##
RESEARCH PROGRAMME ON SEISMIC WAVES IN COMPLEX 3-D STRUCTURES

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 recent 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 the
forward numerical modelling of seismic wave fields in 3-D
inhomogeneous, isotropic and anisotropic structures. Considerable
attention is also devoted to applications involving shear waves,
converted waves, shear wave splitting in anisotropic media, particle
ground motions, etc.

The research programme started on October 1, 1993. The fourth
year of the programme started on October 1, 1996.

### RESEARCH PROGRAMME FOR THE FOURTH YEAR

#### October 1, 1996--September 30, 1997

*1. Revisions of packages MODEL, CRT and NET*

All future revisions of program packages MODEL, CRT, and NET,
delivered to the sponsors in the first year of the project, will be delivered to the
sponsors in the next years of the project.

#### Packages MODEL and CRT:

**Model:**
General 3-D layered and block isotropic structure,
containing isolated bodies,
pinchouts, etc. Inside the layers and blocks, the velocity and
density varies
laterally in all the three dimensions. Dissipation may be
considered.

**Type of waves:**
Arbitrary type of elementary seismic body wave
corresponding to the
zero order ray theory
(P,S, converted).
Arbitrary position of the source.

**Computations:**
Arbitrary position and shape of the source,
two-point ray tracing by the shooting method, initial-value ray
tracing by numerical integration of ray equations, travel time
computation, dynamic ray tracing, paraxial-ray propagator matrix,
geometrical spreading, vectorial amplitudes, polarization vectors.
The package may be applied to the evaluation of the elastodynamic
ray-theory Green function, and to the computation of the synthetic
seismograms.

**Applications:**
Reflection methods, refraction methods, VSP, hole-to-hole.

#### Package NET:

**Model:**
General 3-D layered and block isotropic model. The
medium parameters are specified
at grid points of a 3-D rectangular mesh. The same model as in the
complete seismic
ray tracing may also be used.

**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
computations include also travel time of all non-ray waves (such as
the first arriving diffracted waves in shadow zones, head waves,
etc.). The algorithm of computation is independent on a model
complexity.

**Applications:**
Tomography for an arbitrary source-receiver
configuration, wavefront reconstruction, etc.

* 2. Ray tracing and synthetic wavefields in 3-D inhomogeneous
anisotropic structures*

All future revisions of program package ANRAY, delivered to the
sponsors
in the second year of the project, will be delivered to the sponsors
in
the next years of the project.

#### Package ANRAY:

**Model:**
3-D laterally varying structure containing isotropic and
anisotropic nonvanishing layers. Specification of elastic parameters
inside individual layers either by linear interpolation between
isosurfaces of elastic parameters or by a B-spline interpolation
within a 3-D rectangular grid.

**Types of waves:**
Arbitrary type of elementary seismic body wave
(P, S,
qP, qS1, qS2, any converted wave).

**Computations:**
Arbitrary position of the point source, numerical
integration of ray tracing and dynamic ray tracing equations,
calculation of ray vectorial amplitudes, ray Green function, ray
synthetic seismograms, particle ground motions.

**Applications:**
Reflection methods, refraction methods, VSP
and/or crosshole configuration.

**Main innovations:**
a) testing of the B-spline approximation of
the distribution of elastic parameters inside layers, development of
"interfaces" for specification of simpler types of models; b)
computation of the complete propagator matrix along rays; c)
modification of input data systems; d) further debugging, removing
inconsistencies in the extensive description of the package.

* 3. Sample data for the program packages*

The examples
of input data for the MODEL package, describing or approximating
models delivered by the sponsors or other typical models, will be
prepared. Upon request, also the sample data for programs CRT or NET
to perform calculations in such models will be prepared. The examples
of input data for the ANRAY package, for models delivered by the
sponsors or other typical models, will be prepared too.

* 4. Two-point ray tracing in complex isotropic structures*

The two-point ray tracing code will be tested and
applied to various models. Attention will also be devoted to the
calculation of two-point rays diffracted from edges and corner
points.

* 5. Synthetic seismograms in 3-D isotropic complex structures*

The advantage of the two-point ray tracer will be taken for the computation
of ray synthetic seismograms.

* 6. Ray tracing for common-offset and common-midpoint
configurations*

Study of possibilities of efficient calculation of rays for
common-offset and common-midpoint configurations.

* 7. Seismic wave propagation in weakly anisotropic inhomogeneous
media*

Study of possibilities to calculate *qP* wave Green functions in
general
weakly anisotropic media using the first-order perturbation theory.
Study of *qS* waves signatures in arbitrary weakly anisotropic media.

* 8. Replenishing incomplete sets of anisotropic elastic
parameters*

Study of replenishing incomplete sets of anisotropic elastic
parameters with default values. If all 21 real parts and 21 imaginary
parts of the complex-valued components of the stiffness matrix are not
known, the missing parameters may be supplied in such a way that the
resulting anisotropic medium is in some sense the closest one to the
isotropic medium.

* 9. Synthetic seismograms for sources and receivers situated at
the Earth's surface, at structural interfaces, and close to them*

Development of the code for the calculation of synthetic
seismograms using the radiation patterns of sources situated close to
structural interfaces in 3-D models. The radiation patterns derived
and studied in previous years of the project will be generalized for
this purpose. Attention will be devoted to sources situated at the
sea bottom.

* 10. Fast computation of ray-theory travel times*

Algorithms of fast calculation of ray-theory travel times in dense
rectangular grids will further be investigated. Accuracy and
efficiency
of the interpolation of ray-theory travel times between rays in 3-D
models will be studied and the relevant numerical
algorithms will be proposed. If possible, attention will also be devoted
to the interpolation between different shot and receiver positions.

* 11. Second-order methods in grid travel-time tracing*

Accuracy and efficiency of grid travel-time tracing methods to
evaluate
first-arrival travel times will further be studied.

* 12. Accuracy of seismic modelling*

The research will be concentrated mainly on the accuracy of travel-time
calculations,
on the accuracy of finite-difference modelling of seismic wavefields, and on
the accuracy of other modelling methods designed or studied in the
framework
of the project.

* 13. Seismic tomography*

Development of theory and algorithms applicable in seismic travel-time
tomography with emphasis on the estimation of its accuracy.

* 14. Finite-difference solutions of elastodynamic equations*

The research will concentrate on the hybrid Ray-FD code. The
method will employ the already existing codes for the Ray
and FD methods, respectively, as well as the coupling technique
validated in the hybrid DW-FD method (hereinafter DW=Discrete
Wavenumber).
Testing will start with 2-D models for which the Ray, DW, or FD
synthetic seismograms can be computed separately, without any hybrid
combination. The intention of the tests is to understand the
applicability limits of the method.
Comparisons of the Ray and DW synthetic seismograms will be carried
out for absorbing media with the intention to validate the absorption
tretament in the Ray method.

Investigation of the accuracy of the elastic FD method and DW-FD
method for models with non-planar topography will be finished. An attempt
will be made to evaluate a suite of the benchmark models by several
methods in an international cooperation (E. Priolo - Trieste, S. Gaffet -
France), and to display the results on a ftp server.

First attempts will be made to write a hybrid (DW-FD, or Ray-FD) code
in which a 3-D elastic FD method will be locally applied.

Possible simplifications of the 3-D elastic FD simulation in media
displaying a predominant 2-D structural features will be studied.

* 15. Concluding remarks*

In addition to this programme, we will certainly be responsive to
specific
technical suggestions and recommendations of sponsors within the
general framework of the project.
The research in most directions listed above will continue to the
fifth year of the project.

You may download the ** PostScript ** file
`prog97.ps` (60 kB) with the ** Research Programme**.

SW3D
- main page of consortium ** Seismic Waves in Complex 3-D Structures **.