Report 11 of the Consortium project "Seismic Waves in Complex 3-D Structures" summarizes the work done towards the end of the seventh year and during the eighth year of the project, in the period June, 2000 -- May, 2001. It also includes the compact disk with updated and extended versions of computer programs distributed to the sponsors, and their brief descriptions.
Our group working within the project during the eighth year has consisted of seven research workers: Johana Brokesova, Vaclav Bucha, Petr Bulant, Vlastislav Cerveny, Ludek Klimes, Ivan Psencik and Vaclav Vavrycuk, and of two students: Karel Zacek (MSc), and Xuyao Zheng (PhD). Karel Zacek defended successfully his Diploma Thesis entitled "Smoothing the Marmousi Model and Optimization of the Shape of Gaussian Beams for Gaussian Packet Migration", on June 1, 2001. The Commission decided to propose the work of Karel Zacek for a special award. Rosaria Tondi, a PhD student from Instituto di Ricerca sul Rischio Sismico, Milano, Italy, spent with us four months. Jorge Leonardo Martins from the Campinas University at Brasil continued in his cooperation with our Consortium, even after returning to Brasil (see his article with Ivan in this volume). Steve Horne and Scott Leaney from Schlumberger Gatwick, England, provided data and collaborated with us on a specific problem during the mentioned period (see the article in this volume).
During the eighth year of the project, V. Cerveny received the Ernst Mach Award from the Academy of Sciences of the Czech Republic "for his contribution to physical sciences". Ivan Psencik has been invited to chair the section on "Seismic Modelling" at the 7th international congress of the Brazilian Geophysical Society at Salvador, Brasil, and to present an invited paper there. The relevant paper, entitled "Ray methods in the modelling of seismic wave fields" is included in this volume. Ludek Klimes was invited to give an invited lecture during the three-days workshop "Geometrically Based High Frequency Wave Methods with Applications" at the Institute for Pure and Applied Mathematics, University of California Los Angeles. He presented there a contribution entitled "Meaning and calculation of travel times". I. Psencik and V. Cerveny served as guest editors of the topical issue of Pure and Applied Geophysics, Vol.159 (2002), devoted to the "Workshop Meeting on Seismic Waves in Laterally Inhomogeneous Media V" in Zahradky, Czech Republic, in June 2000.
During the eighth year of the project, considerable effort has been concentrated on the following problems: (a) equations for anisotropic common-ray dynamic ray tracing, which are necessary for including the common-ray approximation into ray tracing programs; (b) derivation of coupling ray theory equations from the elastodynamic equation in order to estimate accuracy of various modifications of the coupling ray theory; (c) numerical algorithm of the linearized tomographic inversion of travel times, including estimation of the model covariance function describing the deviation of the model from the geological structure; (d) decomposition of common-shot time sections into Gaussian packets, necessary for the Gaussian-packet migrations; (e) equations and numerical algorithm of the Gaussian-packet migration. In spite of a considerable progress in solving these problems, we have not reached final results to be published in Report 11, and the research in all these areas has to continue in the next year of the project.
Research Report 11 may be roughly divided into five parts, see the Contents.
The first part, Seismic models and inversion techniques, is devoted to various kinds of inverse problems and to the construction of velocity models suitable for ray tracing and application of ray-based high-frequency asymptotic methods.
In the contribution by X. Zheng and I. Psencik, which is a copy of the article accepted for publication in Pure and Applied Geophysics, Vol.159 (2002), the authors test an inversion scheme for the local evaluation of weak anisotropy (and also elastic) parameters from measurements of slowness and polarization vectors of qP waves recorded in a borehole in a multi-azimuthal multiple-source offset VSP synthetic experiment. The paper contains basic formulae for such an inversion. Effects of varying number of profiles with sources and of noise added to synthetic seismic sections are illustrated.
X. Zheng, I. Psencik, S. Horne and S. Leaney use the same algorithm and apply it to the walkaway VSP synthetic and real data. Since data are collected along a single profile in this case, only some of the weak anisotropy (or elastic) parameters are found, which describe anisotropy within vertical plane containing the profile. Several approaches are used to process the walkaway VSP data collected in the Java region.
In the contribution by J.L. Martins and I. Psencik, which is a copy of the abstract submitted to the 7th international congress of the Brazilian Geophysical Society, the authors analyse limitations of the inversion scheme for the determination of weak anisotropy parameters (or their combinations) from the observed PP reflection coefficients. Effects of number of considered profiles and of noise are studied on a synthetic data set.
Three papers by L. Klimes [1, 2, 3], devoted to the application of the medium covariance functions to travel-time tomography, are the revised versions of the papers published in Reports 4 and 6, accepted for publication in Pure and Applied Geophysics, Vol.159 (2002). The revision includes references to many related papers by other authors.
P. Bulant, in his paper on the construction of velocity models suitable for ray tracing, concisely describes the theory and algorithms of smoothing and illustrates the method on three numerical examples: model Hess (already briefly described in Report 10), SEG/EAGE Salt Model and model Pluto. The paper differs from the analogous paper, accepted for publication in Pure and Applied Geophysics, by extended description of fitting the SEG/EAGE Salt Model and by additional numerical example of model Pluto.
K. Zacek's paper is a revised version of the corresponding paper of Report 10, accepted for publication in Pure and Applied Geophysics, Vol.159 (2002).
The second part, Ray methods in isotropic and anisotropic media, is devoted to the high-frequency methods in general, but does not contain the papers more specifically addressing problems of weak attenuation or dissipative media, which have been postponed to the third and fourth parts.
In the first paper of the second part, K. Zacek describes a brand-new algorithm of simultaneous minimization of the mean squared width of Gaussian beams and squared Sobolev norm describing smoothness of the initial shape of Gaussian beams in dependence on all parameters along the Hamiltonian hypersurface in the phase space (position along the initial surface, take-off angle, and travel time along rays). This algorithm is of principal importance in developing the method of Gaussian packet migrations.
The brief overview of applications of the ray methods in modelling seismic wave fields, by I. Psencik, P. Bulant, V. Cerveny and L. Klimes, has been prepared as an expanded abstract of the invited contribution at the 7th international congress of the Brazilian Geophysical Society.
In the third article, by V. Cerveny, the Fermat's variational principle for anisotropic inhomogeneous media is studied. In seismology, Fermat's variational principle has mostly been used in a parametric form (valid for any parameter used to specify the position of points on curves). It is shown that the Legendre transform cannot be applied to the parametric Lagrangian to determine the relevant Hamiltonian, as the Hessian determinant of the transformation vanishes identically in this case. The Fermat's variational principle has to be modified so that the Hessian determinant is different from zero. Two such modifications are proposed, and the relation between modified Lagrangians and Hamiltonians are discussed in detail. The Finslerian metric tensor is introduced, and its relation to Lagrangians, Hamiltonians, local group velocity surfaces and local slowness surfaces is discussed.
In the fourth article, by V. Cerveny, the reduced Hamiltonian for anisotropic inhomogeneous media is introduced, and the reduced Hamiltonian ray tracing system is derived and discussed. The reduced Hamiltonian is based on the same 6x6 eigenvalue problem which has been successfully exploited in the 6x6 propagator techniques for 1-D anisotropic inhomogeneous media, in the computation of 6x6 R/T coefficients from anisotropic layers, in the factorization of the elastodynamic equation, and in the parabolic equation method for anisotropic inhomogeneous media.
In V. Vavrycuk's contribution, the far-field asymptotic formula for the S-wave elastodynamic Green function at a kiss singularity in homogeneous anisotropic media is derived. In contrast to the standard asymptotics in regular directions the derived formula is more complex and is expressed in the form of a 1-D integral. The integral is specified for the kiss singularity along the symmetry axis in transversely isotropic media and along the fourfold symmetry axes in the tetragonal and cubic media.
The third part, Weak anisotropy, addresses the problems relevant to weakly anisotropic media.
Although the numerical algorithm of the coupling ray theory described in the paper by P. Bulant and L. Klimes applies to general anisotropy, the paper is located in this part because the problem is especially important in weakly anisotropic structures. The paper is a revised version of a paper of Report 7, supplemented with the analysis of various commonly used quasi-isotropic approximations of the coupling ray theory.
I. Psencik and V. Vavrycuk present approximate formulae for the determination of the ray vector (unit vector in the direction of the group velocity) from the phase normal, and vice versa. A simple and transparent formula derived using the first-order perturbation theory is tested on two examples of anisotropic media. Although one of them has anisotropy about 30%, the formula works very well. The formula breaks down in regions of triplication of the group velocity surface. In addition to the mentioned formula, approximate formulae for the qP- and qS-wave phase and group velocities are also presented and tested.
The fourth part, Dissipative media, is devoted to the problems of elastic wave propagation in dissipative media.
The first paper, by V. Cerveny, studies the energy flux of time-harmonic waves in anisotropic dissipative media. It is most common to consider the average energy flux, which is real-valued and time-independent. An extension of this definition is the complex Poynting vector, which is also time-independent, but complex-valued. Recently, also the instantaneous energy flux, which is complex-valued and time-harmonic, has found interesting applications. In the paper, all these energy flux quantities are investigated for general anisotropic media, mainly in relation to the reflection/transmission problem. Energy flux quantities related to the summary wavefield, composed of several waves, are derived. It is shown that, in a dissipative medium, also the interaction energy contributions must be considered in the summary energy flux, in addition to the energy fluxes of the individual waves. The energy conservation equations at a structural interface between two anisotropic dissipative media are derived and studied.
In the second paper, by J. Brokesova and V. Cerveny, the energy flux and relevant energy quantities at a plane interface between two isotropic dissipative media are studied numerically. Large attention is devoted to the energy conservation equations and to interaction contributions. An attempt is done to find whether some energy-based quantities could be used in local criteria for selection of "downgoing" and "upgoing" reflected and transmitted waves. Numerical experiments offer certain interesting conclusions, but no one of those conclusions can be directly applied in the selection criteria. A suitable selection criterion is proposed for a special, but very important case, in which the angle of incidence equals the attenuation angle. This case corresponds to the real-valued lateral slowness vector components.
In the final fifth part, CD-ROM, compact disk SW3D-CD-5 is preceded by the article by V. Bucha discussing the possibilities of displaying 3-D models, rays and other objects using their virtual reality representations, viewed by public-domain VRML browsers or the GOCAD software.
The CD-R compact disk SW3D-CD-5, edited by V. Bucha, P. Bulant and L. Klimes, contains the revised, updated and extended software. A more detailed description can be found directly on the compact disk. Unlike the compact disks of the previous reports, SW3D-CD-5 contains over 100 complete papers from journals and previous Reports, mostly in PostScript, few in PDF or HTML, refer to the copy of the Consortium WWW pages on the compact disk. Compact disk SW3D-CD-5 is included in Report 11 in two versions, as the UNIX disk and DOS disk. The versions differ just by the form of ASCII files.
This Introduction is followed by the list of members of the SW3D Consortium during the eighth year of the project. We have been very pleased to welcome two new consortium members, NORSAR (Kjeller, Norway), and Paradigm (Houston, U.S.A.). We hope they will find the membership in our Consortium profitable.
The Research Programme for the current, eighth year of the Consortium project comes after the list of members. The Research Programme for the next year will be prepared after the discussion on the Consortium meeting, June 18-21, 2001. More detailed information regarding the SW3D Consortium Project is available online at "http://seis.karlov.mff.cuni.cz/consort/main.htm".
We are very grateful to all our sponsors for the financial support. The research has also been partially supported by the Grant Agency of the Czech Republic under Contracts 205/95/1465, 205/00/1350 and 205/01/0927, by the Grant Agency of the Charles University under Contract 237/2001/B-GEO/MFF, by the Ministry of Education of the Czech Republic within Research Project J13/98 113200004, and by the National Natural Science Foundation of China under Contract 49774230.
Prague, June 2001