J. Achenbach, Wave Propagation in Elastic Solids, Journal of Applied Mechanics, vol.41, issue.2, 2012.
DOI : 10.1115/1.3423344

R. A. Adams and J. J. Fournier, Sobolev spaces, 2003.

S. Adjerid, A posteriori finite element error estimation for second-order hyperbolic problems Computer methods in applied mechanics and engineering, pp.4699-4719, 2002.

D. Aggelis, Numerical simulation of surface wave propagation in material with inhomogeneity: Inclusion size effect, NDT & E International, vol.42, issue.6, pp.558-563, 2009.
DOI : 10.1016/j.ndteint.2009.04.005

M. Ainsworth, P. Monk, and W. Muniz, Dispersive and Dissipative Properties of Discontinuous Galerkin Finite Element Methods for the Second-Order Wave Equation, Journal of Scientific Computing, vol.15, issue.2, pp.1-35, 2006.
DOI : 10.1007/978-3-662-04823-8

M. Ainsworth and J. T. Oden, A posteriori error estimation in finite element analysis, 2011.

K. Aki and P. G. Richards, Quantative seismology: Theory and methods, p.801, 1980.

K. Aki and R. Wu, Scattering and attenuation of seismic waves, 1988.

G. R. Arce and S. R. Hasan, Elimination of interference terms of the discrete Wigner distribution using nonlinear filtering, IEEE Transactions on Signal Processing, vol.48, issue.8, pp.482321-2331, 2000.
DOI : 10.1109/78.852013

D. Aubry, D. Lucas, and B. Tie, Adaptive strategy for transient/coupled problems applications to thermoelasticity and elastodynamics, Computer Methods in Applied Mechanics and Engineering, vol.176, issue.1-4, pp.1-441, 1999.
DOI : 10.1016/S0045-7825(98)00329-6

F. Auger, P. Flandrin, P. Gonçalvès, and O. Lemoine, Time-frequency toolbox, p.46, 1996.

T. Baba, Time-Frequency Analysis Using Short Time Fourier Transform, The Open Acoustics Journal, vol.5, issue.1, 2012.
DOI : 10.2174/1874837601205010032

I. Babu?ka and W. C. Rheinboldt, A-posteriori error estimates for the finite element method, International Journal for Numerical Methods in Engineering, vol.15, issue.10, pp.1597-1615, 1978.
DOI : 10.1002/nme.1620121010

I. Babuska, J. Whiteman, and T. Strouboulis, Finite elements: an introduction to the method and error estimation, 2010.

, References

X. Bai, Finite Element Modeling of Ultrasonic Wave Propagation in Polycrystalline Materials, 2017.
URL : https://hal.archives-ouvertes.fr/tel-01483701

X. Bai, B. Tie, J. Schmitt, A. , and D. , Accepted for publication Finite element modeling of grain size effects on the ultrasonic microstructural noise backscattering in polycrystalline materials, Ultrasonics, p.23, 2018.

G. Bal, Kinetics of scalar wave fields in random media, Wave Motion, vol.43, issue.2, pp.132-157, 2005.
DOI : 10.1016/j.wavemoti.2005.08.002

G. Bal, J. B. Keller, G. Papanicolaou, R. , and L. , Transport theory for acoustic waves with reflection and transmission at interfaces, Wave Motion, vol.30, issue.4, pp.303-327, 1999.
DOI : 10.1016/S0165-2125(99)00018-9

G. Bal, T. Komorowski, R. , and L. , Self-Averaging of Wigner Transforms in Random Media, Communications in Mathematical Physics, vol.17, issue.1-2, pp.81-135, 2003.
DOI : 10.1007/BF01014347

G. Bal and O. Pinaud, Accuracy of transport models for waves in random media, Wave Motion, vol.43, issue.7, pp.561-578, 2006.
DOI : 10.1016/j.wavemoti.2006.05.005

P. K. Banerjee and R. Butterfield, Boundary element methods in engineering science, 1981.

W. Bangerth and R. Rannacher, Finite element approximation of the acoustic wave equation: Error control and mesh adaptation, East West Journal of Numerical Mathematics, vol.7, issue.4, pp.263-282, 1999.

R. E. Bank and A. Weiser, Some a posteriori error estimators for elliptic partial differential equations, Mathematics of Computation, vol.44, issue.170, pp.283-301, 1985.
DOI : 10.1090/S0025-5718-1985-0777265-X

I. Baydoun, D. Baresch, R. Pierrat, and A. Derode, Scattering mean free path in continuous complex media: Beyond the Helmholtz equation, Physical Review E, vol.92, issue.3, p.92033201, 2015.
DOI : 10.1121/1.1689960

I. Baydoun, É. Savin, R. Cottereau, D. Clouteau, and J. Guilleminot, Kinetic modeling of multiple scattering of elastic waves in heterogeneous anisotropic media, Wave Motion, vol.51, issue.8, pp.511325-1348, 2014.
DOI : 10.1016/j.wavemoti.2014.08.001
URL : https://hal.archives-ouvertes.fr/hal-01083250

M. Bebendorf, A Note on the Poincar?? Inequality for Convex Domains, Zeitschrift f??r Analysis und ihre Anwendungen, 2003.
DOI : 10.4171/ZAA/1170

A. Bedford and D. Drumheller, Elastic wave propagation, pp.151-165, 1994.

A. Bergam, C. Bernardi, and Z. Mghazli, A posteriori analysis of the finite element discretization of some parabolic equations, Mathematics of Computation, vol.74, issue.251, pp.1117-1138, 2005.
DOI : 10.1090/S0025-5718-04-01697-7
URL : https://hal.archives-ouvertes.fr/hal-00020615

M. Bieterman and I. Babu?ka, The finite element method for parabolic equations, Numerische Mathematik, vol.18, issue.3, pp.373-406, 1982.
DOI : 10.1007/978-3-642-65393-3

J. Blitz and G. Simpson, Ultrasonic methods of non-destructive testing, 1995.

P. Bloomfield, Fourier analysis of time series: an introduction, 2004.
DOI : 10.1002/0471722235

B. Boashash, Time-frequency signal analysis and processing: a comprehensive reference, 2015.

D. M. Boore, Finite Difference Methods for Seismic Wave Propagation in Heterogeneous Materials, Methods in computational physics, vol.11, pp.1-37, 1972.
DOI : 10.1016/B978-0-12-460811-5.50006-4

C. Boutin, Rayleigh scattering of acoustic waves in rigid porous media, The Journal of the Acoustical Society of America, vol.122, issue.4, pp.1888-1905, 2007.
DOI : 10.1121/1.2756755
URL : https://hal.archives-ouvertes.fr/hal-00943752

R. N. Bracewell and R. N. Bracewell, The Fourier Transform and Its Applications, American Journal of Physics, vol.34, issue.8, 1986.
DOI : 10.1119/1.1973431

A. Brougois, M. Bourget, P. Lailly, M. Poulet, P. Ricarte et al., Marmousi, model and data, EAEG Workshop, Practical Aspects of Seismic Data Inversion, 1990.
DOI : 10.3997/2214-4609.201411190

N. Burq and R. Joly, Exponential decay for the damped wave equation in unbounded domains, Communications in Contemporary Mathematics, vol.70, issue.06, p.181650012, 2016.
DOI : 10.1090/gsm/138
URL : https://hal.archives-ouvertes.fr/hal-01058120

Y. Capdeville, L. Guillot, and J. Marigo, 1-D non-periodic homogenization for the seismic wave equation, Geophysical Journal International, vol.25, issue.2, pp.897-910, 2010.
DOI : 10.1017/S0308210500027050
URL : https://hal.archives-ouvertes.fr/hal-00490534

K. Charles, Mother Earth gets undressed, Nature, 2008.
DOI : 10.1038/news.2008.1001

P. G. Ciarlet, The finite element method for elliptic problems, Classics in applied mathematics, vol.40, pp.1-511, 2002.

J. L. Codona, Electromagnetic wave propagation through random media, 1985.

L. Cohen, Time-frequency distributions-a review, Proceedings of the IEEE, pp.941-981, 1989.
DOI : 10.1109/5.30749

J. Corones, R. Dougherty, and H. Mcmaken, A New Parabolic Approximation to the Helmholtz Equation, Review of Progress in Quantitative Nondestructive Evaluation, pp.123-131, 1984.
DOI : 10.1007/978-1-4684-1194-2_11

R. Cottereau, L. Chamoin, and P. Diez, Strict error bounds for linear and nonlinear solid mechanics problems using a patch-based flux-free method, M??canique & Industries, vol.150, issue.3-4, pp.3-4249, 2010.
DOI : 10.1016/S0045-7825(97)00086-8

R. Cottereau and P. Diez, Fast r-adaptivity for multiple queries of heterogeneous stochastic material fields. Computational mechanics, pp.601-612, 2015.
URL : https://hal.archives-ouvertes.fr/hal-01200721

B. Deka, A priori L ??? ( L 2 ) error estimates for finite element approximations to the wave equation with interface, Applied Numerical Mathematics, vol.115, pp.142-159, 2017.
DOI : 10.1016/j.apnum.2017.01.004

L. Demkowicz, J. T. Oden, W. Rachowicz, and O. Hardy, Toward a universal adaptive finite element strategy, part 1. Constrained approximation and data structure, Computer Methods in Applied Mechanics and Engineering, vol.77, issue.1-2, pp.79-112, 1989.
DOI : 10.1016/0045-7825(89)90129-1

A. Dhia, E. Duclairoir, G. Legendre, and J. Mercier, Time-harmonic acoustic propagation in the presence of a shear flow, Journal of Computational and Applied Mathematics, vol.204, issue.2, pp.428-439, 2007.
DOI : 10.1016/j.cam.2006.02.048
URL : https://hal.archives-ouvertes.fr/hal-00876232

S. Dhua and A. Chattopadhyay, Wave propagation in heterogeneous layers of the Earth, Waves in Random and Complex Media, pp.626-641, 2016.
DOI : 10.1017/CBO9780511610127

P. Díez, J. Joséegozcue, and A. Huerta, A posteriori error estimation for standard finite element analysis, Computer Methods in Applied Mechanics and Engineering, vol.163, issue.1-4, pp.1-4141, 1998.
DOI : 10.1016/S0045-7825(98)00009-7

P. Díez, N. Parés, and A. Huerta, Error Estimation and Quality Control, Encyclopedia of Aerospace Engineering, vol.101, issue.4-6, 2010.
DOI : 10.1090/S0025-5718-1985-0777265-X

, References

E. Dormy and A. Tarantola, Numerical simulation of elastic wave propagation using a finite volume method, Journal of Geophysical Research: Solid Earth, vol.51, issue.B2, pp.2123-2133, 1995.
DOI : 10.1190/1.1442147

L. Erd?-os, Linear boltzmann equation as the weak coupling limit of the random schrödinger equation, Mathematical Results in Quantum Mechanics, pp.233-242, 1999.

K. Eriksson and C. Johnson, Adaptive Finite Element Methods for Parabolic Problems I: A Linear Model Problem, SIAM Journal on Numerical Analysis, vol.28, issue.1, pp.43-77, 1991.
DOI : 10.1137/0728003

K. Eriksson and C. Johnson, Adaptive Finite Element Methods for Parabolic Problems IV: Nonlinear Problems, SIAM Journal on Numerical Analysis, vol.32, issue.6, pp.1729-1749, 1995.
DOI : 10.1137/0732078

J. Fish and W. Chen, Space???time multiscale model for wave propagation in heterogeneous media, Computer Methods in Applied Mechanics and Engineering, vol.193, issue.45-47, pp.45-474837, 2004.
DOI : 10.1016/j.cma.2004.05.006

C. Furnas, Evaluation of the Modified Bessel Function of the First Kind and Zeroth Order, The American Mathematical Monthly, vol.208, issue.6, pp.282-287, 1930.
DOI : 10.1080/00029890.1930.11987074

D. Gabor, Theory of communication. Part 1: The analysis of information, Journal of the Institution of Electrical Engineers - Part III: Radio and Communication Engineering, vol.93, issue.26, pp.93429-441, 1946.
DOI : 10.1049/ji-3-2.1946.0074

F. Gatti, L. D. Paludo, A. Svay, R. Cottereau, and D. Clouteau, Investigation of the earthquake ground motion coherence in heterogeneous non-linear soil deposits, Procedia Engineering, vol.199, pp.2354-2359, 2017.
DOI : 10.1016/j.proeng.2017.09.232
URL : https://hal.archives-ouvertes.fr/hal-01586963

E. H. Georgoulis, O. Lakkis, and C. Makridakis, A posteriori L??(L2)-error bounds for finite element approximations to the wave equation, IMA Journal of Numerical Analysis, vol.33, issue.4, pp.1245-1264, 2013.
DOI : 10.1093/imanum/drs057

A. Ghatak and S. Lokanathan, The Dirac Delta Function, Quantum Mechanics: Theory and Applications, pp.3-18, 2004.
DOI : 10.1007/978-1-4020-2130-5_1

O. A. Godin, Rayleigh scattering of a spherical sound wave, The Journal of the Acoustical Society of America, vol.133, issue.2, pp.709-720, 2013.
DOI : 10.1121/1.4774277

P. Gonçalves and R. G. Baraniuk, Pseudo affine Wigner distributions: definition and kernel formulation, IEEE Transactions on Signal Processing, vol.46, issue.6, pp.1505-1516, 1998.
DOI : 10.1109/78.678464

D. Gottlieb and S. A. Orszag, Numerical analysis of spectral methods: theory and applications, 1977.
DOI : 10.1137/1.9781611970425

K. Hadeler, Reaction telegraph equations and random walk systems. Stochastic and spatial structures of dynamical systems, pp.133-161, 1996.

S. L. Hahn, Hilbert transforms in signal processing, 1996.

M. J. Hancock, Solution to problems for the 1-d wave equation, 2006.

F. J. Harris, On the use of windows for harmonic analysis with the discrete Fourier transform, Proceedings of the IEEE, pp.51-83, 1978.
DOI : 10.1109/PROC.1978.10837

S. Hirsekorn, The scattering of ultrasonic waves by polycrystals, The Journal of the Acoustical Society of America, vol.72, issue.3, pp.1021-1031, 1982.
DOI : 10.1121/1.388233

F. Hlawatsch, Interference terms in the wigner distribution, Digital signal processing, vol.84, pp.363-367, 1984.

F. Hlawatsch and G. F. Boudreaux-bartels, Linear and quadratic time-frequency signal representations, IEEE Signal Processing Magazine, vol.9, issue.2, pp.21-67, 1992.
DOI : 10.1109/79.127284

M. Howe, On the Kinetic Theory of Wave Propagation in Random Media, Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences, vol.274, issue.1242, pp.274523-549, 1242.
DOI : 10.1098/rsta.1973.0075

M. Howe, A KINETIC EQUATION FOR WAVE PROPAGATION IN RANDOM MEDIA, The Quarterly Journal of Mechanics and Applied Mathematics, vol.27, issue.2, pp.237-253, 1974.
DOI : 10.1093/qjmam/27.2.237

F. Q. Hu, M. Hussaini, R. , and P. , An Analysis of the Discontinuous Galerkin Method for Wave Propagation Problems, Journal of Computational Physics, vol.151, issue.2, pp.921-946, 1999.
DOI : 10.1006/jcph.1999.6227

T. J. Hughes, The finite element method: linear static and dynamic finite element analysis, Courier Corporation, 2012.

T. J. Hughes and G. M. Hulbert, Space-time finite element methods for elastodynamics: Formulations and error estimates, Computer Methods in Applied Mechanics and Engineering, vol.66, issue.3, pp.339-363, 1988.
DOI : 10.1016/0045-7825(88)90006-0

F. Ibrahima, Estimation d'erreur pour des problèmes de propagation d'ondes en milieux élastiques linéaires hétérogènes, 2011.

A. Ishimaru, Wave propagation and scattering in random media, 1978.
DOI : 10.1109/9780470547045

A. Janssen, Application of the Wigner distribution to harmonic analysis of generalized stochastic processes, Mathematisch Centrum Amsterdam, vol.114, 1979.

E. W. Jenkins, B. Riviaere, and M. F. Wheeler, A Priori Error Estimates for Mixed Finite Element Approximations of the Acoustic Wave Equation, SIAM Journal on Numerical Analysis, vol.40, issue.5, pp.1698-1715, 2002.
DOI : 10.1137/S0036142901388068

A. J. Jerri, The shannon sampling theorem?its various extensions and applications: A tutorial review, Proceedings of the IEEE, pp.1565-1596, 1977.

C. Johnson, Discontinuous Galerkin finite element methods for second order hyperbolic problems, Computer Methods in Applied Mechanics and Engineering, vol.107, issue.1-2, pp.117-129, 1993.
DOI : 10.1016/0045-7825(93)90170-3

S. Kadambe and G. F. Boudreaux-bartels, A comparison of the existence of'cross terms' in the wigner distribution and the squared magnitude of the wavelet transform and the short-time fourier transform, IEEE Transactions on signal processing, issue.10, pp.402498-2517, 1992.

B. Kapralos, M. Jenkin, M. , and E. , Sonel Mapping: A Probabilistic Acoustical Modeling Method, Building Acoustics, vol.41, issue.11, pp.289-313, 2008.
DOI : 10.1121/1.1909343

K. Jr, F. C. Keller, and J. B. , Elastic wave propagation in homogeneous and inhomogeneous media, The Journal of the acoustical society of america, issue.6, pp.31694-705, 1959.

S. Khazaie, Influence of the statistical parameters of a random heterogeneous medium on elastic wave scattering: theoretical and numerical approaches, 2015.
URL : https://hal.archives-ouvertes.fr/tel-01159616

D. Komatitsch and J. Tromp, Introduction to the spectral element method for three-dimensional seismic wave propagation, Geophysical Journal International, vol.88, issue.3, pp.806-822, 1999.
DOI : 10.4294/jpe1952.44.489

, References

P. Ladevèze, Constitutive relation error estimators for time-dependent non-linear FE analysis, Computer Methods in Applied Mechanics and Engineering, vol.188, issue.4, pp.775-788, 2000.
DOI : 10.1016/S0045-7825(99)00361-8

P. Ladeveze and D. Leguillon, Error Estimate Procedure in the Finite Element Method and Applications, SIAM Journal on Numerical Analysis, vol.20, issue.3, pp.485-509, 1983.
DOI : 10.1137/0720033

P. Ladevèze and J. Pelle, Mastering calculations in linear and nonlinear mechanics, 2005.

B. Lapeyre, E. Pardoux, and R. Sentis, Introduction to Monte-Carlo methods for transport and diffusion equations, 2003.

R. J. Leveque, Finite volume methods for hyperbolic problems, 2002.
DOI : 10.1017/CBO9780511791253

A. Li, R. Roberts, F. J. Margetan, and R. Thompson, Influence of forward scattering on ultrasonic attenuation measurement, AIP Conference Proceedings, pp.51-58, 2002.
DOI : 10.1063/1.1472780

X. Li, X. Han, R. Li, and H. Jiang, Geometrical-optics approximation of forward scattering by gradient-index spheres, Applied Optics, vol.46, issue.22, pp.465241-5247, 2007.
DOI : 10.1364/AO.46.005241

X. D. Li, W. , and N. , STRUCTURAL DYNAMIC ANALYSIS BY A TIME-DISCONTINUOUS GALERKIN FINITE ELEMENT METHOD, International Journal for Numerical Methods in Engineering, vol.18, issue.12, pp.392131-2152, 1996.
DOI : 10.1002/eqe.4290160805

W. Lin and C. Wu, The Systems of Second Order Partial Differential Equations with Constant Coefficients, Partial Differential Equations in China, pp.173-181, 1994.
DOI : 10.1007/978-94-011-1198-0_12

P. Lions and T. Paul, Sur les mesures de Wigner, Revista Matem??tica Iberoamericana, vol.9, issue.3, pp.553-618, 1993.
DOI : 10.4171/RMI/143
URL : http://www.ems-ph.org/journals/show_pdf.php?issn=0213-2230&vol=9&iss=3&rank=2

B. Lombard, A. Maurel, and J. Marigo, Numerical modeling of the acoustic wave propagation across a homogenized rigid microstructure in the time domain, Journal of Computational Physics, vol.335, pp.558-577, 2017.
DOI : 10.1016/j.jcp.2017.01.036

B. Lombard and J. Piraux, Numerical treatment of two-dimensional interfaces for acoustic and elastic waves, Journal of Computational Physics, vol.195, issue.1, pp.90-116, 2004.
DOI : 10.1016/j.jcp.2003.09.024
URL : https://hal.archives-ouvertes.fr/hal-00004813

B. Lombard and J. Piraux, Modeling 1-D elastic P-waves in a fractured rock with hyperbolic jump conditions, Journal of Computational and Applied Mathematics, vol.204, issue.2, pp.292-305, 2007.
DOI : 10.1016/j.cam.2006.03.027
URL : https://hal.archives-ouvertes.fr/hal-00095896

J. V. Lorenzo-ginori, An Approach to the 2D Hilbert Transform for Image Processing Applications, International Conference Image Analysis and Recognition, pp.157-165, 2007.
DOI : 10.1007/978-3-540-74260-9_14

W. Lu and Q. Zhang, Deconvolutive short-time fourier transform spectrogram, IEEE Signal Processing Letters, vol.16, issue.7, pp.576-579, 2009.

G. D. Manolis, P. S. Dineva, T. V. Rangelov, and F. Wuttke, Seismic Wave Propagation in Non-Homogeneous Elastic Media by Boundary Elements, 2017.
DOI : 10.1007/978-3-319-45206-7

W. Mansur and C. Brebbia, Numerical implementation of the boundary element method for two dimensional transient scalar wave propagation problems, Applied Mathematical Modelling, vol.6, issue.4, pp.299-306, 1982.
DOI : 10.1016/S0307-904X(82)80038-3

K. J. Marfurt, Accuracy of finite???difference and finite???element modeling of the scalar and elastic wave equations, GEOPHYSICS, vol.49, issue.5, pp.533-549, 1984.
DOI : 10.1190/1.1441689

L. Margerin, M. Campillo, and B. Van-tiggelen, Monte Carlo simulation of multiple scattering of elastic waves, Journal of Geophysical Research: Solid Earth, vol.83, issue.B4, pp.7873-7892, 2000.
DOI : 10.1063/1.1722545

G. S. Martin, R. Wiley, and K. J. Marfurt, Marmousi2: An elastic upgrade for marmousi. The Leading Edge, pp.156-166, 2006.
DOI : 10.1190/1.2172306

F. Mccarthy and M. Hayes, Elastic Wave Propagation, Proceedings of the Second IUTAM-IUPAP, Symposium on Elastic Wave Propagation, 1988.

I. Mozolevski and S. Prudhomme, Goal-oriented error estimation based on equilibrated-flux reconstruction for finite element approximations of elliptic problems, Computer Methods in Applied Mechanics and Engineering, vol.288, pp.127-145, 2015.
DOI : 10.1016/j.cma.2014.09.025
URL : https://hal.archives-ouvertes.fr/hal-00985971

J. T. Oden and S. Prudhomme, Goal-oriented error estimation and adaptivity for the finite element method, Computers & Mathematics with Applications, vol.41, issue.5-6, pp.5-6735, 2001.
DOI : 10.1016/S0898-1221(00)00317-5
URL : https://doi.org/10.1016/s0898-1221(00)00317-5

J. T. Oden and S. Prudhomme, Estimation of Modeling Error in Computational Mechanics, Journal of Computational Physics, vol.182, issue.2, pp.496-515, 2002.
DOI : 10.1006/jcph.2002.7183

J. Paasschens, Solution of the time-dependent Boltzmann equation, Physical Review E, vol.32, issue.1, p.1135, 1997.
DOI : 10.1364/AO.32.004808

L. E. Payne and H. F. Weinberger, An optimal Poincar?? inequality for convex domains, Archive for Rational Mechanics and Analysis, vol.5, issue.1, pp.286-292, 1960.
DOI : 10.2140/pjm.1958.8.551

M. Picasso, Numerical Study of an Anisotropic Error Estimator in the $L^2(H^1)$ Norm for the Finite Element Discretization of the Wave Equation, SIAM Journal on Scientific Computing, vol.32, issue.4, pp.2213-2234, 2010.
DOI : 10.1137/090778249

S. Pikula and P. Bene?, A new method for interference reduction in the smoothed pseudo wigner-ville distribution, Proceedings of 8th International Conference on Sensing Technology, pp.599-603, 2014.

D. Ping, P. Zhao, and B. Deng, Cross-terms suppression in Wigner-Ville distribution based on image processing, The 2010 IEEE International Conference on Information and Automation, pp.2168-2171, 2010.
DOI : 10.1109/ICINFA.2010.5512072

S. Prudhomme and J. T. Oden, On goal-oriented error estimation for elliptic problems: application to the control of pointwise errors, Computer Methods in Applied Mechanics and Engineering, vol.176, issue.1-4, pp.1-4313, 1999.
DOI : 10.1016/S0045-7825(98)00343-0

S. Qian and J. M. Morris, Wigner Distribution decomposition and cross-terms deleted representation, Signal Processing, vol.27, issue.2, pp.125-144, 1992.
DOI : 10.1016/0165-1684(92)90003-F

L. Ryzhik, G. Papanicolaou, and J. B. Keller, Transport equations for elastic and other waves in random media, Wave Motion, vol.24, issue.4, pp.327-370, 1996.
DOI : 10.1016/S0165-2125(96)00021-2
URL : http://math.stanford.edu/~papanico/pubftp/TRANSPORT.pdf

M. Sandsten, Time-frequency analysis of time-varying signals and non-stationary processes, 2016.

H. Sato, M. C. Fehler, and T. Maeda, Seismic wave propagation and scattering in the heterogeneous earth, 2012.
DOI : 10.1007/978-3-540-89623-4
URL : https://link.springer.com/content/pdf/bfm%3A978-3-540-89623-4%2F1.pdf

, References

É. Savin, Transient vibrational power flows in slender random structures: Theoretical modeling and numerical simulations, Probabilistic Engineering Mechanics, vol.28, pp.194-205, 2012.
DOI : 10.1016/j.probengmech.2011.08.012

É. Savin, Kinetic Modeling for Transport of Elastic Waves in Anisotropic Heterogeneous Media, Procedia IUTAM, vol.6, pp.97-107, 2013.
DOI : 10.1016/j.piutam.2013.01.011

F. Seron, F. Sanz, M. Kindelan, and J. Badal, Finite-element method for elastic wave propagation, Communications in Applied Numerical Methods, vol.8, issue.5, pp.359-368, 1990.
DOI : 10.2514/8.1722

P. Sheng, Introduction to wave scattering, localization and mesoscopic phenomena, 1995.

Y. S. Shin and J. Jeon, Pseudo Wigner???Ville Time-Frequency Distribution and Its Application to Machinery Condition Monitoring, Shock and Vibration, vol.1, issue.1, pp.65-76, 1993.
DOI : 10.1155/1993/372086

M. Shinozuka and G. Deodatis, Response Variability Of Stochastic Finite Element Systems, Journal of Engineering Mechanics, vol.114, issue.3, pp.499-519, 1988.
DOI : 10.1061/(ASCE)0733-9399(1988)114:3(499)

M. Shinozuka and G. Deodatis, Simulation of Stochastic Processes by Spectral Representation, Applied Mechanics Reviews, vol.44, issue.4, pp.191-204, 1991.
DOI : 10.1115/1.3119501

M. Shinozuka and G. Deodatis, Simulation of Multi-Dimensional Gaussian Stochastic Fields by Spectral Representation, Applied Mechanics Reviews, vol.49, issue.1, pp.29-53, 1996.
DOI : 10.1115/1.3101883

G. D. Smith, Numerical solution of partial differential equations: finite difference methods, 1985.

W. D. Smith, The Application of Finite Element Analysis to Body Wave Propagation Problems, Geophysical Journal of the Royal Astronomical Society, vol.242, issue.EM3, pp.747-768, 1975.
DOI : 10.1080/00288306.1970.10431342
URL : https://academic.oup.com/gji/article-pdf/42/2/747/1840090/42-2-747.pdf

H. Spohn, Derivation of the transport equation for electrons moving through random impurities, Journal of Statistical Physics, vol.39, issue.6, pp.385-412, 1977.
DOI : 10.1007/BF01014347

F. E. Stanke and G. Kino, A unified theory for elastic wave propagation in polycrystalline materials, The Journal of the Acoustical Society of America, vol.75, issue.3, pp.665-681, 1984.
DOI : 10.1121/1.390577

L. Stankovi´cstankovi´c, T. Alieva, and M. J. Bastiaans, Time???frequency signal analysis based on the windowed fractional Fourier transform, Signal Processing, vol.83, issue.11, pp.2459-2468, 2003.
DOI : 10.1016/S0165-1684(03)00197-X

J. Staudacher, Conservative numerical schemes for high-frequency wave propagation in heterogeneous media, 2013.
URL : https://hal.archives-ouvertes.fr/tel-01005143

L. M. Steffens and P. Díez, A simple strategy to assess the error in the numerical wave number of the finite element solution of the Helmholtz equation, Computer Methods in Applied Mechanics and Engineering, vol.198, issue.15-16, pp.15-161389, 2009.
DOI : 10.1016/j.cma.2008.12.005

E. Süli, A posteriori error analysis and global error control for adaptive finite element approximations of hyperbolic problems, 1995.

E. Süli and P. Houston, Adaptive finite element approximation of hyperbolic problems, Error estimation and adaptive discretization methods in computational fluid dynamics, pp.269-344, 2003.

N. W. Taylor, L. E. Kidder, S. A. Teukolsky, and D. Aubry, Spectral methods for the wave equation in second-order form Adaptive time discontinuous galerkin method for numerical modelling of wave propagation in shell and 3d structures, Physical Review D European Journal of Computational Mechanics, vol.82, issue.156, pp.24037729-757, 2006.

B. Tie, D. Aubry, and A. Boullard, Adaptive computation for elastic wave propagation in plate/shell structures under moving loads, Revue Europ??enne des ??l??ments Finis, vol.33, issue.20, pp.717-736, 2003.
DOI : 10.1002/nme.1620330702

C. Turner, Hilbert transforms, analytic functions, and analytic signals. Retrieved from personal, 2005.

S. Utku and R. J. Melosh, Solution errors in finite element analysis, Computers & Structures, vol.18, issue.3, pp.379-393, 1984.
DOI : 10.1016/0045-7949(84)90058-0

M. Van-der-baan, Acoustic wave propagation in one-dimensional random media: the??wave localization approach, Geophysical Journal International, vol.390, issue.3, pp.631-646, 2001.
DOI : 10.1038/37757

A. Van-pamel, C. R. Brett, P. Huthwaite, and M. J. Lowe, Finite element modelling of elastic wave scattering within a polycrystalline material in two and three dimensions, The Journal of the Acoustical Society of America, vol.138, issue.4, pp.2326-2336, 2015.
DOI : 10.1121/1.4931445

A. Van-pamel, G. Sha, S. Rokhlin, and M. Lowe, Finite-element modelling of elastic wave propagation and scattering within heterogeneous media, Proc. R. Soc. A, p.20160738, 2017.
DOI : 10.1002/nme.2579

K. Van-wijk, M. Haney, and J. A. Scales, 1D energy transport in a strongly scattering laboratory model, Physical Review E, vol.72, issue.3, p.69036611, 2004.
DOI : 10.1103/PhysRevLett.72.633

J. Virieux, wave propagation in heterogeneous media: Velocity???stress finite???difference method, GEOPHYSICS, vol.51, issue.4, pp.889-901, 1986.
DOI : 10.1190/1.1442147

J. Virieux, V. Cruz-atienza, R. Brossier, E. Chaljub, O. Coutant et al., Modelling Seismic Wave Propagation for Geophysical Imaging, 2016.
DOI : 10.5772/30219
URL : https://hal.archives-ouvertes.fr/hal-00682707

M. Vohralík, A posteriori error estimates for efficiency and error control in numerical simulations. Lecture Notes, 2013.

J. V. Wehausen and E. V. Laitone, Surface Waves, Fluid Dynamics / Strömungsmechanik, pp.446-778, 1960.
DOI : 10.1007/978-3-642-45944-3_6

E. W. Weisstein, Boundary conditions. From MathWorld?A Wolfram Web Resource, 2008.

J. E. White, Physics Today, vol.20, issue.2, 1965.
DOI : 10.1063/1.3034162

R. Wu and K. Aki, Introduction: Seismic wave scattering in three-dimensionally heterogeneous earth, Pure and Applied Geophysics PAGEOPH, vol.128, issue.1-2, pp.1-6, 1988.
DOI : 10.1007/BF01772587

M. Yoon, Deep seismic imaging in the presence of a heterogeneous overburden, 2005.

Y. Zeng, Modeling of high-frequency seismic-wave scattering and propagation using radiative transfer theorymodeling of high-frequency seismic-wave scattering and propagation using radiative transfer theory, pp.2948-2962, 2017.

O. C. Zienkiewicz, R. L. Taylor, O. C. Zienkiewicz, T. , and R. L. , The finite element method, 1977.

, References

O. C. Zienkiewicz and J. Z. Zhu, A simple error estimator and adaptive procedure for practical engineerng analysis, International Journal for Numerical Methods in Engineering, vol.7, issue.18, pp.337-357, 1987.
DOI : 10.1016/B978-0-12-747255-3.50049-6

O. C. Zienkiewicz and J. Z. Zhu, The superconvergent patch recovery anda posteriori error estimates. Part 1: The recovery technique, International Journal for Numerical Methods in Engineering, vol.31, issue.7, pp.1331-1364, 1992.
DOI : 10.1002/nme.1620330702

O. C. Zienkiewicz and J. Z. Zhu, The superconvergent patch recovery anda posteriori error estimates. Part 2: Error estimates and adaptivity, International Journal for Numerical Methods in Engineering, vol.8, issue.7, pp.1365-1382, 1992.
DOI : 10.1002/nme.1620330703

H. Zou, X. Lu, Q. Dai, L. , and Y. , Nonexistence of cross-term free time-frequency distribution with concentration of wigner-ville distribution, Science in China Series F: Information Sciences, pp.45174-180, 2002.