Sparse Inversion for Solving the Coupled Marchenko Equations Including Free-surface Multiples

We compare the coupled Marchenko equations without free-surface multiples to the coupled Marchenko equations including free-surface multiples. When using the conventional method of iterative substitution to solve these equations, a difference in convergence behaviour is observed, suggesting that there is a fundamental difference in the underlying dynamics. Both an intuitive explanation, based on an interferometric interpretation, as well as a mathematical explanation, confirm this difference, and suggest that iterative substitution might not be the most suitable method for solving the system of equations including free-surface multiples. Therefore, an alternative method is required. We propose a sparse inversion, aimed at solving an under-determined system of equations. Results show that the sparse inversion is indeed capable of correctly solving the coupled Marchenko equations including free-surface multiples, even when the iterative scheme fails. Using sparsity promotion and additional constraints, it is expected to perform better than iterative substitution when working with incomplete data or in the presence of noise. Also, simultaneous estimation of the source wavelet is a potential possibility.