Identification of some statistical parameters of a random heterogeneous medium using measurements of the surface wavefield

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Identification of some statistical parameters of a random heterogeneous medium using measurements of the

surface wavefield

Shahram Khazaie, R. Cottereau, D. Clouteau

To cite this version:

Shahram Khazaie, R. Cottereau, D. Clouteau. Identification of some statistical parameters of a

random heterogeneous medium using measurements of the surface wavefield. 2nd ECCOMAS Young

Investigators Conference (YIC 2013), Sep 2013, Bordeaux, France. �hal-00855856�

YIC2013 Second ECCOMAS Young Investigators Conference 2–6 September 2013, Bordeaux, France

Identification of some statistical parameters of a random heterogeneous medium using measurements of the surface wavefield

S. Khazaie, R. Cottereau, D. Clouteau

´Ecole Centrale Paris, Laboratoire MSSMat UMR 8579 CNRS, France

∗[email protected]

Abstract. This presentation discusses the identification of some statistical properties of an elastic random medium based on surface measurements. The influence of two fundamental factors in the diffusion regime, i.e. the ratio between the dominant wavelength and the correlation length of the heterogeneities, and the ratio between P and S wave speeds is studied. The influence of the correlation model is also assessed. The global diffusion time of the medium is then used as a key parameter to identify the correlation length of the random medium.

Keywords:Identification, Random medium, Correlation length, Diffusion regime.

Exploration of the structure of the Earth using seismograms recorded at the surface is a classical problem in geo- physics. Its applications range from academic understanding of the interior of the Earth to more industrially-oriented questions related to oil exploration or CO2and nuclear waste sequestration. The seismograms typically consist of a ballistic and coherent part corresponding to the first arrivals of P and S or Rayleigh waves, followed by a wave train made of multiply scattered waves called the Coda, with less definite features, [1]. While most inversion methods use this first-arrivals information, and try to identify for the values of the elastic properties in each point in space, we rather aim at using the information contained in the Coda, and wish only to obtain statistical information about the properties of the medium. The inverse problem we consider is therefore a priori much better posed than the classical one because it uses more information for less parameters. We assume that the behavior of the Coda can be

understood through a model of diffusion of the elastic energy [1, 3].

5.4. Renversement temporel et Propriété conservative du schéma d’intégration en temps

0 50 100 150

0 0.25 0.5 0.75 1

!_x [m]

" [−]

F

. 5.6 – Une réalisation du champ de propriétés élastiques : (à gauche) Cartographie du terme C

et (à droite) Comparaison entre la structure de corrélation théorique (courbes en lignes continues) avec la structure de corrélation observée (courbes en cercles) pour C

dans les directions i

, i

(courbes noires) et dans la direction i

(courbes grises).

F

. 5.7 – Une réalisation du champ stochastique de tenseur d’élasticité. De gauche à droite : les champs correspondant aux termes C

, C

, C

, à la norme �C� et à l’indice d’anisotropie I

.

difficultés majeures réside dans le manque de références analytiques pour valider notre simu- lation numérique. Par contre, on peut se baser sur la propriété réversible de l’équation d’ondes élastiques pour vérifier la propriété non-dissipative du schéma numérique. Pour ce faire, un opérateur de renversement du temps est défini. Il sera appliqué sur l’ensemble du système au bout d’un temps t

que l’on souhaite. D’une façon générale, comme en mécanique classique, un tel opérateur baptisé T peut être défini comme celui transformant tout le déplacement u(x, t) d’un système physique en un autre déplacement u

:= T (u) tel que pour ∀t :

u

(x, t) = u(x, −t) (5.1)

R

5.1. On peut aisément déduire de la relation (5.1) que les dérivées temporelles de ce champ après le renversement sont ainsi donnés par les champs de vitesses et d’accélérations :

˙u

(x, t) = − ˙u(x, −t) (5.2)

¨

u

(x, t) = ¨ u(x, −t) (5.3)

D’une façon imagée, si on tourne un film à partir de l’évolution en temps du système jusqu’à un instant t

, l’opérateur de renversement du temps peut être vu comme l’opérateur de lecture à l’envers, i.e. en commençant par la fin du film.

Dans la pratique de notre simulation, l’implémentation de l’opérateur T de renversement du temps dans le schéma Newmark de vitesse-déplacement se réalise en prenant en compte 163

tel-00585466, version 1 - 13 Apr 2011

Figure 1: Propagation domain as a half-space with PMLs at boundaries (a), and one realization of a random elastic modulus.

We consider the propagation of elastic waves in a half-space (Fig. 1) with random isotropic mechanical properties.

The constitutive random elastic tensor is then parametrized through its average value and variance, a correlation structure, and a correlation length [2]. Fig. 1 also depicts a realization of the first component of the random elastic matrix C11 = λ + 2µwhere λ and µ are the Lam´e coefficients. The tensor is assumed to be statistically homoge- neous. Four types of correlation models, i.e. exponential, Gaussian, squared cardinal sine and Von K´arm´an are used

2 S. Khazaie et al. | Young Investigators Conference 2013

to describe the correlations of the Lam´e coefficients. In the case of an isotropic medium with continuous random heterogeneities, the scattering parameters as well as some parameters of the diffusion regime depend on: 1) the am- plitude of the fluctuations [σ²], 2) the ratio a = λ/`cbetween the dominant wavelength and the correlation length of the heterogeneities, and 3) the ratio K = vP/vS between P and S wave speeds. For example, the total scattering cross-section for P to P mode conversion, normalized by ωa³where ω is the local angular frequency, in the case of an exponentially correlated isotropic random medium with mutually independent elastic parameters and a constant density, is:

Σpp(K, a, σ_λ², σ_µ²)

ωa³ =2(K²− 2)² K⁴(1 + 4a²)σ²_λ

+

 2

3a⁴ +3(1 + 2a²)²

2a⁸ −(1 + 2a²)³

2a¹⁰ ln(1 + 4a²) + 7(1 + 2a²)⁴ 2a⁸(1 + 4a²)

 σ²_µ

K⁴ (1)

In order to evaluate numerically the scattering parameters, especially the scattering and transport mean free paths and the elastic diffusivity of the underlying random medium, we use a high-performance parallel Spectral-Element- based code [2].

The general objective of this research is to perform identification based on a part of the seismic recordings that is seldom used and based on a stochastic model of the medium and a diffusion approximation. We assess here the influence of the basic random model parameters onto the diffusion regime parameters as a first step towards this goal. The next step is to evaluate the scattering parameters using statistical analysis of the Coda waves.

REFERENCES

[1] Aki, K., Chouet, B.. Origin of coda waves: Source, attenuation, and scattering effects, J. Geophys. Res. 80, 3322, 1975.

[2] Ta, Q.A., Clouteau, D. and Cottereau, R.. Modeling of random anisotropic elastic media and inpact on wave propagation., European Journal of Computational Mechanics, 1-13, 2011.

[3] Ryzhik, L., Papanicolaou, G. and Keller, J.B.. Transport Equations for Elastic and Other Waves in Random Media, Wave Motion, 24, 327-370, 1996.

[4] Anache, D., van Tiggelen, B.A. and Margerin, L.. Phase statistics of seismic coda waves, Physical Review Letters 102, 248501, 2009.

[5] Sato, H., Nakahara, H., and Ohtake, M.. Synthesis of scattered energy density for non-spherical radiation from a point shear dislocation source based on the radiative transfer theory, Phys. Earth Planet. Inter., 104, 1-13, 1997.

[6] Weaver, R.L.. Diffusivity of ultrasound in polycrystals, J. Mech. Phys. Solids 38, 55-86, 1990.