May 1997

- 1 Introduction
- 2 The elastodynamic equation and its simple solutions
- 2.1 Linear elastodynamics
- 2.1.1 Stress-strain relations
- 2.1.2 Elastodynamic equation for inhomogeneous anisotropic media
- 2.1.3 Elastodynamic equation for inhomogeneous isotropic media
- 2.1.4 Acoustic wave equation
- 2.1.5 Time-harmonic equations
- 2.1.6 Energy considerations
- 2.2 Elastic plane waves
- 2.2.1 Time-harmonic acoustic plane waves
- 2.2.2 Transient acoustic plane waves
- 2.2.3 Vectorial transient elastic plane waves
- 2.2.4 Christoffel matrix and its properties
- 2.2.5 Elastic plane waves in an anisotropic medium
- 2.2.6 Elastic plane waves in an isotropic medium
- 2.2.7 Energy considerations for plane waves
- 2.2.8 Phase and group velocity surfaces. Slowness surface
- 2.2.9 Elastic plane waves in isotropic and anisotropic media: Differences
- 2.2.10 Inhomogeneous plane waves
- 2.3 Elastic plane waves across a plane interface
- 2.3.1 Acoustic case
- 2.3.2 Isotropic elastic medium
- 2.3.3 Anisotropic elastic medium
- 2.3.4 Transient plane waves
- 2.4 High-frequency elastic waves in smoothly inhomogeneous media
- 2.4.1 Acoustic wave equation
- 2.4.2 Elastodynamic equation for isotropic inhomogeneous media
- 2.4.3 Elastodynamic equation for anisotropic inhomogeneous media
- 2.4.4 Energy considerations for high-frequency waves propagating in smoothly inhomogeneous media
- 2.4.5 High-frequency seismic waves across a smooth interface
- 2.5 Point source solutions of the acoustic and elastodynamic wave equations
- 2.5.1 Point source solutions of the acoustic wave equation
- 2.5.2 Acoustic Green function
- 2.5.3 Point source solutions of the elastodynamic equation
- 2.5.4 Elastodynamic Green function for isotropic homogeneous medium
- 2.5.5 Elastodynamic Green function for anisotropic homogeneous medium

- 3 Seismic rays and travel times
- 3.1 Ray tracing systems in inhomogeneous isotropic media
- 3.1.1 Rays as characteristics of the eikonal equation
- 3.1.2 Relation of rays to wavefronts
- 3.1.3 Rays as extremals of the Fermat's functional
- 3.1.4 Ray tracing system from Snell's law
- 3.1.5 Relation of rays to the energy flux trajectories
- 3.1.6 Physical rays. Fresnel volumes
- 3.2 Rays in laterally varying layered structures
- 3.2.1 Initial conditions for a single ray
- 3.2.2 Rays in layered and block structures
- 3.2.3 Anomalous rays in layered structures
- 3.2.4 Curvature and torsion of the ray
- 3.3 Ray tracing
- 3.3.1 Numerical ray tracing
- 3.3.2 Choice of the integration parameter along the ray
- 3.3.3 Travel-time computations along a ray
- 3.3.4 Ray tracing in simpler types of media
- 3.4 Analytical ray tracing
- 3.4.1 Homogeneous media
- 3.4.2 Constant gradient of the square of slowness,
*V*^{-2} - 3.4.3 Constant gradient of the n-th power of slowness,
*V*^{-n} - 3.4.4 Constant gradient of the logaritmic velocity, ln
*V* - 3.4.5 Polynomial rays
- 3.4.6 More general
*V*^{-2}models - 3.4.7 Cell ray tracing
- 3.4.8 Semi-analytical ray tracing in layered and block structures
- 3.4.9 Approximate ray tracing
- 3.5 Ray tracing in curvilinear coordinates
- 3.5.1 Curvilinear orthogonal coordinates
- 3.5.2 The eikonal equation in curvilinear orthogonal coordinates
- 3.5.3 The ray tracing system in curvilinear orthogonal coordinates
- 3.5.4 Ray tracing in spherical polar coordinates
- 3.5.5 Modified ray tracing systems in spherical polar coordinates
- 3.5.6 Ray tracing in curvilinear non-orthogonal coordinates
- 3.6 Ray tracing in inhomogeneous anisotropic media
- 3.6.1 Eikonal equation
- 3.6.2 Ray tracing system
- 3.6.3 Initial conditions for a single ray in anisotropic inhomogeneous media
- 3.6.4 Rays in layered and block anisotropic structures
- 3.6.5 Ray tracing for simpler types of anisotropic media
- 3.6.6 Ray tracing in factorized anisotropic media
- 3.6.7 Energy considerations
- 3.7 Ray tracing and travel-time computations in 1-D models
- 3.7.1 Vertically inhomogeneous media
- 3.7.2 Analytical solutions for vertically inhomogeneous media
- 3.7.3 Polynomial rays in vertically inhomogeneous media
- 3.7.4 Radially symmetric media
- 3.8 Direct computation of travel times and/or wavefronts
- 3.8.1 Ray-theory travel times and first-arrival travel times
- 3.8.2 Network shortest-path ray tracing
- 3.8.3 Finite-difference method
- 3.8.4 Wavefront construction method
- 3.8.5 Concluding remarks
- 3.9 Perturbation methods for travel times. Frozen rays
- 3.9.1 Isotropic inhomogeneous models
- 3.9.2 Anisotropic inhomogeneous models
- 3.10 Ray fields
- 3.10.1 Ray parameters. Ray coordinates
- 3.10.2 Jacobians of transformations
- 3.10.3 Elementary ray tube. Geometrical spreading
- 3.10.4 Properties and computation of the ray Jacobian J
- 3.10.5 Caustics. Classification of caustics
- 3.10.6 Solution of the transport equation in terms of the ray Jacobian
- 3.11 Boundary-value ray tracing
- 3.11.1 Initial-value and boundary-value ray tracing: a review
- 3.11.2 Shooting methods
- 3.11.3 Bending methods
- 3.11.4 Perturbation methods

- 4 Dynamic ray tracing. Paraxial ray methods
- 4.1 Ray-centered coordinates
- 4.1.1 Ray-centered coordinates: definition, orthogonality
- 4.1.2 Ray-centered basis vectors as polarization vectors
- 4.1.3 Computation of ray-centered basis vectors along ray
- 4.1.4 Local ray-centered Cartesian coordinate system
- 4.1.5 Transformation matrices
- 4.1.6 Ray tracing in ray-centered coordinates. Paraxial ray tracing system
- 4.2 Dynamic ray tracing in ray-centered coordinates
- 4.2.1 Paraxial eikonal equation
- 4.2.2 Matrix
**M**of the second derivatives of the travel time field - 4.2.3 Paraxial travel times
- 4.2.4 Linear dynamic ray tracing systems
- 4.3 Ray propagator matrix
- 4.3.1 Definition of the ray propagator matrix
- 4.3.2 Symplectic properties
- 4.3.3 Determinant of the ray propagator matrix. Liouville's theorem
- 4.3.4 Chain rule
- 4.3.5 Inverse of the ray propagator matrix
- 4.3.6 Solution of the dynamic ray tracing system in terms of the ray propagator matrix
- 4.4 Dynamic ray tracing in layered media
- 4.4.1 Geometry of the interface
- 4.4.2 Matrix
**M**across the interface - 4.4.3 Paraxial slowness vector
- 4.4.4 Transformation of matrices
**Q**and**P**across the interface - 4.4.5 Ray propagator matrix across a curved interface
- 4.4.6 Ray propagator matrix in a layered medium
- 4.4.7 Surface-to-surface ray propagator matrix
- 4.4.8 Chain rules for the minors of the ray propagator matrix. Fresnel zone matrix
- 4.5 Initial conditions for dynamic ray tracing
- 4.5.1 Initial slowness vector at a smooth initial surface
- 4.5.2 Initial values of
**Q**,**P**and**M**at a smooth initial surface - 4.5.3 Special case: Local Cartesian coordinates
*z*as ray parameters_{I} - 4.5.4 Point source
- 4.5.5 Initial line
- 4.5.6 Initial surface with edges and vertexes
- 4.6 Paraxial travel-time field and its derivatives
- 4.6.1 Continuation relations for matrix
**M** - 4.6.2 Determination of matrix
**M**from travel times known along a data surface - 4.6.3 Matrix of curvature of the wavefront
- 4.6.4 Paraxial travel times
- 4.6.5 Paraxial slowness vector
- 4.7 Dynamic ray tracing in Cartesian coordinates
- 4.7.1 Dynamic ray tracing systems in Cartesian coordinates
- 4.7.2 Redundant equations in the systems
- 4.7.3 Reduced dynamic ray tracing systems
- 4.7.4 Determination of the 4*4 ray propagator matrix
- 4.7.5 The 6*6 ray propagator matrix
- 4.7.6 Higher derivatives of the travel-time field
- 4.8 Special cases. Analytical dynamic ray tracing
- 4.8.1 Homogeneous layers separated by curved interfaces
- 4.8.2 Homogeneous layers separated by plane interfaces
- 4.8.3 Layers with a constant gradient of velocity
- 4.8.4 Reflection/transmission at a curved interface
- 4.9 Boundary-value ray tracing for paraxial rays
- 4.9.1 Two-point ray tracing in ray-centered coordinates
- 4.9.2 Two-point ray tracing in Cartesian coordinates
- 4.9.3 Two point eikonal
- 4.9.4 Mixed second derivatives of the travel time field
- 4.10 Geometrical spreading in a layered medium
- 4.10.1 Geometrical spreading in terms of matrices
**Q**(**x**) and**Q^**(**x**) - 4.10.2 Relative geometrical spreading
- 4.10.3 Relation of geometrical spreading to matrices
**M**and**K** - 4.10.4 Factorization of geometrical spreading
- 4.10.5 Determination of the relative geometrical spreading from travel-time data
- 4.11 Fresnel volumes
- 4.11.1 Analytical expressions for Fresnel volumes and Fresnel zones
- 4.11.2 Paraxial Fresnel volumes. Fresnel volume ray tracing
- 4.11.3 Fresnel volumes of first arriving waves
- 4.11.4 Comparison of different methods of calculating Fresnel volumes and Fresnel zones
- 4.12 Phase shift due to caustics. KMAH index
- 4.12.1 Determination of the KMAH index by dynamic ray tracing
- 4.12.2 Decomposition of the KMAH index
- 4.13 Dynamic ray tracing along a planar ray. 2-D models
- 4.13.1 Transformation matrices
**P**and**Q** - 4.13.2 In-plane and transverse ray propagator matrices
- 4.13.3 Matrices
**M**and**K** - 4.13.4 In-plane and transverse geometrical spreading
- 4.13.5 Paraxial travel times
- 4.13.6 Paraxial rays close to a planar central ray
- 4.13.7 Paraxial boundary-value ray tracing in the vicinity of a planar ray. Two-point eikonal
- 4.13.8 Determination of geometrical spreading from the travel time data in 2-D media
- 4.14 Dynamic ray tracing in inhomogeneous anisotropic media
- 4.14.1 Dynamic ray tracing in Cartesian coordinates
- 4.14.2 Reduced dynamic ray tracing system
- 4.14.3 The 4*4 ray propagator matrix in anisotropic inhomogeneous media
- 4.14.4 The 4*4 ray propagator matrix in anisotropic homogeneous media
- 4.14.5 Ray Jacobian and geometrical spreading
- 4.14.6 Matrix of second derivatives of the travel-time field
- 4.14.7 Paraxial travel times, slowness vectors and group velocity vectors
- 4.14.8 Reduced dynamic ray tracing across a structural interface
- 4.14.9 The 4*4 ray propagator matrix in layered anisotropic media
- 4.14.10 Surface-to-surface ray propagator matrix
- 4.14.11 Factorisation of
**Q**_{2}. Fresnel zone matrix - 4.14.12 Boundary-value ray tracing for paraxial rays in anisotropic media
- 4.14.13 Phase shift due to caustics. KMAH index

- 5 Ray amplitudes
- 5.1 Acoustic case
- 5.1.1 Continuation of amplitudes along a ray
- 5.1.2 Point source solutions. Radiation function
- 5.1.3 Amplitudes across an interface
- 5.1.4 Acoustic reflection/transmission coefficients
- 5.1.5 Amplitudes in 3-D layered structures
- 5.1.6 Amplitudes along a planar ray
- 5.1.7 Acoustic ray-theory Green function
- 5.1.8 Receiver on an interface
- 5.1.9 Point source at an interface
- 5.1.10 Final equations for a point source
- 5.2 Elastic isotropic structures
- 5.2.1 Vectorial complex-valued amplitude function of P and S waves
- 5.2.2 Continuation of amplitudes along a ray
- 5.2.3 Point source solutions. Radiation matrices
- 5.2.4 Amplitudes across an interface
- 5.2.5 Amplitudes in 3-D layered structures
- 5.2.6 Elastodynamic ray theory Green function
- 5.2.7 Receiver at an interface
- 5.2.8 Source at an interface
- 5.2.9 Final equations for amplitude matrices
- 5.2.10 Unconverted P waves
- 5.2.11 Compressional waves in liquid media. Particle velocity amplitudes
- 5.2.12 Unconverted S waves
- 5.2.13 Amplitudes along a planar ray. 2-D case
- 5.3 Displacement reflection/transmission coefficients for elastic isotropic media
- 5.3.1 P-SV and SH reflection/transmission coefficients
- 5.3.2 Orientation index
*epsilon* - 5.3.3 Normalized displacement P-SV and SH reflection/transmission coefficients
- 5.3.4 Displacement P-SV and SH R/T coefficients: discussion
- 5.3.5 Displacement reflection/transmission matrices
- 5.3.6 Normalized displacement reflection/transmission matrices
- 5.3.7 Reciprocity of R/T coefficients
- 5.3.8 End-point matrices for the Earth's surface
- 5.4 Elastic anisotropic structures
- 5.4.1 Computation of amplitudes along a ray
- 5.4.2 Point source solution. Radiation functions
- 5.4.3 Amplitudes across an interface
- 5.4.4 Amplitudes in 3-D layered structures
- 5.4.5 Ray theory Green function
- 5.5 Ray amplitudes in weakly dissipative media
- 5.5.1 Non-causal dissipation filters
- 5.5.2 Causal dissipation filters
- 5.5.3 Anisotropic media
- 5.5.4 Waves across interfaces in dissipative media
- 5.6 Ray series method. Acoustic case
- 5.6.1 Scalar ray series. Amplitude coefficients
- 5.6.2 Recurrence system of equation of the ray method
- 5.6.3 Transport equations of higher order and their solutions
- 5.6.4 Reflection and transmission
- 5.6.5 Alternative forms of the scalar ray series
- 5.6.6 Applications of higher-order ray approximations
- 5.6.7 Head waves
- 5.6.8 Modified forms of the ray series
- 5.7 Ray series method. Elastic case
- 5.7.1 Vectorial ray series. Vectorial amplitude coefficients
- 5.7.2 Recurrence system of equations of the ray method
- 5.7.3 Decomposition of vectorial amplitude coefficients
- 5.7.4 Higher-order approximations. Additional components
- 5.7.5 Higher-order approximations. Principal components
- 5.7.6 Reflection and transmission
- 5.7.7 Alternative forms of the vectorial ray series
- 5.7.8 Exact finite vectorial ray series
- 5.7.9 Applications of higher-order ray approximations
- 5.7.10 Seismic head waves
- 5.7.11 Modified forms of the vectorial ray series
- 5.8 Paraxial amplitudes
- 5.8.1 Paraxial ray approximation for the displacement vector
- 5.8.2 Paraxial Gaussian beams
- 5.9 Validity conditions and extensions of the ray method
- 5.9.1 Validity conditions of the ray method
- 5.9.2 Singular regions. Diffracted waves
- 5.9.3 Summation methods
- 5.9.4 Seismic waves of interference character
- 5.9.5 Generalized ray method

- 6 Ray synthetic seismograms
- 6.1 Elementary ray synthetic seismograms
- 6.1.1 Displacement vector of an elementary wave
- 6.1.2 Preservation of the analytical signal along the ray
- 6.1.3 Analytical signal of the elementary wave. Source time function
- 6.1.4 Computation of the elementary synthetic seismograms in the time domain
- 6.1.5 Elementary synthetic seismograms for complex-valued travel times
- 6.1.6 Computation of elementary synthetic seismograms in the frequency domain
- 6.1.7 Fast frequency response (FFR) algorithm
- 6.2 Ray synthetic seismograms
- 6.2.1 Ray expansions
- 6.2.2 Computation of ray synthetic seismograms in the time domain
- 6.2.3 Computation of ray synthetic seismograms for complex-valued travel times
- 6.2.4 Computation of ray synthetic seismograms in the frequency domain
- 6.2.5 Modified frequency-response expansions
- 6.3 Ray synthetic seismograms in weakly dissipative media
- 6.3.1 Dissipation filters
- 6.3.2 Non-causal absorption
- 6.3.3 Causal absorption
- 6.3.4 Constant-Q model
- 6.4 Ray synthetic particle ground motions
- 6.4.1 Polarization plane
- 6.4.2 Polarization equations
- 6.4.3 Polarization of interfering signals
- 6.4.4 Polarization of non-interfering P waves
- 6.4.5 Polarization of non-interfering S waves in a smooth medium
- 6.4.6 Polarization of S waves at structural interfaces
- 6.4.7 Polarization of S waves at the Earth's surface
- 6.4.8 Causes of quasi-elliptical polarization of seismic body waves in isotropic structures
- 6.4.9 Quasi-elliptical polarization of seismic body waves in layered structures
- 6.4.10 Polarization of seismic body waves in anisotropic media

- Appendix A: Fourier transform, Hilbert transform and analytical signals
- A.1 Fourier transform
- A.2 Hilbert transform
- A.3 Analytical signals

- References

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