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Surface-related multiple reflections are often considered noise in the seismic reflection measurements. By seismic processes, such as migration and inversion, they can be mistakenly seen as primary reflections and give an erroneous image of the earth. Therefore, a method is needed to separate the multiples from the primaries. This thesis describes a primary estimation method named estimation of primaries by sparse inversion (EPSI). The interesting aspect of this method is that it does not see the multiples as noise, but uses the multiples to come to a better estimation of the primaries. Other wave equation based primary estimation methods first predict the multiples and then adaptively subtract them from the data. During this adaptive subtraction primary energy may be removed. EPSI tries to explain the total data, both primaries and multiples, in terms of primary impulse responses. By doing so an adaptive subtraction of multiples is avoided. In fact EPSI is a large-scale inversion process that estimates primaries such that they and their corresponding multiples explain the total data. To constrain this process, a sparseness constraint is used, which assumes that our estimated primaries have a certain amplitude distribution (large and small ones). A general characteristic of wave equation based primary estimation methods is that the near-offset data are very important for estimating water column reverberations, especially in shallow water. However, it is not possible to record the near-offset data and, therefore, most wave equation based primary estimation methods have great difficulties with shallow water marine data. A major advantage of EPSI is that it can reconstruct the missing near-offset data from information in the multiples and, therefore, show a good primary estimation result on shallow water marine data. Furthermore, the EPSI method can be extended to other measurement configurations, exploiting a similar relation between primaries and surface multiples. In this thesis both passive and blended seismic data have been considered. For passive seismic data multiples are used to obtain an estimate of the subsurface responses, usually by a cross-correlation process. This cross-correlation process relies on the assumption that the surface has been illuminated uniformly by subsurface sources in terms of incident angles and strength. If this is not the case the cross-correlation process cannot give a true amplitude estimation of the subsurface response. Furthermore, there are cross terms in the cross-correlation result that are not related to actual subsurface inhomogeneities. In this thesis it is demonstrated that, with some modifications to the algorithm, EPSI can obtain true amplitude subsurface responses without the uniform surface illumination assumption. The EPSI method will go beyond the cross-correlation process and will estimate primaries only from the multiples in the available signal. The estimated primary impulse responses, with point sources and receivers at the surface, can be used directly in traditional imaging schemes. This thesis demonstrates that for the situation of blended acquisition, meaning that different sources are shooting in a time-overlapping fashion, multiples can be used to 'deblend' the seismic measurements. With some modifications the EPSI method can be used for blended seismic data. As output EPSI gives unblended primary impulse responses with point sources and receivers at the surface, which can be used directly in traditional imaging schemes. The feasibility of the EPSI method is demonstrated in this thesis by a successful application of the method to two marine field datasets, one with a moderate water depth and one with shallow water. It demonstrates that for deeper water EPSI can compete with the standard surface-related multiple elimination (SRME) method, where for the shallow water EPSI clearly shows better results than SRME. The latter is mainly attributed to the fact that near offset reconstruction, which plays a crucial role in shallow water data, is included in the EPSI method.