Explicit expressions for prestack map time migration in isotropic and VTI media and the applicability of map depth migration in heterogeneous anisotropic media

We present 3D prestack map time migration in closed form for qP-, qSV-, and mode-converted waves in homogeneous transversely isotropic media with a vertical symmetry axis (VTI). As far as prestack time demigration is concerned, we present closed-form expressions for mapping in homogeneous isotropic media, while for homogeneous VTI media we present a system of four nonlinear equations with four unknowns to solve numerically. The expressions for prestack map time migration in VTI homogeneous media are directly applicable to the problem of anisotropic parameter estimation (i.e., the anellipticity parameter η) in the context of time-migration velocity analysis. In addition, we present closed-form expressions for both prestack map time migration and demigration in the common-offset domain for pure-mode (P-P or SS) waves in homogeneous isotropic media that use only the slope 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) domains. 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. Finally, we discuss the condition for applicability of prestack map depth migration and demigration in heterogeneous anisotropic media that allows the formation of caustics and explain that this condition is satisfied if, given a velocity model and acquisition geometry, one can map-depth-migrate without ambiguity in either the migrated location or the migrated orientation of reflectors in the image.