Estimation of primaries by sparse inversion incuding the ghost

Today, the problem of surface-related multiples, especially in shallow water, is not fully solved. Although surface-related multiple elimination (SRME) method has proved to be successful on a large number of data cases, the involved adaptive subtraction acts as a weak link in this methodology, where primaries can be distorted due to their interference with multiples. Therefore, recently, SRME has been redefined as a large-scale inversion process, called estimation of primaries by sparse inversion (EPSI). In this process the multi-dimensional primary impulse responses are considered as the unknowns in a largescale inversion process. By parameterizing these impulse responses as spikes in the space-time domain, and using a sparsity constraint in the update step, the algorithm looks for those primaries that, together with their associated multiples, explain the total input data. As the objective function in this minimization process truly goes to zero, the tendency for distorting primaries is greatly reduced. An additional advantage is that imperfections in the data can be included in the forward model and resolved simultaneously, such as the missing near offsets. In this paper it is demonstrated that the ghost effect can also be included in the EPSI formulation after which a ghost-free primary estimate can be obtained, even in the case the ghost notch is within the desired spectrum.