1. Waves in heterogeneous media: numerical implications
Jean Virieux
Professeur UJF

2. 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 :
FDTD introduction :
ftp://ftp.seismology.sk/pub/papers/FDM-Intro-SPICE.pdf
By P. Moczo, J. Kristek and L. Halada

3. (Very) large-scale problems.
SEISMIC WAVE MODELING FOR SEISMIC IMAGING
(Very) large-scale problems.
Example of hydrocarbon exploration
Target (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: (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: 3x =~2x12= (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.

4. (Very) large-scale problems.
SEISMIC WAVE MODELING FOR SEISMIC IMAGING
(Very) large-scale problems.
Example of the global earth
Target (the earth: sphere of 6370-km radius
Maximum frequency: 0.2 Hz
Hexahedric meshing:
Number of degrees of freedom: (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

5. Triangular meshing for Discontinuous Galerkin method
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)

6. for a realistic full-waveform inversion case study
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.

7. ODE versus PDE formulations
Non-linear
Linear
O.D.E
Ordinary differential Equations
P.D.E
Partial Differential Equations
GOAL : find ways to transform differential operators into algebraic operators in order to use linear algebra at the end
Symmetry between space and time ?

8. An apparent easy way
Spectral methods allow to go directly to this algebraic structure
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

9. One-dimensional scalar wave
The scalar wave equation is verified by the vibration u(t,x)
Homogeneous medium
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
A plane wave is defined by
with the dispersion relation
The phase velocity is for any frequency
If the pulsation w depends on k, we have
and the group velocity is
which is identical to phase velocity for non-dispersive waves

10. First-order hyperbolic equation
stress
Let us define other variables for reducing the derivative order in both time and space
The 2nd order PDE became a 1st order PDE
velocity
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
with
Other choices are possible as displacement-stres instead of velocity-stress.

11. Characteristic variables
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.
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.
The transformation from w to f splits left and right propagating waves

12. Other PDE in physics
The scalar wave equation is a partial differential equation which belongs to second-order hyperbolic system.
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

13. Initial and boundary conditions
Initial conditions u(x,0)
1D string medium
Boundary conditions u(0,t)
Boundary conditions u(L,t)
Dirichlet conditions on u
Neumann conditions on s
f(x,t) Excitation condition
Difficult to see how to discretize the velocity !
Much better for handling heterogeneity

14. Finite-difference discretization
grid interval: h; time interval: t
Basis functions: Dirac comb

15. 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.

16. Exploding-reflector modeling

17. Finite Difference Stencil
Truncations errors :
(Leveque 1992)
Higher-order terms : same procedure but you need more and more points
Second derivative

18. Discretisation and Taylor expansion
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.

19. Second-order accurate central FD approximation and staggered grids
EM (Yee, 1966); Earthquake (Madariaga, 1976), Wave (Virieux, 1986)

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

21. Second-order accurate Central-difference approximation
Leapfrog Second-order accurate Central-difference approximation

22. Leapfrog 2nd-order accurate central-difference approximation
Leapfrog 4th-order accurate central-difference approximation

23. Central-difference approximation of second partial derivative

24. Other expansions
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)

25. Consistency Local error Taylor expansion around (ih,mDt)
FD scheme is consistent with the differential equations (do the same for the other equation)

26. 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.

27. Stability 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)

28. STABLE STENCIL :leap-frog integration
m+1 m
m-1
i i i+1
Harmonic analysis
is real
The solution does not grow with time : STABLE
CFL condition
Courant, Friedrichs & Levy
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

29. 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)

30. Time integration (more theory)
Euler
Backward Euler
Left-side (upwind)
Right-side
Lax-Friedrichs
Leapfrog
Lax-Wendroff
Beam-Warming

31. UNCOUPLED SUBGRID : SAVE MEMORY
RED-BLACK PATTERN
i-1 i
i+1
m-1 m
m+1
The staggered grid
ONLY BOUNDARY CONDITIONS WOULD HAVE COUPLED THEM
UNCOUPLED SUBGRID : SAVE MEMORY
STAGGERED GRID SCHEME
INDICE FORTRAN ?
Second-order in time & in space

32. NUMERICAL DISPERSION Moczo et al (2004)
How small should be h compared to the wavelength to be propagated ?
2ème ordre
4ème ordre

33. PARSIMONIOUS RULE
How to define these discrete values for an heterogeneous medium ?
(especially when considering strong discontinuities)
How to estimate the spatial operator
Do same thing for r

34. FREE SURFACE (Neumann condition)1 2
m-1 m
m+1
Amplitude deficit of wave nearby the free surface 1 2
m-1 m
m+1
We can see that we have amplified by a factor of 2
Antisymmetric stress

35. ESIM procedure SAT procedure 1 2 m-1 m m+11 2
m-1 m
m+1
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

36. Source or grid excitation
Impulsive source
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.
Known solution
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

37. Radiative boundaries
One 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
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).

38. ABC : PML conditions
On conserve des variables à intégrer qui suivent la propagation dans une direction

39. 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)

41. 3D test of PML conditions
Left : finite box with Neuman conditions
Middle : PML
Right : difference between true solution and PML solution

42. STAGGERED GRID : A FATALITY
1D : Yes (for the moment!)
2D & 3D : No (one may use the spatial extension!) X
Trick Z
3D case
FSG
PSG
Combine ?

43. Saenger stencil New staggered grid sxx,szz sxz vz vx
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).

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

45. NUMERICAL ANISOTROPY
PSG
FSG
COMBINE ?

46. 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).

47. 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)

48. 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

49. Finite volume method
Pseudo-flux conservative form

50. Finite volume method

51. Application to the wave equation in 1D homogeneous media

52. exp(jw . t) exp(jk . x) Dispersion:
Dispersion is the variation of wavelength or wavenumber with frequency.
Exemple: f=5 Hz; c=4000m/s
T=0.2 s; l=800m
exp(jw . t)
exp(jk . x)

53. Dispersion relation, phase and group velocities

54. Source excitation

55. Second-order accurate FD discretization

56. 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)

57. S=1 and the magic time step
Verification:

58. Dispersionless simulation - S=1
c=4000 m/s
Code df1d.2.f

59. Dispersive solution - S=0.5
c=4000 m/s
Code df1d.2.f

60. Dispersionless simulation - S=1
c=4000 m/s
Code df1d.2.f

61. Application to the 1D wave equation in heterogeneous media

62. Velocity-stress formulation of the wave equation

63. Second-order accurate staggered-grid discretization

64. The parsimonious second-order accurate staggered-grid discretization

65. Initial and boundary conditions
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

66. Free surface – Particle velocity displacement
S=1 - Milieu homogène c=4000 m/s
Code df1d.2.f

67. Free surface – Stress field S=1 - Milieu homogène c=4000 m/s
Code df1d.2.f

68. Rigid boundary – Particle velocity displacement
S=1 - Milieu homogène c=4000 m/s
Code df1d.2.f

69. Rigid boundary - Stress field S=1 - Milieu homogène c=4000 m/s
Code df1d.2.f

70. Radiation condition

71. Condition de radiation - milieu homogène - S=1
c=4000 m/s
Code df1d.2.f

72. Condition de radiation - milieu homogène - S=0.5
c=4000 m/s
Code df1d.2.f

73. « Sponge » boundary conditions

74. 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

75. Simulation in a two-layer medium
c1=2000 m/s c2=4000 m/s - S=1 in the high-velocity layer
Sponge boundary condition
Code df1d.4.f

76. - 1D velocity-stress wave equation with multiple rhs
Algorithm
- 1D velocity-stress wave equation with multiple rhs
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
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
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
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.

77. 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

78. Seismic propagation in the Angel Bay nearby Nice.
Magnitude 4.9 at a depth of 8 km

79. THANKS YOU !