Migration-velocity Building In Time And Depth From 3d (2d) Common-reflection-surface (crs) Stacking - Theoretical Framework

We demonstrate how the velocity concepts attached to conventional stacking assuming local smooth 1D type of Earth models, are modified within the setting of the Common Reflection Surface (CRS) method to handle lateral velocity variations. The corresponding matrix (scalar) normal moveout (NMO)- and Dix-velocities in 3D (2D) are now linked to a smooth velocity medium in depth sampled along the mapping or normal rays taking into account lateral velocities. This is rather different from the conventional 1D case where the link between the data-driven velocities and the smooth 1D velocity medium are represented by vertical mapping rays. Further and by analogy with the conventional 1D approach, where time-migration velocities are computed from NMO- or Dix-velocities after proper smoothing, its CRS counterpart is established. It is demonstrated that matrix (scalar) time-migration velocities to be used in ray-based 3D (2D) time-migration (TM) can be obtained by proper mapping of the corresponding CRS velocities. Relevant mapping equations valid for the 2D case have earlier been treated in the literature. However, to our knowledge, this is the first time such mapping equations have been derived for the 3D case. This mapping takes into account smooth lateral velocity variations, normally not properly accounted for in the initial model employed in conventional time-migration building based on NMO-velocities. Also, unlike the conventional approach, the use of the CRS method makes it feasible to build a smooth velocity field in depth beyond that of 1D. In this connection, we propose to use Normal Incidence Point (NIP) tomography (alternatively Image Incidence Point (IIP) tomography) to build a macro-velocity model upon which the time-to-depth mapping and the correction factors can be calculated. That step is then followed by construction of an updated smooth replacement medium, better able to capture local changes in velocity. The proposed approach can be used iteratively. Such depth velocities can be employed as an initial model for iterative depth migration. The main idea being that velocity-model building by using time-domain observables ensures more robustness