We often have a problem of truncation. The recording cable is of course finite in length, and perceptible waves generally travel well beyond it. The seismic survey itself has finite dimensions. We also have the problem of gaps. Gaps in seismic data may occur unpredictably, as when a gun misfires or surveyors are denied access to parcels of land in the midst of their survey. In addition we have the problem of spatial aliasing. Because of improving technology, we can expect a substantial reduction in aliasing on the geophone axis, but aliasing on the shot axis will remain. There are only twenty-four hours in a day, and we must wait ten seconds between shots for the echoes to die down. So, given a certain area to survey and a certain number of months to survey it in, we end out with a certain number of shotpoints per square kilometer. With marine data, the spacing in the line of the path of the ship presents no problems compared to the problems presented by data spacing off the line.
Migration provides a mapping from a data space to a model space. This transformation is invertible (in the nonevanescent subspace). When data is missing, the transformation matrix gets broken into two parts. One part operates on the known data values, and the other part operates on the missing values. This book mostly ignores the missing part.
Noisy data can be defined as data that doesn't fit our model. If the missing data were replaced by zeroes, for example, the data would be regarded as complete, but noisy. Data is missing where the signal-to-noise ratio is known to be zero. More general noise models are also relevant, but statistical treatment of partially coherent multidimensional wave fields is poorly developed in both theory and practice.
My prediction is that a major research activity of the next decade will be to try to learn to simultaneously handle both the physics and the statistics of wavefields.