| Description |
1 online resource (xx, 343 pages) : illustrations (some color) |
| Series |
Advances in geophysical and environmental mechanics and mathematics, 1866-8348 |
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Advances in geophysical and environmental mechanics and mathematics.
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| Contents |
Note continued: 6.1.2. Paraxial Approximations as Boundary Conditions for Acoustic Waves -- 6.1.3. High-Order Absorbing Boundary Conditions for Acoustic Waves -- 6.1.4. Generalisation to the Elastic Case -- 6.1.5. Discussion -- 6.2. Gaussian Taper Method -- 6.3. Perfectly Matched Layers (PML) -- 6.3.1. General Development -- 6.3.2. Standard PML -- 6.3.3. Convolutional PML -- 6.3.4. Other Variants of the PML Method -- pt. II Iterative Solution of the Full Waveform Inversion Problem -- 7. Introduction to Iterative Non-linear Minimisation -- 7.1. Basic Concepts: Minima, Convexity and Non-uniqueness -- 7.1.1. Local and Global Minima -- 7.1.2. Convexity: Global Minima and (Non)Uniqueness -- 7.2. Optimality Conditions -- 7.3. Iterative Methods for Non-linear Minimisation -- 7.3.1. General Descent Methods -- 7.3.2. Method of Steepest Descent -- 7.3.3. Newton's Method and Its Variants -- 7.3.4. Conjugate-Gradient Method -- 7.4. Convergence -- 7.4.1. Multi-Scale Approach -- 7.4.2. Regularisation -- 8. Time-Domain Continuous Adjoint Method -- 8.1. Introduction -- 8.2. General Formulation -- 8.2.1. Frechet Kernels -- 8.2.2. Translation to the Discretised Model Space -- 8.2.3. Summary of the Adjoint Method -- 8.3. Derivatives with Respect to the Source -- 8.4. Second Derivatives -- 8.4.1. Motivation: The Role of Second Derivatives in Optimisation and Resolution Analysis -- 8.4.2. Extension of the Adjoint Method to Second Derivatives -- 8.5. Application to the Elastic Wave Equation -- 8.5.1. Derivation of the Adjoint Equations -- 8.5.2. Practical Implementation -- 9. First and Second Derivatives with Respect to Structural and Source Parameters -- 9.1. First Derivatives with Respect to Selected Structural Parameters -- 9.1.1. Perfectly Elastic and Isotropic Medium -- 9.1.2. Perfectly Elastic Medium with Radial Anisotropy |
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Note continued: 9.1.3. Isotropic Visco-Elastic Medium: Q & mu; and Q & kappa; -- 9.2. First Derivatives with Respect to Selected Source Parameters -- 9.2.1. Distributed Sources and the Relation to Time-Reversal Imaging -- 9.2.2. Moment Tensor Point Source -- 9.3. Second Derivatives with Respect to Selected Structural Parameters -- 9.3.1. Physical Interpretation and Structure of the Hessian -- 9.3.2. Practical Resolution of the Secondary Adjoint Equation -- 9.3.3. Hessian Recipe -- 9.3.4. Perfectly Elastic and Isotropic Medium -- 9.3.5. Perfectly Elastic Medium with Radial Anisotropy -- 9.3.6. Isotropic Visco-Elastic Medium -- 10. Frequency-Domain Discrete Adjoint Method -- 10.1. General Formulation -- 10.2. Second Derivatives -- 11. Misfit Functionals and Adjoint Sources -- 11.1. Derivative of the Pure Wave Field and the Adjoint Greens Function -- 11.2. L2 Waveform Difference -- 11.3. Cross-Correlation Time Shifts -- 11.4. L2 Amplitudes -- 11.5. Time-Frequency Misfits -- 11.5.1. Definition of Phase and Envelope Misfits -- 11.5.2. Practical Implementation of Phase Difference Measurements -- 11.5.3. Example -- 11.5.4. Adjoint Sources -- 12. Frechet and Hessian Kernel Gallery -- 12.1. Body Waves -- 12.1.1. Cross-Correlation Time Shifts -- 12.1.2. L2 Amplitudes -- 12.2. Surface Waves -- 12.2.1. Isotropic Earth Models -- 12.2.2. Radial Anisotropy -- 12.3. Hessian Kernels: Towards Quantitative Trade-Off and Resolution Analysis -- 12.4. Accuracy-Adaptive Time Integration -- pt. III Applications -- 13. Full Waveform Tomography on Continental Scales -- 13.1. Motivation -- 13.2. Solution of the Forward Problem -- 13.2.1. Spectral Elements in Natural Spherical Coordinates -- 13.2.2. Implementation of Long-Wavelength Equivalent Crustal Models -- 13.3. Quantification of Waveform Differences |
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Note continued: 13.4. Application to the Australasian Upper Mantle -- 13.4.1. Data Selection and Processing -- 13.4.2. Initial Model -- 13.4.3. Model Parameterisation -- 13.4.4. Tomographic Images and Waveform Fits -- 13.4.5. Resolution Analysis -- 13.5. Discussion -- 13.5.1. Forward Problem Solution -- 13.5.2. Crust -- 13.5.3. Time-Frequency Misfits -- 13.5.4. Dependence on the Initial Model -- 13.5.5. Anisotropy -- 13.5.6. Resolution -- 14. Application of Full Waveform Tomography to Active-Source Surface-Seismic Data -- 14.1. Introduction -- 14.2. Data -- 14.3. Data Pre-conditioning and Weighting -- 14.4. Misfit Functional -- 14.5. Initial Model -- 14.6. Inversion and Results -- 14.7. Data Fit -- 14.8. Discussion -- 15. Source Stacking Data Reduction for Full Waveform Tomography at the Global Scale -- 15.1. Introduction -- 15.2. Data Reduction -- 15.3. Source Stacked Inverse Problem -- 15.4. Validation Tests -- 15.4.1. Parameterisation -- 15.4.2. Experiment Setup and Input Models -- 15.4.3. Test in a Simple Two-Parameter Model -- 15.4.4. Tests in a Realistic Degree-6 Global Model -- 15.5. Towards Real Cases: Dealing with Missing Data -- 15.6. Discussion and Conclusions -- Appendix A Mathematical Background for the Spectral-Element Method -- A.1. Orthogonal Polynomials -- A.2. Function Interpolation -- A.2.1. Interpolating Polynomial -- A.2.2. Lagrange Interpolation -- A.2.3. Lobatto Interpolation -- A.2.4. Fekete Points -- A.2.5. Interpolation Error -- A.3. Numerical Integration -- A.3.1. Exact Numerical Integration and the Gauss Quadrature -- A.3.2. Gauss-Legendre-Lobatto Quadrature -- Appendix B Time-Frequency Transformations |
| Summary |
Recent progress in numerical methods and computer science allows us today to simulate the propagation of seismic waves through realistically heterogeneous Earth models with unprecedented accuracy. Full waveform tomography is a tomographic technique that takes advantage of numerical solutions of the elastic wave equation. The accuracy of the numerical solutions and the exploitation of complete waveform information result in tomographic images that are both more realistic and better resolved. This book develops and describes state of the art methodologies covering all aspects of full waveform tomography including methods for the numerical solution of the elastic wave equation, the adjoint method, the design of objective functionals and optimisation schemes. It provides a variety of case studies on all scales from local to global based on a large number of examples involving real data. It is a comprehensive reference on full waveform tomography for advanced students, researchers and professionals |
| Bibliography |
Includes bibliographical references and index |
| Notes |
English |
| Subject |
Seismic waves -- Mathematical models
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NATURE -- Earthquakes & Volcanoes.
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SCIENCE -- Earth Sciences -- Seismology & Volcanism.
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Environnement.
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Climat.
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Seismic waves -- Mathematical models
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| Form |
Electronic book
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| Author |
Bleibinhaus, Florian.
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Capdeville, Yann.
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| ISBN |
9783642158070 |
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3642158072 |
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9783642158063 |
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3642158064 |
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