Introduction

Data interpolation is one of the most important tasks in geophysical data processing. Its importance is increasing with the development of 3-D seismics, since most of the modern 3-D acquisition geometries carry non-uniform spatial distribution of seismic records. Without a careful interpolation, acquisition irregularities may lead to unwanted artifacts at the imaging step Chemingui and Biondi (1996); Gardner and Canning (1994).

The interpolation problem in geophysics implies interpolating irregularly sampled data to a regular grid. In general, this problem requires a regularized inversion scheme, such as the method of inversion to zero offset Ronen et al. (1991, 1995). Theoretically, it is easier to consider a different (in a sense, the opposite) problem: to find a continuous interpolation function for the data, given on a regular grid. The latter problem has been a traditional subject in computational mathematics. Though its solution is not directly applicable to the handling of irregular acquisition geometries, it can give us some insights into the possible ways of approaching geophysical interpolation.

In this paper, I present a simple mathematical theory of interpolation from a regular grid. I derive the main formulas from a very general idea of function bases. In conclusion, I discuss possible applications of the general theory in geophysics and, in particular, its relation to azimuth moveout Biondi et al. (1996).