Isotropic angle-domain elastic reverse-time migration

Scalar wavefields

For the case of imaging with the acoustic wave equation, the reflection angle corresponding to incidence and reflection of P-wave mode can be constructed after imaging, using mapping based on the relation (Sava and Fomel, 2005)

$$\tan\theta_a = \frac {\vert{\bf k}_ {\boldsymbol{\lambda}} \vert } {\vert{\bf k}_{\bf x}\vert } \;,$$ (11)

where $\theta_a$ is the incidence angle, and ${\bf k}_{\bf x}= {\bf k}_{\bf r}-{\bf k}_{\bf s}$ and ${\bf k}_ {\boldsymbol{\lambda}} = {\bf k}_{\bf r}+{\bf k}_{\bf s}$ are defined using the source and receiver wavenumbers, ${\bf k}_{\bf s}$ and ${\bf k}_{\bf r}$. The information required for decomposition of the reconstructed wavefields as a function of wavenumbers ${\bf k}_{\bf x}$ and ${\bf k}_ {\boldsymbol{\lambda}}$ is readily available in the images ${I}_{} \left ({\mathbf x}, {\boldsymbol{\lambda}} , \tau \right)$ constructed by extended imaging conditions equations  or . After angle decomposition, the image ${I}_{} \left ({\mathbf x},\theta,\phi \right)$ represents a mapping of the image ${I}_{} \left ({\mathbf x}, {\boldsymbol{\lambda}} , \tau \right)$ from offsets to angles. In other words, all information for characterizing angle-dependent reflectivity is already available in the image obtained by the extended imaging conditions.