Stiffnesses perturbations

Most imaging methods described in the literature involve the correlation between scalar wavefields. An exception is the tensorial imaging condition formulated by Karrenbach (1991):

$$\delta c_{jklm}(x,z,t) = \partial_m [\phi^s_l(x,z,t)] \partial_k [\delta \phi^r_j(x,z,t)],$$

where $\phi^s_l$ is the l component of the forward modeled shot wavefield, and $\delta \phi^r_j$ is the reverse propagated difference between the forward modeled wavefield at the receivers and the recorded wavefield. This process is equivalent to one step of a wavefield inversion scheme, and $\delta c_{jklm}$ can be interpreted as the estimated perturbation in the stiffness tensor. In this sense, it can be considered as an extension of Tarantola's (1986) elastic inversion theory to the general anisotropic case.

Hildebrand (1987) used a variation of this principle to image for the acoustic impedance rather than the reflection coefficient, using reverse-time migration to extrapolate the pressure and particle-velocity wavefields. Contrary to the above formulation, Hildebrand's approach obtained a image of the impedance (rather than perturbations), using a recursive depth extrapolation procedure.