Application to true-amplitude migration

The analysis above ignores the v(z) true-amplitude WKBJ Green's functions Stolt and Benson (1986) discussed in section . It also ignores the effect the imaging condition has on dipping events. Sava et al. (2001) discuss this in more detail; but the Jacobian associated with the transformation from temporal frequency to vertical wavenumber causes the slight attenuation of dipping events visible in Figure .

Both of these effects, however, are well-understood, and they can be compensated for in a true-amplitude migration operator, ${\bf A}_{\rm TA}^{\dagger}$. Such a true-amplitude migration operator will no longer be the true adjoint of the forward modeling operator; but we can still build and solve the system of normal equations,  

$${\bf A}_{\rm TA}^\dagger \, {\bf A}_{\rm TA} \; {\tilde {\bf m}} = {\bf A}_{\rm TA}^\dagger \, {\bf d}.$$ (89)

Since ${\bf A}_{\rm TA}^{\dagger}$ replaces ${\bf A}_{\rm TA}'$, the solution to this system will not be equivalent to the solution that minimizes the residual error in fitting goal,

$${\bf A}_{\rm TA} {\bf m} \approx {\bf d}.$$ (90)

However, if a pair of invertible operators (${\bf F}_M$ and ${\bf F}_D$) exist, such that

$${\bf A}_{\rm TA}^\dagger = {\bf F}'_M {\bf F}_M \; {\bf A}_{\rm TA}' \; {\bf F}'_D {\bf F}_D,$$

then the solution to equation () will be the same as the solution that minimizes the residual error in fitting goal,  

$${\bf F}_D \, {\bf A}_{\rm TA}\, {\bf m} \approx {\bf F}_D \, {\bf d}.$$ (91)

The solution will also converge faster under an iterative inversion scheme since ${\bf A}_{\rm TA}^\dagger {\bf A}_{\rm TA} \approx {\bf I}$.