IEEE Signal Processing - March 2018 - 71

0.012
0.011
0.01
0.009
0.008
0.007
0.006
0.005
0.004
0.003
0.002
0.001
0

in the data, the bandwidth, and the order of the approximation used. The computational effort demanded in the process
is dominated either by the size of the transformation for big
problems or by the dimensions of the problem for smaller ones.
The complexity of the algorithm is the same as that of the FFT
algorithm used. Since the computational effort tends to vary,
for bigger data, as usual FFTs, this approach seems to render
the practical difficulties in handling higher-dimensional problems to an acceptable level.

Acknowledgments

0

50 100 150 200 250 300 350 400 450 500

-1
Figure 6. A plot of errors, in decreasing order, of F app
F - I, for the third
(black line), fourth (blue line), and fifth (red line) orders of approximation. The
maximum value in a uniformly distributed random set of ds is 0.25Dx.

0.008
0.007
0.006
0.005

I would like to thank Petrobras, Petróleo Brasileiro SA for
supporting this work and allowing for its publication. Sincere
thanks also go to A.C.P. de Azambuja, A.F. Sardinha de Mattos, and Daniel T. de Paula, for their contribution to the development of the inversion series algorithm.

Author
Adelson Santos de Oliveira (adelson_so@petrobras.com.br)
received his M.Sc. degree in physics from the Universidade
Federal Fluminense, Niterói, Rio de Janeiro, Brazil, in 1986
and his Ph.D. degree in geophysics from the Universidade
Federal da Bahia, Salvador, Bahia, Brazil, in 1990. He has
been a geophysicist at Petrobras, Petróleo Brasileiro SA in
Rio de Janeiro since 1987.

References

0.004

[1] J. W. Cooley and J. W. Tukey, "An algorithm for the machine calculation of complex Fourier series," Math. Comput., vol. 19, no. 90, pp. 297-301, 1965.

0.003

[2] M. T. Heideman, D. H. Johnson, and C. S. Burrus, "Gauss and the history of the
fast Fourier transform," Arch. Hist. Exact Sci., vol. 34, no. 3, pp. 265-277.

0.002

[3] R. N. Bracewell, The Fourier Transform and Its Applications. New York:
McGraw-Hill, 1986.

0.001
0

0

50 100 150 200 250 300 350 400 450 500

-1
Figure 7. A plot of errors, in decreasing order, measured as F app
F - I,

for a maximum displacement of 0.35Dx (black line), 0.25Dx (blue line),
and 0.15Dx (red line). A fifth-order degree of approximation was used in
all three curves.

[4] A. J. W. Duijndam, M. A. Schonewille, and C. O. H. Hindriks, "Reconstruction
of band-limited signals, irregularly sampled along one spatial direction,"
Geophysics, vol. 64, no. 2, pp. 524-538, 1999.
[5] Y. C. Eldar, "Mean-squared error sampling and reconstruction in the presence of
noise," IEEE Trans. Signal Processing, vol. 54, no. 12, pp. 4619-4633, 2006.
[6] A. Guevara and R. Mester, "Signal reconstruction from noisy, aliased, and nonideal samples. What linear MMSE approaches can achieve," in Proc. 18th European
Signal Processing Conf., Aalborg, Denmark, Aug. 23-27, 2010, pp. 1291-1295.
[7] B. Liu and M. D. Sacchi, "Minimum weighted norm interpolation of seismic
records," Geophysics, vol. 69, no. 6, pp. 1560-1568, 2004.
[8] M. Naghizadeh and M. D. Sacchi, "On sampling functions and Fourier reconstruction methods," Geophysics, vol. 75, no. 6, pp. WB137-WB151, 2010.

the identity I. A maximum displacement of 0.25Dx was used
in a uniform distribution of ds. Note that, in the third order of
approximation, the great majority of errors are smaller than
0.002. These differences also vary with the "degree of irregularity," qualitatively related to the maximum displacement.
Figure 7 also shows three curves of errors like in Figure 6,
respectively for 0.35D x, 0.25D x, and 0.15D x maximum displacements, in a fifth-order degree of approximation.

Conclusions
The Fourier spectra of multidimensional irregularly sampled
data can be obtained in a approximate way due to the exponential form of the Fourier transform. The accuracy on the estimated spectra depends on the "degree of irregularities" present

[9] A. Papoulis, The Fourier Integral and Its Applications. New York: McGrawHill, 1962.
[10] E. O. Brigham, The Fast Fourier Transform and Applications. Englewood
Cliffs, NJ: Prentice Hall, 1988.
[11] J. S. Walker, Fast Fourier Transforms, 2nd ed. Boca Raton, FL: CRC, 1996.
[12] D. L. Jagerman and L. J. Fogel, "Some general aspects of the sampling theorem," IRE Trans. Inform. Theory, vol. IT-2, pp. 139-146, Dec. 1956.
[13] D. A. Linden and N. M. Abramson, "A generalization of the sampling theorem," Inf. Control., vol. 3, no. 1, pp. 26-31, 1960.
[14] I. Daubechies, R. Devore, M. Fornasier, and C. S. N. Güntürk, "Iteratively
reweighted least squares minimization for sparse recovery," Commun. Pure Appl.
Math., vol. 63, no. 1, pp. 1-132, 2010.
[15] M. D. Sacchi, "Reweighting strategies in seismic deconvolution," Geophys. J.
Int., vol. 129, no. 3, pp. 651-656, 1997.
[16] A. S. Oliveira, "An approximate representation of the Fourier spectra of irregularly sampled functions," in Proc. 15th Congr. Brazilian Geophysical Society,
2017, pp. 1408-1412.

IEEE Signal Processing Magazine

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