RESEARCH PROGRAMME ON SEISMIC WAVES IN COMPLEX 3-D STRUCTURES
(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.
RESEARCH PROGRAMME FOR THE TWENTIETH YEAR
October 1, 2012 -- September 30, 2013
1. Ray histories and two-point ray tracing in complex 3-D structures
Ray histories are of principal importance not only for
two-point ray tracing and for wavefront tracing,
but also for the summation of Gaussian beams.
Properties of the projection of ray coordinates onto Cartesian
coordinates within individual ray histories will be studied. New
applications of ray histories to numerical algorithms will be proposed.
The two-point ray tracing code will further be tested and applied
to various velocity models.
2. Synthetic seismograms in 3-D isotropic
and anisotropic complex structures
Methods to calculate synthetic seismograms in complex structures will
be studied, mutually compared and combined.
The synthetic seismograms will also be compared with synthetic
seismograms generated by non-ray methods.
For velocity models suggested by the consortium members,
we are ready to perform
ray-synthetic studies illustrating how the wave responses
differ in heterogeneous and anisotropic media
from homogeneous or isotropic media.
Calculation of synthetic seismograms by coupling ray theory
in layered velocity models will be tested.
Emphasis will be put on numerical implementation of Gaussian beams.
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.
3. Gaussian beams in inhomogeneous media
Paraxial Gaussian beams in smoothly varying anisotropic media.
Paraxial Gaussian beams in laterally varying layered anisotropic structures.
Paraxial Gaussian beams in smoothly varying anisotropic attenuating media.
Optimization of the shape of Gaussian beams.
4. Seismic wave propagation in inhomogeneous weakly anisotropic media
In the prevailing-frequency approximation
of the coupling ray theory for S waves,
the wave field is composed of two S waves
expressed in terms of their frequency-independent
coupling-ray-theory travel times and amplitudes.
We shall investigate possibilities to include
the prevailing-frequency approximation of the coupling ray theory
into the interpolation within ray cells in anisotropic media,
which could facilitate
the common-source Kirchhoff prestack depth migration with
coupling-ray-theory S waves in the future.
Theory of coupling S-wave Gaussian packets.
Improving the accuracy of the coupling ray theory for S waves
by including the first-order perturbation of polarization vectors.
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 velocity models with structural 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.
Derivation of P-wave and S-wave moveout expressions
for transverse isotropy with vertical or tilted axis of symmetry
and for lower-symmetry anisotropy
using weak-anisotropy perturbation formulae.
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.
5. Seismic wave propagation in anisotropic attenuating media
Investigation of plane waves
propagating in anisotropic attenuating media.
Both homogeneous and inhomogeneous waves will be considered.
For weakly attenuating media,
the perturbation methods will be used.
Special attention will be devoted to the attenuation vector
and to the quality factor Q,
particularly to its directional dependence.
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.
6. Fast computation of multivalued ray-theory travel times
and Green functions at the nodes of 3-D grids
Controlled initial-value ray tracing is used to cover
the velocity model by ray tubes and ray cells.
Multivalued ray-theory travel times and other quantities
at the nodes of a 3-D grid are then obtained
by interpolation within ray cells.
This algorithm is especially useful for the Born approximation
and for Kirchhoff migrations in heterogeneous 3-D velocity models,
and may be used for nonlinear hypocentre determination.
The current computer code can be applied to
P, S and converted waves in isotropic velocity models,
and to P waves in anisotropic velocity models.
It will further be tested in various velocity models,
including the velocity models delivered by the consortium members.
We intend to extend the algorithm and computer code
to S waves in anisotropic velocity models by including
the coupling-ray-theory S-wave travel times and amplitudes
calculated using perturbations 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 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 shot and receiver positions.
7. Computation of two-point paraxial travel times
Algorithms and codes for fast computation of approximate two-point
paraxial travel times calculated in the vicinities
of reference rays will be studied and tested numerically.
Inhomogeneous, isotropic or anisotropic
velocity models with curved interfaces will be considered.
Principal attention will be devoted to the study
of the accuracy and efficiency of the proposed algorithms.
We shall study calculation of two-point paraxial travel times
from perturbed sources to perturbed receivers
and their use in various applications.
8. Perturbation derivatives and spatial derivatives of travel time
Theoretical and numerical applications of the equations
for transforming the perturbation derivatives
and the third-order and higher-order spatial derivatives
of travel time at structural interfaces.
Generalization of spatial and perturbation derivatives
of travel time to Hamilton-Jacobi equation
with right-hand sides dependent on ray parameters.
Spatial and perturbation derivatives
of two-point travel time (characteristic function).
9. 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 wave fields,
and on the accuracy of other modelling methods designed or studied
within the framework of the project.
The main attention will be devoted to the estimation of the
feasibility and costs of ray tracing,
and to the definition and high-frequency validity of velocity models.
10. Sensitivity of seismic waves to the structure
(sensitivity Gaussian packets)
Continuing investigation how the perturbations of a generally inhomogeneous
isotropic or anisotropic structure manifest themselves
in the wave field, and which perturbations can be detected
within a limited aperture and a limited frequency band.
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.
11. Ray-Born approximation
Applicability of the ray-Born approximation to specific problems:
Study of the nonlinearity of ray-theory seismograms
with respect to perturbations of the velocity model.
Derivation of the weak-contrast reflection-transmission coefficients
from the Born approximation.
12. Lyapunov exponents and velocity model smoothing
Construction and smoothing of velocity models will further
be studied, with emphasis on the application of Sobolev scalar
products and Lyapunov exponents.
Attention will be paid to a possible extension of the estimation
of Lyapunov exponents to smooth 3-D velocity models
and to 2-D velocity models with structural interfaces.
Sobolev scalar products with spatially variable weights
will also be studied.
13. Seismic tomography and related problems
Development of theory, algorithms and programs applicable
in refraction travel-time tomography,
with emphasis on the estimation of the accuracy of the
resulting velocity model compared to the geological structure.
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.
14. Local anisotropy parameter estimation
from vertical-seismic-profiling measurements
Local determination of elastic
parameters from the vertical-seismic-profiling measurements.
Use of P-wave and, possibly, S-wave data.
Inversion for parameters of TI media with tilted axis of symmetry and
for the angles specifying the axis.
Study of possibility to use NMO or AVO data to constrain the inversion.
Applications to real data sets.
15. Migrations
Resolution and accuracy of migrations will be studied. Attention will
be paid to the physical meaning of the migrated sections and to their
sensitivity to the velocity model, including its anisotropy.
Study of possibilities to include coupling ray theory in seismic imaging.
Generalization of Kirchhoff migration to multivalued travel times.
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.
16. Electromagnetic ray theory
Developing the ray theory for electromagnetic waves
in heterogeneous bianisotropic media.
17. Sample data for the program packages
Examples of the input data describing or approximating velocity models
delivered by the consortium members or other typical velocity models
can be prepared.
Examples of the input data to perform calculations in such
velocity models can also be prepared.
18. Concluding remarks
In addition to this programme, we will certainly be responsive to
specific technical suggestions and recommendations of the consortium
members within the general framework of the project.
The research in most directions listed above will continue into the
future years of the project.
SW3D PROGRAM PACKAGES
Package CRT:
Velocity model: Using package MODEL (see below).
Isotropic velocity models with attenuation,
anisotropic velocity models without attenuation.
Type of waves: Arbitrary type of elementary seismic body wave
corresponding to the zero-order ray theory
(P, S, converted, coupled S waves).
Computations: Arbitrary position and shape of the source,
initial-value ray tracing by numerical integration of ray equations,
general isotropic-ray-theory rays,
anisotropic-ray-theory P-wave rays
and anisotropic common S-wave rays in smooth velocity models,
two-point ray tracing by the shooting method based on ray histories,
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 synthetic seismograms,
including the coupling ray theory
along isotropic or anisotropic common S-wave rays
and the response of fine layers at receiver sites
(program package RMATRIX by C.J. Thomson, linked to the CRT package).
Fast computation of multivalued ray-theory travel times,
Green functions and other quantities at the nodes of 3-D grids
by interpolation within ray cells
(P, S and converted waves in general isotropic velocity models,
P waves in smooth anisotropic velocity models).
Least-square travel-time tomography with smoothing using Sobolev
scalar products.
Acquisition schemes: Surface seismics (land and marine),
vertical seismic profiling, cross-hole, ocean bottom.
Planned innovations:
Extension of the anisotropic-ray-theory P-wave and anisotropic
common S-wave ray tracing towards general initial conditions
and velocity models with structural interfaces.
Tracing anisotropic-ray-theory S-wave rays
in smooth velocity models without interfaces.
Coupling ray theory along the anisotropic-ray-theory rays
in smooth velocity models without interfaces.
The package will be extended to solve various inverse problems,
stochastic travel-time tomography in particular.
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) 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:
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.
Package NET:
Velocity model:
Isotropic without attenuation,
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 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:
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.
You may download this Research Programme in
PDF (60 kB).
SW3D
- main page of consortium Seismic Waves in Complex 3-D Structures .