Discussion

A simple linear interpolation theory can be derived from the sole principle of function bases. The choice of a function basis for the interpolated data uniquely defines a linear interpolation operator.

In application to seismic data interpolation, the basis set of functions can be given by the Green functions of an imaging operator, such as prestack migration or DMO. The linear interpolation operator in this case is intimately related to the general formulation of azimuth moveout (AMO). Some of the conclusions that the general theory can supply for AMO are

• In interpolation problems, the accuracy of operators (e.g. taking into account anisotropy, velocity variations, etc.) is of minor importance as long as the operator provides a complete basis set for describing the data.
• Formula (26) stresses the importance of using unitary operators (orthonormal bases) to construct linear interpolation. It suggests that unitary operators are even more important in interpolation problems than ``true-amplitude'' operators. Though applying non-orthogonal bases in interpolation problems is theoretically possible, it requires an intrinsic inversion of the matrix operator $\Psi$, defined in formulas (9) or (24). Such an inversion is rarely feasible in practice. The theory of asymptotic pseudo-unitary operators Fomel (1996) supplies a useful tool for constructing asymptotically orthonormal bases.
• It is also important to seek the most compact set of basis functions, e.g., the fewest number of frequencies in the spectrum. The Green functions may correspond to the solutions of a partial differential equation. The frequencies, actually present in the data, may correspond to the zeroes of the prediction-error filter. More challenging research needs to be done in relating differential equations, prediction filters, and function bases.

How is the mathematical theory of interpolation related to the problem of interpolating irregularly sampled data? The theory provides a linear interpolation operator $\bold{W}$, defined in formula (1) and evaluated in formula (13). What we actually need to consider is a linear equation  

$$\bold{f}_i = \bold{S W f}_n\;,$$ (37)

where $\bold{f}_n$ represents the desired regularly sampled output, $\bold {f}_i$ denotes the recorded irregularly spaced data, and $\bold{S}$ is the sampling operator. Estimating $\bold{f}_n$from (37) requires the art and science of linear inversion, which includes such tools as regularization and preconditioning.