Hilbert transformation is a standard tool in univariate signal-processing. It leaves the information content unaffected for, apart from a change of sign, the iterated transform reproduces the original data. As we know from Fourier transforms, such alternative representations of the same information, notwithstanding their theoretical equivalence with the data, can assist powerfully with extracting and interpreting that information. Although the extension to multivariate data is not so obvious for Hilbert as for Fourier transforms, Nabighian gave a treatment of the bivariale situation in 1984. Fueter, some 50 years earlier, had worked on an analogous extension problem, seeking to generalize complex function theory. On comparing these two developments we learn that, although Nabighian's transforms fit naturally into Fueter's theory, they are only one among many alternative possibilities. This paper presents a general theory, of higher dimensional Hilbert transforms and analytic signals, applicable to data of all dimensions less than eight. The change-of-field-direction fillers used in geophysical data processing are shown to arise as special situations.
This is a preview of subscription content, access via your institution.
Buy single article
Instant access to the full article PDF.
Price includes VAT (USA)
Tax calculation will be finalised during checkout.
Ahlfors, L. V., 1966, Complex analysis (2nd ed.): McGraw-Hill Book Co., New York, 317 p.
Bhattacharyya, B. K., 1964, Magnetic anomalies due to prism-shaped bodies with arbitrary polarization: Geophysics, v. 29, no. 4, p. 517–531.
Bracewell, R., 1965, The Fourier transform and its applications: McGraw-Hill Book Co., New York, 381 p.
Craig, M., 1984, What are quaternions?: Math. Student, v. 50, p. 289–292.
Curtis, C. W., and Reiner, I., 1962, Representation theory of finite groups and associative algebras: John Wiley & Sons, New York, 685 p.
Dickson, L. E., 1919, On quaternions and their generalizations and the history of the eight square theorem: Ann. Math., v. 20, p. 155–171.
Fueter, R., 1934/35, Die Funktionentheorie der Differentialgleichungen Δu=0 und ΔΔu=0 mit vier reellen Variablen: Comment. Math. Helv., v. 7, p. 307–330.
Fueter, R., 1936, Die Theorie der Regulaeren Funktionen einer Quatemionvariablen: Comptes Rendus du congrès int. des mathématiciens, Oslo, p. 75–91.
Gunn, P. J., 1975, Linear transformations of gravity and magnetic fields: Geophys. Prospecting, v. 23, no. 2, p. 300–312.
Hardy, G. H., 1960, A course of pure mathematics: Cambridge Univ. Press, London, 509 p.
Lam, T. Y., 1973, The algebraic theory of quadratic forms: W. A. Benjamin. Reading, Massachusetts, 343 p.
Nabighian, M. N., 1984. Towards a three-dimensional automatic interpretation of potential field data via generalized Hilbert transforms: Fundamental relations: Geophysics, v. 49, no. 6, p. 780–786.
Read, R. R., and Treitel S., 1973, The stabilization of two-dimensional recursive filters via the discrete Hilbert transform: IEEE Trans. Geosci. Electron., GE-11, no. 3, p. 153–160 and Addendum, no. 4, p. 205–207.
Singleton, R. C., 1969, An algorithm for computing the mixed radix fast Fourier transform: IEEE Trans. on Audio and Electroacoustics, v. AU-17, no. 2. p. 93–103.
Stiefel, E., 1952. On Cauchy-Riemann equations in higher dimensions: Jour. Res. Nat. Bur. Stand., v. 48, no. 5, p. 395–398.
Tricomi, F. G., 1985, Integral equations: Dover Publ. Inc., New York, 238 p.
Whittaker, E. T., and Watson, G. N., 1962, A course of modern analysis: Cambridge Univ. Press, London, 608 p.
About this article
Cite this article
Craig, M. Analytic signals for multivariate data. Math Geol 28, 315–329 (1996). https://doi.org/10.1007/BF02083203
- Dirac-Fueter equations
- harmonic function
- Hilbert transform
- magnetic survey
- potential theory