Normalization of Kirchhoff operators

For ideal acquisition geometries, the previous analysis ensures amplitude-preserving operators defined by well-behaved, full-rank matrices. Problems arise in 3D because of the the irregular coverage of seismic surveys which results 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 not balanced and the matrix is ill-conditioned. To improve its condition, I propose two formulations for row and column scaling which I 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 ${\bf F}$. The sum is always positive since Kirchhoff operators are associated with matrices that contain no negative elements.