Using the definition of data-space pseudo inverse, Chemingui and Biondi (1997) presented a new technique to invert for reflectivity models while correcting for the effects of irregular sampling. The final reflectivity model is a two step solution where the data is equalized in a first stage with an inverse filter and an imaging operator is then applied to the equalized data to invert for a model.
We start from the definition for the data-space inverse solution
where is AMO from input offset hi to output offset hj and, is the identity operator (mapping from hi to hi). Conforming to the definition of AMO Biondi et al. (1996), is the adjoint of ; therefore, the filter is Hermitian with diagonal elements being the identity and off-diagonal elements being AMO transforms.