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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.

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Presentation on theme: "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."— Presentation transcript:

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

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

3 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

4 ANATOMY OF GLOBAL-OFFSET DATA Velocity gradient at interfaces : diving waves Anatomy of seismic waves phases From Stéphane Operto

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




9 Critical incidence – total reflection


11 Upgoing conic wave

12 Critical distance



15 Interface wave Conic wave


17 Root wave

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

19 Diving wave

20 Synthetic seismograms Head or conic wave Diving Wave


22 LA PROPAGATION DES ONDES I Tenseur de déformationà partir du déplacement Tenseur de contrainte exprime les forces internes (séismes) Le PFD en présence de forces La loi de Hookeavec les coeffs. élastiques Léquationest dite léquation de lélastodynamique Une rhéologie simple pour des milieux LHI en fonction des coefficients de Lamé !

23 LA PROPAGATION DES ONDES II Léquation élastodynamique en milieu linéaire et élastique en milieu linéaire, élastique. Cest un système de 3 équations du second ordre aux dérivées partielles définissant les composantes u i (x,t). Un système à 9 équations peut aussi être construit à partir des vitesses et des contraintes : où la fonction m ij est non nulle dans les régions sources.

24 LA PROPAGATION DES ONDES III Léquation élastodynamique en milieu linéaire, élastique et isotrope sécrit On utilisera f i ou f i - m ij suivant les besoins. On parlera de systèmes de forces équivalents.

25 LA PROPAGATION DES ONDES IV Dans des milieux liquides, on préfère travailler avec la pression et la vitesse des particules où la fonction q(x,t) sappelle la source volumique en vitesse et est définie par On en déduit les équations donde acoustique

26 LA PROPAGATION DES ONDES V Si on élimine la vitesse des particules, on obtient léquation donde acoustique scalaire pour la pression p(x,t) : avec Si on suppose que la masse volumique est homogène, on a avec qui est léquation donde scalaire que lon retrouve dans différents livres

27 LA PROPAGATION DES ONDES VI Si on élimine la pression, on obtient léquation donde acoustique vectorielle avec la force en vitesse suivante Cette équation est un cas particulier de léquation dérivée de léquation élastodynamique. En général, on ne létudie pas séparément et on ne considère que léquation donde acoustique scalaire.

28 Considérons léquation donde scalaire Si les termes sont nuls, alors lexcitation peut se déduire dun terme excitation en divergence : que nous pouvons mettre sous une forme vectorielle

29 LES FONCTIONS DE GREEN 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 limpulsion est donnée en t0,.tandis que lon calcule la solution au point x et au temps t.

30 Les solutions en milieu homogène Solution 1D Solution 2D Solution 3D Certaines caractéristiques communes mais dautres très différentes comme la trainée à 2D

31 The corner-edge as a complex example

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

33 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

34 3D Elasto-dynamic equations Divide by the density will leave medium properties only on the RHS The previous PDE form is then retrieved

35 P-SV equations Elastic properties No attenuation Medium properties vary from point to point No spatial derivatives of these medium properties

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

37 First-order hyperbolic equation 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 with stress velocity Other choices are possible as displacement-stres instead of velocity-stress.

38 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 f p The curve x 0 + 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. The transformation from w to f splits left and right propagating waves

39 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

40 Initial and boundary conditions Boundary conditions u(0,t) Initial conditions u(x,0) Boundary conditions u(L,t) 1D string medium 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

41 Finite Difference Stencil i-1ii+1 (Leveque 1992) Truncations errors : Second derivative Higher-order terms : same procedure but you need more and more points

42 Discretisation and Taylor expansion 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.

43 Other expansions e i (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)

44 Consistency Local error Taylor expansion around (ih,m t) FD scheme is consistent with the differential equations ( do the same for the other equation )

45 Stability 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 )

46 STABLE STENCIL :leap-frog integration m+1 m m-1 i-1 i i+1 Harmonic analysis is real The solution does not grow with time : STABLE CFL condition Courant, Friedrichs & Levy Magic step t=h/c 0 Characteristic line The time step cannot be larger than the time necessary for propagating over h Von Neuman stability study

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


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


51 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

52 FREE SURFACE (Neumann condition) 0 12 m-1 m m+1 Amplitude deficit of wave nearby the free surface 0 12 m-1 m m+1 We can see that we have amplified by a factor of 2 Antisymmetric stress

53 ESIM procedure 0 12 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

54 Source or grid excitation 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

55 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- t and t-2 t. 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).

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

57 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)


59 3D test of PML conditions Left : finite box with Neuman conditions Middle : PML Right : difference between true solution and PML solution

60 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

61 Saenger stencil vxvx vzvz xx, zz xz New staggered grid 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).

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


64 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).

65 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)

66 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

67 Pseudo-flux conservative form Finite volume method


69 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

70 Propagation sismique dans la baie des anges Seisme de magnitude 4.9 à 8 km de profondeur


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