Non-Periodic Homogenization for Elastic Wave Propagation in Complex Media
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Non-periodic homogenization for elastic wave propagation in complex media Y. Capdeville 1, L. Guillot 2, J.J. Marigo 3 1Laboratoire de Planétologie et de Géodynamique de Nantes, CNRS, France 2Institut de Physique du Globe de Paris, France, 3Laboratoire de Mécanique des Solides, Ecole Polytechnique, France QUEST rst workshop, 22/09/2010 IPGP INSTITUT DE PHYSIQUE DU GLOBE DE PARIS Y. Capdeville , L. Guillot , J.J. Marigo QUEST rst workshop, 22/09/2010 1 Introduction and motivations Introduction and motivations The earth contains inhomogeneities at all scales. ? Scale separation Observations indicate that waves of a given wavelength are sensitive to inhomogeneities of scales much smaller than this wavelength only in an eective way. This is one of the reasons why seismic waves can be used to image the earth. Y. Capdeville , L. Guillot , J.J. Marigo QUEST rst workshop, 22/09/2010 2 Introduction and motivations Introduction and motivations Inverse problem, full waveform inversion (FWI) FWI can only be performed in a limited frequency band. The best that can be recovered is what is seen by the waveeld. It is an eective version (ρ∗; c∗) of the real earth (ρ, c). in that case, we know (at best) (ρ∗; c∗), but we wish to access to (ρ, c). Forward modeling, full waveform modeling full waveform modeling can only be performed in a limited frequency band. In many cases the elastic model (ρ, c) we which to propagate in contains details much smaller than the minimum wavelength. The eective model (ρ∗; c∗) would be enough to propagate the waveeld. In that case, we know (ρ, c), but we would like to have (ρ∗; c∗). Y. Capdeville , L. Guillot , J.J. Marigo QUEST rst workshop, 22/09/2010 3 Introduction and motivations Small scales are an issue for the direct problem : e.g. SEM Two simple but problematic SEM meshes : Global scale Regional scale Mortar PREM 6 6.30 4 densite 6.05 rayon (10^6 m) Vp 2 Vs 0 5.80 gure E. Delavaud 0 5 10 15 0 5 10 Y. CapdevilleVitesse (km/s), densite , L. (1e3Guillot kg/m3) , J.J. Marigo QUEST rst workshop, 22/09/2010 4 Introduction and motivations Introduction and motivations In both cases (Full waveform modeling and inversion) knowing the relation between (ρ, c) and (ρ∗; c∗) would be very useful, and we know since the work of Backus (1962) that the eective medium is not just a low pass ltered version of the original medium. Backus (1962) shown that, form a nely layered media (1-D media), described by the A; C; F ; L; N elastic parameters for TI media, the eective parameters A∗; C ∗; F ∗; L∗; N∗ can be computed as : F ∗2 F 2 1 1 A∗ − = hA − i = h i C ∗ C C ∗ C F ∗ F 1 1 = h i = h i C ∗ C L∗ L N∗ = hNi For higher dimension (2-D, 3-D), the eect of small scales has been studied for long in mechanics with the two scale homogenization of periodic media (e.g. Auriault & Sanchez-Palencia (1977), Sanchez-Palencia (1980) ...) . Y. Capdeville , L. Guillot , J.J. Marigo QUEST rst workshop, 22/09/2010 5 Homogenization 1-D Homogenization 2-D A simple periodic case : wave in a 1-D bar y λm Assumptions : λm : minimum wavelength E elastic modulus and ρ density 0 x ` periodic " = ` << 1 λm 12 11 Wave equation for the set of 10 9 problems (parametrized by "): 8 7 " " " " ρ @tt u − @x σ = f 6 10^3 kg/m^3, km/s " " " 5 σ = E @x u 4 0 0.01 0.02 0.03 0.04 0.05 x (m) Y. Capdeville , L. Guillot , J.J. Marigo QUEST rst workshop, 22/09/2010 6 Homogenization 1-D Homogenization 2-D Classical two scale expansion x 1 Introduction of the fast variable y = " 2 Introduction of ρ(y) = ρ"("y) and E(y) = E "("y) 3 As " ! 0 y and x are treated as independent variables implying @ @ 1 @ @x ! @x + " @y 4 Solutions thought as X i i x u"(x; t) = " u (x; y = ; t) " i≥0 X i i x σ"(x; t) = " σ (x; y = ; t) " i≥−1 5 Series of equation to be solved for each i : i i i+1 i ρ∂tt u + @x σ + @y σ = f i i i+1 σ = E(@x u + @y u ) Y. Capdeville , L. Guillot , J.J. Marigo QUEST rst workshop, 22/09/2010 7 Homogenization 1-D Homogenization 2-D Classical two scale expansion : resolution Cell average : 1 Z λm hhi(x) = h(x; y)dy λm 0 Solving the asymptotic equations one by one gives 1 0 0 0 0 1 σ− = 0, u = hu i, σ = hσ i 1 1 0 1 2 u (x; y) = χ (y)@x u (x) + hu i(x) χ1(y) periodic with hχ1i = 0 solution of cell problem 1 @y E(1 + @y χ ) = 0 0 0 3 u and σ of the order 0 eective wave equation 0 0 0 0 ∗ 0 hρi@tt u − @x σ = f σ = E @x u ∗ 1 with E = hE(1 + @y χ )i 4 ∗ 1 −1 analytical solution of the cell problem gives : E = h E i Y. Capdeville , L. Guillot , J.J. Marigo QUEST rst workshop, 22/09/2010 8 Homogenization 1-D Homogenization 2-D Classical two scale expansion : summary We have found that : " 0 1 1 0 u = hu i + " hu i + χ @x hu i + ::: " 0 1 1 0 σ = hσ i + " hσ i + χσ@x hσ i + ::: χ1 is the rst order periodic corrector with hχ1i = 0 solution of cell problem 1 @ E(1 + @y χ ) = 0 u = hu0i + "hu1i, σ = hσ0i + "hσ1i are solution of the eective equation (here at the order 1) : ∗ hρi@tt u − @x σ = f ; σ = E @x u ∗ 1 with E = hE(1 + @y χ )i is the eective elastic parameter. The nal solution can be obtained with " 1 2 u = u(1 + εχ @x ) + O(" ) " 1 = (1 + @y χ ) + O(") Y. Capdeville , L. Guillot , J.J. Marigo QUEST rst workshop, 22/09/2010 9 Homogenization 1-D Homogenization 2-D Classical two scale expansion : summary Note that, in this simple 1D case, there is an analytical solution to the cell problem leading to 1=E ∗ = h1=Ei. There is not such an analytical solution for higher dimensions. at the order 0 the solutions do not depend on the microscopic scale. This is still true for higher dimension for u0 but not for σ0 ; order > 0 : the boundary condition changes (e.g. Neumann condition becomes DtN) ; order > 1 : the eective equation changes (it is not a classical wave equation anymore) ; Y. Capdeville , L. Guillot , J.J. Marigo QUEST rst workshop, 22/09/2010 10 Homogenization 1-D Homogenization 2-D An example 1 a 0.02 reference b order 0 order 0 residual E average 0.01 0 0 -1 -0.01 2 3 4 5 4 4.1 4.2 0.006 c 0.006 d order0.004 1 residual 0.004 eps=0.15 partial order 2 residual eps=0.075, amp X 4 0.002 0.002 0 0 -0.002 -0.002 -0.004 -0.004 4.05 4.1 4.15 4 4.1 4.2 x (m) x (m) E average means computed with E∗ = hEi Y. Capdeville , L. Guillot , J.J. Marigo QUEST rst workshop, 22/09/2010 11 E (x) 6 periodic homogeniza- 5.5 tion on the whole do- km/s main ? 5 =) 4.5 0 1 2 3 4 5 x (m) homogeneous in random1 0 1 2 time (ms) Homogenization 1-D Homogenization 2-D Non-periodic case What can be done in the non periodic case ? E (x) 6 5.5 km/s 5 4.5 0 1 2 3 4 5 x (m) 1 1.1 Y. Capdeville , L. Guillot , J.J. Marigo QUEST rst workshop, 22/09/2010 12 E (x) 6 5.5 km/s 5 4.5 0 1 2 3 4 5 x (m) homogeneous in random1 0 1 2 time (ms) Homogenization 1-D Homogenization 2-D Non-periodic case What can be done in the non periodic case ? E (x) 6 5.5 periodic homogeniza- tion on the whole do- km/s 5 main ? =) 4.5 0 1 2 3 4 5 x (m) 1 1.1 Y. Capdeville , L. Guillot , J.J. Marigo QUEST rst workshop, 22/09/2010 13 E (x) 6 5.5 km/s 5 4.5 0 1 2 3 4 5 x (m) homogeneous in random1 0 1 2 time (ms) Homogenization 1-D Homogenization 2-D Non-periodic case What can be done in the non periodic case ? E (x) 6 5.5 periodic homogeniza- tion on the whole do- km/s 5 main ? =) 4.5 0 1 2 3 4 5 x (m) 1 1.1 Y. Capdeville , L. Guillot , J.J. Marigo QUEST rst workshop, 22/09/2010 14 Homogenization 1-D Homogenization 2-D Non-periodic case : an intuitive solution A dierent spatial lter : "0 h x R h x 0 w x x 0 dx 0 F ( )( ) = ( ) "0 ( − ) An example of wavelet w : "0 Intuitive order 0 solution, using : λcuto 1 1 "0 = (x) = F "0 (x) λmin E ∗ E Y. Capdeville , L. Guillot , J.J. Marigo QUEST rst workshop, 22/09/2010 15 Homogenization 1-D Homogenization 2-D Testing this intuitive idea E (x) 6 6 km/s 5 5 0 1 2 3 4 5 x (m) 1 1.1 random1 random1 f1 0 0.5 1 1.5 2 time (ms) Y.