Kirchhoff imaging and irregular geometry

For idealized geometry, the previous analysis could ensure amplitude-preserved operators and well behaved matrices. Problems arise in 3D multichannel recording where the reality of seismic acquisition causes seismic data to be sampled in a sparse and irregular fashion. These irregularities are often observed in the form of variations in fold coverage resulting in an abundance of seismic traces in some bins and missing data in others.

Considering an imaging operator ${\bf F}$ (for instance $\bf F = \bf L^{-1}$), each row of ${\bf F}$ corresponds to an output bin and each column corresponds to a data trace. Due to the irregular coverage, the columns and rows of ${\bf F}$ are badly scaled and the matrix is ill-conditioned. Its condition can be improved by column scaling Ronen (1994).

Based on a similar approach, we propose two formulations for row and column normalization which we refer to as image normalization and data normalization. They involve pre- and post-multiplying the operator ${\bf F}$ by a diagonal matrix whose diagonal entries are the inverse of the sum of the rows or columns of ${\bf F}$.

Since Kirchhoff operators are associated with matrices that contain no negative elements, it is safe to use the sum of the elements. In case of negative entry values, we can sum the absolute values of the elements or compute the norms of the rows or columns. Similarly for the case of complex values, we should use an L₂ norm to compute the diagonal entries of the normalization operator; i.e., the square root of the sum of elements squared.