Conclusions

We have presented new developments for accurate implementations of discrete Kirchhoff operators as matrix-vector multiplication. The main process is based on the normalization of the Kirchhoff matrix by a diagonal transformation using the sum of the rows (summation surfaces) and columns (impulse responses). The normalization operator is designed in consistency with the numerical implementation of the Kirchhoff operator as pull (sum) or push (spray) operator. The final image is normalized by a reference model that is the operator's response to an input vector with all components equal to one (flat event).

We also presented an explicit formulation of a data covariance matrix for the solution of two-step inversion of irregularly sampled data. This data covariance is an AMO matrix that measures the correlations among data elements and corrects the imaging operator for the effects of irregular sampling.

Beyond the fold normalization, the diagonal transformations have proved to be a suitable preconditioner for the Inversion to Common Offset (ICO). It accelerates the convergence of the iterative solution and, therefore, enables a cost effective technique for 3D dealiasing inversion.