Angle domain common image gathers for steep reflectors

Angle gathers by plane-wave migration in tilted coordinates

Reverse-time migration solves the two issues in downward continuation migration in generating CIGs for steep reflectors, but it is well known that it is expensive to apply reverse-time routinely. Plane-wave migration in tilted coordinates has been demonstrated useful imaging technology for steep reflectors (Shan and Biondi, 2004; Shan et al., 2007). In plane-wave migration in tilted coordinates, the propagation direction of the waves illuminating steeply dipping reflectors is usually close to the extrapolation direction and thus they can be imaged correctly. In this section, we discuss how to generate angle domain CIGs by plane-wave migration in tilted coordinates and show that it can also produce reliable CIGs for steep reflectors. We start with CIGs in the conventional plane-wave migration.

As with shot-profile migration, offset domain CIGs in plane-wave migration are formed as follows:

$$I(x,z,h_x)=\int\int \omega S^*(p_x,x-h_x,z,\omega)R(p_x,x+h_x,z,\omega)d\omega dp_x ,$$ (9)

where $h_x$ is the horizontal subsurface offset, $S(p_x,x,z,\omega)$ and $R(p_x,x,z,\omega)$ are the source and receiver wavefields corresponding to the ray parameter $p_x$, respectively. Notice that the imaging condition in equation 9 is the cross-correlation between the source and receiver wavefields weighted with the angular frequency $\omega$, which is also called $\rho$-filter in Radon transform literature. As with the conventional zero-subsurface-offset image, offset domain CIGs defined in equation 9 are equivalent to those obtained by shot-profile migration. Offset domain CIGs are transformed to angle domain CIGs by local slant-stacking (equation 2).

bpvel
Figure 1. Velocity model of the BP velocity Benchmark.

imagetilt
Figure 2. Image obtained by plane-wave migration in tilted coordinates. Both steep salt flank and near-flat sediments are present in this area.

Given a plane-wave source corresponding to the ray parameter $p_x$, we use the tilted coordinates $(x^\prime,z^\prime)$ with a tilting angle $\theta$. The subsurface offset domain CIGs for this plane-wave source are formed by:

$$I_{p_x}(x^\prime,z^\prime,h_{x^\prime})=\int \omega S^*(p_x... ...me,\omega)R(p_x,x^\prime+h_{x^\prime},z^\prime,\omega)d\omega.$$ (10)

where the subsurface offset $h_{x^\prime}$ parallels the $x^\prime$ axis. In plane-wave migration in tilted coordinates, the subsurface offset direction is not necessary the geologic dip direction, but is usually closer to the dip direction for steeply dipping reflectors, than the conventional horizontal subsurface offset. As for the transformation in the conventional plane-wave migration, we can transform offset domain CIGs $I_{p_x}(x^\prime,z^\prime,h_{x^\prime})$ of plane-wave source corresponding to $p_x$ to angle domain CIGs $I_{p_x}(x^\prime,z^\prime,\gamma)$ in tilted coordinates by applying

$$\tan\gamma=-\frac{k_{h_{x^\prime}}}{k_{z^\prime}},$$ (11)

where $k_{h_{x^\prime}}$ and $k_{z^\prime}$ are wavenumbers corresponding to $h_{x^\prime}$ and $z^\prime$, respectively. For each angle $\gamma$, we rotate the image $I_{p_x}(x^\prime,z^\prime,\gamma)$ back to vertical Cartesian coordinates. The angle domain CIGs of all possible plane-wave sources are then stacked in vertical Cartesian coordinates.

We can also transform the subsurface offset CIGs obtained by plane-wave migration in tilted coordinates into horizontal offset and vertical offset CIGs, and merge them using equation 8 after transforming them into angle domain CIGs, similarly to reverse-time migration. Equations 6 and 7 are the relationships linking the geologic offset $h_0$, horizontal offset $h_x$ and vertical offset $h_z$. The horizontal and vertical offsets are two special cases and the relationship can be generalized to a general-direction offset. If the angle between the general-direction offset $\bar{h}$ and geologic offset $h_0$ is $\beta$, the relationship between them is

$$\bar{h}=\frac{h_0}{\cos\beta}.$$ (12)

The angle $\beta$ in equation 12 for $\bar{h}=h_x$ is $\alpha$ and for $\bar{h}=h_z$ is $90^\circ-\alpha$. From equation 12, the geologic offset $h_0$ is the optimal offset to generate angle domain CIGs and the further the offset direction is from the dip direction, the larger the subsurface offset we need given the same opening angle. For the tilted coordinate system $(x^\prime,z^\prime)$, the angle between the subsurface offset and geologic offset is $\theta-\alpha$. Therefore, the subsurface offset $h_{x^\prime}$ in tilted coordinates and the geologic offset $h_0$ can be linked by the following relationship:

$$h_{x^\prime}=\frac{h_0}{\cos(\theta-\alpha)}.$$ (13)

From equations 13, 6 and 7, the subsurface offset in tilted coordinates $h_{x^\prime}$, vertical offset $h_z$ and horizontal offset $h_x$ are linked by the following relationship:

$$h_x=\frac{ h_{x^\prime}\cos(\theta-\alpha) }{\cos\alpha},$$ (14)

$$h_z=\frac{ h_{x^\prime}\cos(\theta-\alpha) }{\sin\alpha}.$$ (15)

By equations 14 and 15, the offset domain CIGs in tilted coordinates $I_{x^\prime}(x^\prime,z^\prime,h_{x^\prime})$ can be decomposed into horizontal offset CIGs and vertical offset CIGs. Vertical offset domain CIGs and horizontal offset domain CIGs of all possible plane-wave sources are stacked after being rotated back to vertical Cartesian coordinates. Being transformed to angle domain CIGs, they are merged using equation 8, as with reverse-time migration.

hxgathers Figure 3. Horizontal offset domain CIGs with the true velocity obtained by reverse-time migration. (a) For relatively flat sediments, the energy focuses well at zero offset; (b) For steep salt flanks, the energy leaks to far offsets and the frequency is low.

hzgather
Figure 4. Vertical offset domain CIGs with the true velocity obtained by reverse-time migration. For the steeply dipping salt flank at $x=33.2$ km, the energy focuses well at zero offset. For the near-flat sediments at $x=24.5$ km, the energy leaks to far offsets.

Angle domain common image gathers for steep reflectors