Elastic imaging with exact wavefield extrapolation for application to ocean-bottom 4C seismic data

A central component of imaging methods is receiver-side wavefield backpropagation or extrapolation in which the wavefield from a physical source scattered at any point in the subsurface is estimated from data recorded by receivers located near or at the Earth’s surface. Elastic reverse-time migration usually accomplishes wavefield extrapolation by simultaneous reversed-time ‘injection’ of the particle displacements (or velocities) recorded at each receiver location into a wavefield modeling code. Here, we formulate an exact integral expression based on reciprocity theory that uses a combination of velocity-stress recordings and quadrupole-dipole backpropagating sources, rather than the commonly used approximate formula involving only particle velocity data and dipole backpropagating sources. The latter approximation results in two types of nonphysical waves in the scattered wavefield estimate: First, each arrival contained in the data is injected upward and downward rather than unidirectionally as in the true time-reversed experiment; second, all injected energy emits compressional and shear propagating modes in the model simulation (e.g., if a recorded P-wave is injected, both P and S propagating waves result). These artifacts vanish if the exact wavefield extrapolation integral is used. Finally, we show that such a formula is suitable for extrapolation of ocean-bottom 4C data: Due to the fluid-solid boundary conditions at the seabed, the data recorded in standard surveys are sufficient to perform backpropagation using the exact equations. Synthetic examples provide numerical evidence of the importance of correcting such errors.