Closed-form expressions for map time-migration in VTI media and the applicability of map depth-migration in the presence of caustics

Provided the velocity of the medium is known and the the medium does not allow different reflectors to have identical surface seismic measurements that persist under small perturbations of the medium, map migration achieves a one-to-one mapping from surface seismic data to the subsurface seismic image by using the slopes in addition to the location (and time) of the events in the data. In this paper we present 3D pre-stack map time-migration in closed form for P-waves in homogeneous isotropic and qP-waves in VTI media, and discuss the condition for applicability of pre-stack map depth-migration and demigration in the presence of caustics. As far as pre-stack time-demigration is concerned, we present closed-form expressions for the mapping in isotropic homogeneous media, while for homogeneous VTI media we derive a system of four nonlinear equations with four unknowns that needs to be solved numerically. In addition we present closed-form expressions for both pre-stack map time-migration and demigration in the common-offset domain for homogeneous isotropic media that use only the slopes in the common-offset domain as opposed to slopes in both the common-shot and common-receiver (or equivalently the common-offset and common-midpoint) domain. All time-migration and demigration equations presented can be used in media with mild lateral and vertical velocity variations, provided the velocity is replaced with the local RMS velocity. The expressions for pre-stack map time-migration in VTI homogeneous media can be used for anisotropic parameter estimation (i.e., the anellipticity parameter η) in the context of time-migration velocity analysis.