Summary
A new approximation of the velocity-depth distribution in a vertically inhomogeneous medium is suggested. This approximation guarantees the continuity of velocity and of its first and second derivatives and does not generate false low-velocity zones. It is very suitable for the computations of seismic wave fields in vertically inhomogeneous media by ray methods and its modifications, as it removes many false anomalies from the travel-time and amplitude-distance curves of seismic body waves. The ray integrals can be evaluated in a closed form; the resulting formulae for rays, travel times and geometrical spreading are very simple. They do not contain any transcendental functions (such asln (x) orsin −1, (x)) like other approximations; only the evaluation of one square root and of certain simple arithmetic expressions for each layer is required. From a computational point of view, the evaluation of ray integrals and of geometrical spreading is only slightly slower than for a system of homogeneous parallel layers and even faster than for a piece-wise linear approximation.
Similar content being viewed by others
References
V. Červený, I. A. Molotkov, I. Pšenčík: Ray Method in Seismology. Univ. Karlova, Praha 1977.
V. Červený, V. Pretlová: Application of Smoothed Splines in the Computation of Ray Amplitudes of Seismic Body Waves. Proc. XV. Assembly of the ESC, Kraków 1976; Publ. Inst. Geoph., Pol. Acad. Sc. A-4 (115), 187, PWN, Warszawa-Lódž 1977.
V. Červený, V. Pretlová: Computation of Ray Amplitudes of Seismic Body Waves in Vertically Inhomogeneous Media. Studia geoph. et geod., 21 (1977), 248.
V. Červený, R. Ravindra: The Theory of Seismic Head Waves. Toronto Univ. Press, Toronto 1971.
C. H. Chapman: On the Computation of Seismic Ray Travel-Times and Amplitudes. Bull. Seism. Soc. Am., 61 (1971), 1267.
C. B. Moler, L. P. Solomon: Use of Splines and Numerical Integration in Geometrical Acoustics. J. Acoust. Soc. Am., 48 (1970), 739.
Ch. H. Reinsch: Smoothing by Spline Functions. Numer. Math., 10 (1967), 177.
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Červený, V. A new approximation of the velocity-depth distribution and its application to the computation of seismic wave fields. Stud Geophys Geod 24, 17–27 (1980). https://doi.org/10.1007/BF01628377
Received:
Published:
Issue Date:
DOI: https://doi.org/10.1007/BF01628377