Project Report (define problem, set up and solve inverse)
Chapter 1. Introduction to the inverse problem
The implicit/explicit non-linear/linear forms
Describing Inverse Problems, earth science applications, atmospheric science applications
Chapter 2. The character of Digital Data
Digitizing data, dynamic range, sampling, aliasing, resolution, incomplete etc.
Variance, probability density functions, correlation, confidence estimators
Vector length, cross correlation, autocorrelation function
The Gaussian distribution
Testing the assumption of Gaussian statistics
Relation between inverse theory and filter theory
Causality, convolution
Fourier Transform. (Integral form, Periodogram, Discrete Fourier Transform (DFT))
Radon transform
z-transform representation of digital trace, minimum phase and maximum phase
Convolution, in integral, digital, z-transform
Moving average and auto regressive filters
All Pass Filter (the phase shift filter)
Filter design, (Weiner or optimal filters)
Spiking filter-develop optimum delay for shaping filter
Stability, use of future time to stabilize
Prediction filters, Prediction error Burg spectra
Auto regressive spectral estimates
Data interpolation and missing data restoration
Cross Talk - introduction to multi-channel filter operation
Chapter 3. Solution to the linear Gaussian Inverse Problem
Example, joint seismic traveltime and gravity inversion
*Seismic tomography (ART, SIRT, application to Kirchhoff migration)
*Simulated Annealing
*Finding the inverse (Gauss-Jordan Elimination, conjugate Gradient Method)
*Fractal Inversion (an application of genetic algorithms)
*Wavelet transforms? Atmospheric applications - satellite data retrievals
Chapter 4. Generalized Inverses
With application to Atmospheric absorption
Chapter 5. Maximum Likelihood Methods
Chapter 6. Nonuniqueness and localized averages
Chapter 7. Applications of vector spaces
Singular value decomposition, empirical orthogonal functions
Multivariate regression analysis
Kalman filtering and data assimilation
Newton solution techniques
*Neural networks
Presentations of students' papers
*sequence may be changed