Full waveform inversion (FWI) is very powerful in estimating the subsurface properties by minimizing the difference between the modelled data and observed data iteratively. Multi-parameter FWI can also be employed to inverse the properties of the naturally fractured reservoirs, when assuming that the wavelength is much larger than the fracture size. Estimating the fracture properties using multi-parameter FWI will be challenging for several difficulties, one of which is the cross-talk problem in multi-parameter FWI. And it refers to that the seismic wavefields responses by different parameters' perturbations are coupled together. This difficulty also gives rise to the parameterization issue for multi-parameter FWI. The Fréchet derivative serves as the inversion sensitivity for the least-squares inverse problem. Furthermore, the Fréchet derivative controls the trade-off among different parameters and the amplitude variations with varying the scattering angle and azimuthal angle (in 3D) (radiation pattern) can help identify the efficiency of the parameterization and design optimal acquisition geometry. In this research, we focus on studying the inversion sensitivities of elastic constants in fractured media. We, first, review the general principle of FWI and then give the explicit Fréchet derivative for the general anisotropic media with Born approximation. The 3D Fréchet derivative with respect to different elastic constants for parallel vertical fractured media (described as HTI model) are provided. And then, we analyze the scattering characteristics due to the fractured inclusion and discuss the inversion sensitivities with varying the scattering angle and azimuthal angle.
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