Waves in heterogeneous media: numerical implications

Jean Virieux

Professeur UJF

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

SEISMIC WAVE MODELING FOR SEISMIC IMAGING

(Very) large-scale problems.

Example of hydrocarbon explorationTarget (oil & gas exploration): 20 km x 20 km x 8 km

Maximum frequency: 20 Hz

Minimum P and S wave velocities: 1.5 km/s and 0.7 km/s

FD discretization in the acoustic approximation:

h=1500/20/4 ~18 m. Dimensions of the FD grid: 1110 x 1110 x 445

Number of degrees of freedom: 550.106 (for the scalar pressure wavefield P)

FD discretization in the elastic approximation:

h=700/20/4~9 m. Dimensions of the FD grid: 2220 x 2220 x 890

Number of degrees of freedom: 3x4.4.109=~2x12=13.109 (for the vectorial velocity wavefield Vx, Vy, Vz)

If the wave equation is solved in the frequency domain (implicit scheme), sparse linear system should be solved, the dimension of which is the number of unknowns.

Need of efficient parallel algorithms on large-scale distributed-memory platforms. Both time and memory management are key issues.

SEISMIC WAVE MODELING FOR SEISMIC IMAGING

(Very) large-scale problems.

Example of the global earthTarget (the earth: sphere of 6370-km radius

Maximum frequency: 0.2 Hz

Hexahedric meshing:

Number of degrees of freedom: 14.6 109 (for the vectorial velocity wavefield Vx, Vy, Vz)

Simulation length: 60 minutes (50000 time steps with a time interval of 72 ms)

Simulation on 1944 processors of the Eart Simulator (Japan):

2.5 Terabytes of memory.

MPI (Message Passing Intyterface) parallelism.

Performance: 5 Teraflops.

Wall-clock time for one simulation: 15 hours.

From D. Komatitsch, J. Ritsema and J. Tromp, The spectral-element method, Beowulf computing, and global seismology, Science, 298, 2002, p. 1737-1742.

SEISMIC WAVE MODELING FOR SEISMIC IMAGING

q Modeling in heterogeneous media: Need of sophisticated numerical schemes on unstructured meshes (finite-element-based method).

Example of a shallow-water target with soft sediments on the near surface.

Triangular meshing for Discontinuous Galerkin method

From Brossier et al. (2009)

SEISMIC WAVE MODELING FOR SEISMIC IMAGING

q Multi-r.h.s simulations in the context of large 3D surveys and non linear iterative optimization.

Building a model by full-waveform inversion requires at least to solve three times per inversion iteration the forward problem (seismic wave modeling) for each source of the acquisition survey.

Number of simulations

for a realistic full-waveform inversion case study

Let’s consider a coarse wide-aperture/wide-azimuth survey with a network of land stations.

DR=400 m. Number of sensors (processed as sources in virtue of reciprocity of Green functions: 50x50=2500 r.h.s.

The forward problem must be solved 2500 x 3=7500 per inversion iterations.

Few hundreds to few thounsands of inversion iterations may be required to converge towards final models.

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

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 w, wavelength l, 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 w 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

1

1 2( , ...., )n

D R R

DR R

with diag

)()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+lp 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 s

Finite-difference discretization

grid interval: h; time interval: t

Basis functions: Dirac comb

Physical justification of the FD method

Huygen’s principle

Each point of the FD grid acts as a secondary source. The envelope of the elementary diffractions provides the seismic response of the continuous medium if this latter is sufficiently-finelly discretized.

Exploding-reflector modeling

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 Dx,Dt 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.

Second-order accurate central FD approximation and staggered grids

EM (Yee, 1966); Earthquake (Madariaga, 1976), Wave (Virieux, 1986)

Higher-order accurate centred FD scheme: the 4th-order example

Second-order accurate Central-difference approximation

Leapfrog Second-order accurate Central-difference approximation

Leapfrog 2nd-order accurate central-difference approximation

Leapfrog 4th-order accurate central-difference approximation

Central-difference approximation of second partial derivative

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,mDt)

21( , ) ( , ) ( ) ( )

0 , 0i

vL ih m t ih m t O t O h

t x

L L when h t

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

Stability analysis

A numerical scheme is stable if it provides a bounded solution for a bounded source excitation.

A numerical scheme is unstable if it provides an unbounded solution for a bounded source excitation.

A numerical scheme is conditionnally stable if it is stable provided that the time step verifies a particular condition.

Stability

)exp(

)exp(

jkihtmjBT

jkihtmjAVm

i

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

w 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 integrationm+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 & Levy i

ii

i Ecwith

c

ht

.. Magic step Dt=h/c0

Characteristic line

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

Von Neuman stability study

)exp(

)exp(

jkihtmjBT

jkihtmjAVm

i

mi

Long-term stability

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

Long-term stability is difficult to analyze and comes from glass modes or numerical noise associated with finite discretisation Which could amplified constructively these coherent noises.

Finally

CONSISTENCE + STABILITY = CONVERGENCE

(not always to the physical solution)

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

v

UNCOUPLED 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 ordre10

h

5

h

acf

vh

10min

acf

v

2min

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

vE

xt

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

im

ii

mi

mi TT

hVV

t

1 / 2 3/ 2 1/ 2 1/ 2 1/ 21/ 2 1 1/ 2 12 2

1/ 21/ 2 1/ 2

1 1( 2 ) (

2 ( ))

m m m m mi i i i i i i

mi i i

V V V E V E Vt h

V E E

Do same thing for r

xE

x

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-Dt and t-2Dt.

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

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

ABC : PML conditions

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

sxx,szz

sxz

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

Application to the wave equation in 1D homogeneous media

Dispersion:

Dispersion is the variation of wavelength or wavenumber with frequency.

Exemple: f=5 Hz; c=4000m/s

T=0.2 s; l=800mexp(j .w t) exp(jk . x)

Dispersion relation, phase and group velocities

Source excitation

Second-order accurate FD discretization

Explicit time-marching algorithm

Explicit scheme: wavefield solution at position j and time n+1 in the left-hand side are inferred from the wavefield solutions at previous times in the right-hand side.

Implicit scheme: wavefield solutions at several positions (j=1,…) and time n+1 in the left hand side are inferred from the wavefield solutions at previous times in the righ hand side.

Memory storage: wavefield at 3 different times (out-of-place algorithm)

S=1 and the magic time step

Verification:

Dispersionless simulation - S=1

c=4000 m/s

Code df1d.2.f

Dispersive solution - S=0.5

c=4000 m/s

Code df1d.2.f

Dispersionless simulation - S=1

c=4000 m/s

Code df1d.2.f

Application to the 1D wave equation in heterogeneous media

Velocity-stress formulation of the wave equation

Second-order accurate staggered-grid discretization

The parsimonious second-order accurate staggered-grid discretization

Initial conditions

Boundary conditions

Reflection and transmission coefficients at an interface between two media

Z=rc is the impedance; the opposition of a medium to the propagation of an acoustic wave. It is given by the ratio between pressure and the local particle displacement velocity.

Free-surface condition Rigid condition

Initial and boundary conditions

Free surface – Particle velocity displacement

S=1 - Milieu homogène c=4000 m/s

Code df1d.2.f

Free surface – Stress field

S=1 - Milieu homogène c=4000 m/s

Code df1d.2.f

Rigid boundary – Particle velocity displacement

S=1 - Milieu homogène c=4000 m/s

Code df1d.2.f

Rigid boundary - Stress field

S=1 - Milieu homogène c=4000 m/s

Code df1d.2.f

Radiation condition

Condition de radiation - milieu homogène - S=1

c=4000 m/s

Code df1d.2.f

Code df1d.2.f

Condition de radiation - milieu homogène - S=0.5

c=4000 m/s

« Sponge » boundary conditions

Simulation in a two-layer medium

c1=2000 m/s c2=4000 m/s - S=1 in the high-velocity layer

Radiation boundary condition

Code df1d.3.f

Code df1d.4.f

Simulation in a two-layer medium

c1=2000 m/s c2=4000 m/s - S=1 in the high-velocity layer

Sponge boundary condition

Do l=1,Nsource !Loop over sources

Do i=1,Nx

v(i)=0;t(i)=0 !Initial conditions

End do

Do n=1,Nt !Loop over time steps

Do i=1,Nx !Loop over spatial steps

v(i)=v(i)+(b(i).dt/h)[t(i+1/2)-t(i-1/2)] !In-place update of v

End do

Implementation of boundary condition for v at t=(n+1).dt

v(is)=v(is)+f ! Application of source

Do i=1,Nx !Loop over spatial steps

t(i+1/2)=t(i+1/2)+(E(i+1/2).dt/h)[v(i+1)-v(i)] !In-place update of t

End do

Implémentation of boundary conditions for t at t=(n+3/2).dt

write v à t=(n+1).dt and t à t=(n+3/2).dt

End do

End do

Algorithm

- 1D velocity-stress wave equation with multiple rhs

Remarks:

q The time complexity linearily increases with the number of sources.

q In-place algorithm: the velocity and stress fields are stored in core only at one time.

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

Seismic propagation in the Angel Bay nearby Nice.

Magnitude 4.9 at a depth of 8 km

THANKS YOU !