Multichannel inversion

Multichannel recording results in an abundance of seismic traces at every CMP bin, whereas the goal of imaging is to invert 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 data is generally an over-determined problem. However, in many geophysical applications the size of the model space is not fixed but rather determined according to a desired resolution, computational costs and anti-aliasing criteria. Moreover, whenever gaps in seismic coverage occur, the inversion problem becomes locally under-determined. Therefore, the problem is never genuinely over-determined as often perceived. At best, the least-squares solutions for the seismic inverse problem can be distinguished by the space where the filtering is applied. Consequently for the remaining of the chapter I simply refer to the solution in equations equ4 and equ5 as data-space and model-space inverse.

The most important question one may ask, is when should we use one solution or the other. An important factor to take into account is the characteristics of $\bf L \bf L^T$ and $\bf L^T \bf L$, and in particular how difficult it is to compute their inverses. In most of the practical cases, both $\bf L \bf L^T$ and $\bf L^T \bf L$ are singular and care must be taken when approximating their inverse.