(SW3D)

The research is focused primarily on the fundamental issues of
high-frequency wave propagation in complex 3-D isotropic
and anisotropic structures, which go beyond the traditional approaches.
The ray method and its extensions, as well as its combination
with other methods are mainly applied and investigated.
The emphasis is put on new, stable, more efficient and flexible
algorithms for both forward numerical modelling and inversion
of wave fields in 3-D inhomogeneous,
isotropic or anisotropic, non-attenuating or attenuating structures.
Considerable attention is devoted to applications involving
S waves or electromagnetic waves, converted waves,
wave splitting and coupling in anisotropic media,
particle ground motions, etc.
Much more detailed information can be obtained at
"http://sw3d.cz".
The research programme was begun on October 1, 1993,
and was devoted to seismic waves as suggested by its title.
In the recent years,
the research has been extended to electromagnetic waves.

We shall continue in extending the algorithm and computer code of interpolating the Green function within ray cells to S waves or electromagnetic waves in smooth anisotropic velocity models by including the coupling-ray-theory travel times and amplitudes calculated using the prevailing-frequency approximation of the coupling ray theory along common anisotropic rays.

Algorithms of fast calculation of ray-theory travel times in dense rectangular grids will be investigated further. The accuracy and efficiency of the interpolation of ray-theory travel times within ray cells in 3-D velocity models will be studied, and the relevant numerical algorithms will be improved, or new ones will be proposed. Attention will also be devoted to the interpolation between different source and receiver positions.

Integral superposition of paraxial Gaussian beams. Integral superposition of paraxial Gaussian packets. Applications of the integral superpositions.

Optimization of the shape of Gaussian beams.

Anisotropic common ray approximation of the coupling ray theory: extension of the anisotropic-ray-theory P-wave and anisotropic common S-wave and electromagnetic-wave ray tracing towards general initial conditions and velocity models with structural interfaces.

A quantitative criterion whether the SH and SV reference rays are better than the anisotropic common reference rays in a given velocity model, or vice versa. Tracing the SH and SV reference rays in velocity models with generally heterogeneous dependence of the reference symmetry vector on the spatial coordinates. The SH and SV reference rays for the coupling ray theory: initial conditions, structural interfaces.

Study of the problems related to anisotropic-ray-theory ray tracing in a vicinity of S-wave or electromagnetic-wave singularities. Theoretical and numerical studies of the phase-slowness and ray-velocity surfaces.

Derivation of coupling ray theory from the elastodynamic or Maxwell equations concentrated on the study of errors due to neglected terms in order to estimate the accuracy of coupling ray theory. Study of coupling ray series.

Numerical comparison of various approximations of coupling ray theory with more accurate methods.

The results obtained for plane waves propagating in homogeneous media will be generalized to non-planar waves propagating in smoothly varying anisotropic weakly attenuating media using perturbation methods based on the ray theory. A special attention will be devoted to waves generated by point sources. Together with the attenuation properties, also the dispersion properties of waves propagating in attenuating media will be studied.

Developing the algorithms and computer programs for calculating the sensitivity Gaussian packets. Developing the corresponding algorithm for linearized inversion based on wave-field sensitivity to structural Gabor functions. Sensitivity Gaussian packets should offer explicit correspondence between the time and depth sections. Attention is paid to the optimization of the shape of Gaussian packets.

Stochastic travel-time tomography: Developing an algorithm for calculating geometrical covariances of travel times. Developing an algorithm for calculating geometrical covariances between rays and B-splines.

Determination of the medium correlation functions from well logs. Calculation of sonic-log travel times in anisotropic media. Estimation of uncertainty of sonic-log travel times. Estimation of attenuation from vertical-seismic-profiling travel times.

Estimation of anisotropy parameters from various experiments (vertical seismic profiling, reflection seismics, etc.): travel-time inversion, inversion of moveout formulae, local parameter estimation from vertical-seismic-profiling measurements. Use of P-wave and, possibly, S-wave data (travel times, polarizations, etc.). Tests with synthetic and, if available, real data sets.

The numerical algorithm of the common-source Kirchhoff prestack depth migration will be extended to three-component seismograms.

Possibilities to include coupling ray theory in seismic imaging will be studied. Kirchhoff migration will be generalized to multivalued travel times.

Amplitude preserving Kirchhoff migration in anisotropic media: Attention will be paid to the synthetic study of possibilities and limitations to recover reflection coefficients from data measured in inhomogeneous anisotropic media. Common-source Kirchhoff prestack depth migration with S waves will be generalized from isotropic to anisotropic velocity models. Estimation of the misplacement of the interface due to an incorrectly known medium using the travel-time perturbations will be compared numerically with migrated sections.

**Package ANRAY:**

*Velocity model*:
3-D laterally varying structures containing isotropic and
anisotropic non-vanishing layers. Specification of elastic parameters
inside individual layers either by linear interpolation between
isosurfaces of elastic parameters, or by B-spline interpolation
within a 3-D rectangular grid of elastic parameters.
VRML and GOCAD visualization.

*Types of waves*: Arbitrary type of elementary seismic body wave
(P, S, any converted wave, coupled S waves).

*Computations*: Numerical integration of ray tracing and dynamic
ray tracing equations, calculation of ray vectorial amplitudes,
ray-theory Green function including the Green function in the quasi-isotropic
approximation for S waves, ray synthetic seismograms, particle
ground motions.

*Acquisition schemes*: Surface seismics (land and marine),
vertical seismic profiling, cross-hole, ocean bottom.

*Planned innovations*:
(a) Incorporation of effects of weak attenuation.
(b) Ray tracing and dynamic ray tracing
in media with rotated higher-symmetry anisotropy
(transverse isotropy, orthorhombic symmetry).
(c) First-order ray tracing of P waves and coupled S waves
in layered inhomogeneous weakly anisotropic media.
(d) Simplified dynamic ray tracing in Cartesian coordinates.
(e) Calculation of KMAH index in anisotropic media.
(f) Removal of problems of P-wave reflections/transmissions
in a vicinity of S-wave singularities.
(g) Extension of applicability of the ANRAY package
to Gaussian beam summation or Maslov method.
(h) Further debugging, completion, and removal of inconsistencies
in the description of the package.

**Package SEIS:**

*Velocity model*:
2-D laterally varying isotropic structures composed of layers
separated by curved interfaces. Any interface may form edges.
It may also coincide with a neighbouring interface(s) in some region.
Thus, the velocity models with isolated bodies and pinchouts can be considered.
Inside the layers, the velocities of P and S waves may vary
in two directions.
Weak attenuation and non-planar topography can be considered.

*Types of waves*:
Arbitrary type of elementary seismic body wave
(P,S, any converted or multiply reflected wave).

*Computations*:
Arbitrary position of a point source,
numerical integration of 2-D ray tracing and dynamic ray
tracing equations, computation of ray vectorial amplitudes
or Green functions of individual elementary waves,
ray synthetic seismograms, particle ground motions.

*Acquisition schemes*:
Surface seismics (land and marine), vertical seismic profiling, cross-hole.

*Planned innovations*:
Alternative computation of synthetic seismograms in the frequency domain.
Ocean bottom configuration.
Extended visualization. Extension of test examples.

**Package MODEL:**

*Velocity model*: General 3-D layered and block isotropic or anisotropic
structures, containing isolated bodies, pinchouts, etc.
Inside the layers and blocks, the elastic parameters may vary
in all three dimensions.
Attenuation and non-planar topography can be considered.
Possibility of velocity model smoothing, data fitting by inversion
including fitting and smoothing GOCAD models,
conversion of velocity model parametrization,
triangulation of structural interfaces,
VRML and GOCAD visualization.

*Types of waves*:
Seismic waves.

**Package NET:**

*Velocity model*:
Isotropic without attenuation,
using package MODEL or using gridded velocities.

*Types of waves*:
Wave-independent, first arrivals, constrained first arrivals.

*Computations*: Arbitrary position and shape of the source.
First-arrival travel times in the whole velocity model are computed.
The algorithm of computation is independent of the velocity model's
complexity.

*Acquisition schemes*: Surface seismics (land and marine),
vertical seismic profiling, cross-hole, ocean bottom.

**Package FORMS:**

*Types of waves*:
Wave-independent or seismic waves.

*Computations*:
Subroutines used by other program packages including
data input and output subroutines,
management and plotting of synthetic seismograms,
2-D and 3-D graphics including 3-D virtual reality
with VRML and GOCAD visualization,
manipulation and calculation with gridded data (data cubes),
programs for matrix and vector operations necessary for inversion,
other general-purpose seismic software.

*Planned innovations*:
Nonlinear hypocentre determination.
Program for computation of plane-wave reflection/transmission coefficients
at planar interfaces separating arbitrary anisotropic media.

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SW3D - main page of consortium