# Calculation of wave fields in a layered half-space with absorption based on the Thomson-Haskell method

2018;
: 40-49

Revised: December 24, 2018
Accepted: December 26, 2018

R. Pak, O. Lanets, "Calculation of wave fields in a layered half-space with absorption based on the Thomson-Haskell method", Ukrainian Journal of Mechanical Engineering and Materials Science, vol. 4, no. 2, pp. 40-49, 2018.

Authors:
1
Hetman Petro Sahaidachnyi National Army Academy
2
Lviv Polytechnic National University

This article is devoted to the development of calculation method of seismic waves on the free surface of a horizontally-layered half-space, which are perturbed by local sources. Previous obtained relations for scalar potentials of direct waves P, SV and SH from the simple force in a homogeneous environment are used for this purpose. It let us derive formulas for a complete wave field in the horizontally-layered isotropic elastic half-space, which is perturbed by the point source in the form of the simple time-depend force by using modified Thomson-Haskell’s matrix method. Exact expressions for three-dimensional displacement vector on the free surface are developed. Obtained results are generalized on the case of absorbing environments.

The algorithm is built and the computer program for calculation of three-component synthetic seismograms in the horizontally-layered isotropic environment with absorption is written on the base of developed method of direct problem solving.

In order to verify the efficiency and stability of the algorithm, full synthetic seismograms were calculated  on the test examples.

[1] L.A. Molotkov, Matrychnyi metod v teoii rasprostranenija voln v sloystykh, upruhikh i zhydkikh sredakh [“The matrix method in the theory of wave propagation in layered, elastic and liquid environment.”]. Leningrad, Russia: Nauka Publ., 1984. [in Russian].

[2] J.W. Dunkin, “ Computation of modal solution in layered elastic media at high frequencies,” Bulletin of the Seismological Society of America, vol. 55, no. 2, pp. 335–358, 1965.

[3] A. Abo‐Zena, “Dispersion function computations for unlimited frequency values,” Geophys. J. R. Astron. Soc., vol. 58, no. 1, pp. 91–105, Jul. 1979. https://doi.org/10.1111/j.1365-246X.1979.tb01011.x

[4] B. Kennet, Seismic Wave Propagation in Stratified Media. Canberra, Australia: ANU E Press, 2011.

[5] G. Muller, “The reflectivity method: a tutorial,” Journal of Geophysics, vol. 58, issue 1-3, pp. 153-174, 1985.

[6] M. Bouchon, “A review of the discrete wavenumber method,” Pure Appl. Geophys., vol. 160, no. 3–4, pp. 445–465, 2003. https://doi.org/10.1007/PL00012545

[7] K. Aki, and P. Richards, Quantitative seismology‎. Sausalito, CA: University Science Book, 2002.

[8] Yu. V. Rohanov, R.M. Pak “Predstavlenie potentsiala ot tochechnykh istochnikov dlia odnorodnoi izotropnoi sredy v vide intehralov Besselia-Mellina” [“Representation of potential from point sources for a homogeneous isotropic medium in the form of Bessel-Mellin integrals”], Heofyzychnyi zhurnal [Journal of Geophysics], vol. 35, no 2., pp. 163–167, 2013. [in Russian].

[9] L. Knopoff, “A matrix method for elastic wave problem,” Bulletin of the Seismological Society of America, vol. 55, no. 1, pp. 431-438, 1964.

[10] M. Baumbach, etc., “Study of the Foreshocks and Aftershocks of the Intraplate Latur Earthquake of September 30, 1993, India”, in Latur earthquake, (Memoir / Geological Society of India; 35), Geological Society of India, pp. 33–63, 1994.