The algorithm for solving the direct kinematic problem of seismics in three-dimensional heterogeneous isotropic media

  • Galaktionova Anastasiia A., a.galaktionova@g.nsu.ru Institute of Computational Mathematics and Mathematical Geophysics, 6 Acad. Lavrentiev Avenue, Novosibirsk 630090, Russia; Novosibirsk State University, 1 Pirogov Street, Novosibirsk 630090, Russia
  • Belonosov Andrey S., white@sscc.ru Institute of Computational Mathematics and Mathematical Geophysics, 6 Acad. Lavrentiev Avenue, Novosibirsk 630090, Russia; Novosibirsk State University, 1 Pirogov Street, Novosibirsk 630090, Russia
Keywords: ray-based method, ray tracing, 3D shooting

Abstract

The solution of the direct kinematic problem of seismics is an important stage of the seismic data processing in seismology and seismic explorations, e.g., while solving the inverse kinematic problem of seismics by the method of nonlinear seismic tomography, in the method of the space-time Kirchhoff’s migration, etc. There are several approaches now to the solution of this problem: 1) methods based on the solution of two-point boundary value problems; 2) the wavefront construction method; 3) finite-difference methods for solving eikonal differential equations, etc. [2–3]. The method proposed in this paper relates to ray tracing methods and overcomes some of the limitations inherent in the above methods.

The basis of the approach is the iterative Newton’s method. The algorithm of threedimensional shooting is simple to implement and allows "almost" linear speedup from parallelization. Questions of convergence of the Newton method applied to this problem are not considered in this paper. In addition, we do not consider the applicability of the method in the presence of caustics, shadow zones, and diffraction.

References


[1]
Yurchenko M. A., Belonosov A. S., and Belonosova A. V., “On an algorithm to solve an inverse kinematic problem of seismics,” Sib. Electron. Math. Rep., 10, 74–86 (2013).

[2]
Gjoystdal H., Iversen E., Laurain R. et al., “Review of ray theory applications in modelling and imaging of seismic data,” Stud. Geoph. Geod., 46, 113–164 (2002).

[3]
Cerveny V., Seismic Ray Theory, Camb. Univ. Press, New York (2001).

[4]
Rawlinson N., Hauser J., and Sambridge M., “Seismic ray tracing and wavefront tracking in laterally heterogeneous media,” Adv. Geophys., 49, 203–273 (2007).

[5]
Alekseev A. S. and Gelchinsky B. Ya., “Ray-tracing method for the calculation of wave fields in inhomogeneous media with curved interfaces [in Russian],” in: Problems in Dynamic Theory of Seismic Wave Propagation, No. 3, 107–159 (1959).

[6]
Rabiner L. R. and Gold B., Theory and Application of Digital Signal Processing, ch. 5, pp. 300–302, Prentice Hall (1975).

[7]
Pogorelov A. V., Differential Geometry [in Russian], Nauka, Moscow (1974).

[8]
Lapidus L., Numerical Solution of Ordinary Differential Equations, Acad. Press, New York (1971).

[9]
Tsetsokho V. A., Belonosova A. V., and Belonosov A. S., “Calculation formulas of linear geometrical spreading at ray tracing in a 3D block-inhomogeneous gradient medium [in Russian],” Sib. Zh. Vychisl. Mat., 12, No. 3, 325–339 (2009).

[10]
Galaktionova A. and Belonosov A., “Computation of seismic wave field kinematics in a threedimensional heterogeneous isotropic medium,” in: Application of Mathematics in Technical and Natural Sciences, AIP Conf. Proc., 120004, 1895 (2017).
How to Cite
Galaktionova, A. and Belonosov, A. (2020) “The algorithm for solving the direct kinematic problem of seismics in three-dimensional heterogeneous isotropic media”, Mathematical notes of NEFU, 27(1), pp. 53-68. doi: https://doi.org/10.25587/SVFU.2020.12.61.004.
Section
Mathematical Modeling