Aliasing in image space

 

The simplest type of aliasing related to imaging operator is image-space aliasing . It occurs when the spatial sampling of the image is too coarse to represent adequately the steeply dipping reflectors that the imaging operator attempts to build during the imaging process. In other words, image-space aliasing is caused by too coarse sampling of the image space, and consequent aliasing of the migration ellipsoid. A simple way of avoiding image aliasing is to make the spatial sampling of the image denser. But for a given spatial sampling of the image, to avoid image aliasing we need to limit, during the migration process, the frequency content of the image as a function of reflectors' dips. This goal can be accomplished by performing a dip-dependent temporal lowpass filtering of the input data during the summation process. An efficient method to lowpass time-domain data with variable frequency is the triangle-filters method described in Basic Earth Imaging Claerbout (1995). An alternative method is to precompute lowpassed versions of the input traces, and select the appropriate input data during summation Gray (1992). This second method is potentially more accurate than the triangle filtering, and it is more computationally efficient when each input trace is summed into many output traces, as happens for 3-D migrations Abma (1998). However, it may require storing many versions of the input data. To reduce the storage requirements, without affecting the accuracy, I linearly interpolate the lowpassed traces along the frequency axis during the summation.

The anti-aliasing constraints to avoid image aliasing can be easily derived from basic sampling theory. For the case of time migration, when the coordinates of the image space are ${\bf \vec\xi}=\left(\tau_\xi,x_\xi,y_\xi\right)$,the pseudo-depth frequency $\omega^{\xi}_\tau$ must fulfill the following inequalities:

$$\begin{eqnarray} \omega^{\xi}_\tau & \leq & \frac{\pi}{\Delta x_\xi p^{{\rm \xi}... ...mega^{\xi}_\tau & \leq & \frac{\pi}{\Delta y_\xi p^{{\rm \xi}}_y};\end{eqnarray}$$ (1)

where $\Delta x_\xi$ and $\Delta y_\xi$are respectively the image sampling rate of the $x_\xi$ and $y_\xi$ axes, and $p^{{\rm \xi}}_x$ and $p^{{\rm \xi}}_y$, are the reflector dip components.

As discussed above, it is convenient to apply an anti-aliasing filter as a band-pass filter of the input traces, before summing their contributions to the image into the output. Therefore, we need to recast the constraints on the pseudo-depth frequency $\omega^{\xi}_\tau$ of equations (1) into constraints on the input data frequency $\omega^{D}_t$.This distinction is important because the frequency content of the seismic wavelet changes during the imaging process because of stretching, or compression, of the time axis. In particular, during migration the wavelet is always stretched; neglecting this stretch would lead to anti-aliasing constraints that are too stringent. Notice, that the seismic wavelet may get compressed, instead of stretched, by other imaging operators, such as inverse DMO and AMO Biondi et al. (1998). The pseudo-depth frequency of the image and the temporal frequency of the data are thus linked by the wavelet-stretch factor  ${dt_{D}}/{d\tau_\xi}$, as $\omega^{\xi}_\tau = \omega^{D}_t{dt_{D}}/{d\tau_\xi}$.Taking into account the wavelet-stretch factor, we can write the constraints on the data frequency that control image aliasing, as a function of the image sampling rates $\Delta x_\xi$ and $\Delta y_\xi$,the image dips $p^{{\rm \xi}}_x$ and $p^{{\rm \xi}}_y$,and the wavelet-stretch factor ${dt_{D}}/{d\tau_\xi}$,

$$\begin{eqnarray} \omega^{D}_t& \leq & \frac{\pi}{\Delta x_\xi p^{{\rm \xi}}_x \f... ...\frac{\pi}{\Delta y_\xi p^{{\rm \xi}}_y \frac{dt_{D}}{d\tau_\xi}}.\end{eqnarray}$$ (2)