THE CONJUGATE

Peter Mora (1986) showed the conjugate operation to the elastic forward modeling and used it to find iteratively the inverse of elastic wave propagation. With his notation, the solution for perturbations to wave equation (1) can formally be written in terms of Green's functions:  

$$\delta ~u_i = \int_V G_{ij}*\delta~f_j~-\int_V \partial_t~G_... ...partial_k~G_{ij})*(\delta~c_{jklm}~\partial_m~u_l) \delta~u_i .$$ (2)

When ignoring perturbations in the source function or in density, the adjoint (transpose) can be written in terms of material parameters:

$$\delta~c_{jklm}~=~-\int~dt~ \partial_k~G_{ij}*\partial_m~u_l ~ \delta~u_i .$$ (3)

The above equation is intuitively more understandable when written in terms of wavefields:

$$\delta~c_{jklm}~=~\int~dt~ [\partial_m~u_l(x,t)] ~ [\partial_k~\delta~\psi_j(x,t)] ,$$ (4)

where u is the forward-propagated shot wavefield and $\psi$ is the backward-propagated data (in the case of prestack migration) or the residual data (in the case of Mora's inversion).

Changes in the material parameters can be estimated by crosscorrelating the backward-propagated modeled strain field and the backward-propagated data strain field. Both fields are tensor fields and what are crosscorrelated are actually the strain tensor components. In terms of prestack migration this crosscorrelation is the imaging condition. In the following I use this imaging condition to find perturbations of elastic stiffness parameters for a given starting model. One might ask: why image stiffnesses and not velocities? For isotropic media the imaging of velocities or wave potential works well, because the wave types are separable into a P wave and S wave potential and those potentials are decoupled. In the case of anisotropic media such a decoupling is only possible in a few rare cases of special symmetry. Imaging stiffness naturally evolves out of the anisotropic wave equation and wave types don't have to be decoupled in order to perform imaging. Some stiffness components, however, may not be be very well resolved, depending strongly on the acquisition geometry and source radiation pattern illuminating the subsurface. The above stated imaging condition reduces to the well-known U/D principle (Claerbout, 1986) in the case of an isotropic scalar wavefield.