When is anti-aliasing needed in Kirchhoff migration?

OPERATOR ALIASING

Operator aliasing most often occurs when operator moveout across adjacent traces exceeds the time sampling rate. Cycle skips can occur when the operator is aliased. For a moveout curve with slope $dt/dx$, and data with a spatial Nyquist frequency of $k_n$, temporal frequencies above

$$\omega = {k_n\over{dt/dx}}$$

are aliased. In terms of the mesh spacing $\triangle x$ and operator slope $dt/dx$, operator aliasing will occur for all frequencies above $f_{op}$, where $f_{op}$ is given by:

$$f_{op} \; = \; \frac{1}{2 (\frac{dt}{dx}) \triangle x}.$$ (1)

Defining the maximum stepout as $p = \delta x /\delta t$, the highest dip frequency in the data is given by

$$f_d \; = \; \frac{1}{2 p \triangle x}.$$ (2)

When the stepout is captured by the mesh spacing, $\delta x = \triangle x$, and $\delta t = \triangle t$, the highest unaliased dip frequency is equal to the Nyquist frequency $f_n = 1/2\triangle t$. In areas of economic interest, steep dips are often present in the data and $f_d > f_n$.

Anti-aliasing is called for when the frequency content of the data, $f_s$, falls between

$$f_{op} \; \leq \; f_s \; \leq \; f_d.$$ (3)

This situation is illustrated in Figure 1.

spectrum Figure 1. Operator aliasing, event dip, and frequency content of the data.

When is anti-aliasing needed in Kirchhoff migration?