Isotropic angle-domain elastic reverse-time migration

Imaging with vector displacements

Assuming vector recorded data, wavefield extrapolation using a vector wave equation reconstructs source and receiver wavefields ${\bf u}_s\left ({\bf x}, t \right)$ and ${\bf u}_r\left ({\bf x}, t \right)$ at every location $\mathbf{x}$ in the subsurface. Here, ${\bf u}_s$ and ${\bf u}_r$ represent displacement fields reconstructed from data recorded by multicomponent geophones at the surface boundary. Using the vector extrapolated wavefields ${{\bf u}}_s=\{{u_s}_x,{u_s}_y,{u_s}_z\}$ and ${{\bf u}}_r=\{{u_r}_x,{u_r}_y,{u_r}_z\}$, an imaging condition can be formulated as a straightforward extension of equation  by cross-correlating all combinations of components of the source and receiver wavefields. Such an imaging condition for vector displacements can be formulated mathematically as

$${I}_{ij}\left ({\bf x}\right)= \int {W_s}{i}\left ({\bf x}, t \right){W_r}{j}\left ({\bf x}, t \right)dt \;,$$ (3)

where the quantities $u_i$ and $u_j$ stand for the Cartesian components ${x,y,z}$ of the vector source and receiver wavefields, ${\bf u}\left ({\bf x}, t \right)$. For example, ${I}_{zz}\left ({\bf x}\right)$ represents the image component produced by cross-correlating of the $z$ components of the source and receiver wavefields, and ${I}_{zx}\left ({\bf x}\right)$ represents the image component produced by cross-correlating of the $z$ component of the source wavefield with the $x$ component of the receiver wavefield, etc. In general, an image produced with this procedure has nine components at every location in space.

The main drawback of applying this type of imaging condition is that the wavefield used for imaging contains a combination of P- and S-wave modes. Those wavefield vectors interfere with one-another in the imaging condition, since the P and S components are not separated in the extrapolated wavefields. The crosstalk between various components of the wavefield creates artifacts and makes it difficult to interpret the images in terms of pure wave modes, e.g. PP or PS reflections. This situation is similar to the case of imaging with acoustic data contaminated by multiples or other types of coherent noise which are mapped in the subsurface using an incorrect velocity.

Isotropic angle-domain elastic reverse-time migration