Antialiasing Filter for Reverse-time Migration

If $G({\bf {x}}\vert{\bf {x}}')$ is computed by a finite-difference solution to the wave equation, then equation  defines the RTM formula. Its traditional implementation (Stolt and Benson, 1986) is to backpropagate the data and take the zero-lag correlation of it with the forward propagated field to get $m({\bf {x}})$ . An alternative implementation+interpretation of RTM is provided by generalized diffraction-stack migration (GDM). Schuster (2002) showed that the RTM image at ${\bf {x}}$ can be implemented as a dot-product of the kernel fingerprint ${\mathcal F}^{-1}[G({\bf {r}}\vert{\bf {x}})G({\bf {x}}\vert{\bf {s}})]$ with that of the CSG fingerprint ${\mathcal F}^{-1}[D({\bf {r}}\vert{\bf {s}})]$ , where ${\mathcal F}^{-1}[~]$ represents the inverse Fourier transform. In either the RTM or GDM implementations, the resulting migration images are identical.

As an example, Figure b shows that the kernel fingerprint consists of several hyperbolas for a specified image point ${\bf {x}}$ . The early arriving hyperbola is for the primary scattering and the later one is associated with a multiple. Therefore, the antialiasing strategy for RTM consists of the following:

1. Define $g({\bf {r}},t\vert{\bf {x}},0)={\mathcal F}^{-1} [G({\bf {r}}\vert{\bf {x}})]$ . Use a finite-difference solution to the space-time wave equation to compute $g({\bf {r}},t\vert{\bf {x}},0)$ for a source at ${\bf {x}}$ and receiver at ${\bf {r}}$ . Here ${\bf {x}}$ is in the model space.
2. Convolve $g({\bf {r}},t\vert{\bf {x}},0)$ with $g({\bf {s}},t\vert{\bf {x}},0)$ to get the kernel fingerprint (RTM migration kernel) $\mathcal G({\bf {r}},{\bf {s}},{\bf {x}},t)=g({\bf {r}},t\vert{\bf {x}},0)*g({\bf {s}},t\vert{\bf {x}},0)$ , where $*$ denotes temporal convolution. The convolution results are the colored-line hyperbolas shown in Figure b.
3. Apply local low-pass filters to every time sample of this kernel so that the antialiasing condition  is satisfied. Denote this filtered kernel fingerprint as $\tilde \mathcal G({\bf {r}},{\bf {s}},{\bf {x}},t)$ .
4. Take dot-products of the migration kernel $\tilde \mathcal G({\bf {r}},{\bf {s}},{\bf {x}},t)$ with the CSG in the time domain $d({\bf {r}}, t\vert{\bf {s}},0)$ to get the prestack migration image $m({\bf {x}})$ for a single shot at ${\bf {s}}$ . The resulting image will be free of aliasing artifacts.