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Summary Seismic marine data inversion is a very heavy process, especially for the 3D seismic case. Often, approximations are made to limit the number of physical parameters or to speed up the forward modeling. Because the data are often dominated by unconverted P waves, one popular approximation is to consider the earth as purely acoustic: no shear modulus; even sometimes with constant density. Non-linear waveform seismic inversion consists in iteratively minimizing the misfit between the amplitudes of the measured and the modeled data. Approximations, such as acoustic medium, lead to model incorrectly the amplitudes of the seismic waves, especially with respect to offset (AVO) and therefore have a direct impact on the inversion results. For evaluation purposes, we performed series of inversions with different approximations and different constraints where the synthetic data set to recover is computed for a 1D elastic medium. The different physical approximations go from an acoustic medium to a fully elastic medium. The different geometrical constraints go from fixed interface positions to full 2D inversion and for the data: from near to far offset. As expected, the acoustic approximation is not minor and it impacts the inversion results, in some cases drastically. Introduction Interest in non-linear waveform inversion has been increasing rapidly since the late 90s as the advances in computer technology make the process affordable. Seismic marine data inversion is still very large, especially for the 3D seismic case. If, in the early 90s, several attempts have been made to invert for elastic parameters, to recover elastic impedance parameters, it is becoming a common practice to reduce the computational time needed by inverting for the acoustic parameters for retrieving P velocities of the medium; and, for 3D, the few synthetic attempts are made in the acoustic approximation as well. If the seismic marine data are dominated by unconverted P waves, the acoustic approximation holds for the kinematics but not for the wave amplitudes except for longitudinal waves having a normal incidence. Classical non-linear waveform inversions are based on minimizing iteratively the misfit between the modeled and the observed data in time domain (Tarantola, 1987) or in frequency domain (Pratt, 1996). Therefore, the modeling of the wave propagation should be as exact as possible, or otherwise, one has to take into account the errors due to the approximations in the theory (Tarantola, 1987). In practice, it is a difficult task to evaluate the errors in the theory; however, it is easy and enlightening to evaluate the impacts of the different approximations on the inversion results by running series of inversion of synthetic data computed with no approximations (in our case elastic medium). These numerical experiments allow characterizing the effects in the inversion process and artifacts in the inversion results when using the acoustic approximation while data are modeled in an elastic medium. Numerical Experiment Settings The true subsurface models are elastic horizontally layered model (1D model). The word ''true'' indicates that the synthetic seismic data used as observed data during the inversion process are generated with this model.
Summary Modeling of wave propagation in a realistic geological environment needs a numerical scheme able to handle complex shapes and geometries. Finite difference scheme based on a generalization of the rotated staggered grid method can be used for modeling of elastic waves on curvilinear grid. This scheme has been validate with classical analytical solutions and used to simulate elastic wave propagation in complex geometries. The proposed method is simple and computationally performing. Introduction Numerical modeling of wave propagation through irregular interfaces between layers, especially for the case of the sea floor having complex shapes is a fundamental problem in seismology. The methods which allow modeling correctly the boundary effects (e.g. Rayleigh waves on the free surface with complex topography, scattering of elastic waves on fractures in rocks, etc) are of great interest. For such complex geological structures, finite or, more recently, spectral element methods (FEM/SEM) proved to be well adapted [Komatitsch and Vilotte, 1998; Seriani, 1998]; However, these methods are more complex and computationally more expensive than classical staggered finite difference methods. Moreover; in the case of FEM/SEM methods special care should be taken when dealing with fluid-solid interfaces as spurious modes are generated in fluid regions [Komatitsch et al., 2000]. In this article we describe an alternative method for the wave propagation problem. It combines the simplicity of finite difference methods and the flexibility of FEM/SEM to model complex geometries without the need of making any special treatments for the fluid-solid interfaces. The method can be seen as a generalization of the rotated operators finite difference method [Saenger et al., 2000]- well known in the geophysical literature; or, it can be referred as the HEMP method well known in the rock mechanic literature [Wilkins, 1999]. Our purpose is to demonstrate that our finite difference scheme on a curvilinear grid correctly models the propagation of Rayleigh waves and is capable to model the wave propagation through the surface between the liquid environment and an elastic body. For comparison purposes with the SEM method, we reproduced the synthetic tests with the same geometry and the same physical parameters described in the papers [Komatitsch and Vilotte, 1998; Komatitsch et al., 2000] where the authors demonstrated the efficiency of the SEM method. Method To define the spatial Cartesian x and y derivatives in the case of curvilinear mesh system, let''s consider the new coordinate system (?,?) being the direction of diagonals of a convex quadrangular grid cell as shown in (Figure 1). Analogously, the relations for the average stress spatial derivative can be derived. For the case of rectangular cells the described method of discretization degenerates to the rotated staggered grid method [Saenger et al., 2000]. At the same time, the scheme can be considered as a particular case of the HEMP scheme used by Wilkins [Wilkins, 1999] for modeling of 3D finite elastic body deformations. Numerical tests To check the accuracy of the solution obtained by means of our scheme, we performed three series of numerical tests. Test #1 . First, Lamb''s classical problem was considered to check the scheme on an irregular grid.