Geophysical Inverse Theory
File : pdf, 1.7 MB, 208 pages
TOC
1 What Is Inverse Theory
1.1 Too many models
1.2 No unique answer
1.3 Implausible models
1.4 Observations are noisy
1.5 The beach is not a model
1.6 Summary
1.7 Beach Example
2 A Simple Inverse Problem that Isn’t
2.1 A First Stab at rho
2.1.1 Measuring Volume
2.1.2 Measuring Mass
2.1.3 Computing rho
2.2 The Pernicious Effects of Errors
2.2.1 Errors in Mass Measurement
2.3 What is an Answer?
2.3.1 Conditional Probabilities
2.3.2 What We’re Really (Really) After
2.3.3 A (Short) Tale of Two Experiments
2.3.4 The Experiments Are Identical
2.4 What does it mean to condition on the truth?
2.4.1 Another example
3 Example: A Vertical Seismic Profile
3.0.2 Travel time fitting
4 A Little Linear Algebra
4.1 Linear Vector Spaces
4.1.1 Matrices
4.1.2 Matrices With Special Structure
4.2 Matrix and Vector Norms
4.3 Projecting Vectors Onto Other Vectors
4.4 Linear Dependence and Independence
4.5 The Four Fundamental Spaces
4.5.1 Spaces associated with a linear system Ax = y
4.6 Matrix Inverses
4.7 Eigenvalues and Eigenvectors
4.8 Orthogonal decomposition of rectangular matrices
4.9 Orthogonal projections
4.10 A few examples
5 SVD and Resolution in Least Squares
5.0.1 A Worked Example
5.0.2 The Generalized Inverse
5.0.3 Examples
5.0.4 Resolution
6 A Summary of Probability and Statistics
6.1 Sets
6.1.1 More on Sets
6.2 Random Variables
6.2.1 A Definition of Random
6.2.2 Generating random numbers on a computer
6.3 Bayes’ Theorem
6.4 Probability Functions and Densities
6.4.1 Expectation of a Function With Respect to a Probability Law
6.4.2 Multi-variate probabilities
6.5 Random Sequences
6.5.1 The Central Limit Theorem
6.6 Expectations and Variances
6.7 Bias
6.8 Correlation of Sequences
6.9 Random Fields
6.10 Probabilistic Information About Earth Models
6.11 Other Common Analytic Distributions
6.12 Computer Exercise
7 Linear Inverse Problems With Uncertain Data
7.0.1 Model Covariances
7.1 The World’s Second Smallest Inverse Problem
7.1.1 The Damped Least Squares Problem
8 Tomography
8.1 The X-ray Absorber
8.1.1 The Forward Problem
8.1.2 Linear Absorption
8.1.3 Model Representation
8.1.4 Some Numerical Results
8.2 Travel Time Tomography
8.3 Computer Example: Cross-well tomography
9 From Bayes to Weighted Least Squares
10 Bayesian versus Frequentist Methods of Inference
10.0.1 Bayesian Inversion in Practice
10.0.2 Bayes vs Frequentist
10.1 What Difference Does the Prior Make?
10.1.1 Bayes Risk
10.1.2 What is the Most Conservative Prior?
10.2 Example: A Toy Inverse Problem
10.2.1 Bayes Risk
10.2.2 The Flat Prior is Informative
10.3 Priors in High Dimensional Spaces: The Curse of Dimensionality
11 Iterative Linear Solvers
11.1 Classical Iterative Methods
11.2 Conjugate Gradient
11.2.1 Inner Products
11.2.2 Quadratic Forms
11.2.3 Quadratic Minimization
11.2.4 Computer Exercise: Steepest Descent
11.2.5 The Method of Conjugate Directions
11.2.6 The Method of Conjugate Gradients
11.2.7 Finite Precision Arithmetic
11.2.8 CG Methods for Least-Squares
11.2.9 Computer Exercise: Conjugate Gradient
11.3 Practical Implementation
11.3.1 Sparse Matrix Data Structures
11.3.2 Data and Parameter Weighting
11.3.3 Regularization
11.3.4 Jumping Versus Creeping
11.3.5 How Smoothing Affects Jumping and Creeping
11.4 Sparse SVD
11.4.1 The Symmetric Eigenvalue Problem
11.4.2 Finite Precision Arithmetic
11.4.3 Explicit Calculation of the Pseudo-Inverse
12 More on the Resolution-Variance Tradeoff
12.1 A Surfer’s Guide to Backus-Gilbert Theory
12.2 Using the SVD
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