Abstract
In this paper, linearized tomography and the Herglotz-Wiechert inverse formulation are compared. Tomographic inversions for 2-D or 3-D velocity structure use line integrals along rays and can be written in terms of Radon transforms. For radially concentric structures, Radon transforms are shown to reduce to Abel transforms. Therefore, for straight ray paths, the Abel transform of travel-time is a tomographic algorithm specialized to a one-dimensional radially concentric medium. The Herglotz-Wiechert formulation uses seismic travel-time data to invert for one-dimensional earth structure and is derived using exact ray trajectories by applying an Abel transform. This is of historical interest since it would imply that a specialized tomographic-like algorithm has been used in seismology since the early part of the century (seeHerglotz, 1907;Wiechert, 1910). Numerical examples are performed comparing the Herglotz-Wiechert algorithm and linearized tomography along straight rays. Since the Herglotz-Wiechert algorithm is applicable under specific conditions, (the absence of low velocity zones) to non-straight ray paths, the association with tomography may prove to be useful in assessing the uniqueness of tomographic results generalized to curved ray geometries.
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References
Aki, K. (1977),Three Dimensional Seismic Velocity Anomalies in the Lithosphere, J. Geophys.43, 235–242.
Aki, K., Christoffersson, A., andHusebye, E. S. (1976),Determination of the Three-Dimensional Seismic Structure of the Lithosphere, J. Geophys. Res.82, 277–296.
Aki, K., andRichards, P.,Quantitative Seismology: Theory and Methods (W. H. Freeman and Co., San Francisco 1980).
Beylkin, G. (1983),Inversion of the Generalized Radon Transform, Proc. SPIE, Inverse Optics413, 32–40.
Beylkin, G. (1984),The Inversion Problem and Applications of the Generalized Radon Transform, Commun. Pure Appl. Math.37, 579–599.
Bôcher, M.,An Introduction to the Study of Integral Equations (Cambridge University Press 1909).
Bracewell, R. N. (1956),Strip Integration in Radio Astronomy, Aust. J. Phys.9, 198–201.
Bracewell, R. N.,The Fourier Transform and Its Applications (McGraw Hill, New York 1978).
Chapman, C. H.,The Radon transform and seismic tomography, inSeismic Tomography (ed. Nolet, G.) (Reidel Publishing Co. 1987) pp. 25–47.
Cormack, A. M. (1963),Representation of a Function by Its Line Integrals, with Some Radiological Applications, J. Appl. Phys.34, 2722–2727.
Deans, S. R.,The Radon Transform and Some of Its Applications (Wiley-Interscience, New York 1983).
Deans, S. R.,The Radon Transform, inMathematical Analysis of Physical Systems (ed. Mickens, R.E.) (Van Nostrand Reinhold, New York 1985).
Herglotz, G. (1907),Über das Benndorfsche Problem der Fortpflanzungsgeschwindigkeit der Erdbebenstrahlen, Zeitschr. für Geophys.8, 145–147.
McMechan, G. A. andOttolini, R. (1980),Direct Observation of a p-tau Curve in a Slant Stacked Wavefield, Bull. Seismol. Soc. Am.70, 775–789.
Radon, J. (1917),Über die Bestimmung von Funktionen durch ihre Integralwerte längs gewisser Mannigfaltigkeiten, Berichte Sächsische Akadamie der Wissenschaften, Leipzig, Math.-Phys. Kl.69, 262–267.
Vest, C. M. (1974),Formation of Images from Projections: Radon and Abel Transforms, J. Optical Soc. Am.64, 1215–1218.
Vest, C. M.,Holographic Interferometry (J. Wiley & Sons, New York 1979).
Wiechert, E. (1910),Bestimmung des Weges der Erdbebenwellen im Erdinnern. I. Theoretisches, Phys. Z.11, 294–304.
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Nowack, R.L. Tomography and the Herglotz-Wiechert inverse formulation. PAGEOPH 133, 305–315 (1990). https://doi.org/10.1007/BF00877165
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DOI: https://doi.org/10.1007/BF00877165