Cornelius Lanczos Collected Published Papers with Commentaries

CORNELIUS LANCZOS COLLECTED PUBLISHED PAPERS WITH COMMENTARIES

EDITED BY William R. Davis (Genera.! Editor) Moody T. Chu Patrick Dolan James R. McConnell Larry K. Norris Eduardo Ortiz Robert J. Plemmons Don Ridgeway B. K. P. Scaife William J. Stewart James W. York, Jr. Wesley O. Doggett (Associate Editor) Barbara M. Gellai (Associate Editor) Andre A. Gsponer (Associate Editor) Carmine A. Prioli . (ConSUlting Editor)

FOREWORD .BY George Marx.

TRANSLATORS Jozsef Illy Don Ridgeway Laurent Choquet Judith Kontsag Mesko

EDITORIAL ASSISTANTS Mary Ga.y Vicki Grantham Paresh Kenkare Jayakrishnan Ra.gha.van

COLLEGE OF PHYSICAL AND MATHEMATICAL SCIENCES NORTH CAROLINA STATE UNIVERSITY Raleigh, North Caroliua 27695 1998 APPLICATIONS OF THE FFT IN GEOPHYSICS

Robert L. Nowack Department of Earth and Atmospheric Sciences, Purdue University, West Lafayette, IN 47907, USA

Abstract

In recent years, the fast Fourier transform, or FFT, has had a major impact on most scientific fields. In this note, severa.! applications of the FFT in the geophysica.! sciences are presented. Also, since one of the earliest 'recent' discoveries of the FFT was by Danielson a.nd Lanczos (1942a.,b), severa.! comments on this work are also given.

Introduction One of the more important developments in scientific computing has been the ability to compute Fourier transforma using the FFT algorithm. An important ramification of the existence of the FF1' is the increased emphasis on the discrete Fourier transform (DFT). In contrast to Z-transforma, which use a discrete time variable, the discrete Fourier transform uses both discrete time and frequency. As a result, DFT's are restricted to finite impulse responses. Also, convolution using the DFT must be approximated by periodic convolution (Oppenheim and Schafer, 1975, Chap. 3). Even with various restrictions, the discrete Fourier transform is often the method of choice, primarily because of the efficient implementation using the fast Fourier transform algorithm. The number of algebraic operations using the FFT is proportional to N log2 N for a time series of length N, as opposed to N2 for a direct evaluation of the discrete Fourier transform (Oppenheim and Schafer, 1975, Chap. 6). As an example, for a seismic data trace having a length of 1024 points, the FFT involves 0(104 ) algebraic operations as compared to 0(106 ) for direct evaluation. The difference becomes even more extreme for longer time series or in higher dimensions. Various specialized FFT computer algorithma are given by DSPC (1979). Within the geophysical sciences, the fast Fourier transform has revolutionized the processing, synthesis and inversion of geophysical data.

The Collaboration of Danielson and Lanczos

The use of the FIT has been made widespread primarily through the work of Cooley and Tukey (1965). However, shortly after this work, Rudnick (1966) described a similar algorithm which had been previously implemented and based on the work of Danielson and Lanczos (1942a,b) (see also, Cooley et aI., 1967). A geophysical implementation of the FFT prior to 1965 has also been noted by Cla.erbout (1976, pp. 12). The review by Heideman et al. (1984) placed the roots of the FFT back to Gauss. However, the reviews by Cooley (1987, 1990) subsequently gave more emphasis to the work of Danielson and Ln.nczos (1942a,b), and suggested that Danielson and Lanczos independently discovered the doubling algorithm implicit in the FFT. Cooley (1990, pp. 137) noted that, "it appears that Lanczos had the FFT algorithm; a.nd if he had had an electronic computer, he would have been able to write a progra.m permitting him to go to arbitrarily high N." There were several early applications of the Danielson and Lanczos algorithm. It was noted as a very rapid method for Fourier analysis in a 1954 conference proceedings of the X-ray Analysis Group of the (British) Institute of Physics (Robinson and Steward, 1955). Kolsky (1956) used it to perform a '64-point Fourier synthesis for stress pulses in a viscoelastic solid. Breen, et al. (1957) used an early digital, electronic computer to compute a 128-point Fourier inversion for Neutron scattering using the Danielson and Lanczos algorithm. Nonetheless, to a large extent, the FIT algorithm was brought to the attention of the general scientific community by Cooley and Tukey (1965) (see the special issues on the FFT in IEEE, 1967, 1969).

R. L. NOWACK, COMMENTARY ON LANCZOS 1942a,b 3-511 Danielson and Lanczos performed their work in the late 1930's at Purdue University where Lanczos was a professor of mathematical physics from 1931-46 (for biographical information on Lanczos, see Scaife (ed.), 1974; Gellai, this volume). Gordon C. Danielson (1913--83) was a graduate student in physics at Purdue working on the application of Fourier analysis to X-ray scattering from liquids. From 1941-42, he was an assistant professor at the University ofIdaho. During the war years, Danielson worked at the Radiation Lab at MIT, and from 1946-48 at Bell Labs. After 1949, he went to Iowa State University where he later became a distinguished professor of physics in 1964. . Mrs. Dorothy Danielson, G.C. Danielson's wife, recently commented that her husband came to know Lanczos as a graduate student in one of his mathematics classes, and that Lanczos suggested that they work together on an idea for rapid evaluation in Fourier analysis (personal communication, 1990). This became part of Danielson's Ph.D. thesis (Danielson, 1940). It also resulted in several of the rare collaborative works of Lanczos (Lanczos and Danielson, 1939; Danielson and Lanczos, 1942a,b). As noted by Press et al. (1986), the work of Danielson and Lanczos was one of the earliest and clearest "recent" discoveries of the FFT.

Geophysical Applications of the FFT

The fast Fourier transform has.had a major influence in the geophysical sciences. Geophysical ap­ plications of the FFT can be divided into the areas of forward modeling, data processing and geophysical inversion. In forward modeling, synthetic results are obtained on the computer for later comparison with observed data. In the modeling of seismic waves, transient results are often synthesized using the FFT from frequency domain calculations (Aki and Richards, 1980; Kennett, 1983). In other cases, general forcing functions can be incorporated using FFT based convolution. A number of computer programs for seismic modeling are included in Doornbos (ed.) (1988). A recent geophysical modeling procedure, known as the Fourier method, computes.synthetic results by evaluating spatial derivatives using the FFT (Kosloff and Baysal, 1982). This approach can often be competitive with finite difference methods (Fornberg, 1987; Daudt et al., 1989). The processing of geophysical data has benefited from the efficiency of the FFT algorithm. Time series analysis of geophysical data using the FFT has been described by Cla.erbout (1976, 1991) and Kanasewich (1981). Power spectra can also be estimated using the fast Fourier transform (Oppenheim and Schafer, 1975, Chap. 11). Another example of processing using the FFT is the analysis of dispersive waves for phase and group velocity (Landisman, et at, 1969; Dziewonski and Hales, 1972). Deconvolution can be utilized to remove certain effects in observed geophysical data. For example, unwanted reverberations in seismic data can be approximately removed by deconvolution, as well as the effect of source functions. Deconvolution methods based on the FFT and using damped least squares are discussed by Claerbout (1991). Seismic deconvolution using homomorphic signal processing has been described by Ulrych (1971) and Tribolet (1979). The migration of seismic reflection data is a processing technique which collapses diffractions and repositions reflectors in processed seismic data (Gardner (ed.), 1985). Seismic migration can be consid­ ered as an approximate inverse method for mapping seismic reflectors. FFT implementations of seismic migration can be very efficient and have been described by Stolt (1976), Gazdag (1978) and Cla.erbout (1985). Another processing technique, known as slant stacking, is used to decompose seismic data into plane wave components and can also be implemented using the FFT (McMechan and Ottolini, 1980; Gardner and Lu (eds.), 1991). Finally, a developing area of a.pplications for the FFT is in the tomographic inversion of geophysical data (Lines (ed.), 1988). In tomographic inversion, physical parameters of the Earth's interior are inferred from remotely recorded geophysical data. Tomographic inversions can often be formulated in terms of Radon transforms, which can be evaluated using Fourier transforms and the FFT (Deans, 1983; Chapman, 1987). Diffraction tomographic imaging has been described by Devaney (1984) and Kak and Slaney (1988).

3-512 3.3.3 FACSIMILES &. COMMENTARIES Conclusion

In this paper, a brief overview of the fast Fourier transform algorithm and its applications in the geophysical sciences has been presented. Since one of the earliest modern discoveries of the FFT was by Danielson and Lanczos (1942a,b), several comments on this have also been given.

Acknowledgements

The author wishes to thank Dr. Barbara Gellai for her encouragement. Thanks are also given to Mrs. Dorothy Danielson for her informal comments. This work was supported in part by NSF Grants No. EAR~8904169 and EAR-9018217.

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R. L. NOWACK. COMMENTARY ON LANCZOS 1942",b 3-515