E-Book Overview
Covering spectral analysis as well as inversion of geophysical data, David Gubbins introduces the necessary theory and techniques and then demonstrates how they may be practically applied to interpret various types of geophysical signals, including seismic, magnetic and gravity data. Featuring summary boxes, extensive mathematical and computing exercises (solutions and software available on the Internet), and a set of mathematical appendices, this textbook will prove invaluable to geophysics students and instructors.
E-Book Content
Contents 1 1.1 1.2 1.3 1.4 1.5 2 2.1 2.2 2.3 2.4 3 3.1 3.2 3.3 3.4 4 4.1 4.2 4.3 5 5.1 5.2 5.3 6 6.1 Introduction The digital revolution Digital Recording Processing Inversion About this book Part one: PROCESSING Mathematical Preliminaries: the TVU and Discrete Fourier Transforms The T -transform The Discrete Fourier Transform Properties of the discrete Fourier transform DFT of random sequences Practical estimation of spectra Aliasing Aliasing Spectral leakage and tapering Examples of Spectra Processing of time sequences Filtering Correlation Deconvolution Processing two-dimensional data The 2D Fourier Transform 2D Filtering Travelling waves Part two: INVERSION Linear Parameter Estimation The linear problem 6 9 9 11 13 15 18 23 25 25 29 34 43 47 47 51 51 57 65 65 71 73 82 82 84 87 93 95 95 Contents 6.2 6.3 6.4 6.5 7 7.1 7.2 7.3 7.4 7.5 7.6 8 8.1 8.2 8.3 8.4 8.5 9 9.1 9.2 9.3 9.4 9.5 9.6 9.7 10 10.1 10.2 10.3 11 11.1 11.2 11.3 12 12.1 12.2 12.3 12.4 Least squares solution of over-determined problems Weighting the data Model Covariance Matrix and the Error Ellipsoid “Robust methods” The Underdetermined Problem The null space The Minimum Norm Solution Ranking and winnowing Damping and the Trade-off Curve Parameter covariance matrix The Resolution Matrix Nonlinear Inverse Problems Methods available for nonlinear problems Earthquake Location: an Example of Nonlinear Parameter Estimation Quasi-linearisation and Iteration for the General Problem Damping, Step-Length Damping, and Covariance and Resolution Matrices The Error Surface Continuous Inverse Theory A linear continuous inverse problem The Dirichlet Condition Spread, Error, and the Trade-off Curve Designing the Averaging Function Minimum Norm Solution Discretising the Continuous Inverse Problem Parameter Estimation: the Methods of Backus and Parker Part three: APPLICATIONS Fourier Analysis as an inverse problem The Discrete Fourier Transform and Filtering Wiener Filters Multitaper Spectral Analysis Seismic Travel Times and Tomography Beamforming Tomography Simultaneous Inversion for Structure and Earthquake Location Geomagnetism Introduction The Forward Problem The Inverse Problem: Uniqueness Damping 7 99 102 110 113 120 120 122 123 125 127 131 139 139 141 144 145 146 154 154 155 157 159 160 163 165 173 175 175 177 180 185 185 192 198 203 203 204 206 208 8 Contents 12.5 The Data 12.6 Solutions along the Trade-off Curve 12.7 Covariance, Resolution, and Averaging Functions 12.8 Finding Fluid Motion in the Core Appendix 1 Fourier Series Appendix 2 The Fourier Integral Transform Appendix 3 Shannon’s Sampling Theorem Appendix 4 Linear Algebra Appendix 5 Vector Spaces and the Function Space Appendix 6 Lagrange Multipliers and Penalty Parameters Appendix 7 Files for the Computer Exercises References Index 212 216 218 221 228 234 240 242 250 257 260 261 265 List of Illustrations 1.1 1.2 1.3 2.1 2.2 2.3 2.4 2.5 3.1 3.2 3.3 3.4 3.5 3.6 3.7 4.1 4.2 4.3 4.4 4.5 5.1 5.2 5.3 5.4 5.5 6.1 Fig 1.1. BB seismogram Nov 1 Fig 1.2. First arrival Gravity anomaly The convolution process In the Argand diagram, our choice for , , lies always on the unit circle because . Discretisation places the p