Residual moveout in the Fourier domain

Conventional residual moveout is usually performed in the space domain. It assumes that the reflectors are flat, and thus image points only need to move vertically to flatten the ADCIGs. However, in dip-dependent residual moveout, the image points are moved along the direction normal to the reflector to flatten the ADCIGs.

It is difficult to calculate an accurate dip map for an image in the space domain, but the dip map can be obtained easily in the Fourier domain by the following relation:  

$$\tan \alpha=\frac{k_x}{k_z},$$ (3)

where $\alpha$ is the dip angle, and k_x and k_z are the wave numbers of x and z direction, respectively. In the local versions of equation (1) and (2), the shift along the direction normal to the reflectors depends only on the dip angle $\alpha$ and the opening angle $\gamma$. In the Fourier domain, the dip angle $\alpha$ is calculated by equation (3), and the opening angle $\gamma$ by

$$\tan \gamma=\frac{k_h}{k_z},$$ (4)

where k_h is the wave number of offset h Sava and Fomel (2003). Let $\Delta {n}$ be the shift along the direction normal to the reflectors. From the geometric relation, the shift along the normal direction is equivalent to a horizontal shift $\Delta x=\sin \alpha \Delta {n}$ followed by a vertical shift $\Delta z=\cos \alpha \Delta {n}$. Figure shows, for an ADCIG at a reflection point with a dip angle of $\alpha$, the relation between normal direction shift $\Delta {n}$, horizontal shift $\Delta x$ and vertical shift $\Delta z$.

 

geometry
Figure 2 Geometric relation between the shift in normal direction $\Delta {n}$, the horizontal shift $\Delta x$ and the vertical shift $\Delta z$.

A shift in the space domain is equivalent to a phase shift in the Fourier domain. Let I(x,z,h) be the image cube obtained by migration, and I(k_x,k_z,k_h) is its Fourier transformation. Then in the space domain, the shift $\Delta {n}$ along the direction normal to the reflector is equivalent to a phaseshift of k_x followed by a phaseshift of k_z:

$$I^{RMO}(k_x,k_z,k_h)=I(k_x,k_z,k_h)\cdot e^{ik_x\Delta x}\cdot e^{ik_z\Delta z}$$ (5)

in the Fourier domain.