    Forward interpolation  Next: Continuous case and seismic Up: Interpolation with Fourier basis Previous: Continuous Fourier basis

## Discrete Fourier basis

Assuming that the range of the variable is limited in the interval from to , the discrete Fourier basis (Fast Fourier Transform) employs a set of orthonormal periodic functions (23)

where the discrete frequency index also ranges, according to the Nyquist sampling criterion, from to . The interpolation function is computed from equation (15) to be     (24)

An interpolation function equivalent to (24) has been found by Muir (Popovici et al., 1993,1996; Lin et al., 1993). It can be considered a tapered version of the sinc interpolant (21) with smooth tapering function Unlike most other tapered-sinc interpolants, Muir's interpolant (24) satisfies not only the obvious property (2), but also properties (3) and (19), where the interpolation function should be set to zero for outside the range from to . The form of this function is shown in Figure 11. ma-sinc
Figure 11.
The left plots show the sinc interpolation function. Note the slow decay in . The middle shows the effective tapering function of Muir's interpolation; the right is Muir's interpolant. The top is for (5-point interpolation); the bottom, (13-point interpolation).   The development of the mathematical wavelet theory (Daubechies, 1992) has opened the door to a whole universe of orthonormal function bases, different from the Fourier basis. The wavelet theory should find many useful applications in geophysical data interpolation, but exploring this interesting opportunity would go beyond the scope of the present work.

The next section carries the analysis to the continuum and compares the mathematical interpolation theory with the theory of seismic imaging.    Forward interpolation  Next: Continuous case and seismic Up: Interpolation with Fourier basis Previous: Continuous Fourier basis

2014-02-21