Basic Earth Imaging Jon F. Claerbout c November 17, 2010 Contents 1 Field recording geometry 1 1.1 RECORDING GEOMETRY . 1 1.1.1 Fast ship versus slow ship . 6 1.2 TEXTURE . 6 1.2.1 Texture of horizontal bedding, marine data . 7 1.2.2 Texture of land data: near-surface problems . 8 2 Adjoint operators 11 2.1 FAMILIAR OPERATORS . 12 2.1.1 Adjoint derivative . 14 2.1.2 Zero padding is the transpose of truncation . 15 2.1.3 Adjoints of products are reverse-ordered products of adjoints . 16 2.1.4 Nearest-neighbor coordinates . 17 2.1.5 Data-push binning . 17 2.1.6 Linear interpolation . 18 2.1.7 Causal integration . 19 2.2 ADJOINTS AND INVERSES . 21 2.2.1 Dot product test . 21 3 Waves in strata 23 3.1 TRAVEL-TIME DEPTH . 23 3.1.1 Vertical exaggeration . 24 3.2 HORIZONTALLY MOVING WAVES . 24 3.2.1 Amplitudes . 24 3.2.2 LMO by nearest-neighbor interpolation . 28 CONTENTS 3.2.3 Muting . 28 3.3 DIPPING WAVES . 30 3.3.1 Rays and fronts . 30 3.3.2 Snell waves . 31 3.3.3 Evanescent waves . 33 3.3.4 Solution to kinematic equations . 33 3.4 CURVED WAVEFRONTS . 33 3.4.1 Root-mean-square velocity . 34 3.4.2 Layered media . 35 3.4.3 Nonhyperbolic curves . 37 3.4.4 Velocity increasing linearly with depth . 37 3.4.5 Prior RMS velocity . 38 4 Moveout, velocity, and stacking 39 4.1 INTERPOLATION AS A MATRIX . 39 4.1.1 Looping over input space . 40 4.1.2 Looping over output space . 40 4.1.3 Formal inversion . 40 4.2 THE NORMAL MOVEOUT MAPPING . 41 4.3 COMMON-MIDPOINT STACKING . 44 4.3.1 Crossing traveltime curves . 45 4.3.2 Ideal weighting functions for stacking . 47 4.3.3 Gulf of Mexico stack and AGC . 47 4.4 VELOCITY SPECTRA . 48 4.4.1 Velocity picking . 53 4.4.2 Stabilizing RMS velocity . 54 5 Zero-offset migration 57 5.1 MIGRATION DEFINED . 57 5.1.1 A dipping reflector . 57 5.1.2 Dipping-reflector shifts . 58 5.1.3 Hand migration . 59 5.1.4 A powerful analogy . 60 CONTENTS 5.1.5 Limitations of the exploding-reflector concept . 61 5.2 HYPERBOLA PROGRAMMING . 62 5.2.1 Tutorial Kirchhoff code . 63 5.2.2 Fast Kirchhoff code . 65 5.2.3 Kirchhoff artifacts . 66 5.2.4 Sampling and aliasing . 68 5.2.5 Kirchhoff migration of field data . 70 6 Waves and Fourier sums 73 6.1 FOURIER TRANSFORM . 73 6.1.1 FT as an invertible matrix . 74 6.1.2 The Nyquist frequency . 75 6.1.3 Laying out a mesh . 75 6.2 INVERTIBLE SLOW FT PROGRAM . 77 6.2.1 The simple FT code . 78 6.3 CORRELATION AND SPECTRA . 79 6.3.1 Spectra in terms of Z-transforms . 79 6.3.2 Two ways to compute a spectrum . 80 6.3.3 Common signals . 80 6.4 SETTING UP THE FAST FOURIER TRANSFORM . 82 6.4.1 Shifted spectrum . 84 6.5 SETTING UP 2-D FT . 85 6.5.1 Basics of two-dimensional Fourier transform . 85 6.5.2 Guide through the 2-D FT of real data . 88 6.5.3 Signs in Fourier transforms . 88 6.5.4 Simple examples of 2-D FT . 89 6.5.5 Magic with 2-D Fourier transforms . 91 6.5.6 Passive seismology . 93 6.5.7 The Stolt method of migration . 95 6.6 THE HALF-ORDER DERIVATIVE WAVEFORM . 95 6.6.1 Hankel tail . 96 6.7 References . 97 CONTENTS 7 Downward continuation 99 7.1 MIGRATION BY DOWNWARD CONTINUATION . 99 7.1.1 Huygens secondary point source . 99 7.1.2 Migration derived from downward continuation . 102 7.2 DOWNWARD CONTINUATION . 103 7.2.1 Continuation of a dipping plane wave. 104 7.2.2 Downward continuation with Fourier transform . 105 7.2.3 Linking Snell waves to Fourier transforms . 106 7.3 PHASE-SHIFT MIGRATION . 107 7.3.1 Pseudocode to working code . 108 7.3.2 Kirchhoff versus phase-shift migration . 111 7.3.3 Damped square root . 111 7.3.4 Adjointness and ordinary differential equations . 112 7.3.5 Vertical exaggeration example . 114 7.3.6 Vertical and horizontal resolution . 115 7.3.7 Field data migration . 117 8 Dip and offset together 119 8.1 PRESTACK MIGRATION . 119 8.1.1 Cheops’ pyramid . 120 8.1.2 Prestack migration ellipse . 121 8.1.3 Constant offset migration . 122 8.2 INTRODUCTION TO DIP . 125 8.2.1 The response of two points . 126 8.2.2 The dipping bed . 126 8.3 TROUBLE WITH DIPPING REFLECTORS . 129 8.3.1 Gulf of Mexico example . 129 8.4 SHERWOOD’S DEVILISH . 131 8.5 ROCCA’S SMEAR OPERATOR . 133 8.5.1 Push and pull . 134 8.5.2 Dip moveout with v(z) . 136 8.5.3 Randomly dipping layers . 136 CONTENTS 8.5.4 Many rocks on a shallow water bottom . 137 8.6 GARDNER’S SMEAR OPERATOR . 139 8.6.1 Residual NMO . 141 8.6.2 Results of our DMO program . 142 9 Finite-difference migration 145 9.1 THE PARABOLIC EQUATION . 145 9.2 SPLITTING AND SEPARATION . 147 9.2.1 The heat-flow equation . 147 9.2.2 Splitting . 147 9.2.3 Full separation . 147 9.2.4 Splitting the parabolic equation . 148 9.2.5 Validity of the splitting and full-separation concepts . 149 9.3 FINITE DIFFERENCING IN (omega,x)-SPACE . 151 9.3.1 The lens equation . 151 9.3.2 First derivatives, explicit method . ..