Geomatic methods for the analysis of data in the earth sciences
An overview of data analysis methods in geomatics ............................... 1
A . Dermanis. F . Sansb. A . Griin
Data analysis methods in geodesy ................................................................. 17
A . Dermanis and R . Rumrnel
1. Introduction ......................................................................................................... 17
2 . The art of modeling ............................................................................................. 19
3 . Parameter estimation as an inverse problem ........................................................ 24
3.1. The general case: Overdetermined and underdetermined system without full rank
(r<rnin(n.m ). ..............................................................................................................2.9
3.2. The regular case (r=m=n) .........................................................................................3..9
3.3. The full-rank overdetermined case (r=rn<n) ...........................................................4.. 0
3.4. The full-rank underdetermined case (rs=n<rn). ..........................................................4..1
3.5. The hybrid solution (Tikhonov regularization) ........................................................4..3
3.6. The full rank factorization ........................................................................................4..6
4 . The statistical approach to parameter determination: Estimation and
prediction ............................................................................................................ 47
5 . From finite to infinite-dimensional models (or from discrete to continuous
models) ..............................................................................................................5 3
5.1. Continuous observations without errors ...................................................................5..8
5.2. Discrete observations affected by noise ...................................................................6..5
5.3. The stochastic approach ...........................................................................................7..3
6 . Beyond the standard formulation: Two examples from satellite geodesy ........... 75
6.1. Determination of gravity potential coefficients ........................................................7..5
6.2. GPS observations and integer unknowns .................................................................7..8
References ............................................................................................................... 83
Appendix A: The Singular Value Decomposition ................................................... 86
Linear and nonlinear inverse problems ..................................................... 93
R . Snieder and J . Trampert
1 . Introduction ......................................................................................................... 93
2 . Solving finite linear systems of equations ........................................................... 96
2.1. Linear model estimation .........................................................................................9..6..
2.2. Least-squares estimation .........................................................................................9..9.
2.3. Minimum norm estimation ....................................................................................1..0. 0
2.4. Mixed determined problems ..................................................................................... 102
2.5. The consistency problem for the least-squares solution ...........................................1 03
2.6. The consistency problem for the minimum-norm solution. ......................................1 06
2.7. The need for a more general regularization ............................................................1..0 8
2.8. The transformation rules for the weight matrices ..................................................... 110
2.9. Solving the system of linear equations ....................................................................1. 12
2.9.1. Singular value decomposition ......................................................................1..1 3
2.9.2. Iterative least-squares ..................................................................................1..1. 7
3 . Linear inverse problems with continuous models ............................................. 120
3.1. Continuous models and basis functions ...................................................................1. 22
3.2. Spectral leakage, the problem .................................................................................1..2 3
3.3. Spectral leakage, the cure .......................................................................................1..2 7
3.4. Spectral leakage and global tomography .................................................................1. 29
4 . The single scattering approximation and linearized waveform inversion ......... 131
4.1. The Born approximation .......................................................................................1..3. 1
4.2. Inversion and migration .........................................................................................1..3 3
4.3. The Born approximation for transmission data ......................................................1..3 6
4.4. Surface wave inversion of the structure under North-America ................................1 39
5 . Rayleigh' s principle and perturbed eigenfrequencies ........................................ 141
5.1. Rayleigh-Schrodinger perturbation theory ..............................................................1. 41
5.2. The phase velocity perturbation of Love waves .....................................................1..4 3
6 . Fermat' s theorem and seismic tomography ....................................................... 145
6.1. Fermat's theorem, the eikonal equation and seismic tomography ............................ 146
6.2. Surface wave tomography ........................................................................................1 48
7 . Nonlinearity and ill-posedness .......................................................................... 150
7.1. Example 1: Non-linearity and the inverse problem for the Schrodinger equation .... 151
7.2. Example 2: Non-linearity and seismic tomography. ................................................1. 53
8. Model appraisal for nonlinear inverse problems ............................................... 155
8.1. Nonlinear Backus-Gilbert theory ...........................................................................1..5 5
8.2. Generation of populations of models that fit the data ...............................................1 57
8.3. Using different inversion methods .......................................................................... 159
9 . Epilogue ............................................................................................................ 159
References ............................................................................................................. 160
Image Preprocessing for Feature Extraction in Digital
Intensity. Color and Range Images ............................................................ 165
W . Forstner
1. Motivation ......................................................................................................... 165
2 . The image model ............................................................................................... 167
2.1. Intensity images ...................................................................................................1..6..8
2.2. Color images .........................................................................................................1..6. 9
2.3. Range images ...........................................................................................................1 69
3 . Noise variance estimation .................................................................................1. 71
3.1. Estimation of the noise variance in intensity images ...............................................1. 72
3.2. Noise estimation in range images ...........................................................................1..7 5
4 . Variance equalization ........................................................................................ 176
4.1 . Principle ..............................................................................................................1..7..6.
4.2. Linear variance function. ........................................................................................1..7 7
4.3. General variance function ........................................................................................1 77
5 . Information preserving filtering ........................................................................ 177
5.1 . The Wiener filter ...................................................................................................1..7. 7
5.2. Approximation of the auto covariance function ......................................................1. 78
5.3. An adaptive Wiener filter for intensity images ........................................................1.7 9
5.4. An adaptive Wiener filter for range images .............................................................1. 81
6 . Fusing channels: Extraction of linear features ................................................... 182
...
Vlll
6.1. Detecting edge pixels ............................................................................................1..8. 2
6.2. Localizing edge pixels ............................................................................................1..8 7
7 . Outlook .............................................................................................................. 187
References ............................................................................................................. 188
Optimization-Based Approaches to Feature Extraction
from Aerial Images ........................................................................................... 190
P . Fua. A . Gruen and H . Li
1. Introduction ...................................................................................................... 190
2 . Dynamic programming ...................................................................................... 191
2.1. Generic road model ..............................................................................................1..9..2
2.2. Road delineation ...................................................................................................1..9. 3
3. Model based optimization ................................................................................. 196
3.1. Generalized snakes ................................................................................................1..9. 8
3.2. Enforcing consistency ...........................................................................................2..0. 9
3.3. Consistent site modeling ......................................................................................2..1..2
4 . LSB-snakes ........................................................................................................ 215
4.1. Photometric observation equations ........................................................................2..1. 5
4.2. Geometric observation equations ..........................................................................2..1. 8
4.3. Solution of LSB-snakes ............................................................................................ 219
4.4. LSB-snakes with multiple images ..........................................................................2..2 0
4.5. Road extraction experiments .................................................................................2..2. 2
5 . Conclusion ...................................................................................................... 225
References .......................................................................................................2 2 6
Diffraction tomography through phase back-projection .................. 229
S . Valle. F . Rocca and L . Zanzi
1 . Introduction ..................................................................................................... 229
2 . Born approximation and Fourier diffraction theorem ................................. 231
3 . Diffraction tomography through phase back-projection .................................... 235
3.1. Theory .................................................................................................................2..3..5.
4 . Diffraction tomography and pre-stack migration ........................................ 239
4.1. Diffraction tomography wavepath ..........................................................................2..3 9
4.2. Migration wavepath ...............................................................................................2..4.1
4.3. Diffraction tomography and migration: wavepath and inversion process
comparison ........................................................................................................2...4.. 5
5 . Numerical and experimental results .................................................................. 246
5.1. Data pre-processing ...............................................................................................2..4. 6
5.2. Numerical examples ..............................................................................................2..4. 7
5.3. Laboratory model and real case examples ...............................................................2. 48
Appendix A: The Green Functions ........................................................................ 253
Appendix B: Implementation details .....................................................................2 54
Appendix C: DT inversion including the source/receiver directivity function ...... 254
References ............................................................................................................ 2 5 5