Wave propagation in solid medium in time UMR Géosciences Azur – CNRS – IRD – UPMC - UNSA Wave propagation in solid medium in time By J. Virieux and S. Operto Ecole thématique CNRS-CGG-UNSA SEISCOPE 11-15 septembre 2006

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

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

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

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

Critical incidence – total reflection

Upgoing conic wave

Critical distance

Conic wave Interface wave

Root wave

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

Diving wave

Synthetic seismograms Diving Wave Head or conic wave

LA PROPAGATION DES ONDES I Tenseur de contrainte exprime les forces internes (séismes) Tenseur de déformation à partir du déplacement en présence de forces Le PFD La loi de Hooke avec les coeffs. élastiques L’équation est dite l’équation de l’élastodynamique Une rhéologie simple pour des milieux LHI en fonction des coefficients de Lamé !

LA PROPAGATION DES ONDES II L’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.

LA PROPAGATION DES ONDES III L’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.

LA PROPAGATION DES ONDES IV Dans des milieux liquides, on préfère travailler avec la pression et la vitesse des particules On en déduit les équations d’onde acoustique où la fonction q(x,t) s’appelle la source volumique en vitesse et est définie par

LA PROPAGATION DES ONDES V Si 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

LA PROPAGATION DES ONDES VI Si 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.

Considérons l’équation d’onde scalaire Si les termes sont nuls, alors l’excitation peut se déduire d’un terme excitation en divergence : que nous pouvons mettre sous une forme vectorielle

LES FONCTIONS DE GREEN 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. où c(x) est la vitesse et la distribution dirac est notée par d et peut se voir comme une fonction de valeur infinie en zéro.

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

The corner-edge as a complex example

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 ?

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

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

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

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

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.

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

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

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

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

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.

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)

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

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)

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 Dt=h/c0 Characteristic line The time step cannot be larger than the time necessary for propagating over h Von Neuman stability study

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

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

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

NUMERICAL ANISOTROPY PSG FSG COMBINE ?

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

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

ESIM procedure SAT procedure 1 2 m-1 m m+1 1 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

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

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

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

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

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).

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

Finite volume method Pseudo-flux conservative form

Finite volume method

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

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

THANKS YOU !