REPORT 5
May 1997

Seismic Ray Theory

Vlastislav Cerveny

Contents

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, 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

3.6 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_I 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

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 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 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

SW3D