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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
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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
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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
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ANATOMY OF GLOBAL-OFFSET DATA
Velocity gradient at interfaces : diving waves
Anatomy of seismic waves phases
From Stéphane Operto
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Anatomy of global-offset seismograms:Continuous sampling of apertures from transmission to reflection
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Critical incidence – total reflection
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Upgoing conic wave
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Critical distance
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Interface wave
Conic wave
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Root wave
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Asymptotic « convergence » between direct and super-critical reflected waves
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Diving wave
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Synthetic seismograms
Head or conic wave
Diving Wave
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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 !
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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.
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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.
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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
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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
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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.
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),(),(),(
)(
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
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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.
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Les solutions en milieu homogène
Solution 1D Solution 2D Solution 3D
Certaines caractéristiques communes mais d’autres très différentes comme la trainée à 2D
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The corner-edge as a complex example
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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 ?
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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
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3D Elasto-dynamic equations
Divide by the density will leave medium properties only on the RHS
The previous PDE form is then retrieved
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P-SV equations
Elastic properties
No attenuation
Medium properties vary from point to point
No spatial derivatives of these medium properties
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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
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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.
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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
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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
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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
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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
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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.
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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)
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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)
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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)
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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
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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
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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 ?
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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
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NUMERICAL ANISOTROPY
PSG FSG
COMBINE ?
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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
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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
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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
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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
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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).
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ABC : PML conditions
On conserve des variables à intégrer qui suivent la propagation dans une direction
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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)
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3D test of PML conditions
Left : finite box with Neuman conditions
Middle : PML
Right : difference between true solution and PML solution
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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
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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).
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FSG versus PSG
PSG should be preferred when one needs all components at a single node (anisotropy, plasto-elastic formulation …)
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NUMERICAL ANISOTROPY
PSG FSG
COMBINE ?
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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).
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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)
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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
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Pseudo-flux conservative form
Finite volume method
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Finite volume method
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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
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Propagation sismique dans la baie des anges
Seisme de magnitude 4.9 à 8 km de profondeur
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THANKS YOU !