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 and anisotropic 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 "". The research programme was begun on October 1, 1993.


October 1, 2003 - September 30, 2004

1. Program packages

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, structural interface triangulation, VRML and GOCAD visualization.

Package CRT:
Model: Using package MODEL.
Type of waves: Arbitrary type of elementary seismic body wave corresponding to the zero-order ray theory (P,S, converted).
Computations: Arbitrary position and shape of the source, initial-value ray tracing by numerical integration of ray equations, 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 for weak anisotropy 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.
Aquisition schemes: Surface seismics (land and marine), VSP, cross-hole, OBS, OBC.
Planned innovations: Anisotropic common-ray approximation of the coupling ray theory for S waves. The package will be extended to solve various inverse problems, 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. Possibility of VRML visualization.
Types of waves: Arbitrary type of elementary seismic body wave (P, S, qP, qS1, qS2, any converted wave, coupled qS 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 qS waves, ray synthetic seismograms, particle ground motions.
Aquisition schemes: Surface seismics (land and marine), VSP, cross-hole, OBS, OBC.
Planned innovations: (a) Common ray approximation based on a ray in an anisotropic medium. (b) Generalization of the quasi-isotropic approximation for layered media. (c) Weak attenuation. (d) Calculation of KMAH index in anisotropic media. (e) Stabilization of ray tracing in a vicinity of shear-wave singularities. (f) Further debugging, removal of inconsistencies in the description of the package.

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.
Aquisition schemes: Surface seismics (land and marine), VSP, cross-hole, OBS, OBC.

Package FD:
Model: Using package MODEL.
Type of waves: Complete elastic wave field.
Computations: Presently 2-D, without fluids.
Aquisition schemes: Surface seismics (land).

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.

2. 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.

3. Two-point ray tracing in complex isotropic 3-D structures

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.

4. 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.

5. 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. 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.

6. Seismic wave propagation in weakly anisotropic inhomogeneous media

Anisotropic common ray approximation of the coupling ray theory. Derivation of coupling ray theory from the elastodynamic equation concentrated on the study of errors due to neglected terms. 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. Study of accuracy of coupling ray theory in a vicinity of S-wave singularities by comparison with more precise methods like finite differences, reflectivity, etc. Estimation of frequencies at which the quasi-isotropic common-ray or anisotropic common-ray approximation of the coupling ray theory should be replaced by the anisotropic ray theory, or by the coupling ray theory along the anisotropic ray theory rays.

Study of models obeying ellipsoidal P--wave weak anisotropy. Development of a P--wave ray tracer for inhomogeneous weakly anisotropic 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, quasi-isotropic approximation, 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 other more precise methods (exact solutions, finite differences, etc.).

8. Seismic wave propagation in anisotropic dissipative media

Computation of slowness vectors, polarisation vectors, energy and pseudoenergy fluxes of P, S1 and S2 plane waves propagating in general anisotropic viscoelastic media. Both homogeneous and inhomogeneous plane waves will be considered. Behaviour of attributes of these waves in various types of viscoelastic anisotropic media will be studied both analytically and numerically. Perturbation method will be used to find approximate formulae for the attributes.

9. 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.

10. 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.

11. Sensitivity of seismic waves to the structure

We shall study how the perturbations of a generally heterogeneous isotropic or anisotropic structure manifest themselves in the wavefield, and which perturbations can be detected within a limited aperture and a limited frequency band.

12. 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.

13. Seismic tomography

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.

14. Local anisotropy parameter estimation from VSP measurements

Development of algorithms for the local determination of elastic parameters from the VSP measurements. Use of P- and S-wave data. Inversion for parameters of TI media with tilted axis of symmetry and for the angles specifying the axis. Attempts to answer the question whether is it possible to discriminate between VTI and TI with tilted axis of symmetry. Study of possibility to use NMO or AVO data to constrain the inversion.

15. Decomposition of a wave field into Gabor wavelets

Discretized integral decomposition of a spatial wave field at a fixed time or a time-dependent wavefield 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.

16. 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 be very efficient and 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.

17. Finite-difference solutions of elastodynamic equations

The main attention will be paid to the hybrid DW-FD and Ray-FD methods.

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.
You may download PostScript file (47 kB) with the Research Programme.
SW3D - main page of consortium Seismic Waves in Complex 3-D Structures .