Aliasing and anti-aliasing

From the discretized data space, the discretized model space, and their relation formula,  

$$\textbf{d}\left(m_{x_{i}},m_{y_{j}},h_{x_{k}},h_{x_{l}},z_{n... ...t) =L\textbf{m}\left(m_{x_{i}},m_{y_{j}}, n\triangle z \right),$$ (23)

some causes of aliasing can be clearly seen. The sources of aliasing can be divided into the following three types:

(1) Overly coarse sampling intervals. For example, $\triangle \vec {s}$ and/or $\triangle \vec {g}$, $\triangle \vec {m}$ and/or $\triangle \vec {h}$ are too coarse, where $\triangle \vec {s}$ and $\triangle \vec {g}$ are the shot-point and receiver-point intervals respectively, and $\triangle \vec {m}$ and $\triangle \vec {h}$ are the CMP and half-offset intervals, respectively.

(2) Unsuitable modeling or imaging operators, such as the integrated DMO operator (), or the Kirchhoff integral operator (, , , ).

(3) Insufficient output resolution, where the output spatial intervals, such as $\triangle \vec {m}$ and $\triangle z$ are too coarse. We can analyze the aliasing in the cases of shot-gather migration, receiver-gather migration, common-offset gather migration, and midpoint half-offset domain migration.

Aliasing in shot gather migration is described by the following equation:  

$$I\left(\vec{m}, z\right)=\sum_{S_{k}} \sum_{\omega}S^{*}\lef... ...ec{m},z\vert S_{k} \right)R\left(\vec{m},z \vert S_{k}\right) ,$$ (24)

where $S^{*}\left(\vec{m},z\vert S_{k}\right)$ is the conjugate of the extrapolated shot wave-field; $R\left(\vec{m},z \vert S_{k}\right)$ is the extrapolated receiver wave-field. S_k stands for the K^th common-shot gather. In general, single common-shot-gather imaging presents no aliasing, because the receiver-point interval is easily arranged regularly and small enough. However, imaging an entire 2D line or 3D area will present severe aliasing problems because of the irregular and large shot-point intervals.

Aliasing in receiver-gather migration is described by the following equation:  

$$I\left(\vec{m}, z\right)=\sum_{R_{k}} \sum_{\omega}S_{sort}^... ... R_{k} \right)R_{sort}\left(\vec{m^{k}},z \vert R_{k} \right) ,$$ (25)

where $S_{sort}^{*}\left(\vec{m^{k}},z \vert R_{k} \right)$ is the extrapolated so-called shot wavefield, which corresponds to a specific receiver point; $R_{sort}\left(\vec{m^{k}},z \vert R_{k} \right)$ is the extrapolated receiver wavefield, which is sorted from shot gathers. R_k stands for the K^th common-receiver-gather. In general, single common-receiver-gather imaging profiles are susceptible to aliasing.

Aliasing in midpoint half-offset domain migration follows equation ():  

$$I\left(\vec{m}, z\right)=\sum_{h} \sum_{\omega}U\left(\vec{m},\vec{h}, \omega, z \right) ,$$ (26)

where $U\left(\vec{m},\vec{h}, \omega, z \right)$ is the extrapolated wavefield with a double-square-root equation. Aliasing will result if $\triangle \vec {m}$ and/or $\triangle \vec {h}$ are too coarse.

Aliasing in common-offset-gather migration is described by the following equation:  

$$I\left(\vec{m}, z\right)=\sum_{h_{k}} \sum_{\omega}U\left(\vec{m},h_{k}, \omega, z \right),$$ (27)

where $U\left(\vec{m},h_{k}, \omega, z \right)$ is the extrapolated wavefield with a double-square-root equation. Aliasing will result if $\triangle \vec {m}$ is too coarse.

In the case of rugged topography, the geometry of the acquisition configuration becomes more and more irregular, especially the shot-point coordinates. Therefore, antialiasing processing is necessary in redatuming, seismic data regularization and migration imaging for land-data imaging.