The seismic ray method has found broad applications in the numerical calculation of seismic wavefields in complex 3-D, isotropic and anisotropic, laterally varying layered structures and in the solution of forward and inverse problems of seismology and seismic exploration for oil. This chapter outlines the basic features of the seismic ray method, and reviews its possibilities and recent extensions. Considerable attention is devoted to ray tracing and dynamic ray tracing of S waves in heterogeneous anisotropic media, to the coupling ray theory for S waves in such media, to the summation of Gaussian beams and packets, and to the selection of models suitable for ray tracing.

- 1. Introduction
- 2. Seismic ray theories for isotropic and anisotropic media
- 2.1. Seismic rays and travel times. Initial-value ray tracing
- 2.2. Ray histories, two-point ray tracing, wavefront tracing
- 2.2.1. Model and ray histories
- 2.2.2. Controlled initial-value ray tracing
- 2.2.3. Two-point ray tracing
- 2.2.4. Other applications of controlled initial-value ray tracing
- 2.2.5. Wavefront tracing
- 2.2.6. Interpolation within ray cells
- 2.3. Dynamic ray tracing. Paraxial ray methods
- 2.3.1. Dynamic ray tracing in Cartesian coordinates
- 2.3.2. Dynamic ray tracing in ray-centred coordinates
- 2.3.3. DRT propagator matrix
- 2.3.4. Second-order spatial derivatives of travel time
- 2.3.5. Third-order and higher-order spatial derivatives of travel time
- 2.3.6. Applications of DRT propagator matrix. Paraxial ray methods
- 2.4. Ray-theory amplitude
- 2.4.1. Determination of the scalar amplitude
- 2.4.2. Polarization in a non-degenerate case
- 2.4.3. Polarization in the vicinity of S-wave singularities or in isotropic media
- 2.5. Effects of interfaces
- 2.6. Ray-theory Green function
- 2.6.1. Elementary ray-theory Green function
- 2.6.2. Reciprocity of the ray-theory Green function
- 2.6.3. Phase shift of the Green function due to caustics. KMAH index
- 2.6.4. Example of phase shifts
- 2.7. Ray-theory seismograms

- 3. Ray-theory perturbations
- 3.1. Perturbation parameters
- 3.2. Perturbation of travel time
- 3.2.1. First-order perturbation derivatives of travel time
- 3.2.2. First-order perturbation derivatives of the travel-time gradient
- 3.2.3. Second-order perturbation derivatives of travel time
- 3.3. Optimizing model updates during linearized inversion of travel times

- 4. Coupling ray theory for S waves
- 4.1. Coupling equation for S waves
- 4.2. Coupling-ray-theory S-wave propagator matrix
- 4.3. Quasi-isotropic approximations of the coupling ray theory
- 4.3.1. Selection of the reference ray
- 4.3.2. Quasi-isotropic projection of the Green function
- 4.3.3. Quasi-isotropic approximation of the Christoffel matrix
- 4.3.4. Quasi-isotropic perturbation of travel times
- 4.4. Numerical examples

- 5. Summation of Gaussian beams and packets
- 5.1. Gaussian beams
- 5.1.1. Gaussian beam
- 5.1.2. Paraxial Gaussian beam
- 5.1.3. Equations for the second derivatives of travel time in Cartesian coordinates
- 5.1.4. Equations for the second derivatives of travel time in ray-centred coordinates
- 5.1.5. Solving the equation for a Gaussian beam
- 5.1.6. Amplitude of a Gaussian beam
- 5.2. Gaussian packets
- 5.2.1. Gaussian packet and the space-time eikonal equation
- 5.2.2. Paraxial Gaussian packet
- 5.2.3. Equations for the second derivatives of phase function in Cartesian coordinates
- 5.2.4. Equations for the second derivatives of phase function in ray-centred coordinates
- 5.2.5. Solving the equations for a Gaussian packet
- 5.2.6. Amplitude of a Gaussian packet
- 5.3. Optimization of the shape of Gaussian beams or packets
- 5.4. Asymptotic summation of Gaussian beams and packets
- 5.4.1. Asymptotic decomposition into Gaussian beams
- 5.4.2. Asymptotic decomposition into Gaussian packets
- 5.4.3. Discretization error
- 5.4.4. Linear canonical transforms
- 5.4.5. Coherent-state transforms
- 5.4.6. Maslov methods
- 5.5. Decomposition of a general wavefield into Gaussian packets or beams
- 5.6. Sensitivity of waves to heterogeneities
- 5.7. Migrations
- 5.7.1. Gaussian packet migrations
- 5.7.2. Gaussian beam migrations

- 6. Ray chaos, Lyapunov exponents, models suitable for ray tracing
- 6.1. Lyapunov exponents and rotation numbers
- 6.1.1. Lyapunov exponents for ray tracing
- 6.1.2. Rotation number for 2-D ray tracing
- 6.1.3. Approximation of the positive Lyapunov exponent in 2-D models without interfaces
- 6.1.4. Approximation of the rotation number in 2-D models without interfaces
- 6.1.5. Lyapunov exponent and rotation number for a system of finite rays
- 6.1.6. Average Lyapunov exponent for the model and average rotation number for the model
- 6.1.7. Numerical example
- 6.2. Models suitable for ray tracing
- 6.2.1. Application of Sobolev scalar products to smoothing models

- 7. Other topics related to the ray method
- 7.1. Higher-order ray approximations
- 7.2. Direct computation of first-arrival travel times
- 7.3. Ray method with complex eikonal
- 7.4. Hybrid methods
- 7.5. Several other extensions of the ray method

- Acknowledgements
- References

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