Preliminary Investigation of Wavefield Depth Extrapolation by Two-Way Wave Equations

Most of the wavefield downward continuation migration approaches are relying on one-way wave equations, which move the seismic energy always in one direction along depth. The one-way downward continuation migrations only use the primaries for imaging and do not treat secondary reflections recorded on the surface correctly. In this paper, we investigate wavefield depth extrapolators based on the full acoustic wave equations, which can propagate wave components to opposite directions. Several two-way wavefield downward continuation propagators are numerically tested in this study. Recursively implementing of the depth extrapolator makes it necessary and important to eliminate the unstable wave modes, that is, evanescent waves. For the laterally varying velocity media, distinction between the propagating and evanescent wave mode is less clear. We demonstrate that the spatially localized two-way beamlet propagator is an effective way to remove the evanescent waves while maintain the propagating mode in laterally inhomogeneous media. 1. Introduction Downward continuation migration calculates the wavefield at greater depth based on the existing wavefield at the shallower depth. For each frequency component, the wavefield can be downward continued recursively from surface to target depth. These algorithms have the flexibility of migrating the seismic data sequentially in depth and frequency, which leads to substantial reduction of both computational and memory requirements. It is particularly advantageous in a migration-velocity-analysis procedure, since one can analyze one layer for each iteration (layer stripping). It also leads to the definition of another useful family of seismic imaging methods—survey-sinking migration [1]. Although many different downward continuation algorithms have been proposed, most of them are based on solving the one-way wave equation. The common ground for those methods is splitting the full wave equation into two one-way wave equations that allow for downward or upward wave propagation separately. The directional splitting of the operator suppresses the up-going propagating waves, thus making it difficult, if not impossible, to use the secondary reflections correctly for imaging [2]. The wavefield downward continuation scheme based on the full acoustic wave equation was first introduced by Kosloff and Baysal [3]. For a background with depth-dependent velocity and zero offset source-receiver configuration, the full wave partial differential equation is changed to a second-order ordinary differential equation, and they solved the