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Migration and dip‐moveout seismic processes expressed as an invariant two‐dimensional convolution

Migration and dip‐moveout (partial prestack migration) are two‐dimensional (space and time) processes of fundamental importance in digital processing of seismic data. Their goal is to make the seismic section appear similar to the geological image along the seismic line subsurface. In this article we study migration and dip‐movement processes from the point of view of their impulse responses. The constant‐velocity migration process is a space‐invariant and time‐variant operator. We demonstrate and illustrate by examples of synthetic and real seismic sections that by transforming the time axis with a square function the constant‐velocity migration operator becomes temporally stationary as well as spatially stationary, that is, it can be expressed as an invariant two‐dimensional convolution. The dip‐moveout seismic process can be applied to constant‐offset seismic sections or to shot seismic profiles. The application of dip‐moveout process to constant‐offset sections is also a temporally varying and spatially stationary operator. By transforming the time axis with a logarithmic function the constant‐offset dip‐moveout operator becomes temporally stationary as well as spatially stationary. The shot dip‐moveout operator is space variant and time variant. After a logarithmic transformation of both the time and the space coordinates, it becomes time invariant and space invariant. Therefore the dip‐moveout seismic processes can be also expressed as an invariant two‐dimensional convolution. We illustrate this by examples of synthetic seismic sections.