

cTk

222

A plane wave is defined by )(),( kxtietxu

The scalar wave equation is verified by the vibration u(t,x)

with the dispersion relation

222 kcThe phase velocity is for any frequency c

kVp

If the pulsation depends on k, we have kcdk

d 2 and the group velocity is

cc

c

dk

dVg

.

2

which is identical to phase velocity for non-dispersive waves

Homogeneous medium

First-order hyperbolic equation

t

uv

x

u

x

v

t

xc

t

v

2

x

uE

xt

u

2

2

Let us define other variables for reducing the derivative order in both time and space

The 2nd order PDE became a 1st order PDE

This is true for any order differential equations: by introducing additionnal variables, one can reduce the level of differentiation. Among these different systems, one has a physical meaning

which becomes

x

vE

t

xt

v

1

E

c 2with

stress

velocity

Other choices are possible as displacement-stres instead of velocity-stress.

Characteristic variables

)....,,(. 21

1

ndiagwith

RDR

RRD

)()0,(

0

0 xwxwx

wD

t

w

npx

f

t

fx

f

t

f

wRf

pp

p ,...,1;0

0

1

Consider an linear system is defined by

If the matrix A could be diagonalizable with real eigenvalues, the system is hyperbolic.If eigenvalues are positive, the system is strictly hyperbolic.

)0,(),( txftxf ppp

The system could be solved for each component fp

The curve x0+p t is the p-characteristic

The scalar wave introduces w=(v,s) and the following matrix w(u,d) where u design the upper solution and d the downgoing solution.

corc

cwith

EA

..

0

0..

0

10

The transformation from w to f splits left and right propagating waves

Other PDE in physics

x

u

t

u2

2

x

uc

t

u2

22

2

2

ukx

u 22

2

0

The scalar wave equation is a partial differential equation which belongs to second-order hyperbolic system.

x

u2

2

0

x

u

t

u2

22

2

2

x

u

t

u2

2

Wave Equation

Fluid Equation

Diffusion Equation

Laplace Equation

Fractional derivative Equation

Time is involved in all physical processes except for the Laplace equation related to Newton law and mass distribution.

Poisson equation could be considered as well when mass is distributed inside the investigated volume

Poisson Equation

Initial and boundary conditions

Boundary conditions u(0,t)

Initial conditions u(x,0)

Boundary conditions u(L,t)

1D string medium

fx

uc

t

u

2

22

2

2

x

vE

t

fxt

v

1

Difficult to see how to discretize the velocity !

f(x,t) Excitation condition

Much better for handling heterogeneity

Dirichlet conditions on u

Neumann conditions on

Finite Difference Stencil

i-1 i i+1

(Leveque 1992)

centeredh

UUUD

backwardh

UUUD

forwardh

UUUD

iii

iii

iii

211

0

1

1

Truncations errors : 0h

Second derivative

iii UDUDDUDD 200

)2(1

1122

iiii UUUh

UD

Higher-order terms : same procedure but you need more and more points

x

ux

x

ux

x

ux

x

uxuuxxu

nininininini 4

44

,3

33

,2

22

,,,1 2462

)(

x

ux

x

ux

x

ux

x

uxuuxxu

nininininini 4

44

,3

33

,2

22

,,,1 2462

)(

x

ux

x

uxuuu

nininini 4

44

,2

22

,,1,1 122

Discretisation and Taylor expansion

)(2 2

2

,,1,1

,2

2

xx

uuu

x

u ninini

ni

Assuming an uniform discretisation x,t on the string, we consider interpolation upto power 4

by summing, we cancel out odd terms

neglecting power 4 terms of the discretisation steps. We are left with quadratic interpolations, although cubic terms cancel out for precision.

Other expansions

)()('

)()(' xeuxu

xeuxu

ii

ii

ei(x) could be any basis describing our solution model and for which we can compute easily and accurately either analytical or numerical compute derivatives

A polynomial expansion is possible and coefficients of the polynome could be estimated from discrete values of u: linear interpolation, spline interpolation, sine functions, chebyshev polynomes etc

Choice between efficiency and accuracy (depends on the problem and boundary conditions essentially)

Consistency

x

vE

t

xt

v

1

)(2

1)(

1

)(2

11)(

1

111

111

mi

mii

mi

mi

mi

mi

i

mi

mi

VVh

ETTt

TTh

VVt

Local error

),(1

),(

)(2

11)(

111

1

tmihx

tmiht

vL

TTh

VVt

L

i

mi

mi

i

mi

mi

Taylor expansion around (ih,mt)

0,0

)()(),(1

),( 2

thwhenLL

htOtmihx

tmiht

vL

i

FD scheme is consistent with the differential equations (do the same for the other equation)

Stability

)exp(

)exp(

jkihtmjBT

jkihtmjAVmi

mi

khjAh

tEtjB

khjBh

ttjA

i

i

sin22

1)exp(

sin22

1)exp(

222 )(sin)()1)(exp( khh

tEtj

i

i

1sin)(1)exp( 2/1

khh

tEjtj

i

i

Harmonic analysis in space and in time

is complex : the solution grows exponentially with time : UNSTABLE

Local stability # long-term stability (finite domain validity)

CONSISTENCE + STABILITY = CONVERGENCE (not always to the physical

solution)

STABLE STENCIL :leap-frog integration

m+1

m

m-1

i-1 i i+1)(

2

1)(

2

1

)(2

11)(

2

1

1111

1111

mi

mii

mi

mi

mi

mi

i

mi

mi

VVh

ETTt

TTh

VVt

Harmonic analysis

khjAh

tEtBj

khjBh

ttAj

i

i

sin2sin2

sin2sin2

khh

tEt

khh

tEt

i

i

i

i

sin)(sin

)(sin)()(sin

2/1

222

th

tE

i

i

sin1)( 2/1

is real

The solution does not grow with time : STABLE

CFL condition

Courant, Friedrichs & Levyi

ii

i Ecwith

c

ht

.. Magic step t=h/c0

Characteristic line

The time step cannot be larger than the time necessary for propagating over h

Von Neuman stability study

Time integration (more theory)0

2111

ni

nin

ini

uu

h

kauu

02

11

111

ni

nin

ini

uu

h

kauu

02

11

11

ni

nin

ini

uu

h

kauu

02

1111

ni

nin

ini

uu

h

kauu

022

1 1111

1

ni

nin

ini

ni

uu

h

kauuu

0)2(22 11

22

2111

ni

ni

ni

ni

nin

ini uuua

h

kuu

h

kauu

02

1111

ni

nin

ini

uu

h

kauu

0)2(22

4321

22

2211

ni

ni

ni

ni

ni

nin

ini uuua

h

kuuu

h

kauu

Euler

Backward Euler

Left-side (upwind)

Right-side

Lax-Friedrichs

Leapfrog

Lax-Wendroff

Beam-Warming

RED-BLACK PATTERN

i-1 i i+1m-1

m

m+1The staggered grid

vUNCOUPLED SUBGRID :

SAVE MEMORY

ONLY BOUNDARY CONDITIONS WOULD HAVE COUPLED THEM

STAGGERED GRID SCHEME)()(

1

)(11

)(1

2/12/11

2/112/12/1

2/12/12/12/1

mi

mi

imi

mi

mi

mi

i

mi

mi

VVh

ETT

t

TTh

VVt

2sin)()

2sin( 2/12/1 kh

h

tEt

i

i

Second-order in time & in spaceINDICE FORTRAN ?

NUMERICAL DISPERSION

Moczo et al (2004)

22sin

22sin

khkh

tt

02/12/1 )( c

E

k i

i

How small should be h compared to the wavelength to be propagated ?

2/120

0

0

))sin(1(

cos

)sinarcsin(

hht

c

hc

kv

h

h

tc

htk

h

kc

gridg

grid

2ème ordre 4ème ordre

10

h

5

h

acf

vh

10min

acf

v

2min

NUMERICAL ANISOTROPY

PSG FSG

COMBINE ?

PARSIMONIOUS RULE

))2/1((

)(

2/11 hiE

ihi

How to define these discrete values for an heterogeneous medium ?

(especially when considering strong discontinuities)

x

vE

t

xt

v

1

x

vExt

v

1

2

2

How to estimate the spatial operator

)()(

)(11

)(1

2/11

2/122/12/12/1

122/1

12/1

12/1

2/12/12

mi

mi

i

imi

mi

i

i

mi

mi

i

mi

mi

VVh

EVV

h

E

TTth

VVt

)(11

)(1

2/12/12/12/1 m

imi

i

mi

mi TT

hVV

t

))(2

(1

)2(1

2/12/12/1

2/112/1

2/112/12

2/12/32/12

iimi

mii

mii

mi

mi

mi

EEV

VEVEh

VVVt

Do same thing for

xEx

2/1

2/12/1

2/1

1

2/)(

1

i

ii

i

Ei

EEi

Ei

1

1

i

i

i

FREE SURFACE (Neumann condition)

0 1 2m-1

m

m+1

v

)(11

)(1

02/32/1

12/1

1 TTh

VVt

m

i

mm

Amplitude deficit of wave nearby the free surface

0 1 2m-1

m

m+1

v

m

i

mm

i

mm

Th

TTh

VVt

2/3

2/12/32/1

12/1

1

21

)(11

)(1

We can see that we have amplified by a factor of 2Antisymmetric stress

ESIM procedure

0 1 2m-1

m

m+1

v

Predict by extrapolation values outside the domain for keeping the finite difference stencil while verifying solutions on the boundary

SAT procedure Modify the stencil when hitting the boundary for keeping same accuracy while using only values on one-side of the boundary

SAT has a mathematical background while ESIM has not

)3/13(11

)(1

2/52/32/1

12/1

1mm

a

mm TTh

VVt

12/1 a

Source or grid excitation

fx

uc

t

u

2

22

2

2

ni

ni

ni

ni

ni

ni

ftuu

ftuu

2/12

11

2/12

000

000

Impulsive source

Known solution

The source is a term which should be added to the equation. Because it is related to acceleration, we denote it as an impulsive excitation.

A particular solution of the wave equation is injected into the medium or the grid. Typically an incident plane wave is applied at each grid point along a given line.

Explosive source

A very popular excitation is the explosive source, which requires either applications of opposite sign forces on two nodes or a fictious force between two nodes. Once integration has been performed, we should add

20 )(..)( tteofsderivativetf

Radiative boundariesOne may assign boundary conditions as if the medium was infinite, also known as radiative conditions. These conditions may be very complex to design if the medium is heterogeneous.

For the 1D case, we may simply say that

),)1((),(

),(),0(

1

21

c

xtxLutxLu

c

xtxutu

LL

which again is exactly verified for the magic step of characteristics. For other time steps, interpolation between t-t and t-2t.

In 2D and 3D, the shape of the wavefront must be introduced in an attempt for absorbing waves along boundaries and we shall see that other techniques rather radiative conditions may be considered (p-characteristics).

The Perfeclty Matched Layer concept turns out to be very efficient (Berenger, 1994).

ABC : PML conditions

On conserve des variables à intégrer qui suivent la propagation dans une direction

Energy balance

PML absorption is better than absorption by other methods at any angle of incidence (at the expense of a cost in time domain)

3D test of PML conditions

Left : finite box with Neuman conditions

Middle : PML

Right : difference between true solution and PML solution

STAGGERED GRID : A FATALITY

3D case

1D : Yes (for the moment!)

2D & 3D : No (one may use the spatial extension!)

Trick

Combine ?

FSG

X

Z

PSG

Saenger stencil

vx

vz

xx,zz

xz

New staggered grid

)(2

1

)(2

1

1,11,11,11,1

1,11,11,11,1

jijijiji

jijijiji

uuuuz

u

uuuux

u

Local coupling between x and z directions: new staggered grid and velocity components define at a single node (as for the stress). Expected better behaviour for the interaction with the free surface (it has been verified).

FSG versus PSG

PSG should be preferred when one needs all components at a single node (anisotropy, plasto-elastic formulation …)

NUMERICAL ANISOTROPY

PSG FSG

COMBINE ?

All you need is there•We have all ingredients for resolving partial differential equations in the FDTD domain.•Loop over time k = 1,n_max t=(k-1)*dt•Loop over stress field i=1,i_max x=(i-1)*dx

compute stress field from velocity field: apply stress boundary conditions; end•Loop over velocity field i=1,i_max x=(i-1)*dx

compute velocity field from stress field: apply velocity boundary conditions; end•Set external sources effects

compute by replacing OR by adding external values at specific points. If we replace, the input should be a solution of the wave equation.•End loop over time

Exercice : write the same organigram in the frequency domain.

Exercice : write a fortran program to solve the 1D equation (should be done in a WE).

COLLOCATION

• FD method : discrete equations exact at nodes (strong formulations)

• FE method : equations verified on the average over an element (to be defined with respect to nodes) (weak formulation)

• FV method : equations verified on the average over an volume (only flux between volumes)

COLLOCATION

FD dirac cumb

FE method : elements share nodes !

FV method : elements share edges !

FV method requires simpler meshing as well as simpler message communications …. Usually this is the standard extension of FD modeling in mechanics

Pseudo-flux conservative form

Finite volume method

Finite volume method

CONCLUSION

• Efficient numerical methods for propagating seismic waves

• Time integration versus frequency integration

• Competition between FE & FV for modelling

• FD an efficient tool for imaging

Propagation sismique dans la baie des anges

Seisme de magnitude 4.9 à 8 km de profondeur

THANKS YOU !