Seismic Ray Theory

Vlastislav Cerveny


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
2.4.6 Space-time ray method
2.5 Point-source solutions. Green functions
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 media
2.5.5 Elastodynamic Green function for anisotropic homogeneous media
2.6 Application of Green functions to the construction of more general solutions
2.6.1 Representation theorems
2.6.2 Scattering integrals. First-order Born approximation
2.6.3 Line-source solutions
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. Ray codes
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, lnV
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
3.5.7 Comments to ray tracing in curvilinear 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 Solution of the eikonal equation by separation of variables
3.8.3 Network shortest-path ray tracing
3.8.4 Finite-difference method
3.8.5 Wavefront construction method
3.8.6 Concluding remarks
3.9 Perturbation methods for travel times
3.9.1 First-order perturbation equations for travel times in smooth media
3.9.2 Smooth isotropic medium
3.9.3 Smooth anisotropic medium
3.9.4 Degenerate case of qS waves in anisotropic media
3.9.5 Travel time perturbations in layered media
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 Methods based on structural perturbations
3.12 Surface-wave ray tracing
3.12.1 Surface waves along a surface of a laterally varying structure
3.12.2 Dispersion relations and surface-wave ray tracing
3.12.3 Surface-wave ray tracing along a surface of an isotropic structure
4 Dynamic ray tracing. Paraxial ray methods
4.1 Dynamic ray tracing in 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 Ω
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
4.1.7 Dynamic ray tracing system in ray-centered coordinates
4.1.8 Paraxial travel times
4.2 Hamiltonian approach to dynamic ray tracing
4.2.1 Cartesian rectangular coordinates
4.2.2 Wavefront orthonormal coordinates
4.2.3 Orthonomic system of rays
4.2.4 Curvilinear coordinates
4.3 Propagator matrices of dynamic ray tracing systems
4.3.1 Definition of the propagator matrix
4.3.2 Symplectic properties
4.3.3 Determinant of the propagator matrix. Liouville's theorem
4.3.4 Chain rule
4.3.5 Inverse of the propagator matrix
4.3.6 Solution of the dynamic ray tracing system in terms of the propagator matrix
4.3.7 6x6 propagator matrices
4.3.8 Inhomogeneous dynamic ray tracing system
4.4 Dynamic ray tracing in isotropic 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
4.4.9 Backward propagation
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 zI as ray parameters
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. Parabolic and hyperbolic 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 6x6 propagator matrix in a layered medium
4.7.3 Transformation of the interface propagator matrix
4.7.4 Ray perturbation theory
4.7.5 Second order travel-time perturbation
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 Analytical dynamic ray tracing in Cartesian coordinates
4.8.5 Reflection/transmission at a curved interface
4.9 Boundary-value ray tracing for paraxial rays
4.9.1 Paraxial two-point ray tracing in ray-centered coordinates
4.9.2 Paraxial two-point ray tracing in Cartesian coordinates
4.9.3 Paraxial two point eikonal
4.9.4 Mixed second derivatives of the travel time field
4.9.5 Boundary-value problems for surface-to-surface ray tracing
4.9.6 Concluding remarks
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
4.10.6 Determination of the 4x4 propagator matrix from travel-time data
4.10.7 Exponentially increasing geometrical spreading. Chaotic behavior of rays
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 Q and P
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 Dynamic ray tracing in wavefront orthonormal coordinates
4.14.3 The 4x4 ray propagator matrix in anisotropic inhomogeneous media
4.14.4 The 4x4 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 Dynamic ray tracing across a structural interface
4.14.9 The 4x4 ray propagator matrix in layered anisotropic media
4.14.10 Surface-to-surface ray propagator matrix
4.14.11 Factorisation of Q2. 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 pressure reflection/transmission coefficients
5.1.5 Amplitudes in 3-D layered structures
5.1.6 Amplitudes along a planar ray
5.1.7 Pressure 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
5.1.11 Initial ray-theory amplitudes at a smooth initial surface
5.1.12 Initial ray-theory amplitudes at a smooth initial line
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. Conversion coefficients
5.2.8 Source at an interface
5.2.9 Final equations for amplitude matrices
5.2.10 Unconverted P waves
5.2.11 P waves in liquid media. Particle velocity amplitudes
5.2.12 Unconverted S waves
5.2.13 Amplitudes along a planar ray. 2-D case
5.2.14 Initial ray-theory amplitudes at a smooth initial surface in a solid medium
5.2.15 Initial ray-theory amplitudes at a smooth initial line in a solid medium
5.3 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 P-SV and SH conversion coefficients
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
5.4.6 Quasi-isotropic ray theory. qS wave coupling
5.4.7 R/T coefficients and R/T matrices
5.4.8 Initial ray-theory amplitudes at a smooth initial surface. Elastic Kirchhoff integrals
5.5 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 equations 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 ray approximations. Additional components
5.7.5 Higher-order ray 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. Two-term ray method
5.7.10 Seismic head waves
5.7.11 Modified forms of the vectorial ray series
5.8 Paraxial displacement vector. Paraxial Gaussian beams
5.8.1 Paraxial ray approximation for the displacement vector
5.8.2 Paraxial Gaussian beams
5.8.3 Summation methods
5.8.4 Superposition integrals
5.8.5 Maslov-Chapman integrals
5.8.6 Summation in 2-D models
5.8.7 Alternative versions of the superposition integral
5.8.8 Phase shift due to caustics. Derivation
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 Inhomogeneous waves
5.9.4 Summation methods
5.9.5 Waves propagating in a preferred direction
5.9.6 Generalized ray theory
6 Ray synthetic seismograms
6.1 Elementary ray synthetic seismograms
6.1.1 Displacement vector of an elementary wave
6.1.2 Conservation 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
6.2.6 Time-domain versions of integral solutions
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

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Cambridge Univ. Press, Cambridge, 2001, ISBN 0-521-36671-2 hardback (2001), ISBN 0-521-01822-6 paperback (2005).
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