Isotropic angle-domain elastic reverse-time migration

Extended elastic imaging conditions

The conventional imaging condition from equation  discussed in the preceding section uses zero space- and time-lags of the cross-correlation between the source and receiver wavefields. This imaging condition represents a special case of a more general form of an extended imaging condition (Sava and Fomel, 2006b)

$${I}_{} \left ({\mathbf x}, {\boldsymbol{\lambda}} , \tau \r... ...ft ({\mathbf x}+ {\boldsymbol{\lambda}} ,t+\tau \right)dt \;,$$ (8)

where ${\boldsymbol{\lambda}} =\{\lambda_x,\lambda_y,\lambda_z\}$ and $\tau$ stand for cross-correlation lags in space and time, respectively. The imaging condition from equation  is equivalent to the extended imaging condition from equation  for ${\boldsymbol{\lambda}} =\mathbf{0}$ and $\tau =0$.

The extended imaging condition has two main uses. First, it characterizes wavefield reconstruction errors, since for incorrectly reconstructed wavefields, the cross-correlation energy does not focus completely at zero lags in space and time. Sources of wavefield reconstruction errors include inaccurate numeric solutions to the wave-equation, inaccurate models used for wavefield reconstruction, inadequate wavefield sampling on the acquisition surface, and uneven illumination of the subsurface . Typically, all these causes of inaccurate wavefield reconstruction occur simultaneously and it is difficult to separate them after imaging. Second, assuming accurate wavefield reconstruction, the extended imaging condition can be used for angle decomposition. This leads to representations of reflectivity as a function of angles of incidence and reflection at all points in the imaged volume (Sava and Fomel, 2003). Here, we assume that wavefield reconstruction is accurate and concentrate on further extensions of the imaging condition, such as angle decomposition.

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Figure 1. Local wave vectors of the converted wave at a common image point location in 3D. The plot shows the conversion in the reflection plane in 2D. ${\bf p}_{\bf s}$, ${\bf p}_{\bf r}$, ${\bf p}_{\bf x}$ and ${\bf p}_ {\boldsymbol{\lambda}}$ are ray parameter vectors for the source ray, receiver ray, and combinations of the two. The length of the incidence and reflection wave vectors are inversely proportional to the incidence and reflection wave velocity, respectively. Vector ${\bf n}$ is the normal of the reflector. By definition, ${\bf p}_{\bf x}={\bf p}_{\bf r}-{\bf p}_{\bf s}$ and ${\bf p}_ {\boldsymbol{\lambda}} ={\bf p}_{\bf r}+{\bf p}_{\bf s}$ .

Isotropic angle-domain elastic reverse-time migration