Relation of missing data to inversion

We take data space to be a uniform mesh on which some values are given and some are missing. We rarely have missing values on a time axis, but commonly have missing values on a space axis, i.e., missing signals. Missing signals (traces) happen occasionally for miscellaneous reasons, and they happen systematically because of aliasing and truncation. The aliasing arises for economic reasons--saving instrumentation by running receivers far apart. Truncation arises at the ends of any survey, which, like any human activity, must be finite. Beyond the survey lies more hypothetical data. The traces we will find for the missing data are not as good as real observations, but they are closer to reality than supposing unmeasured data is zero valued. Making an image with a single application of an adjoint modeling operator amounts to assuming that data vanishes beyond its given locations. Migration is an example of an economically important process that makes this assumption. Dealing with missing data is a step beyond this. In inversion, restoring missing data reduces the need for arbitrary model filtering.