On the computation of the Fréchet derivatives for seismic waveform inversion in 3D general anisotropic, heterogeneous media

We present a perturbation method and a matrix method for formulating the explicit Fréchet derivatives for seismic body-wave waveform inversion in 3D general anisotropic, heterogeneous media. Theoretically, the two methods yield the same explicit formula valid for any class of anisotropy and are completely equivalent if the model parameterization in the inversion is the same as that used in the discretization scheme (unstructured or structured mesh) for forward modeling. Explicit formulas allow various model parameterization schemes that try to match the resolution capability of the data and possibly reduce the dimensions of the Jacobian matrix. Based on the general expressions, relevant formulas for isotropic and 2.5D and 3D tilted transversely isotropic (TTI) media are derived. Two computational schemes, constant-point and constant-block parameterization, offer effective and efficient means of forming the Jacobian matrix from the explicit Fréchet derivatives. The sensitivity patterns of the displacement vector to the independent model parameters in a weakly anisotropic medium clearly convey the imaging capability possible with seismic waveform inversion in such an anisotropic medium.