My model of the world

In your ears now are sounds from various directions. From moment to moment the directions change. Momentarily, a single direction (or two) dominates. Your ears sample only two points in x-space. Earthquake data is a little better. Exploration data is much better and sometimes seems to satisfy the Nyquist requirement, especially when we forget that the world is 3-D.

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. For noisy regions there is no escaping the simple form of the Nyquist limitation. For good regions 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 is prospective. 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 as we did in chapter . If we can deal with tiny pieces of data, then we can adapt to rapid spatial and temporal variations. This chapter will show such tiny windows of data. We will begin with missing-data problems in one dimension. Because these are somewhat artificial, we will move on to two dimensions, where the problems are genuine.