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I have presented an efficient algorithm for interpolating irregularly sampled,
one dimensional data. This algorithm can also be used to compute
the discrete Fourier transform of irregularly sampled data. Examples show that,
if the given data samples are not aliased and are of
finite length, the algorithm gives satisfactory results. Two potential problems
are not addressed in this paper. The first one is the truncation effects
at the beginning and ending points of the interpolated signal. Because the
signal is modeled with its Fourier series expansion, it behaves periodically.
The ending point of one period of the signal is connected with the
starting point of the next period. Any discontinuity at the
juncture cannot be modeled by a finite number of frequencies. This
problem results in interpolation errors near the beginning and ending points
of the signal. This problem can be partially solved by padding
free samples at the end of the signal to be interpolated.
The second problem is related to noises and bad samples. To prevent
isolated noisy samples from affecting far away interpolated samples,
Hale (1980) suggested that a local interpolation operator should be used.
The operator defined in equation (2) contains periodic functions;
hence it is a global operator. It is not obvious how one can localize
this operator. Nevertheless, the output of the algorithm is insensitive
to random noises outside of the signal band.

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Stanford Exploration Project

12/18/1997