Methods and apparatuses for processing marine seismic data with a process of combined deghosting and sparse τ-p transformation. The process is formulated as an optimization problem. The optimization problem has an objective function that is a weighted sum of two norms: one norm is an Lp norm of the differences between the modeled data and acquired survey wherein the modeled data are derived from a model and a set of adaptive filters; the other norm is an Lq norm of the model; and the optimization variables and solutions are the coefficients of the model and coefficients of the adaptive filters.