A different approach, on which the elastic modeling and migration method described in Cunha (1992) are based, is to solve only the elastic wave equation, distinguishing the solid and liquid regions only by their different elastic parameters. The application of the migration method is based on the assumption that the particle displacement vector field is known at the surface. However, a significant part of seismic reflection data collected nowadays comes from offshore surveys, where the seismic waves propagating in the water are recorded by pressure-sensitive phones arranged along a cable.
To apply the migration method to practical problems it is imperative that a vector displacement wavefield be obtained from the recorded pressure field. I show that the conversion of the scalar recorded field into the elastic (vector) wavefield at the cable depth can be achieved by the application of a simple linear filter in the frequency-spatial-wavenumber domain.
One of the critical points of the vectorization is the implicit separation of the downgoing and upcoming waves at the cable depth. In the most general case this separation cannot be done for standard marine data and the use of two cables at different depth has been advocated by several authors as a way to solve the separation problem (see e.g. Monk (1990) for a general discussion). I show that with the help of a few assumptions it is possible to solve the wavefield separation problem for standard marine surveys. Although the particular application described here (vectorization of the recorded field) does not seem to be critically sensitive to these assumptions, other applications such as deghosting may prove to be unfeasible under this separation scheme. The assumptions are that the water surface is nearly horizontal and that the cable depth is a smooth function of the receiver position. The cancellation of the total wavefield at some vertical wavenumbers introduces a set of singular strings in the vectorizer operator. However, it is possible to remove these singularities and to design a stable operator with a small loss of resolution in events with apparent velocities near the water velocity.