Convergence analysis of a coherency-constrained inversion for the separation of blended data

Blended or simultaneous sources have been the focus of a good deal of interest recently. In conventional acquisition the time intervals between successive sources are large enough to avoid interference in time. In blended acquisition, temporal overlap between source responses is allowed. The procedure of retrieving data as if they were acquired in the conventional way is called deblending. This is an essential step if standard processing flows are to be applied. Several inversion techniques have been proposed for solving this ill-posed problem. We study the properties of an iterative estimation and subtraction algorithm that integrates a coherency-pass filter in a dedicated iteration. We begin by stating the problem we wish to solve and develop a new, more general, interpretation of the method. We then apply an algebraic analysis of the iteration to establish the convergence characteristics of the algorithm. In order to facilitate this analysis, the notion of leakage subspace is introduced, i.e., a subspace where energy that cannot be uniquely assigned to one of the sources resides. We find that a unique solution exists, if, and only if, there is no leakage subspace. If a unique solution does not exist, then the iteration converges to a least-norm solution contaminated by the projection of the initial guess onto the leakage subspace. The insights gained by this analysis lead us to the development of a simple tool that can provide valuable information during the design of a blended survey. Finally we present results from the application of this method to real blended marine data and then draw our conclusions.