(SW3D)

The research is focused primarily on the fundamental issues of
high-frequency seismic 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 seismic wave fields in 3-D inhomogeneous,
isotropic or anisotropic, elastic or attenuating structures.
Considerable attention is also devoted to applications involving
S waves, converted waves, S-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.

Gaussian-beam synthetic seismograms: Deriving the discretization error of the uneven summation of Gaussian beams. Developing an algorithm for sampling the ray-parameter domain for Gaussian beams.

Optimization of the shape of Gaussian beams.

Research into the prevailing-frequency approximation of the coupling ray theory. This prevailing-frequency approximation could facilitate coupling-ray-theory migrations in the future.

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

Coupling ray theory along the anisotropic-ray-theory rays in smooth models without interfaces.

Derivation of coupling ray theory from the elastodynamic equation 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. Theory of coupling S-wave Gaussian packets. Definition of coupling ray theory travel times and amplitudes, and study of frequency dispersion of S-wave coupling. Quantification of the relevance of anisotropic S-wave coupling to velocity analysis and imaging.

Development of the code for modelling P- and coupled S-wave propagation based on the first-order ray tracing and dynamic ray tracing (FORT and FODRT) in layered inhomogeneous weakly anisotropic media.

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 seismic body waves propagating in attenuating media will be studied.

We intend to extend the algorithm and computer code to S waves in anisotropic velocity models by including the perturbations of S-wave travel times from common S-wave 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 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 shot and receiver positions.

Developing the corresponding algorithm for linearized inversion based on wave-field sensitivity to structural Gabor functions.

Derivation of the weak-contrast reflection-transmission coefficients from the Born approximation.

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.

Two alternative decompositions will be studied and compared: (a) discretized analytical integral decomposition, (b) numerical discrete decomposition. Whereas the discretized analytical integral decomposition has been applied to Gaussian-packet prestack depth migration, the numerical discrete decomposition can also be applied to linearized inversion based on wave-field sensitivity to structural Gabor functions.

Algorithm of the Gaussian-packet prestack depth migration is being developed. Gaussian packets should offer explicit correspondence between the time and depth sections. Attention is paid to the optimization of the shape of Gaussian packets.

Study of possibilities to include coupling ray theory in seismic imaging.

Amplitude preserving Kirchhoff migration in anisotropic media: 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.

**Package ANRAY:**

*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) Computation of the 6×6 propagator matrix 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) Further debugging, completion, and removal of inconsistencies
in the description of the package.

**Package SEIS:**

*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 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:**

*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 model smoothing, data fitting by inversion
including fitting and smoothing GOCAD models,
conversion of model parametrization,
triangulation of structural interfaces,
VRML and GOCAD visualization.

**Package NET:**

*Model*: Using package MODEL or using gridded velocities.

*Types of waves*: First arrivals, constrained first arrivals.

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

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

**Package FORMS:**

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

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