Angle domain common image gathers for steep reflectors

Angle domain CIGs by reverse-time migration

Reverse-time migration, based on the two-way wave equation, handles high-angle energy and overturned waves naturally. In downward continuation migration, source and receiver wavefields are extrapolated along the $z$-axis and the subsurface offset direction (the horizontal direction) is normal to the extrapolation direction (the vertical direction). In contrast, in reverse-time migration the source wavefield $S=S(s_x,x,z,t)$ and the receiver wavefield $R=R(s_x,x,z,t)$ are extrapolated along the time axis, where $s_x$ is the source location, $x$ is the horizontal location, $z$ is the vertical location and $t$ is the travel-time. There is no functional difference between the $x$-axis and $z$-axis. Therefore, we can obtain general-direction subsurface offset CIGs in reverse-time migration and conventional horizontal offset and vertical offset are only two special cases (Biondi and Shan, 2002). As with downward continuation migration, in reverse-time migration horizontal offset domain CIGs are formed by cross-correlating source and receiver wavefields with a horizontal shift $h_x$ as follows:

$$I_x(x,z,h_z)=\int\int S(s_x,x-h_x,z,t)R(s_x,x+h_x,z,t) dtds_x,$$ (3)

where the shift $h_x$ is called horizontal subsurface offset. Similarly, vertical offset domain CIGs are formed by cross-correlating source and receiver wavefields with a vertical shift $h_z$ as follows:

$$I_z(x,z,h_z)=\int\int S(s_x,x,z-h_z,t)R(s_x,x,z+h_z,t) dtds_x,$$ (4)

where the shift $h_z$ is called vertical subsurface offset.

As with downward continuation migration, we can apply equation 2 to transform the horizontal offset domain CIGs $I_x(x,z,h_x)$ to angle domain CIGs $I_x(x,z,\gamma)$. Similarly, we can also transform the vertical offset domain CIGs $I_z(x,z,h_z)$ to angle domain CIGs $I_z(x,z,\gamma)$ as follows:

$$\tan\gamma=-\frac{k_{h_z}}{k_x},$$ (5)

where $k_{h_z}$ and $k_x$ are wavenumbers corresponding to $h_z$ and $x$, respectively. Horizontal CIGs work well for flat reflectors but they are not reliable for steep reflectors, while vertical CIGs are good for steep reflectors. Both vertical and horizontal CIGs are not robust for an area with complex geology, where reflectors have a full range of dips. For a image point, the subsurface offset that parallels the dip direction of the reflector is called geologic offset. CIGs would be robust if we used geologic offset for each image point. However, it is too expensive to generate geologic offset CIGs directly. Biondi and Symes (2004) demonstrate that the geologic offset $h_0$, horizontal offset $h_x$, and vertical offset $h_z$ can be linked by the following relationships:

$$h_{x} = \frac{h_0}{\cos(\alpha)},$$ (6)

$$h_{z} = \frac{h_0}{\sin(\alpha)},$$ (7)

where $\alpha$ is the dip angle of the reflector. The relationships (equations 6 and 7) also show why horizontal CIGs fail at steeply dipping reflectors. Large horizontal subsurface offset is needed to get reliable angle domain CIGs for a steep reflector. For the extreme case that the reflector is vertical, from equation 6 we need infinite horizontal subsurface offset.

Although neither vertical nor horizontal CIGs are robust, robust angle domain CIGs can be constructed by merging them as follows (Biondi and Symes, 2004):

$$I(x,z,\gamma)=\cos^2 \alpha(x,z) I_x(x,z,\gamma)+\sin^2 \alpha(x,z)I_z(x,z,\gamma),$$ (8)

where $\alpha(x,z)$ is the dip angle at the location $(x,z)$. Equation 8 is performed in the Fourier domain $(k_x,k_z)$, in which the dip angle of the reflector can be calculated accurately.

Angle domain common image gathers for steep reflectors