Summary

Often anelastic parameters need to be taken into account for a consistent seismic full waveform inverse problem. The Fréchet derivatives allow solving the inverse problem with a high number of parameters; however, for the anelastic case, the Fréchet derivatives do not have simple expressions. This is due to the time convolution in the general linear constitutive relation. By using viscoelastic model based on the superposition of relaxation Zener mechanisms, we showed that it was possible to derive simple Fréchet derivative expressions for a quasi constant quality factor. We illustrated the feasibility of our approach on simple synthetic inversion experiments. To validate on real data, we have chosen an offset VSP data set from North Sea displaying highly attenuated phases that we identified as P to S converted waves from the sea floor. By inverting simultaneously P-wave and S-wave velocities, density, S-wave Q-factor but also the source function, we have been able to find an Earth model reproducing fairly well the real data. As expected, the highly attenuating anomaly is located at the sea floor. The fairly good inversion results on this data set demonstrate the feasibility of our proposed method.

Introduction

Elastic approximation of Earth properties for seismic wave is limited as waves undergo attenuation and dispersion. Often anelastic parameters need to be taken into account for a consistent seismic full waveform inverse problem. For highly heterogeneous media, the most efficient method for modeling wave propagation is in time domain (Carcione, 1990); therefore, it would be a great advantage to use a formulation of the inverse problem for viscoelastic parameters in the time domain. Tarantola (1988) derived the expression of the Fréchet derivatives for a general linear viscoelastic medium in the time domain; however, this formulation leads to expressions having time convolution integral. As the numerical computation for such integral is not tractable, the formulation of the viscoelasticity has to be simplified: for the direct problem, the linear viscoelasticity constitutive law is modeled by the superposition of relaxation mechanisms, classically called Zener or standard viscoelastic bodies. The key concept of this technique is the replacement of the time convolution between relaxation rates and strain by a set of first order temporal partial differential equations, storing the strain history interactions with the medium trough new fields called strain memory variables (Carcione, 1990).

Based on this new viscoelastic constitutive relation, we derived (Charara et al., 2000 and Barnes et al., 2004), the Fréchet derivatives for viscoelastic parameters without time convolutions. The feasibility and practicality of this new approach has been demonstrated on several noise free synthetic data: for a 1D offset VSP numerical inversion experiment (Barnes et al., 2004) and for a 2D crosswell numerical inversion experiment (Charara et al., 2004).

In this paper, in order to validate our approach on a real data set, we have chosen an offset VSP data set from North Sea displaying highly attenuated phases; mostly S-waves. The inverted model is 2D and the inverted parameters are P-wave and S-wave velocities, density and the S-wave Q-factor. The source time function is inverted as well.

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