Angle domain common image gathers for steep reflectors

Angle domain CIGs by downward continuation migration

Both shot-profile and source-receiver migrations can generate subsurface offset domain CIGs that are equivalent to each other (Shan and Zhang, 2003). In shot-profile migration, the source wavefield $S(s_x,x,z,\omega)$ and receiver wavefield $R(s_x,x,z,\omega)$ are extrapolated into all depths, where $s_x$ is the source location, $x$ is the horizontal location, $z$ is the vertical location, and $\omega$ is the angular frequency. Offset domain CIGs $I(x,z,h_x)$ are formed by cross-correlating the source and receiver wavefields with a horizontal shift $h_x$ as follows (Rickett and Sava, 2002):

$$I(x,z,h_x)=\int\int S^*(s_x,x-h_x,z,\omega)R(s_x,x+h_x,z,\omega)d\omega ds_x$$ (1)

where the horizontal shift $h_x$ is called horizontal subsurface offset, and $S^*$ is the conjugate complex of the source wavefield $S$. And $I(x,z,h_x=0)$ is the conventional image. Offset domain CIGs $I(x,z,h_x)$ are transformed to angle domain CIGs $I(x,z,\gamma)$ by applying local slant-stacking in the space domain or radial-trace transform in the Fourier domain using the relationship as follows (Sava and Fomel, 2003):

$$\tan\gamma=-\frac{k_{h_x}}{k_z},$$ (2)

where $k_{h_x}$ and $k_z$ are wavenumbers corresponding to $h_x$ and $z$, respectively.

For near-flat reflectors, angle domain CIGs obtained by downward continuation migration using equations 1 and 2 are reliable. However, there are two issues in downward continuation migration that impose difficulty in obtaining reliable CIGs of steeply dipping reflectors. First, steeply dipping reflectors are mainly illuminated by high angle and overturned waves, but these waves cannot be modeled accurately by downward continuation migration. Second, because of the stretch of the horizontal subsurface offset at steeply dipping reflectors, we cannot obtain reliable angle domain CIGs from horizontal offset domain CIGs. Given an opening angle, the steeper the reflector is, the larger the horizontal subsurface offset is needed to get a reliable angle domain CIG. However, the length of the subsurface offset is limited in shot-profile migration to save the cost. Therefore, we cannot get useful angle domain CIGs of steep reflectors from horizontal offset domain CIGs.

In the next two sections, we discuss how the two issues in downward continuation migration are solved by reverse-time migration and plane-wave migration in tilted coordinates.

Angle domain common image gathers for steep reflectors