We explore the classic signal and noise separation problem of removing linear events from shot-gathers through several inversion schemes using a combined modeling operator composed of both hyperbolic and linear radon transforms. Data are inverted simultaneously for both linear and hyperbolic moveout which provides two model-space outputs. These are forward modeled seperately to provide an output data-space devoid of one of the model-space components. We employ this approach to analyze the removal of direct arrivals and ground-roll from shot-gathers. Inversion schemes used imnclude: 1) bound-constrained, 2) Cauchy norm regularization, 3) Huber norm approximating the l1 norm, and 4) the l2 norm using linear least-squares. Synthetic tests and four field shot-gathers are used to demonstrate the approach. Methods 1, 2, and 3 are designed to provide sparse model-space inversions. In the real data examples, the least-squares solution is able to better achieve the signal to noise separation goal despite its model-space being often less appealing.