Abstract
We consider the inverse kinematic problem of seismics (IKPS) with internal sources. It consists in determining the velocities of the longitudinal and transverse waves by the travel times from earthquake sources in the focal zone to a group of seismic stations. We propose an algorithm for the numerical solution of the problem which bases on the eikonal equation and the technology of smoothing multidimensional splines, which give an approximation of the velocity structure of the focal zone. The paper presents some theoretical results that substantiate the algorithm for solving the problem by approximation methods on using smoothing with multidimensional splines from data on irregular grids. We describe the results of the numerical solution of the problem, the calculations with real data on earthquakes in the focal zone, and give the estimates of the velocity and elastic parameters of a medium.
Similar content being viewed by others
REFERENCES
G. Herglotz, “Über das Benndorfsche Problem der Fortpflanzungsgeschwindigkeit der Erdbebenstrahlen,” Zeitschr. für Geophys. 8, 145–147 (1907).
E. Wiechert, “Bestimmung des Weges der Erdbebenwelleh on Erdinnern. L Theoretisches,” Phys. Zeitschr. 11, 294–304 (1910).
M. M. Lavrentyev, V. G. Romanov, and S. P. Shishatskii, Ill-Posed Problems of Mathematical Physics and Analysis (Nauka, Moscow, 1980; Amer. Math. Soc., Providence, 1986).
V. G. Romanov, Inverse Problems of Mathematical Physics (Nauka, Moscow, 1984; VNU Science Press, Utrecht, 1987).
S. V. Goldin, Geometrical Seismics (IPGG SB RAS, Novosibirsk, 2017) [in Russian].
G. Nolet, Seismic Tomography (D. Reidel Publ. Co., Dordrecht, 1987).
D. Royer and E. Dieulesaint, Elastic Waves in Solids (Nauka, Moscow, 1982; Berlin, Springer, 1996).
V. K. Andreev, Mathematical Models of Continuum Mechanics (Lan, St. Petersburg, 2015) [in Russian].
Yu. A. Kravtsov and Yu. I. Orlov, Geometrical Optics of Inhomogeneous Media. Wave Phenomena (Nauka, Moscow, 1980; Springer, Berlin, 1990).
N. V Butenin and N. A. Fufaev, Introduction to Analytical Mechanics (Nauka, Moscow, 1991) [in Russian].
B. A. Dubrovin, S. P. Novikov, and A. T. Fomenko, Modern Geometry. Methods and Applications (Springer-Verlag, GTM 93, Part 1, 1984; GTM 104, Part 2, 1985; GTM 124, Part 3, 1990; Nauka, Moscow, 1986).
M. L. Gerver and V. M. Markushevich, “Properties of the Hodograph from a Surface Source,” in Some Direct and Inverse Problems of Seismology (Nauka, Moscow, 1968), pp. 15–63.
V. M. Markushevich, “Characteristic Properties of the Hodograph from a Deep Source,” in Some Direct and Inverse Problems of Seismology (Nauka, Moscow, 1968), pp. 64–77.
Yu. E. Anikonov, “On One Problem of Determining the Riemannian Metric,” Dokl. Akad. Nauk SSSR 204 (6), 1287–1288 (1972).
Yu. E. Anikonov, Some Methods for Investigating Multivariate Inverse Problems for Differential Equations (Nauka, Novosibirsk, 1978) [in Russian].
P. P. Belinskii, “On Continuity of Spatial Quasiconformal Mappings and Liouville’s Theorem,” Dokl. Akad. Nauk SSSR 147 (5), 1003–1004 (1962).
Yu. G. Reshetnyak, “On Stability in Liouville’s Theory of Conformal Mappings of Space,” Dokl. Akad. Nauk SSSR 152 (2), 286–287 (1963).
A. V. Pogorelov, Extrinsic Geometry of Convex Surfaces (Nauka, Moscow, 1969) [in Russian].
Yu. E. Anikonov, N. B. Pivovarova, and L. B. Slavina, “Three-Dimensional Velocity Field of the Focal Zone of Kamchatka,” in Mathematical problems of geophysics, No. 5, Part 1 (Vychisl. Tsentr, Sibir. Otdel. Akad. Nauk SSSR, Novosibirsk, 1974), pp. 92–117.
A. V. Kabannik, Yu. A. Orlov, and V. A. Cheverda, “Numerical Solution of the Problem of Linear Seismic Tomography on Transmitted Waves: The Case of Incomplete Data,” Sibir. Zh. Ind. Mat. 7 (2), 54–67 (2004).
H. Wendland, Scattered Data Approximation (Cambridge Univ. Press, Cambridge, 2005).
A. I. Rozhenko, Theory and Algorithms of Variational Spline Approximation (Inst. Vychisl. Mat. Mat. Geofiz., Novosibirsk, 2005) [in Russian].
M. I. Ignatov and A. B. Pevny, Natural Splines of Many Variables (Nauka, Leningrad, 1991) [in Russian].
R. Schaback, Native Hilbert Spaces for Radial Basis Functions. I. New Developments in Approximation Theory (Birkhäuser, Basel, 1999), pp. 255–282.
A. I. Rozhenko, “Comparison of Radial Basis Functions,” Sibir. Zh. Vychisl. Mat. 21 (3), 273–292 (2018) [Numer. Anal. Appl. 11 (3), 220–235 (2018)].
Yu. S. Volkov and V. L. Miroshnichenko, “Construction of a Mathematical Model of the Universal Characteristic of a Radial-Axial Hydraulic Turbine,” Sibir. Zh. Ind. Mat. 1 (1), 77–88 (1998).
Yu. E. Anikonov, V. V. Bogdanov, E. Yu. Derevtsov, V. L. Miroshnichenko, N. B. Pivovarova, and L. B. Slavina, “Some Approaches to Numerical Solution for the Multidimensional Inverse Kinematic Problem of Seismics with Inner Sources,” J. Inverse Ill-Posed Probl. 17 (3), 209–238 (2009).
V. V. Bogdanov, W. V. Karsten, V. L. Miroshnichenko, and Yu. S. Volkov, “Application of Splines for Determining the Velocity Characteristic of a Medium from a Vertical Seismic Survey,” Central European J. Math. 11 (4), 779–786 (2013).
A. I. Rozhenko and T. S. Shaidorov, “On Spline Approximation with a Reproducing Kernel Method,” Sibir. Zh. Vychisl. Mat. 16 (4), 365–376 (2013) [Numer. Analysis Appl. 6 (4), 314–323 (2013)].
J. O. Aasen, “On the Reduction of a Symmetric Matrix to Tridiagonal Form,” BIT 11 (3), 233–242 (1971).
Funding
The authors were supported within the framework of the State Contract of the Sobolev Institute of Mathematics (projects Nos. 0314–2019–0011 and 0314–2019–0013) and partially by the Russian Foundation for Basic Research (RFBR) and the German Science Foundation (DFG) (joint German–Russian research project 19–51–12008).
Author information
Authors and Affiliations
Corresponding authors
Rights and permissions
About this article
Cite this article
Anikonov, Y.E., Bogdanov, V.V., Volkov, Y.S. et al. On the Determination of the Velocity and Elastic Parameters of a Medium in the Focal Zone from Earthquake Hodographs. J. Appl. Ind. Math. 15, 569–585 (2021). https://doi.org/10.1134/S1990478921040013
Received:
Revised:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1134/S1990478921040013