True-amplitude migration

We define ``true-amplitude'' migration as the process of recovering the amplitude of the reflectivity vector given perfect data, infinite bandwidth and aperture.

The expression for the true-amplitude migration operator is  

$$\bold L_{t}^{*}=\bold G^{-1}\bold \AA^{-1}\bold W^{-1}\bold L^{*},$$ (12)

which we can immediate verify to be inverse to modeling using Equation (9):

$$\bold L_{t}^{*}\bold d = \bold G^{-1}\bold \AA^{-1}\bold W^{-1}\bold L^{*}\bold L\bold \AA\bold G\bold r = \bold r.$$ (13)

Even in the simple case of layered media, the operator $\bold \AA$ is singular for evanescent waves propagating at the surface, and null, thus not invertible, for evanescent waves at depth. As discussed in the preceding section, we are not interested in the evanescent energy, and thus we are not even trying to invert $\bold \AA$for those wavefield components.