On Choosing Fourier Transforms for Practical Geoscience Applications

Author(s)
David Boteler

ABSTRACT

The variety of definitions of Fourier transforms can create confusion for practical applications. This paper examines the choice of formulas for Fourier transforms and determines the appropriate choices for geoscience applications. One set of Discrete Fourier Transforms can be defined that approximate Fourier integrals and provide transforms between sampled continuous functions in both domains. For applications involving transforms between a continuous function and a discrete function a second set of Discrete Fourier Transforms is needed with different scaling factors. Two classes of application are presented: those where either form of transforms can be used and those where it is necessary to use a particular transform to obtain the correct results.

The variety of definitions of Fourier transforms can create confusion for practical applications. This paper examines the choice of formulas for Fourier transforms and determines the appropriate choices for geoscience applications. One set of Discrete Fourier Transforms can be defined that approximate Fourier integrals and provide transforms between sampled continuous functions in both domains. For applications involving transforms between a continuous function and a discrete function a second set of Discrete Fourier Transforms is needed with different scaling factors. Two classes of application are presented: those where either form of transforms can be used and those where it is necessary to use a particular transform to obtain the correct results.

Cite this paper

D. Boteler, "On Choosing Fourier Transforms for Practical Geoscience Applications,"*International Journal of Geosciences*, Vol. 3 No. 5, 2012, pp. 952-959. doi: 10.4236/ijg.2012.325096.

D. Boteler, "On Choosing Fourier Transforms for Practical Geoscience Applications,"

References

[1] [R. N. Bracewell, “The Fourier Transform and Its Applications,” McGraw-Hill, New York, 1978.

[2] J. W. Cooley and J. W. Tukey, “An Algorithm for the Machine Computation of Complex Fourier Series,” Mathematics of Computation, Vol. 19, 1965, pp. 297-301. doi:10.1090/S0025-5718-1965-0178586-1

[3] E. O. Brigham, “The Fast Fourier Transform,” Prentice- Hall, Upper Saddle River, 1974.

[4] J. G. Proakis and D. G. Manolakis, “Digital Signal Processing, Principles, Algorithms, and Applications,” 3rd Edition, Prentice Hall, Upper Saddle River, 1996.

[5] W. H. Press, S. A. Teukolsky, W. T. Vetterling and B. P. Flannery, “Numerical Recipes in Fortran 77: The Art of Scientific Computing,” 2nd Edition, Cambridge University Press, Cambridge, 1999.

[6] A. T. Price, “The Theory of Magnetotelluric Methods When the Source Field Is Considered,” Journal of Geophysical Research, Vol. 67, No. 5, 1962, pp. 1907-1918.doi:10.1029/JZ067i005p01907

[7] J. R. Wait, “Electromagnetic Waves in Stratified Media,” 2nd Edition, Pergamon Press, Oxford, 1970.

[8] S. H. Ward and G. W. Hohmann, “Electromagnetic Theory for Geophysical Applications, in Electromagnetic Methods in Applied Geophysics—Theory,” Society of Exploration Geophysicists, Tulsa, Vol. 1, 1988, pp. 131- 311.

[9] A. D. Chave and P. Weidelt, “The Theoretical Basis for Electromagnetic Induction, in The Magnetotelluric Method: Theory and Practice,” Cambridge University Press, Cambridge, 2012.

[10] D. H. Boteler, R. M. Shier, T. Watanabe and R. E. Horita, “Effects of Geomagnetically Induced Currents in the BC Hydro 500 kV System,” IEEE Transactions on Power Delivery, Vol. 4, No. 1, 1989, pp. 818-823. doi:10.1109/61.19275

[1] [R. N. Bracewell, “The Fourier Transform and Its Applications,” McGraw-Hill, New York, 1978.

[2] J. W. Cooley and J. W. Tukey, “An Algorithm for the Machine Computation of Complex Fourier Series,” Mathematics of Computation, Vol. 19, 1965, pp. 297-301. doi:10.1090/S0025-5718-1965-0178586-1

[3] E. O. Brigham, “The Fast Fourier Transform,” Prentice- Hall, Upper Saddle River, 1974.

[4] J. G. Proakis and D. G. Manolakis, “Digital Signal Processing, Principles, Algorithms, and Applications,” 3rd Edition, Prentice Hall, Upper Saddle River, 1996.

[5] W. H. Press, S. A. Teukolsky, W. T. Vetterling and B. P. Flannery, “Numerical Recipes in Fortran 77: The Art of Scientific Computing,” 2nd Edition, Cambridge University Press, Cambridge, 1999.

[6] A. T. Price, “The Theory of Magnetotelluric Methods When the Source Field Is Considered,” Journal of Geophysical Research, Vol. 67, No. 5, 1962, pp. 1907-1918.doi:10.1029/JZ067i005p01907

[7] J. R. Wait, “Electromagnetic Waves in Stratified Media,” 2nd Edition, Pergamon Press, Oxford, 1970.

[8] S. H. Ward and G. W. Hohmann, “Electromagnetic Theory for Geophysical Applications, in Electromagnetic Methods in Applied Geophysics—Theory,” Society of Exploration Geophysicists, Tulsa, Vol. 1, 1988, pp. 131- 311.

[9] A. D. Chave and P. Weidelt, “The Theoretical Basis for Electromagnetic Induction, in The Magnetotelluric Method: Theory and Practice,” Cambridge University Press, Cambridge, 2012.

[10] D. H. Boteler, R. M. Shier, T. Watanabe and R. E. Horita, “Effects of Geomagnetically Induced Currents in the BC Hydro 500 kV System,” IEEE Transactions on Power Delivery, Vol. 4, No. 1, 1989, pp. 818-823. doi:10.1109/61.19275