Feasibility study for an anisotropic full waveform inversion of cross-well seismic data

Anisotropy is often observed due to the thin layering or aligned micro-structures, like small fractures. At the scale of cross-well tomography, the anisotropic effects cannot be neglected. In this paper, we propose a method of full-wave inversion for transversely isotropic media and we test its robustness against structured noisy data. Optimization inversion techniques based on a least-square formalism are used. In this framework, analytical expressions of the misfit function gradient, based on the adjoint technique in the time domain, allow one to solve the inverse problem with a high number of parameters and for a completely heterogeneous medium. The wave propagation equation for transversely isotropic media with vertical symmetry axis is solved using the finite difference method on the cylindrical system of coordinates. This system allows one to model the 3D propagation in a 2D medium with a revolution symmetry. In case of approximately horizontal layering, this approximation is sufficient. The full-wave inversion method is applied to a crosswell synthetic 2-component (radial and vertical) dataset generated using a 2D model with three different anisotropic regions. Complex noise has been added to these synthetic observed data. This noise is Gaussian and has the same amplitude f-k spectrum as the data. Part of the noise is localized as a coda of arrivals, the other part is not localized. Five parameter fields are estimated, (vertical) P-wave velocity, (vertical) S-wave velocity, volumetric mass and the Thomsen anisotropic parameters epsilon and delta. Horizontal exponential correlations have been used. The results show that the full-wave inversion of cross-well data is relatively robust for high-level noise even for second-order parameters such as Thomsen epsilon and delta anisotropic parameters.