Forward interpolation |
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
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). |
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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 |