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.
SW3D PROGRAM PACKAGES
Package CRT:
Model: Using package MODEL (see below).
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 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).
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 models with structural interfaces.
Tracing anisotropic-ray-theory S-wave rays
in smooth models without interfaces.
Coupling ray theory along the anisotropic-ray-theory rays
in smooth models without interfaces.
The package will be extended to solve various inverse problems,
stochastic travel-time tomography in particular.
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 S-wave common-ray computations
for inhomogeneous weakly anisotropic media.
(d) First-order S-wave common-ray computations for layered media.
(e) Computation of the complete propagator matrix in Cartesian coordinates.
(f) Calculation of KMAH index in anisotropic media.
(g) Stabilization of ray tracing in a vicinity of S-wave singularities.
(h) Removal of problems of P-wave reflections/transmissions
in a vicinity of S-wave singularities.
(i) 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 dissipation 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.
New documentation. 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.
Dissipation 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.
RESEARCH PROGRAMME FOR THE SIXTEENTH YEAR
October 1, 2008 -- September 30, 2009
1. Sample data for the program packages
Examples of the input data describing or approximating models
delivered by the consortium members or other typical models
will be prepared.
Examples of the input data to perform calculations in such
models will also be prepared.
2. 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 models.
3. Paraxial Fresnel edge waves
Contributions of single diffractions to the Green function may be
approximated by paraxial approximations of Fresnel edge waves in
vicinities of boundary rays between ray histories.
4. 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 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 models will further 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.
5. Gaussian beams in inhomogeneous anisotropic 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 dissipative media.
6. Seismic wave propagation in inhomogeneous weakly anisotropic media
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.
Continuing development of an algorithm and a code for modelling
P- and S-wave propagation (including coupling)
based on the first-order ray tracing (FORT)
and dynamic ray tracing (FODRT) in smooth
laterally varying weakly anisotropic media
with varying axes or planes of symmetry.
Derivation of formulae and development of a code
to apply FORT and FODRT to layered media.
7. Applicability of the high-frequency asymptotics in the
vicinity of S-wave singularities
Anisotropic S-wave ray tracing in a vicinity of singularities
in inhomogeneous anisotropic media. Investigation of accuracy
of various high-frequency approximations (zeroth-order anisotropic
ray approximation, coupling ray theory, higher-order
anisotropic ray approximations, Gaussian beams, etc.) for S waves,
propagating in strongly or weakly anisotropic
homogeneous or inhomogeneous media in the vicinity of singularities,
by comparison with more precise methods
(exact solutions, finite differences, etc.).
8. Seismic wave propagation in anisotropic dissipative media
Investigation of plane waves
propagating in anisotropic viscoelastic media.
Both homogeneous and inhomogeneous waves will be considered.
For weakly dissipative 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 great attention will be devoted to waves generated by point sources.
9. Reflection/transmission coefficients
Effective reflection coefficients for spherical waves
incident at a planar interface between two homogeneous
isotropic dissipative media.
Perturbation of reflection/transmission coefficients
with respect to elastic moduli cijkl.
Estimating influence of attenuation on the
reflection/transmission coefficients.
10. Computation of ray-theory travel times, amplitudes
and other quantities at the nodes of 3-D grids
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.
11. Perturbations of travel time
Generalization of the equations for the second-order and higher-order
perturbations of travel time to models with 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).
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 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.
13. Sensitivity of seismic waves to the structure
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 corresponding algorithm for
linearized inversion based on wave-field sensitivity
to structural Gabor functions.
14. Lyapunov exponents and model smoothing
Construction and smoothing of velocity macro 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 models and to 2-D models with
structural interfaces.
Sobolev scalar products with spatially variable weights
will also be studied.
15. Seismic tomography and related problems
Development of theory, algorithms and programs applicable
in seismic travel-time tomography and inversion of the coherency
panels, with emphasis on the estimation of the accuracy of the
resulting 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.
16. Seismic sources and hydrofracture monitoring
Studying the inaccuracy of the absolute source location
and the inaccuracy of the relative source location due
to the inaccurate velocity model.
Forward and inverse problems for moment tensors
of seismic sources in isotropic and anisotropic media.
17. 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.
18. Decomposition of a wave field into Gabor wavelets
Discretized integral decomposition of a spatial wave field
at a fixed time or a time-dependent wave field along a smooth
surface into Gabor wavelets in 2-D or 3-D.
The Gabor wavelets may be frequency-dependent and their
shape may be smoothly varying in space and time, which
requires much more general decomposition than the integral
Gabor transform.
The decomposition into Gabor wavelets may be useful for the
decomposition of a general wave field into Gaussian beams or packets.
19. 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.
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.
20. 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.
You may download PostScript file
prog09.ps (58 kB) with the Research Programme.
SW3D
- main page of consortium Seismic Waves in Complex 3-D Structures .