waves in heterogeneous media: numerical implications
DESCRIPTION
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 - PowerPoint PPT PresentationTRANSCRIPT
Waves in heterogeneous media: numerical implications
Jean VirieuxProfesseur 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
Ayydtd
yAydtd
)(
Dyty
yDty
)(
O.D.EOrdinary differential Equations
P.D.EPartial Differential Equations
Linear
Non-linear
Symmetry between space and time ?
An apparent easy waySpectral methods allow to go directly to this algebraic structure
xuc
tu
2
22
2
2
xucu 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 strategyGo to the dual space only for computing spatial derivatives and goes back to the standard space for equations and conditionsFrequency approach of Pratt : the opposite way around
One-dimensional scalar wave
xuc
tu
2
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 l, wavenumber k and frequency f and period T. We have the following relations
ccf
cTk
l
222
A plane wave is defined by )(),( kxtietxu
The scalar wave equation is verified by the vibration u(t,x)
with the dispersion relation222 kc
The phase velocity is for any frequency ck
Vp
If the pulsation depends on k, we have kcdkd 2 and the group velocity is
cc
cdkdVg
.2
which is identical to phase velocity for non-dispersive waves
Homogeneous medium
First-order hyperbolic equation
tuv
xu
xv
t
xc
tv
2
xuE
xtu
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
xvE
t
xtv
1
Ec 2
with
stress
velocity
Other choices are possible as displacement-stres instead of velocity-stress.
Characteristic variables
1
1 2( , ...., )n
D R RDR Rwith diag l l l
)()0,(
0
0 xwxwxwD
tw
npxf
tf
xf
tf
wRf
pp
p ,...,1;0
0
1
l
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 l
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
l ..
00
..0
10
The transformation from w to f splits left and right propagating waves
Other PDE in physics
xu
tu
2
2
xuc
tu
2
22
2
2
ukxu 2
2
2
0
The scalar wave equation is a partial differential equation which belongs to second-order hyperbolic system.
xu
2
2
0
xu
tu
2
22
2
2
xu
tu
2
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
fxuc
tu
2
22
2
2
xvE
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 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)
centeredhUUUD
backwardhUUUD
forwardh
UUUD
iii
iii
iii
211
0
1
1
Truncations errors : 0h
Second derivative
iii UDUDDUDD 200
)2(1112
2iiii UUU
hUD
Higher-order terms : same procedure but you need more and more points
xux
xux
xux
xuxuuxxu
nininininini 4
44
,3
33
,2
22
,,,1 2462
)(D
D
D
DD
xux
xux
xux
xuxuuxxu
nininininini 4
44
,3
33
,2
22
,,,1 2462
)(D
D
D
DD
xux
xuxuuu
nininini 4
44
,2
22
,,1,1 122
D
D
Discretisation and Taylor expansion
)(2 2
2,,1,1
,2
2
xx
uuuxu ninini
niD
D
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
xvE
t
xtv
1
)(21)(1
)(211)(1
111
111
mi
mii
mi
mi
mi
mi
i
mi
mi
VVh
ETTt
TTh
VVt
D
D
Local error
),(1),(
)(211)(1
111
tmihx
tmihtvL
TTh
VVt
L
i
mi
mi
i
mi
mi
D
D
D
D
Taylor expansion around (ih,mDt)21( , ) ( , ) ( ) ( )
0 , 0i
vL ih m t ih m t O t O ht x
L L when h t
D
D
D D D
D
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
D
D
khjAh
tEtjB
khjBh
ttjA
i
i
sin22
1)exp(
sin22
1)exp(
DD
DD
222 )(sin)()1)(exp( khhtEtj
i
i DD
1sin)(1)exp( 2/1 D
D khhtEjtj
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 integrationm+1mm-1
i-1 i i+1)(
21)(
21
)(211)(
21
1111
1111
mi
mii
mi
mi
mi
mi
i
mi
mi
VVh
ETTt
TTh
VVt
D
D
Harmonic analysis
khjAh
tEtBj
khjBhttAj
i
i
sin2sin2
sin2sin2
DD
DD
khhtEt
khhtEt
i
i
i
i
sin)(sin
)(sin)()(sin
2/1
222
DD
DD
thtE
i
i DD
sin1)( 2/1
is realThe solution does not grow with time : STABLE
CFL conditionCourant, Friedrichs & Levy i
ii
i Ecwith
cht
D .. 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
D
D
Long-term stabilityLocal 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
uuhkauu
02
11
111
ni
nin
ini
uuhkauu
02
11
11
ni
nin
ini
uuhkauu
02
1111
ni
nin
ini
uuhkauu
022
1 1111
1
ni
nin
ini
ni
uuhkauuu
0)2(22 11
22
2111
ni
ni
ni
ni
nin
ini uuua
hkuu
hkauu
02
1111
ni
nin
ini
uuhkauu
0)2(22
4321
22
2211
ni
ni
ni
ni
ni
nin
ini uuua
hkuuu
hkauu
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
D
D
mi
mi
imi
mi
mi
mi
i
mi
mi
VVh
ETTt
TTh
VVt
2sin)()
2sin( 2/12/1 kh
htEt
i
i D
D
Second-order in time & in spaceINDICE FORTRAN ?
NUMERICAL DISPERSION
Moczo et al (2004)
22sin
22sin
khkh
tt
D
D
02/12/1 )( cE
k i
i
How small should be h compared to the wavelength to be propagated ?
2/120
0
0
))sin(1(
cos
)sinarcsin(
ll
ll
hhtc
hc
kv
hhtc
htkh
kc
gridg
grid
D
DD
2ème ordre 4ème ordre10l
h5l
h
acfvh
10min
acfv2
min
PARSIMONIOUS RULE
))2/1(()(
2/11 hiEihi
How to define these discrete values for an heterogeneous medium ?(especially when considering strong discontinuities)
xvE
t
xtv
1
xvE
xtv
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
D
D
mi
mi
i
imi
mi
i
i
mi
mi
i
mi
mi
VVh
EVVh
E
TTth
VVt
)(11)(12/12/1
2/12/1 mi
mi
i
mi
mi TT
hVV
t
D
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 hV E E
D Do same thing for
xE
x
2/1
2/12/1
2/1
12/)(
1
i
ii
i
EiEEi
Ei
1
1
i
i
i
FREE SURFACE (Neumann condition)
0 1 2m-1
m
m+1
v
)(11)(102/3
2/11
2/11 TT
hVV
tm
i
mm D
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
D
We can see that we have amplified by a factor of 2Antisymmetric stress
ESIM procedure
0 1 2m-1
m
m+1
vPredict 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)(12/52/3
2/11
2/11
mm
a
mm TTh
VVt
D
12/1 a
Source or grid excitation
fxuc
tu
2
22
2
2
ni
ni
ni
ni
ni
ni
ftuu
ftuu
2/12
11
2/12
000
000
D
D
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
cxtxLutxLu
cxtxutu
LLD
DD
DD
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 conditionsLeft : finite box with Neuman conditionsMiddle : PMLRight : 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
uuuuzu
uuuuxu
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 timeExercice : 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/sT=0.2 s; l=800m
exp(j. 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=1c=4000 m/s
Code df1d.2.f
Dispersive solution - S=0.5c=4000 m/s
Code df1d.2.f
Dispersionless simulation - S=1c=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 conditionsReflection and transmission coefficients at an interface between two media
Z=c 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 displacementS=1 - Milieu homogène c=4000 m/s
Code df1d.2.f
Free surface – Stress fieldS=1 - Milieu homogène c=4000 m/s
Code df1d.2.f
Rigid boundary – Particle velocity displacementS=1 - Milieu homogène c=4000 m/s
Code df1d.2.f
Rigid boundary - Stress fieldS=1 - Milieu homogène c=4000 m/s
Code df1d.2.f
Radiation condition
Condition de radiation - milieu homogène - S=1c=4000 m/s
Code df1d.2.f
Code df1d.2.f
Condition de radiation - milieu homogène - S=0.5c=4000 m/s
« Sponge » boundary conditions
Simulation in a two-layer mediumc1=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 mediumc1=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;(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)[(i+1/2)-(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
(i+1/2)=(i+1/2)+(E(i+1/2).dt/h)[v(i+1)-v(i)] !In-place update of
End do
Implémentation of boundary conditions forat t=(n+3/2).dt
write v à t=(n+1).dt and 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 !