Isotropic angle-domain elastic reverse-time migration

Imaging with scalar and vector potentials

An alternative to the elastic imaging condition from equation  is to separate the extrapolated wavefield into P and S potentials after extrapolation and image using cross-correlations of the vector and scalar potentials (Dellinger and Etgen, 1990). Separation of scalar and vector potentials can be achieved by Helmholtz decomposition, which is applicable to any vector field ${\bf u}\left ({\bf x}, t \right)$:

$${\bf u}= \nabla { {\Phi}} + \nabla \times { {\boldsymbol{\Psi}}} \;,$$ (4)

where ${\Phi}\left ({\bf x}, t \right)$ represents the scalar potential of the wavefield ${\bf u}\left ({\bf x}, t \right)$ and ${\boldsymbol{\Psi}}\left ({\bf x}, t \right)$ represents the vector potential of the wavefield ${\bf u}\left ({\bf x}, t \right)$, and $\nabla \cdot { {\boldsymbol{\Psi}}}=0$. For isotropic elastic wavefields, equation  is not used directly in practice, but the scalar and vector components are obtained indirectly by the application of the divergence ( $\nabla \cdot {}$) and curl ( $\nabla \times {}$) operators to the extrapolated elastic wavefield ${\bf u}\left ({\bf x}, t \right)$:

$$P = \nabla \cdot {{\bf u}} = \nabla^2 { {\Phi}} \;,$$ (5)

$$\mathbf S = \nabla \times {{\bf u}} = -\nabla^2 { {\boldsymbol{\Psi}}} \;.$$ (6)

For isotropic elastic fields far from the source, quantities $P$ and $\textbf S$ describe compressional and transverse components of the wavefield, respectively (Aki and Richards, 2002). In 2D, the quantity $\textbf S$ corresponds to SV waves that are polarized in the propagation plane.

Using the separated scalar and vector components, we can formulate an imaging condition that combines various incident and reflected wave modes. The imaging condition for vector potentials can be formulated mathematically as

$${I}_{ij}\left ({\bf x}\right)= \int {\alpha_s}_{i}\left ({\bf x}, t \right){\alpha_r}_{j}\left ({\bf x}, t \right)dt \;,$$ (7)

where the quantities $\alpha_i$ and $\alpha_j$ stand for the various wave modes $\alpha=\{P, S\}$ of the vector source and receiver wavefields ${\bf u}\left ({\bf x}, t \right)$. For example, ${I}_{PP}\left ({\bf x}\right)$ represents the image component produced by cross-correlating of the $P$ wave mode of the source and receiver wavefields, and ${I}_{PS}\left ({\bf x}\right)$ represents the image component produced by cross-correlating of the $P$ wave mode of the source wavefield with the $S$ wave-mode of the receiver wavefield, etc. In isotropic media, an image produced with this procedure has four independent components at every location in space, similar to the image produced by the cross-correlation of the various Cartesian components of the vector displacements. However, in this case, the images correspond to various combinations of incident P or S and reflected P- or S-waves, thus having clear physical meaning and being easier to interpret for physical properties.

Isotropic angle-domain elastic reverse-time migration