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 *R*_{PP} 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 "http://seis.karlov.mff.cuni.cz/consort/main.htm".

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