You have just opened Report 7 of the Consortium Project "Seismic Waves in Complex 3-D Structures", Faculty of Mathematics and Physics, Charles University, Prague. The report summarizes the work done during the fifth year of the project in the period October 1, 1997 - September 30, 1998. It also includes a compact disk with updated and extended versions of all computer programs distributed to the sponsors, and their brief description.

During the fifth year, our group working within the project, consisted of seven research workers: J. Brokesova, V. Bucha, P. Bulant, V. Cerveny, L. Klimes, I. Psencik and J. Zahradnik, and three PhD students: I. Oprsal, M. Pakzad and V. Plicka. One of the former PhD students, P. Bulant, defended successfully his PhD Thesis "Calculation of multivalued ray theory travel times in 3-D structures" and continues in his work in the Consortium Project as a postdoc.

Several other colleagues collaborated with us on specific problems. Professor Ctirad Matyska of the Charles University worked on the problem of chaotic behaviour of rays and on Lyapunov exponents. Dr. Vaclav Vavrycuk of the Geophysical Institute, Acad. Sci., Czech Republic has been studying behaviour of S waves near singular directions in anisotropic media, and R/T coefficients in weakly anisotropic media. Dr. Leonardo Martins of Universidade Estadual do Norte Fluminense, Macae, RJ, Brasil, is currently staying with us as a postdoc. He is working on problems of seismic waves in anisotropic inhomogeneous media with I. Psencik. Similarly, Dr. X. Zheng of Res. Center of Exploration Geophysics, State Seismological Bureau, Zhengzhou, China, is visiting us for three months and works with I. Psencik on similar problems. Especially fruitful was a several weeks long visit of Professor Colin Thomson of Dept. of Geol. Sci., Queen's Univ., Kingston, Canada. We appreciate very much Colin's suggestions in the field of seismic wave field propagation in complex structures. He has been recently interested in the problems of wavefield extrapolation, and has agreed to include the manuscript of his paper related to this subject in this report. We expect that the article will be of great interest to our sponsors, as it fills the gap between the ray method and "full" one-way wavefield extrapolation. Colin Thomson also agreed to include his program package Rmatrix into our CD-ROM. The program is designed to find the seismic plane wave response of a stack of anisotropic layers. Companies, which wish to use these programs, are kindly asked to contact C. Thomson for licence conditions (

Among other visitors of our group, let us name Dr. Jean-David Benamou, of INRIA, Rocquencourt, France; Mr. Marcos Hexsel Grochau, Universidade Estadual de Campinas, Campinas, Brazil; Dr. Liu Jinsong, Inst. of Geophysics, Chinese Acad. Sci., Beijing, China and Prof. Alexei P. Kiselev, LOMI, Russian Acad. Sci., St. Petersburg, Russia.

Research Report 7 may be roughly divided into six parts, see the Contents.

The first part, Ray Methods, is devoted to the ray method in general. The first article by P. Bulant and L. Klimes describes the method of interpolation of ray-theory travel times within the ray cells. The relevant algorithm and program is now included in the computer package CRT. The second article by P. Bulant and L. Klimes describes the test model "98" which was constructed by L. Klimes during the technical part of the Consortium Meeting, Prague, June 1998, according to proposals of its participants. The model was constructed with the use of the program package MODEL and then immediately, during the meeting, used for ray computations using the program package CRT. An important research subject, devoted to the chaotic behaviour of rays, was opened in the fifth year of the project. We feel that the problem of chaotic behaviour of rays will play a very important role in the application of seismic ray methods in complex geological structures. A more detailed investigation is planned for the sixth year of the project, and relevant results will be published in the next report.

The second part, Reflection-Transmission Coefficients, is devoted to the plane-wave reflection/transmission coefficients on plane structural interfaces and on stacks of thin layers situated between homogeneous halfspaces. This part includes four articles. The first article, by J. Brokesova and V. Cerveny, is devoted to dissipative models. The article contains systematic parameteric computations of R/T coefficients for realistic isotropic dissipative models. Two articles by C. Thomson are devoted to the R/T coefficients on a stack of anisotropic non-dissipative layers and to the relevant program package Rmatrix. Article by L. Klimes describes a numerical experiment, in which the Colin's package Rmatrix was used together with the ray method (hybrid ray-matrix method). Synthetic seismograms at receivers, situated on the top of a thin layer over a homogeneous halfspace, were computed for a wave incident from the halfspace, and compared with finite-difference computations.

The third part, Weak anisotropy, is devoted to weakly anisotropic media. The research in this field was iniciated in the fourth year of the project by Ivan Psencik, see Report 6. Revised forms of his articles presented in Report 6 have appeared in Geophysics, Geophys. J. Int., Pageoph and SEG Expanded Abstracts. The paper by V. Cerveny presents a general theory of quasi-shear wave coupling in inhomogeneous weakly anisotropic media and introduces the quasi-isotropic propagator matrices for the coupling equations. It also introduces a suitable method for numerical computation of the propagator matrices (method of mean coefficients). The article is intended as an extension of the manuscript on "Seismic Ray Theory", distributed to the SW3D sponsors as Report 5. Two following articles by P. Bulant and L. Klimes and by I. Psencik are also devoted to the coupling ray theory and present numerical computations of S waves in weakly anisotropic media. Certain slight differences between the results of I. Psencik and of P. Bulant and L. Klimes will be investigated next year. Two papers, the first by I. Psencik, Leonardo Martins and V. Vavrycuk, and the second by V. Vavrycuk, are devoted to the computation of R/T coefficients on a plane, weak contrast interface between two weakly anisotropic media. I. Psencik and X. Zheng propose an algorithm for solving the inverse problem of the determination of parameters of weakly anisotropic media from P wave particle ground motions. V. Vavrycuk investigates the behaviour of S waves near a kiss singularity in a homogeneous anisotropic medium.

The fourth part, Wavefield extrapolation, includes the article by C. Thomson, mentioned before. In the article, the "full" one-way wavefield extrapolation in anisotropic inhomogeneous media is derived. It is shown how the resulting equations can be reduced to the standard ray equations.

The fifth part, Finite differences, is devoted to the finite-difference method. The article by I. Oprsal, M. Pakzad and J. Zahradnik describes a new algorithm for stable FD modelling in sedimentary basins. The article by J. Brokesova, J. Zahradnik and P. Paraskevopoulos gives comparisons of ray and FD modelling of CDP seismic sections for shallow lignite deposits. During the fifth year of the project, the 2-D FD codes were modified so that they conform with program packages FORMS and MODEL. Thus, the FD computations may be simply performed for the same model description as the ray computation in the program package CRT. This modification will be very useful in future investigations, as the ray method computations may be immediately compared with FD computations and vice versa. The computations performed by L. Klimes in the article "Comparison ...", included in the second part of this report, represent a first example. Other comments and examples will be given in the next report.

The final, but most important sixth part, CD-ROM, contains the compact disk SW3D-CD-2, edited by L. Klimes, containing the revised, updated and extended software. Two articles by L. Klimes and by I. Psencik briefly describe the most important modifications. A more detailed description can be found directly on the CD-ROM.

Research Report 7 also includes the list of members of the SW3D Consortium Project (during the fifth year), and the Research Programme for the sixth year. More detailed information regarding the SW3D Consortium project is available at "".


We are very grateful to all our sponsors for the financial support. The research has been also partially supported by the Grant Agency of the Czech Republic under Contract 205/96/0968, by the Grant Agency of the Academy of Sciences of the Czech Republic under Contract A3012601, by the European Commission within the framework of the INCO-Copernicus Project IC15 CT96 200, by the Brasilian Ministry of Science and Technology within the framework of Pronex-Engenharia de Petroleo, by CNPq Brasil (postdoc. scholarship for J.L.M., No. 200466/93-3), and by Czech-French programme Barrande under Contract 97078.

Prague, November 1998

Vlastislav Cerveny
Ludek Klimes
Ivan Psencik

In: Seismic Waves in Complex 3-D Structures, Report 7, pp. 5-7, Dep. Geophys., Charles Univ., Prague, 1998.
SW3D - main page of consortium Seismic Waves in Complex 3-D Structures .