Angle-Domain CIGs in Tilted Cartesian coordinates

In shot-profile migration, offset-domain CIGs can be obtained by cross-correlating the source and receiver wavefields with a horizontal shift Rickett and Sava (2002). The offset-domain CIGs can be transformed into ADCIGs by slant-stack Sava and Fomel (2003). In reverse-time migration, in addition to horizontal offset-domain CIGs, we have vertical offset-domain CIGs, which are obtained by cross-correlating the source and receiver wavefields with a vertical shift Biondi and Shan (2002). Both horizontal and vertical offset-domain CIGs can be transformed into ADCIGs and merged into dip-dependent CIGs, by transforming the horizontal and vertical offsets into apparent geological offset as follows Biondi and Symes (2003):

$$\begin{eqnarray} h_x&=&\frac{h_0}{\cos\alpha},\\ h_z&=&\frac{h_0}{\sin\alpha},\end{eqnarray}$$ (2) (3)

where $\alpha$ is the dip angle of the subsurface reflector, h_x is the horizontal offset, h_z is the vertical offset and h₀ is the apparent geological offset.

 

mar.green
Figure 2 The Marmousi model: (a) wavefields obtained by downward continuation with FFD; (b) wavefields obtained by plane-wave decomposition and extrapolation in tilted coordinates.

In the tilted coordinates, wavefields are extrapolated in the $z^\prime$ direction. The offset-domain CIGs are generated by cross-correlating the source and receiver wavefields with an $x^{\prime}$ direction shift. So the subsurface offset is in the $x^{\prime}$ direction. As with the apparent geological offset h₀, the $x^{\prime}$ direction offset $h_{x^{\prime}}$ can be transformed to horizontal and vertical offsets as follows:

$$\begin{eqnarray} h_x&=&\frac{h_{x^\prime}}{\cos\theta},\\ h_z&=&\frac{h_{x^\prime}}{\sin\theta},\end{eqnarray}$$ (4) (5)

where $\theta$ is the tilting angle in Figure 1. As for reverse-time migration, horizontal and vertical offset-domain CIGs can be transformed into ADCIGs and merged into dip-dependent ADCIGs. A simple way to merge them is with the following weights:

$$\begin{eqnarray} w_{h_x}&=&\cos^2\alpha, \\ w_{h_z}&=&\sin^2\alpha,\end{eqnarray}$$ (6) (7)

where $\alpha$ is the apparent dip angle of the reflector. Dip-dependent residual moveout Shan and Biondi (2003) can be used to analyze dip-dependent ADCIGs to provide useful moveout information for velocity analysis.

In tilted coordinates, the direction of subsurface offset is close to that of apparent geological offset, since the extrapolation direction of the wavefields is close to the propagation direction of the waves. Within a limited length of subsurface offsets, we can obtain much more accurate CIGs at steeply dipping reflectors than with standard downward continuation.