INTRODUCTION TO ALIASING

In its simplest form, the Nyquist condition says that we can have no frequencies higher than two points per wavelength. In migration, this is a strong constraint on data collection. It seems there is no escape. Yet, in applications dealing with a CMP gather (such as in Figure  or ), we see data with spatial frequencies that exceed Nyquist and we are not bothered, because after NMO, these frequencies are OK. Nevertheless, such data is troubling because it breaks many of our conventional programs, such as downward continuation with finite differences or with Fourier transforms. (No one uses focusing for stacking.) Since NMO defies the limitation imposed by the simple statement of the Nyquist condition, we revise the condition to say that the real limitation is on the spectral bandwidth, not on the maximum frequency. Mr. Nyquist does not tell us where that bandwidth must be located. Further, it seems that precious bandwidth need not be contiguous. The signal's spectral band can be split into pieces and those pieces positioned in different places. Fundamentally, the issue is whether the total bandwidth exceeds Nyquist. Noncontiguous Nyquist bands are depicted in Figure 1.

 

nytutor Figure 1 Hypothetical spatial frequency bands. Top is typical. Middle for data skewed with $\tau=t-px$. Bottom depicts data with wave arrivals from three directions.

Noncontiguous bandwidth arises naturally with two-dimensional data where there are several plane waves present. There the familiar spatial Nyquist limitation oversimplifies real life because the plane waves link time and space.

The spatial Nyquist frequency need not limit the analysis of seismic data because the plane-wave model links space with time.