INTRODUCTION TO THE MATHEMATICS OF INVERSION IN REMOTE SENSING AND INDIRECT MEASUREMENTS

By F. LIERE

1 INTRODUCTION 2 REGULARIZATION THEORY 3 CYCLING, TIKHONOV REGULARIZATION AND 3DVAR 4 ERROR ANALYSIS 5 BAYESIAN APPROACH TO INVERSE PROBLEMS 6 4DVAR 7 KALMAN FILTER AND KALMAN SMOOTHER 8 ENSEMBLE METHODS 9 NUMERICAL EXAMPLES 91 DATA ASSIMILATION FOR AN ADVECTION-DIFFUSION SYSTEM 92 DATA ASSIMILATION FOR THE LORENZ-95 SYSTEM 10 CONCLUDING REMARKS VARIATIONAL DATA ASSIMILATION FOR VERY LARGE ENVIRONMENTAL PROBLEMS 2 THEORY OF VARIATIONAL DATA ASSIMILATION 21 INCREMENTAL VARIATIONAL DATA ASSIMILATION 3 PRACTICAL IMPLEMENTATION 31 MODEL DEVELOPMENT 32 BACKGROUND ERROR COVARIANCES 33 OBSERVATION ERRORS 34 OPTIMIZATION METHODS 35 REDUCED ORDER APPROACHES 36 ISSUES FOR NESTED MODELS 37 WEAK-CONSTRAINT VARIATIONAL ASSIMILATION 4 SUMMARY AND FUTURE PERSPECTIVES ENSEMBLE FILTER TECHNIQUES FOR INTERMITTENT DATA ASSIMILATION 1 BAYESIAN STATISTICS 11 PRELIMINARIES 12 BAYESIAN INFERENCE 13 COUPLING OF RANDOM VARIABLES 14 MONTE CARLO METHODS 2 STOCHASTIC PROCESSES 21 DISCRETE TIME MARKOV PROCESSES 22 STOCHASTIC DIFFERENCE AND DIFFERENTIAL EQUATIONS 23 ENSEMBLE PREDICTION AND SAMPLING METHODS 3 DATA ASSIMILATION AND FILTERING 31 PRELIMINARIES 32 SEQUENTIALMONTE CARLO METHOD 33 ENSEMBLE KALMAN FILTER (ENKF) 34 ENSEMBLE TRANSFORM KALMAN–BUCY FILTER 35 GUIDED SEQUENTIAL MONTE CARLO METHODS 36 CONTINUOUS ENSEMBLE TRANSFORM FILTER FORMULATIONS 4 CONCLUDING REMARKS INVERSE PROBLEMS IN IMAGING 1 MATHEMATICALMODELS FOR IMAGES 2 EXAMPLES OF IMAGING DEVICES 21 OPTICAL IMAGING 22 TRANSMISSION TOMOGRAPHY 23 EMISSION TOMOGRAPHY 24 MR IMAGING 25 ACOUSTIC IMAGING 26 ELECTROMAGNETIC IMAGING 3 BASIC IMAGE RECONSTRUCTION 31 DEBLURRING AND POINT SPREAD FUNCTIONS 32 NOISE 33 RECONSTRUCTION METHODS 4 MISSING DATA AND PRIOR INFORMATION 41 PRIOR INFORMATION 42 UNDERSAMPLING AND SUPERRESOLUTION 43 INPAINTING 44 SURFACE IMAGING 5 CALIBRATION PROBLEMS 51 BLIND DECONVOLUTION 52 NONLINEAR MR IMAGING 53 ATTENUATION CORRECTION IN SPECT 54 BLIND SPECTRAL UNMIXING 6 MODEL-BASED DYNAMIC IMAGING 61 KINETIC MODELS 62 PARAMETER IDENTIFICATION 63 BASIS PURSUIT 64 MOTION AND DEFORMATION MODELS 65 ADVANCED PDE MODELS THE LOST HONOR OF ?2-BASED REGULARIZATION 2 ?1-BASED REGULARIZATION 3 POOR DATA 4 LARGE, HIGHLY ILL-CONDITIONED PROBLEMS 41 INVERSE POTENTIAL PROBLEM 42 THE EFFECT OF ILL-CONDITIONING ON L1 REGULARIZATION 43 NONLINEAR, HIGHLY ILL-POSED EXAMPLES