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Image Estimation By Example - IMAGE ESTIMATION BY EXAMPLE...

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IMAGE ESTIMATION BY EXAMPLE: Geophysical Soundings Image Construction Multidimensional autoregression Jon F. Claerbout Cecil and Ida Green Professor of Geophysics Stanford University with Sergey Fomel Stanford University c February 2, 2011
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Contents 1 Basic operators and adjoints 1 1.0.1 Programming linear operators . . . . . . . . . . . . . . . . . . . . . . 3 1.1 FAMILIAR OPERATORS . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 1.1.1 Adjoint derivative . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 1.1.2 Transient convolution . . . . . . . . . . . . . . . . . . . . . . . . . . 8 1.1.3 Internal convolution . . . . . . . . . . . . . . . . . . . . . . . . . . . 10 1.1.4 Zero padding is the transpose of truncation . . . . . . . . . . . . . . 11 1.1.5 Adjoints of products are reverse-ordered products of adjoints . . . . 12 1.1.6 Nearest-neighbor coordinates . . . . . . . . . . . . . . . . . . . . . . 12 1.1.7 Data-push binning . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13 1.1.8 Linear interpolation . . . . . . . . . . . . . . . . . . . . . . . . . . . 14 1.1.9 Spray and sum : scatter and gather . . . . . . . . . . . . . . . . . . 15 1.1.10 Causal and leaky integration . . . . . . . . . . . . . . . . . . . . . . 16 1.1.11 Backsolving, polynomial division and deconvolution . . . . . . . . . 19 1.1.12 The basic low-cut filter . . . . . . . . . . . . . . . . . . . . . . . . . 20 1.1.13 Smoothing with box and triangle . . . . . . . . . . . . . . . . . . . . 22 1.1.14 Nearest-neighbor normal moveout (NMO) . . . . . . . . . . . . . . . 24 1.1.15 Coding chains and arrays . . . . . . . . . . . . . . . . . . . . . . . . 26 1.2 ADJOINT DEFINED: DOT-PRODUCT TEST . . . . . . . . . . . . . . . . 28 1.2.1 Definition of a vector space . . . . . . . . . . . . . . . . . . . . . . . 28 1.2.2 Dot-product test for validity of an adjoint . . . . . . . . . . . . . . . 29 1.2.3 The word “adjoint” . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 1.2.4 Matrix versus operator . . . . . . . . . . . . . . . . . . . . . . . . . . 31 1.2.5 Inverse operator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32
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CONTENTS 1.2.6 Automatic adjoints . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 2 Model fitting by least squares 35 2.1 HOW TO DIVIDE NOISY SIGNALS . . . . . . . . . . . . . . . . . . . . . 35 2.1.1 Dividing by zero smoothly . . . . . . . . . . . . . . . . . . . . . . . . 35 2.1.2 Damped solution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 2.1.3 Smoothing the denominator spectrum . . . . . . . . . . . . . . . . . 36 2.1.4 Imaging . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 2.1.5 Formal path to the low-cut filter . . . . . . . . . . . . . . . . . . . . 40 2.1.6 The plane-wave destructor . . . . . . . . . . . . . . . . . . . . . . . . 40 2.2 MULTIVARIATE LEAST SQUARES . . . . . . . . . . . . . . . . . . . . . 45 2.2.1 Inside an abstract vector . . . . . . . . . . . . . . . . . . . . . . . . . 45 2.2.2 Normal equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48 2.2.3 Differentiation by a complex vector . . . . . . . . . . . . . . . . . . . 49 2.2.4 From the frequency domain to the time domain . . . . . . . . . . . . 49 2.3 KRYLOV SUBSPACE ITERATIVE METHODS . . . . . . . . . . . . . . . 51 2.3.1 Sign convention . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51 2.3.2 Method of random directions and steepest descent . . . . . . . . . . 52 2.3.3 The meaning of the gradient . . . . . . . . . . . . . . . . . . . . . . 53 2.3.4 Null space and iterative methods . . . . . . . . . . . . . . . . . . . . 54 2.3.5 Why steepest descent is so slow . . . . . . . . . . . . . . . . . . . . . 55 2.3.6 Conjugate direction . . . . . . . . . . . . . . . . . . . . . . . . . . . 55 2.3.7 Routine for one step of conjugate-direction descent . . . . . . . . . . 57 2.3.8 A basic solver program . . . . . . . . . . . . . . . . . . . . . . . . . . 58 2.3.9 The modeling success and the solver success . . . . . . . . . . . . . . 60 2.3.10 Measuring success . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61 2.3.11 Roundoff . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61 2.3.12 Why Fortran 90 is much better than Fortran 77 . . . . . . . . . . . . 62 2.3.13 Test case: solving some simultaneous equations . . . . . . . . . . . . 62 2.4 INVERSE NMO STACK . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64 2.5 FLATTENING 3-D SEISMIC DATA . . . . . . . . . . . . . . . . . . . . . . 65 2.5.1 Gulf of Mexico Salt Piercement Example (Jesse Lomask) . . . . . . 67
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CONTENTS 2.6 VESUVIUS PHASE UNWRAPPING . . . . . . . . . . . . . . . . . . . . . 69 2.6.1 Estimating the inverse gradient . . . . . . . . . . . . . . . . . . . . . 71 2.6.2 Digression: curl grad as a measure of bad data . . . . . . . . . . . . 73 2.6.3 Discontinuity in the solution . . . . . . . . . . . . . . . . . . . . . . 74 2.6.4 Analytical solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . 75 2.7 THE WORLD OF CONJUGATE GRADIENTS . . . . . . . . . . . . . . . 76 2.7.1 Physical nonlinearity . . . . . . . . . . . . . . . . . . . . . . . . . . . 76 2.7.2 Statistical nonlinearity . . . . . . . . . . . . . . . . . . . . . . . . . . 77 2.7.3 Coding nonlinear fitting problems . . . . . . . . . . . . . . . . . . . 77 2.7.4 Standard methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78 2.7.5 Understanding CG magic and advanced methods . . . . . . . . . . . 79 2.8 REFERENCES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83 3 Empty bins and inverse interpolation 85 3.1 MISSING DATA IN ONE DIMENSION . . . . . . . . . . . . . . . . . . . . 86 3.1.1 Missing-data program . . . . . . . . . . . . . . . . . . . . . . . . . . 89 3.2 WELLS NOT MATCHING THE SEISMIC MAP . . . . . . . . . . . . . . . 94 3.3 SEARCHING THE SEA OF GALILEE . . . . . . . . . . . . . . . . . . . . 99 3.4 INVERSE LINEAR INTERPOLATION . . . . . . . . . . . . . . . . . . . . 101 3.4.1 Abandoned theory for matching wells and seismograms . . . . . . . 105 3.5 PREJUDICE, BULLHEADEDNESS, AND CROSS VALIDATION . . . . . 106 4 The helical coordinate 109 4.1 FILTERING ON A HELIX . . . . . . . . . . . . . . . . . . . . . . . . . . . 109 4.1.1
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