Multichannel recording results in an abundance of seismic traces at every CMP bin. Imaging aims at inverting for a reflectivity model using the entire prestack volume. The model is regularly sampled at the nominal CMP spacing. Therefore, considering multiple records at every CMP bin to present redundant information, the inversion for a reflectivity model from multi-offset seismic input is generally an overdetermined problem.
The reality of seismic acquisition results into variations in CMP locations within the nominal bin spacing and, therefore, could introduce extra degrees of freedom if the model space were allowed to be sampled at the resolution of these spatial variations. Moreover, whenever gaps in seismic coverage occur, the inversion problem becomes locally underdetermined. Therefore, the problem is never genuinely overdetermined as often perceived.
In the context of preserving amplitudes by adapting the data to fit the imaging operator, we seek a solution in which the input is first regularized to correct for the interdependencies between data elements, then an imaging operator (the adjoint of L) is applied to solve for the final model. This type of solution lends itself readily to the definition of a data-space inverse. The problem then reduces to estimating an inverse for the cross-product matrix LLT, which we shall first define, and explain its properties.