You have just opened Report 10 of the Consortium Project "Seismic Waves in Complex 3-D Structures", Charles University, Faculty of Mathematics and Physics, Department of Geophysics, Prague. The report summarizes the work done towards the end of the sixth year and during the seventh year of the project, in the period June, 1999 -- May, 2000. It also includes the compact disk with updated and extended versions of all computer programs distributed to the sponsors, and their brief description.

Our group working within the project during the seventh year has consisted of seven research workers: J. Brokesova, V. Bucha, P. Bulant V. Cerveny, L. Klimes, I. Psencik and V. Vavrycuk, and two students: K. Zacek (MS) and Xuyao Zheng (PhD). Jorg Leonardo Martins, Jingsong Liu, J. Dellinger and R.P. Young collaborated with us on specific problems during the above mentioned period.

Considerable effort has been concentrated on the algorithms and codes to solve inverse problems. Preliminary versions of the programs related to inversion have been revised, and about 20 of new Fortran programs are devoted to model fitting, smoothing and inversion. We have concentrated on the construction of models suitable for ray methods, taking advantage of Sobolev scalar products and Lyapunov exponents. We have simultaneously concentrated on the development of the algorithms for the Gaussian-packet prestack depth migration, which we estimate to be more efficient than the Kirchhoff migration. To move forward to the travel-time inversion using the medium correlation functions and to advance in the development of the Gaussian-packet migration belong to our most important tasks for the next year of the project.

Research Report 10 may be roughly divided into five parts, see the Contents.

The first part, Seismic models and inversion techniques, is devoted to the construction of sufficiently smooth velocity models and to the inversion of travel times. Since the research related to various inversion methods is in progress, the papers collected in this part do not cover all fields of the research performed in the respective period and most of them just express preliminary results and will be subject to future revisions.

In the first paper, by L. Klimes, the Sobolev scalar products are defined and discussed from the point of view of the application to model fitting and smoothing.

The second paper, by K. Zacek, and the third paper, by L. Klimes, describe the construction of a velocity model by smoothing gridded velocities of the "Marmousi model and dataset" using the Sobolev scalar products. The smooth velocity model should be suitable for ray tracing and for Gaussian-packet migrations.

The fourth paper, by P. Bulant, is devoted to an example how to fit the gridded velocities and the gridded interfaces by a 2-D model with interfaces, suitable for ray tracing.

The article by I. Psencik, J. Liu & R.P. Young contains an extension of the work of the first two authors on travel time inversion in inhomogeneous generally anisotropic media. In contrast to the last year, when the authors tested the inversion on synthetic data, this year they use a unique experimental data set from the Underground Research Laboratory in Manitoba, Canada. The data set yields a very good spatial and angular coverage of the volume, in which the elastic parameters are sought. The authors study the effects of the choice of the reference model, orientation of the model box and number of used grid points on the results of the inversion.

The second part, Ray methods in isotropic media, is devoted to the ray method in general, applied to isotropic media.

P. Bulant explains in his short contribution that the difference of the computational times between the bilinear travel-time interpolation within ray cells and much more accurate bicubic interpolation is negligible.

The second paper, by L. Klimes, is devoted to the numerical calculation of the point-source geometrical spreading from gridded travel times or slowness vectors in 2-D. The geometrical spreading is obtained by finite differencing the slowness vectors corresponding to two different positions of a point source.

The third part Ray methods in anisotropic media, is devoted to the extensions and modifications of the standard ray method in inhomogeneous anisotropic media.

The first contribution, by V. Cerveny, studies the extensions, like summation of paraxial ray approximations, summation of paraxial Gaussian beams, Maslov-Chapman method, etc. Approximate high-frequency solutions of the elastodynamic equation for a 3-D inhomogeneous anisotropic layered medium, based on the summation of paraxial ray approximations and of paraxial Gaussian beams, are derived. The wave under consideration may be generated at an initial surface with the variable initial travel time along it, by a point source with an arbitrary radiation function, and so forth. By the proper choice of the radiation function, relevant summation Green functions are obtained. Individual quantities in the superposition integrals can be computed by dynamic ray tracing in wavefront orthonormal coordinates or in Cartesian coordinates. The receiver may be situated arbitrarily in the model, including structural interfaces and the Earth's surface. The summation of paraxial ray approximations includes the Maslov-Chapman integrals as a special case.

The second contribution, by V. Vavrycuk, is devoted to the ray tracing in anisotropic media with S-wave singularities. The paper studies a problem of tracing rays in anisotropic media in singular directions and their vicinities, where standard ray-tracing techniques fail. The paper shows that these problems are not essential and can be overcome by following the behaviour of polarization vectors along the ray. Reliability of the ray tracer depends on the quality of the used eigenvalue/eigenvector solver used in the computation. A possibility of splitting of rays in singularities is discussed.

The third contribution, by V. Vavrycuk, is devoted to the anisotropic Green functions in simple types of anisotropy. The paper shows that exact forms of the Green function for simple types of anisotropic media can be derived by calculation of higher-order ray approximations. The derived Green functions are in some cases even simpler than the Green function for isotropic media. The paper demonstrates applicability of higher-order ray approximations in problems of wave propagation in anisotropic media.

The fourth part, Weak anisotropy, is devoted to the extension of the ray method for inhomogeneous, weakly anisotropic media, and to the accuracy of relevant computations. The article by I. Psencik & J. Dellinger represents a final version of the paper on quasi-isotropic approach, which should appear in Geophysics. In addition to many refinements, the paper contains a new section on comparison of synthetics generated by the quasi-isotropic (QI) approach with the synthetics generated by the reflectivity method for anisotropic media (program Anivec developed by Mallick and Frazer was used for this purpose). Despite differences in the specification of the model and signal, the reflectivity and QI seismograms resemble each other very well. Let us note that the reflectivity synthetics required several thousand times greater computational effort than the QI synthetics.

The final, but very important fifth part, CD-ROM, contains the compact disk SW3D-CD-4, edited by V. Bucha, P. Bulant and L. Klimes, containing the revised, updated and extended software. A more detailed description can be found directly on the CD-ROM.

Research Report 10 also includes the list of members of the SW3D Consortium Project (during the seventh year), and the Research Programme of the Consortium Project for the seventh year. More detailed information regarding the SW3D Consortium Project is available at "".


P. Bulant received the 1999 Van Weelden Award from the EAGE (European Association of Geoscientists and Engineers) "for his contribution to the calculation of 3-D travel times".

P. Bulant received the award of the General Conference of the European Physical Society EPS11 (London 1999) for his contribution to the theoretical geophysics.

V. Cerveny received in 1999 the Conrad Schlumberger Award from the EAGE (European Association of Geoscientists and Engineers) "in recognition of his many contributions to asymptotic wave theory and its applications to seismic modeling".


We are very grateful to all our sponsors for the financial support. The research has been also partially supported by the Grant Agency of Czech Republic, GACR 205/00/1350, by CNPq Brasil (postdoc. scholarship for J.L. Martins, No. 200.466/93-3) and by the National Nature Science Foundation of China No. 49774230 (X. Zheng).

Prague, May 2000

Vlastislav Cerveny
Ludek Klimes
Ivan Psencik

In: Seismic Waves in Complex 3-D Structures, Report 10, pp. 5-7, Dep. Geophys., Charles Univ., Prague, 2000.
SW3D - main page of consortium Seismic Waves in Complex 3-D Structures .