Analytic expressions for the velocity sensitivity to the elastic moduli for the most general anisotropic media

For non-linear kinematic inversion of elastic anisotropy parameters and related investigations of the sensitivity of seismic data, the derivatives of the wavespeed (phase velocity and group velocity) with respect to the individual elastic moduli are required. This paper presents two analytic methods, called the eigenvalue and eigenvector methods, to compute the derivatives of the wavespeeds for wave propagation in a general anisotropic medium, which may be defined by up to 21 density-normalized elastic moduli. The first method employs a simple and compact form of the eigenvalue (phase velocity) and a general form of the group velocity, and directly yields general expressions of the derivatives for the three wave modes (qP, qS(1), qS(2)). The second method applies simple eigenvector solutions of the three wave modes and leads to other general forms of the derivatives. These analytic formulae show that the derivatives are, in general, functions of the 21 elastic moduli as well as the wave propagation direction, and they reflect the sensitivity of the wavespeeds to the individual elastic moduli. Meanwhile, we give results of numerical investigations with some examples for particular simplified forms of anisotropy. They show that the eigenvalue method is suitable for the qP-, qS(1)- and qS(2)-wave computations and mitigates the singularity problem for the two quasi-shear waves. The eigenvector method is preferable to the eigenvalue method for the group velocity and the derivative of the phase velocity because it involves simpler expressions and independent computations, but for the derivative of the group velocity the derivative of the eigenvector is required. Both methods tackle the singularity problem and are applicable to any degree of seismic anisotropy for all three wave modes.