Spatial aliasing and scale invariance

Spatial aliasing and scale invariance

Jon Claerbout

Landforms are not especially predictable. Therefore, crude PEF approximations are often satisfactory. Wavefields are another matter. Consider the ``shape'' of the acoustic wavefronts at this moment in the room you are in. The acoustic wavefield has statistical order in many senses. If the 3-D volume is filled with waves emitted from a few point sources, then (with some simplifications) what could be a volume of information is actually a few 1-D signals. When we work with wavefronts we can hope for more dramatic, even astounding, results from estimating properly.

The plane-wave model links an axis that is not aliased (time) with axes (space) that often are.

We often characterize data from any region of $(t,x)$ -space as ``good'' or ``noisy'' when we really mean it contains ``few'' or ``many'' plane-wave events in that region. Where regions are noisy, there is no escaping the simple form of the Nyquist limitation. Where regions are good we may escape it. Real data typically contains both kinds of regions. Undersampled data with a broad distribution of plane waves is nearly hopeless. Undersampled data with a sparse distribution of plane waves offer us the opportunity to resample without aliasing. Consider data containing a spherical wave. The angular bandwidth in a plane-wave decomposition appears huge until we restrict attention to a small region of the data. (Actually a spherical wave contains very little information compared to an arbitrary wave field.) It can be very helpful in reducing the local angular bandwidth if we can deal effectively with tiny pieces of data. If we can deal with tiny pieces of data, then we can adapt to rapid spatial and temporal variations. This chapter shows such tiny windows of data.

Spatial aliasing and scale invariance