• Theoretical and Computational Seismology
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Theoretical and Computational Seismology

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An authoritative, self-contained reference text on theoretical and computational seismologyOver the past several decades, computational advances have revolutionized seismology, making it possible t...
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  • Publication Date: 01 July 2025
  • ISBN: 9780691267968
  • Pages: 600
  • Imprint: Princeton University Press
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An authoritative, self-contained reference text on theoretical and computational seismology

Over the past several decades, computational advances have revolutionized seismology, making it possible to simulate seismic wave propagation in complex Earth models and create detailed images of the planet’s interior. This cutting-edge text introduces students and scholars to the fundamentals, techniques, and applications of this exciting field of research and discovery.

After establishing a strong foundation in continuum mechanics, the book presents the fundamentals of theoretical seismology, providing a basis for subsequent forward and inverse modeling grounded in numerical methods, and then focuses on computational seismology, investigating numerical solutions to seismic wave equations. The adjoint-state method is covered next, along with applications of this technique to waveform inversions across scales, after which the book concludes with a set of appendixes that provide a primer to differential geometry and tensor calculus, which are used throughout the book to explain the fundamental concepts of deformation, strain, and stress from both Eulerian and Lagrangian perspectives. Including over 150 student-tested exercises, the book is an essential resource for motivated students and scholars seeking to master the state of the art of theoretical and computational seismology.

  • Establishes a strong foundation through a geometric analysis of continuum mechanics
  • Reveals how linearizing the resulting equations of motion enables the simulation of seismic wave propagation across nine decades of frequencies and wavelengths
  • Demonstrates how to leverage the capabilities of simulations to create detailed tomographic images from the information embedded in seismographic recordings
  • Covers diverse application areas, including seismology, helioseismology, underwater acoustics, medical imaging, and nondestructive testing
  • Features a wealth of exercises (with online solutions)
  • Includes a comprehensive set of appendixes on differential geometry and tensor calculus
  • An ideal textbook for graduate students studying theoretical seismology, computational seismology, or optimization and inverse problems
  • An essential reference for researchers and scholars
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Price: $90.00
Pages: 600
Publisher: Princeton University Press
Imprint: Princeton University Press
Publication Date: 01 July 2025
ISBN: 9780691267968
Format: Hardcover
Jeroen Tromp is the Blair Professor of Geology and Professor of Applied and Computational Mathematics at Princeton University. His books include A Geometrical Introduction to Tensor Calculus and (with F. A. Dahlen) Theoretical Global Seismology (both Princeton).
  • Preface
  • How to Use This Book
  • I CONTINUUM MECHANICS
  • 1 Kinematics
    • 1.1 Motion
      • 1.1.1 Compatibility
    • 1.2 Vectors: Material Velocity
    • 1.3 One-Forms
      • 1.3.1 Duality Product
    • 1.4 Tensors
    • 1.5 Covariant Derivative
      • 1.5.1 Evolution of Connection Coefficients
    • 1.6 Metric
      • 1.6.1 Covariant Derivative of the Metric
    • 1.7 Deformation Rate and Vorticity
    • 1.8 Lie Derivative
    • 1.9 Euler Derivative
    • 1.10 Material Derivative
    • 1.11 Corotational Material Derivative
    • 1.12 Levi-Civita Density and Capacity
    • 1.13 Levi-Civita Pseudotensor and Volume Form
    • 1.14 Pullback and Pushforward
    • 1.15 Volumes
    • 1.16 Jacobian of the Motion
    • 1.17 Surfaces
    • 1.18 Reynolds Transport Theorem
      • 1.18.1 Lagrangian Version
    • 1.19 Conservation of Mass
      • 1.19.1 Lagrangian Version
    • 1.20 Strain
      • 1.20.1 Deformation Gradient Tensor
      • 1.20.2 Cauchy--Green Tensor
      • 1.20.3 Stretch Tensor
      • 1.20.4 Lagrangian or Material Strain Tensor
      • 1.20.5 Eulerian or Almansi Strain Tensor
      • 1.20.6 Logarithmic or Hencky Strain Tensor
      • 1.20.7 Seth--Hill Strain Tensors
      • 1.20.8 Arguments for Logarithmic Strain
      • 1.20.9 Logarithmic Strain Rate
      • 1.20.10 Strain as a Two-Vector
  • 2 Dynamics
    • 2.1 Stress
      • 2.1.1 Cauchy Stress Tensor
      • 2.1.2 Kirchhoff Stress Tensor
      • 2.1.3 Second Piola--Kirchhoff Stress
      • 2.1.4 First Piola--Kirchhoff Stress
    • 2.2 Thermodynamics
      • 2.2.1 First Law of Thermodynamics
      • 2.2.2 Second Law of Thermodynamics
      • 2.2.3 Helmholtz Free Energy
      • 2.2.4 Elasticity
      • 2.2.5 Material Frame Indifference
    • 2.3 Constitutive Relationships
      • 2.3.1 Truesdell Stress Rate
      • 2.3.2 Logarithmic Stress Rate
      • 2.3.3 Viscosity
    • 2.4 Hamilton’s Principle
    • 2.5 Conservation of Linear Momentum
    • 2.6 Noether’s Theorem
      • 2.6.1 Conservation of Linear Momentum (Revisited)
      • 2.6.2 Conservation of Angular Momentum
      • 2.6.3 Conservation of Energy
    • 2.7 Rotation
      • 2.7.1 Centrifugal Potential
    • 2.8 Self-Gravitation
      • 2.8.1 Poisson’s Equation
      • 2.8.2 Gravitational Energy
      • 2.8.3 Action Due to Gravity
    • 2.9 Navier--Stokes Equations
    • 2.10 Equations of Motion in Terms of Displacement
  • Continuum Mechanics Glossary
  • 3 Defects
    • 3.1 Integrability
    • 3.2 Compatible Motion
    • 3.3 Tetrad Formalism
    • 3.4 Connection
    • 3.5 Metric
    • 3.6 Torsion and Curvature
      • 3.6.1 Eulerian Description
      • 3.6.2 Lagrangian Description: Incompatibilities
    • 3.7 Nonmetricity
    • 3.8 Application to Defects
      • 3.8.1 Lagrangian Dynamics of Defects
      • 3.8.2 Mixed Dynamics of Defects
      • 3.8.3 Four-Dimensional Kinematics of Defects
      • 3.8.4 Incompatibilities
    • 3.9 Alternative Approach Based on a Referential Manifold
    • 3.10 Three-Dimensional Dynamics and Kinematics of Defects
      • 3.10.1 Three-Dimensional Incompatibilities
      • 3.10.2 Burgers and Frank Vectors
    • 3.11 Variational Approach
      • 3.11.1 Properties of the 4D Volume Form
      • 3.11.2 Material Geometry Action
      • 3.11.3 Defect Action
      • 3.11.4 Defect Field Equations
      • 3.11.5 Anatomy of the Defect Hypermomentum Current
      • 3.11.6 Anatomy of the Lagrangian Connection Coefficients
      • 3.11.7 Defect Dynamics
      • 3.11.8 Metricity
      • 3.11.9 Linear Defect Equations
      • 3.11.10 Three-Dimensional Defect Equations
    • 3.12 Continuum Mechanics with Spin
      • 3.12.1 Anatomy of the Stress-Energy and Spin Tensors
      • 3.12.2 Classical Conservation Laws with Spin
      • 3.12.3 Stress and Couple-Stress Gluts
  • II Seismology
  • 4 Linearized Equations of Motion
    • 4.1 Simplified Notation
    • 4.2 Infinitesimal Strain
    • 4.3 Constitutive Relationship without Prestress
    • 4.4 Elastic Wave Equation
      • 4.4.1 Plane-Wave Solutions
      • 4.4.2 Weak Form
    • 4.5 Acoustic Wave Equation
      • 4.5.1 Weak Form
    • 4.6 Earthquake Fault
      • 4.6.1 Ideal Fault
      • 4.6.2 Earthquake Source Contribution
    • 4.7 Betti Reciprocal Relation
      • 4.7.1 Reciprocity
      • 4.7.2 Volterra Representation Theorem
      • 4.7.3 Fault Slip
    • 4.8 Moment Tensor
      • 4.8.1 Ideal Fault
      • 4.8.2 Beach Balls
      • 4.8.3 Source-Time Function
    • 4.9 Seismology with Spin
      • 4.9.1 Alternative Approach
      • 4.9.2 Homogeneous Medium
      • 4.9.3 Couple-Moment Tensor
    • 4.10 Equilibrium State
      • 4.10.1 Hydrostatic Earth Model
      • 4.10.2 Spherically Symmetric Earth Model
      • 4.10.3 Ellipticity
    • 4.11 Density Perturbations
    • 4.12 Gravity Perturbations
    • 4.13 Constitutive Relationship with Prestress
    • 4.14 Displacement Variational Principle
    • 4.15 Displacement-Potential Variational Principle
    • 4.16 Elastic Tensor Selection
    • 4.17 Fluid Regions
      • 4.17.1 Potential Formulation
    • 4.18 Quasi-Hydrostatic Approximation
    • 4.19 Generalized Betti Reciprocal Relation
      • 4.19.1 Generalized Reciprocity
      • 4.19.2 Generalized Volterra Representation Theorem
    • 4.20 Idealized Seismometer Response
    • 4.21 Weak Global Equations of Motion
    • 4.22 Cowling Approximation
    • 4.23 Ocean-Load Approximation
    • 4.24 Global Body-Wave Propagation
    • 4.25 Seismic Noise
      • 4.25.1 Noise Cross-Correlation
  • 5 Anelasticity and Attenuation
    • 5.1 Creep and Stress Relaxation Functions
    • 5.2 Springs and Dashpots
    • 5.3 Standard Linear Solid
    • 5.4 Linear Combination of Standard Linear Solids
    • 5.5 Linear Viscoelasticity
      • 5.5.1 Maxwell Rheology
    • 5.6 Constant-Q Absorption Band Model
    • 5.7 Viscoacoustics
  • Seismology Glossary
  • III Forward Problems
  • 6 Strong Methods
    • 6.1 Finite-Difference Method
      • 6.1.1 Taylor Series and Finite Differences
      • 6.1.2 Homogeneous Wave Equation
      • 6.1.3 Inhomogeneous Wave Equation
      • 6.1.4 Grid Dispersion
      • 6.1.5 Staggered Grids
      • 6.1.6 Shallow-Water Waves
      • 6.1.7 Grid Anisotropy
      • 6.1.8 Heat Equation
    • 6.2 Pseudospectral Method
      • 6.2.1 Fourier Transform
      • 6.2.2 Velocity-Stress Wave Equation
      • 6.2.3 Grid Dispersion
  • 7 Weak Methods
    • 7.1 Rayleigh--Ritz Method
      • 7.1.1 Coupled-Mode Method
      • 7.1.2 Direct-Solution Method
    • 7.2 Boundary-Element Method
    • 7.3 Finite-Element Method
      • 7.3.1 Static Heat Equation
      • 7.3.2 Dynamic Heat Equation
      • 7.3.3 Generalized Trapezoidal Time Scheme
      • 7.3.4 Local-Element Method
      • 7.3.5 General Finite-Element Method
    • 7.4 Spectral-Element Method
      • 7.4.1 Dynamic Heat Equation
      • 7.4.2 Wave Equation
      • 7.4.3 Newmark Time Scheme
      • 7.4.4 General Spectral-Element Method
      • 7.4.5 3D Seismic Wave Equation
      • 7.4.6 Absorbing Boundary Conditions
      • 7.4.7 Attenuation
      • 7.4.8 Compact Notation
      • 7.4.9 Hexahedral Meshing
      • 7.4.10 Cubed Sphere
      • 7.4.11 Global Wave Propagation Simulations
      • 7.4.12 Normal-Mode Benchmarks
    • 7.5 Discontinuous Galerkin Method
    • 7.6 Infinite-Element Method
    • 7.7 Spectral-Infinite-Element Method
      • 7.7.1 Self-Gravitation
      • 7.7.2 Coseismic and Post-Earthquake Deformation
      • 7.7.3 Examples
      • 7.7.4 Full-Gravity Global Wave Propagation Simulations
      • 7.7.5 Idealized Seismometer Response
  • IV Inverse Problems
  • 8 Adjoint-State Method
    • 8.1 Born Approximation
      • 8.1.1 Unperturbed Equations of Motion
      • 8.1.2 Perturbed Equations of Motion
    • 8.2 Waveform Tomography
    • 8.3 Adjoint Equations
    • 8.4 Lagrange Multiplier Method
    • 8.5 Traveltime Tomography
      • 8.5.1 Banana-Doughnut Kernels
      • 8.5.2 Cross-Correlation Traveltime Misfit Kernels
      • 8.5.3 Differential-Traveltime Tomography
    • 8.6 Amplitude Tomography
    • 8.7 Attenuation
    • 8.8 Generic Tomography
    • 8.9 Point-Source Perturbations
    • 8.10 Topography on Internal Discontinuities
    • 8.11 Source Encoding
      • 8.11.1 Encoded Forward Wavefield
      • 8.11.2 Source-Encoded Inversion
      • 8.11.3 Fréchet Derivatives
      • 8.11.4 Attenuation
      • 8.11.5 Laplace-Domain Source Encoding
    • 8.12 Interferometry
      • 8.12.1 Noise Cross-Correlation Tomography
  • 9 Optimization
    • 9.1 Preliminaries
    • 9.2 Model Parameter Selection
    • 9.3 Objective Function
    • 9.4 Bayesian Inference
      • 9.4.1 Linear Inverse Problems
    • 9.5 Local Optimization
      • 9.5.1 Secant Method
      • 9.5.2 Steepest Descent Method
      • 9.5.3 Conjugate-Gradient Method
      • 9.5.4 Variable Metric Method
      • 9.5.5 DFP Method
      • 9.5.6 BFGS Method
      • 9.5.7 L-BFGS Method
    • 9.6 Preconditioning
    • 9.7 Regularization
      • 9.7.1 Tikhonov Regularization
      • 9.7.2 Total Variation Regularization
      • 9.7.3 Projection
      • 9.7.4 Smoothing
      • 9.7.5 Level-Set Methods
    • 9.8 Multiscale Inversion
    • 9.9 Line Search
      • 9.9.1 Bracketing Line Search
      • 9.9.2 Backtracking Line Search
    • 9.10 Point-Spread Function
    • 9.11 Uncertainty Quantification
      • 9.11.1 Exploration Seismology
      • 9.11.2 Global Seismology
  • Introduction to the Appendices
  • A Linear Spaces and Transformations
    • A.1 Properties of Linear Spaces
    • A.2 Vector Spaces
    • A.3 Linear Transformations
  • B Differentiable Manifolds
    • B.1 Charts and Coordinates
    • B.2 Definition
    • B.3 Local Coordinate Changes
    • B.4 Functions on Manifolds
    • B.5 Orientable Manifolds
  • C Vectors and One-Forms
    • C.1 Vectors
      • C.1.1 Vectors as Tangents to Curves
      • C.1.2 Bases and Coordinates
      • C.1.3 Vector Field
      • C.1.4 Transformations
    • C.2 One-Forms
      • C.2.1 Duality
      • C.2.2 Bases
      • C.2.3 Transformations
    • C.3 Alternative Perspective
    • C.4 Lie Bracket
  • D Tensors
    • D.1 Definition
    • D.2 Operations on Tensors
      • D.2.1 Addition
      • D.2.2 Tensor Product
      • D.2.3 Contraction
      • D.2.4 Transpose of (2,0) and (0,2) Tensors
      • D.2.5 Transpose of a (1,1) Tensor
    • D.3 Transformations
      • D.3.1 Tetrad Formalism
      • D.3.2 Pseudotensors
    • D.4 Kronecker or Identity Tensor
    • D.5 Logarithms and Exponentials of (1,1) Tensors
    • D.6 Tensor Densities and Capacities
      • D.6.1 Pseudotensor Densities and Capacities
    • D.7 Levi-Civita Density and Capacity
      • D.7.1 Cross Product
    • D.8 Determinant of Rank-2 tensors
    • D.9 Inverse of Rank-2 Tensors
    • D.10 Metric Tensor
      • D.10.1 Formulation
      • D.10.2 Geometrical Meaning
      • D.10.3 Norm of Vectors and One-Forms
      • D.10.4 Metric in Tetrads
    • D.11 Adjoint of a (1,1) Tensor
    • D.12 Tensor Densities and Capacities Revisited
    • D.13 Levi-Civita Pseudotensor
      • D.13.1 Cross Product
    • D.14 Kronecker Determinants
    • D.15 Rotations
      • D.15.1 Euler Angles
      • D.15.2 Rodrigues’s Formula
  • E Maps between Manifolds
    • E.1 Maps
    • E.2 Maps between Manifolds of Different Dimensions
      • E.2.1 Pullback
      • E.2.2 Pushforward
    • E.3 Maps between Manifolds of the Same Dimensions
  • F Differentiation on Manifolds
    • F.1 Covariant Derivative
      • F.1.1 Formulation
      • F.1.2 Transformation of Connection Coefficients
      • F.1.3 Divergence
      • F.1.4 Parallel Transport
      • F.1.5 Torsion and Curvature Tensors
      • F.1.6 Bianchi Identities
      • F.1.7 Torsion-Free Connection
      • F.1.8 Covariant Derivative of the Metric Tensor
      • F.1.9 Mixed Covariant Derivative in Tetrad Basis
      • F.1.10 Spin Connection
      • F.1.11 Contracted Bianchi Identities
      • F.1.12 Covariant Derivative of Tensor Densities and Capacities
      • F.1.13 Nonmetricity
    • F.2 Euler Derivative
    • F.3 Lie Derivative
      • F.3.1 Lie Derivative of Vectors
      • F.3.2 Geometrical Interpretation
      • F.3.3 Autonomous Lie Derivative
      • F.3.4 Lie Derivative of One-Forms
      • F.3.5 Lie Derivative of (p,q) Tensors
      • F.3.6 Lie Derivative of Functions
      • F.3.7 Lie Derivative of Metric Tensors
      • F.3.8 Lie Derivative of Levi-Civita Tensor
  • G Differential Forms
    • G.1 Definition
    • G.2 Operations on Forms
      • G.2.1 Addition
      • G.2.2 Exterior Product
      • G.2.3 Interior Product
    • G.3 k-Vectors
    • G.4 Hodge Dual
    • G.5 Volumes
      • G.5.1 Properties
    • G.6 Surfaces
    • G.7 Exterior Derivative
      • G.7.1 Coordinate-Free Definition
      • G.7.2 Cartesian Examples
      • G.7.3 Exact Forms
      • G.7.4 Commutativity with Pullback and Pushforward
    • G.8 Lie Derivative of a Form
    • G.9 Vector- and Tensor-Valued Forms
      • G.9.1 Transformations of Tensor-Valued Forms
      • G.9.2 Operations on Tensor-Valued Forms
      • G.9.3 Connection One-Forms
      • G.9.4 Torsion Two-Forms
      • G.9.5 Exterior Covariant Derivative
      • G.9.6 Covariant Lie Derivative
      • G.9.7 Curvature Two-Forms
      • G.9.8 Commutator of Covariant Lie and Exterior Covariant Derivatives
      • G.9.9 Bianchi Identities Revisited
      • G.9.10 Nonmetricity Revisited
    • G.10 Integration of Forms
      • G.10.1 Line Integrals
      • G.10.2 Surface Integrals
      • G.10.3 Volume Integrals
    • G.11 Generalized Stokes’s Theorem
      • G.11.1 Fundamental Theorem of Calculus
      • G.11.2 Green’s Theorem
      • G.11.3 Gauss’s Theorem
      • G.11.4 Stokes’s Theorem
      • G.11.5 Variational Principles
      • G.11.6 Noether’s Theorem
  • Appendix Glossary
  • Bibliography
  • Author Index
  • Index