Kirchhoff imaging techniques have been widely used on 3D datasets since they can be applied to any subset of the full prestack data and, presumably, can handle irregular geometry. Mathematical derivations of integral operators assume continuous wavefields. In practice, the resulting imaging algorithms are only applied to discretely sampled seismic data and their implementations simply reduce to a matrix-vector multiplication. Due to irregular coverage, the matrix is ill-conditioned and the linear systems that need to be solved may be badly scaled. It is therefore advisable to normalize the discrete summations (, ).
Chemingui and Biondi proposed a simple technique to compensate for the effects of irregular fold distributions. The method extends the multiplicity concept in CMP stacking to wave equation processes. It normalizes each input trace in the prestack process according to the local stacking fold of its corresponding bin. The technique has the advantage of being independent of the imaging operator and depends only on the acquisition geometry. However, it cannot account for the time and space variability of the operators, their limited aperture, or for varying velocity functions. In this paper we propose two alternative approaches for fold normalization depending on the numerical implementation of the algorithms. We relate the weights to the concept of ``fold'' in Kirchhoff imaging and show that the technique is essentially equivalent to calibrating the image by the response of a flat event.
Based on the definition of data-space pseudo inverse (), we apply the normalization technique to estimate an inverse for the cross-product operator in the two-step solution of Chemingui and Biondi (). The approximate solution enables an efficient computation for a data covariance matrix, and thus, for the fold equalization problem. This covariance matrix corrects the imaging operator for interdependencies between data-elements. It is computationally more attractive than the numerical solution based on iterative solvers.
To go beyond the normalized pseudo inverse, we can use the diagonal transformation as a preconditioner for iterative solutions. Chemingui and Biondi presented an inversion technique that is suitable for sparse and uneven sampling and takes advantage of the abundance of seismic traces in multifold seismic data to interpolate beyond aliasing. The technique, which we now refer to as Inversion to Common Offset (ICO), is based on least-squares theory and wave-equation interpolation for missing data using the azimuth-moveout operator (). To guide the iterative optimization to the desired solution, we precondition the inversion by the normalization operator. The diagonal transformation proves to be a suitable preconditioner for the dealiasing inversion.
In this paper we first discuss discrete Kirchhoff methods and their associated algorithms. We relate the implementations to the concepts of accuracy of operators and proper handling of the irregular geometry in true-amplitude imaging. Next we present the two formulations for column and row normalization which we refer to as data and model normalization. Finally we demonstrate the efficiency of the diagonal transformation for normalizing the fold, providing better approximation of a pseudo-inverse and defining an operator that is as unitary as possible and well preconditioned for linear solvers.