Geophysical Inverse Theory and Regularization Problems

摘要: Preface. I. Introduction to Inversion Theory. 1. Forward and inverse problems in geophysics. 1.1 Formulation of forward for different geophysical fields. 1.2 Existence uniqueness the problem solutions. 1.3 Instability solution. 2. Ill-posed methods their 2.1 Sensitivity resolution methods. 2.2 well-posed ill-posed problems. 2.3 Foundations regularization 2.4 Family stabilizing functionals. 2.5 Definition parameter. II. Methods Solution Inverse Problems. 3. Linear discrete 3.1 least-squares inversion. 3.2 purely under determined problem. 3.3 Weighted method. 3.4 Applying principles probability theory a linear 3.5 Regularization 3.6 The Backus-Gilbert 4. Iterative solutions 4.1 operator equations solution by iterative 4.2 A generalized minimal residual 4.3 method 5. Nonlinear inversion technique. 5.1 Gradient-type 5.2 Regularized gradient-type nonlinear 5.3 5.4 Conjugate gradient re-weighted optimization. III. Geopotential Field Inversion. 6. Integral representations modeling gravity magnetic 6.1 Basic 6.2 potential fields based on functions complex variable. 7. data. 7.1 Gradient 7.2 Gravity field migration. 7.3 anomaly 7.4 Numerical modeling. IV. Electromagnetic 8. electromagnetic theory. 8.1 equations. 8.2 energy flow. 8.3 Uniqueness 8.4 Green's tensors. 9. 9.1 equation 9.2 integral approximations field. 9.3 non-linear higher orders. 9.4 numerical dressing. 10. 10.1 10.2 10.3 Quasi-linear 10.4 Quasi-analytical 10.5 Magnetotelluric (MT) data 11. migration imaging. 11.1 frequency domain. 11.2 time 12. Differential 12.1 as boundary-value 12.2 Finite difference approximation 12.3 element 12.4 differential V. Seismic 13. Wavefield 13.1 elastic waves. 13.2 wavefield 13.3 Kirchhoff formula its analogs. 13.4 14. 14.1 acoustic analysis. 14.2 wavefield. 14.3 Method vector 14.4 15. 15.1 15.2 15.3 15.4 Principles 15.5 Elastic A. Functional spaces models A.1 Euclidean space. A.2 Metric A.3 spaces. A.4 Hilbert A.5 Complex A.6 Examples B. Operators B.1 functional B.2 operators. B.3 B.4 Some B.5 Gram - Schmidt orthogonalization process. C. Functionals models. C.1 norms. C.2 Riesz representation theorem. C.3 an D. operators functionals revisited. D.1 Adjoint D.2 Differentiation D.3 Concepts variational calculus. E. formulae rules from matrix algebra. E.1 operation matrices. E.2 Eigenvalues eigenvectors. E.3 Spectral decomposition symmetric matrix. E.4 Singular value (SVD). E.5 spectral Lanczos F. tensor F.1 functions. F.2 Tensor statements Gauss formulae. F.3 Lame Laplace Bibliography. Index.