You have just opened Report 8 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 sixth year of the project, in the period November 1998 -- May 1999. It also includes the compact disk with updated and extended versions of all computer programs distributed to the sponsors, and their brief description.

Two previous research reports of the consortium project covered one-year period, starting from October 1 of one year, and ending on September 30 of the next year. According to Research Agreements, the relevant one-year research reports were distributed within two months after September 30. This was, however, inconvenient with respect to Consortium meetings, organized regularly in June. It was recommended by the sponsors during the 1998 meeting to prepare the research reports before the relevant meetings, to be available at the meeting. For this reason, Report 8 actually covers only sixth months of work, starting in November 1998 (when Report 7 was distributed) and ending in May 1999. The future research reports will be available at the June meetings and will cover work for a period of one year (reports will thus be issued sooner than specified in the Research Agreements).

Our group working within the project during the sixth year consisted of nine research workers: J. Brokesova, V. Bucha, P. Bulant, V. Cerveny, L. Klimes, C. Matyska, I. Psencik, V. Vavrycuk and J. Zahradnik, and two PhD. students: I. Oprsal and V. Plicka. Jorge Leonardo Martins and Xuyao Zheng collaborated with us on specific problems during the above mentioned period. JingSong Liu (China) spent 3 months with us.

Research Report 8 may be roughly divided into five parts, see the Contents.

The first part, Seismic models and inversion techniques, is devoted to the construction, usage and visualization of seismic models and to the inversion of seismic data, because the inversions are closely related to the models. The properties of the models of geological structures are estimated from seismic data and the accuracy of the models considerably influences the accuracy of subsequent inversions, like migrations or hypocentre determinations. The first paper, by P. Bulant, presents the algorithm for the triangulation of structural interfaces and velocity sections in the models. Since the interfaces are defined by implicit functions, the triangulation of structural interfaces is very useful to display and check the interfaces in models. The triangulation enables to display structural interfaces together with velocity distributions, rays, sources, receivers and other objects of interest in three dimensions using the Virtual Reality Modeling Language (VRML). The programs and examples are included on compact disk SW3D-CD-3. In the next paper, by J. Liu & I. Psencik, an iterative tomographic procedure based on the qP wave travel times and aimed at determination of variation of all 21 elastic parameters specifying a studied structure is proposed. No a priori assumptions are made concerning anisotropy and inhomogeneity of the structure except that the variation of elastic parameters within the studied model is tri-linear. Preliminary results of a synthetic test based on the presented formulae are also presented. Last two papers of the first part, are devoted to the problems of local determination of weak anisotropy parameters from the qP wave slowness and particle motion measurements in laterally homogeneous and laterally inhomogeneous media. In the first paper, by X. Zheng & I. Psencik, results of synthetic tests based on formulae derived in Report 7 are presented. In the second paper, by I. Psencik & X. Zheng, formulae are proposed, which could be used for solving the same problem as above but for laterally inhomogeneous media. The use of results of the particle motion measurements in inverting for local parameters of a medium brings new independent data into the problem. On the other hand, an open question is how reliable can be particle motion measurements in practice. Tests with noisy data give certain hope in this respect.

The second part, Ray methods, is devoted to the ray method in general. The first brief contribution, by P. Bulant, is devoted to the interpolation of travel times inside ray cells. Performance of the bicubic interpolation inside prismatic ray cells is numerically compared with the decomposition of the ray cell into three tetrahedra and the interpolation inside the tetrahedra. The second paper of this part, by C. Matyska, discusses various possibilities how to study chaotic behaviour of rays. In the third paper, by L. Klimes, the present, early state of studying the relation of the ray chaos to the properties of the seismic model is briefly summarized. The fourth paper of this part, by L. Klimes, presents the equations for the linear paraxial approximation of the polarization vectors and for the variation of the polarization vectors with a velocity perturbation in heterogeneous isotropic media.

The third part, Weak anisotropy, is devoted to weakly anisotropic media. The first three papers of this part concern with the propagation of plane S waves along the axis of spirality in very simple 1-D anisotropic "twisted crystal" model. The exact analytical expressions for the one-way propagator matrices are presented in the first paper, by L. Klimes, together with the analytical solutions of the equations for zero-order isotropic and anisotropic ray theories, coupling ray theory and quasi-isotropic approximation in the comparable form. The second paper, by P. Bulant, L. Klimes & I. Psencik, demonstrates the applicability and accuracy of the ray methods with respect to the exact solution numerically. The third paper, by V. Vavrycuk, is devoted to the zero-order to high-order approximations of the anisotropic ray theory and to the finite differences in the same model as in the second paper, but for a different signal in the time domain. In the last contribution, by I. Psencik & J.L. Martins, properties of weak contrast PP R/T coefficients for weakly anisotropic media are studied. It is shown that introduction of an arbitrary isotropic background into the expressions for coefficients can make applicability of the formulae broader because accurate estimates of parameters describing vertical propagation can be made. By comparing formulae specified for media with anisotropy of higher symmetry with formulae used in literature, it is shown that an improper use of some weak anisotropy parameters can lead to unnecessary decrease of accuracy. Sensitivity of the approximate RPP coefficients to basic weak anisotropy parameters is presented.

The fourth part, Finite differences, is devoted to the finite-difference method. The corresponding paper, by V. Bucha & L. Klimes, introduces Fortran package FD for 2-D P-SV elastic second-order finite differences and illustrates its usage on two numerical examples. Package FD is based on Fortran finite-difference code written by J. Zahradnik. The differences between 2-D finite-difference seismograms and 3-D ray-theory seismograms are demonstrated and discussed in the second example.

The final, but very important fifth part, CD-ROM contains the compact disk SW3D-CD-3, edited by V. Bucha and L. Klimes, containing the revised, updated and extended software. A more detailed description can be found directly on the CD-ROM.

Research Report 8 also includes the list of members of the SW3D Consortium Project (during 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 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. Martins, No. 200.466/93-3), by the National Nature Science Foundation of China No. 49774230 (X. Zheng), by the Charles University grant 170/1998/B-GEO/MFF and by Czech-French programme Barrande under Contract 97078.

Prague, June 1999

Vlastislav Cerveny
Ludek Klimes
Ivan Psencik

In: Seismic Waves in Complex 3-D Structures, Report 8, pp. 5-7, Dep. Geophys., Charles Univ., Prague, 1999.
SW3D - main page of consortium Seismic Waves in Complex 3-D Structures .