Book contents
- Frontmatter
- Contents
- Preface to the first edition
- Preface to the second edition
- Acknowledgment
- 1 Introduction
- 2 Stress and strain
- 3 The seismic wave equation
- 4 Ray theory: Travel times
- 5 Inversion of travel time data
- 6 Ray theory: Amplitude and phase
- 7 Reflection seismology
- 8 Surface waves and normal modes
- 9 Earthquakes and source theory
- 10 Earthquake prediction
- 11 Instruments, noise, and anisotropy
- Appendix A The PREM model
- Appendix B Math review
- Appendix C The eikonal equation
- Appendix D Fortran subroutines
- Appendix E Time series and Fourier transforms
- Bibliography
- Index
5 - Inversion of travel time data
- Frontmatter
- Contents
- Preface to the first edition
- Preface to the second edition
- Acknowledgment
- 1 Introduction
- 2 Stress and strain
- 3 The seismic wave equation
- 4 Ray theory: Travel times
- 5 Inversion of travel time data
- 6 Ray theory: Amplitude and phase
- 7 Reflection seismology
- 8 Surface waves and normal modes
- 9 Earthquakes and source theory
- 10 Earthquake prediction
- 11 Instruments, noise, and anisotropy
- Appendix A The PREM model
- Appendix B Math review
- Appendix C The eikonal equation
- Appendix D Fortran subroutines
- Appendix E Time series and Fourier transforms
- Bibliography
- Index
Summary
In the preceding chapter we examined the problem of tracing rays and calculating travel time curves from a known velocity structure. We derived expressions for ray tracing in a one-dimensional (1-D) velocity model in which velocity varies only with depth; ray tracing in general three-dimensional (3-D) structures is more complex but follows similar principles. We now examine the case where we are given travel times obtained from observations and wish to invert for a velocity structure that can explain the data. As one might imagine, the inversion is much more complicated than the forward problem. The main strategy used by seismologists, both in global and crustal studies, has generally been to divide the problem into two parts:
A 1-D “average” velocity model is determined from all the available data. This is generally a non-linear problem but is tractable since we are seeking a single function of depth. Analysis often does not proceed beyond this point.
If sufficient 3-D ray coverage is present, the 1-D model is used as a reference model and a travel time residual is computed for each datum by subtracting the predicted time from the observed time. A 3-D model is obtained by inverting the travel time residuals for velocity perturbations relative to the reference model. If the velocity perturbations are fairly small, this problem can be linearized and is computationally feasible even for large data sets. This is the basis for tomographic inversion techniques.
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- Introduction to Seismology , pp. 103 - 138Publisher: Cambridge University PressPrint publication year: 2009