Full waveform inversion sensitivities in anisotropic viscoelastic media
Shahpoor Moradi, Kristopher A. Innanen
We analyze the scattering of seismic waves from both anisotropic and viscoelastic inclusions in an attenuative isotropic background. There are mainly two methods used in investigation of seismic wave scattering, the method of so-called Aki-Richards approximation based on the linearization of the exact solutions of the Zoeppritz equation, and, alternatively the Born approximation which is based on the ﬁrst order perturbation theory. Solution of Zoeppritz equation even for elastic medium has a complicated form and these coefﬁcients should be linearized with respect to the changes in elastic properties. For anisotropic viscoelastic media the situation is more complicated. Born approximation overcome this difﬁculty as we don’t need to solve the Zoeppritz equation and linearized the the reﬂection coefﬁcients. Instead by having the perturbed stiffness tensor and polarizations in the background medium we can derived the linearized reﬂection coefﬁcients. The resulted scattering amplitudes are called scattering potentials which can be transformed to the weak reﬂection coefﬁcients by proper transformations. We consider to the Vertical Transverse Isotropic (VTI) and orthorhombic media with low loss attenuation and weak anisotropy such that the second and higher orders of quality factors and Thompson parameters are neglected. In a viscoelastic medium we have P-wave, SI-wave and SII waves, all with complex slowness vectors. We derived the expressions for potentials of scattering of P-to-P, P-to-SI, SI-to-SI and SII-to-SII. We show that how our results cover the previously derived scattering potentials for elastic/viscoelastic media. The resulting expressions for scattering potentials are sensitivity kernels related to the Frchet derivatives which linearly link data and parameters perturbations.