BASIC EARTH IMAGING (Version 2.4)

BASIC EARTH IMAGING (Version 2.4) Jon F. Claerbout with James L. Black c October 31, 2005 Contents 1 Field recording geometry 1 1.1 RECORDINGGEOMETRY .......................... 1 1.2 TEXTURE ................................... 7 2 Adjoint operators 11 2.1 FAMILIAROPERATORS ........................... 12 2.2 ADJOINTSANDINVERSES .. ..... .... .... ..... .... 21 3 Waves in strata 23 3.1 TRAVEL-TIMEDEPTH ............................ 23 3.2 HORIZONTALLYMOVINGWAVES . 24 3.3 DIPPINGWAVES ............................... 29 3.4 CURVEDWAVEFRONTS ........................... 34 4 Moveout, velocity, and stacking 41 4.1 INTERPOLATIONASAMATRIX . 41 4.2 THENORMALMOVEOUTMAPPING . 44 4.3 COMMON-MIDPOINTSTACKING . 46 4.4 VELOCITYSPECTRA............................. 51 5 Zero-offset migration 59 5.1 MIGRATIONDEFINED ............................ 59 5.2 HYPERBOLAPROGRAMMING . 64 CONTENTS 6 Waves and Fourier sums 73 6.1 FOURIERTRANSFORM ........................... 73 6.2 INVERTIBLESLOWFTPROGRAM . 77 6.3 CORRELATIONANDSPECTRA. 79 6.4 SETTINGUPTHEFASTFOURIERTRANSFORM . 82 6.5 SETTINGUP2-DFT.............................. 85 6.6 THEHALF-ORDERDERIVATIVEWAVEFORM . 94 6.7 References.................................... 95 7 Downward continuation 97 7.1 MIGRATION BY DOWNWARD CONTINUATION . 97 7.2 DOWNWARDCONTINUATION . 101 7.3 PHASE-SHIFTMIGRATION . 105 8 Dip and offset together 121 8.1 PRESTACKMIGRATION . 121 8.2 INTRODUCTIONTODIP . 127 8.3 TROUBLEWITHDIPPINGREFLECTORS . 133 8.4 SHERWOOD’SDEVILISH . 135 8.5 ROCCA’SSMEAROPERATOR. 137 8.6 GARDNER’SSMEAROPERATOR. 140 8.7 DMOINTHEPROCESSINGFLOW . 142 9 Finite-difference migration 147 9.1 THEPARABOLICEQUATION . 147 9.2 SPLITTINGANDSEPARATION . 149 9.3 FINITE DIFFERENCING IN (omega,x)-SPACE . 153 9.4 WAVEMOVIEPROGRAM. 161 9.5 HIGHERANGLEACCURACY . 170 10 Antialiased hyperbolas 177 CONTENTS 10.1 MIMICINGFIELDARRAYANTIALIASING . 179 10.2 MIGRATIONWITHANTIALIASING . 186 10.3 ANTIALIASEDOPERATIONSONACMPGATHER . 191 11 Imaging in shot-geophone space 197 11.1 TOMOGRAPYOFREFLECTIONDATA . 197 11.2 SEISMIC RECIPROCITY IN PRINCIPLE AND IN PRACTICE . 205 11.3 SURVEYSINKINGWITHTHEDSREQUATION . 207 11.4 THEMEANINGOFTHEDSREQUATION . 213 12 RATional FORtran == Ratfor 215 13 Seplib and SEP software 217 13.1 THEDATACUBE ............................... 218 13.2 THEHISTORYFILE.............................. 219 13.3 MEMORYALLOCATION. 220 13.4 SHAREDSUBROUTINES. 221 13.5 REFERENCES ................................. 222 Index 223 CONTENTS Themes The main theme of this book is to take a good quality reflection seismic data set from the Gulf of Mexico and guide you through the basic geophysical data processing steps from raw data to the best-quality final image. Secondary themes are to introduce you (1) to cleaned up but real working Fortran code that does the job, (2) to the concept of “adjoint operator”, and (3) to the notion of electronic document. What it does, what it means, and how it works A central theme of this book is to merge the abstract with the concrete by linking mathematics to runnable computer codes. The codes are in a consistent style using nomenclature that resembles the accompanying mathematics so the two illuminate each other. The code shown is exactly that used to generate the illustrations. There is little or no mathematics or code that is not carried through with examples using both synthetic and real data. The code itself is in a dialect of Fortran more suitable for exposition than standard Fortran. (This "ratfor" dialect easily translates to standard Fortran). Some codes have been heavily tested while others have only been tested by the preparation of the illustrations. Imaging with adjoint (conjugate-transpose) operators A secondary theme of this book is to develop in the reader an understanding of a universal linkage beween forward modeling and data processing. Thus the codes here that incarnate linear operators are written in a style that also incarnates the adjoint (conjugate-transpose) operator thus enabling both modeling and data processing with the same code. This style of coding, besides being concise and avoiding redundancy, ensures the consistency required for estimation by conjugate-gradient optimization as described in my other books. Adjoint operators link the modeling activity to the model estimation activity. While this linkage is less sophisticated than formal estimation theory (“inversion”), it is robust, easily available, and does not put unrealistic demands on the data or imponderable demands on the interpreter. i ii CONTENTS Electronic document A goal that we met with the 1992 CD-ROM version of this book was to give the user a full copy, not only of the book, but of all the software that built the book including not only the seismic data processing codes but also the word processing, the data, and the whole super- structure. Although we succeeded for a while having a book that ran on machines of all the major manufacturers, eventually we were beaten down by a host of incompatibilities. This struggle continues. With my colleagues, we are now working towards having books on the World Wide Web where you can grab parts of a book that generates illustrations and modify them to create your own illustrations. Acknowledgements I had the good fortune to be able to establish a summer 1992 collaboration with Jim Black of IBM in Dallas who, besides bringing fresh eyes to the whole undertaking, wrote the first version of chapter 8 on dip moveout, made significant contributions to the other chapters, and organized the raw data. In this book, as in my previous (and later) books, I owe a great deal to the many students at the Stanford Exploration Project. The local computing environment from my previous book is still a benefit, and for this I thank Stew Levin, Dave Hale, and Richard Ottolini. In preparing this book I am specially indebted to Joe Dellinger for his development of the intermediate graphics language vplot that I used for all the figures. I am grateful to Kamal Al-Yahya for converting my thinking from the troff typesetting language to LATEX. Bill Harlan offered helpful suggestions. Steve Cole adapted vplot to Postscript and X. Dave Nichols introduced our multivendor environment. Joel M. Schroeder and Matthias Schwab converted from cake to gmake. Bob Clapp expanded Ratfor for Fortran 90. Martin Karrenbach got us started with CD-ROMs. Sergey Fomel upgraded the Latex version to “2e” and he implemented the basic changes taking us from CD-ROM to the WWW, a process which continues to this day in year 2000. Jon Claerbout Stanford University October 31, 2005 Chapter 1 Field recording geometry The basic equipment for reflection seismic prospecting is a source for impulsive sound waves, a geophone (something like a microphone), and a multichannel waveform display system. A survey line is defined along the earth’s surface. It could be the path for a ship, in which case the receiver is called a hydrophone. About every 25 meters the source is activated, and the echoes are recorded nearby. The sound source and receiver have almost no directional tuning capability because the frequencies that penetrate the earth have wavelengths longer than the ship. Consequently, echoes can arrive from several directions at the same time. It is the joint task of geophysicists and geologists to interpret the results. Geophysicists assume the quantitative, physical, and statistical tasks. Their main goals, and the goal to which this book is mainly directed, is to make good pictures of the earth’s interior from the echoes. 1.1 RECORDING GEOMETRY Along the horizontal x-axis we define two points, s, where the source (or shot or sender) is located, and g, where the geophone (or hydrophone or microphone) is located. Then, define the midpoint y between the shot and geophone, and define h to be half the horizontal offset between the shot and geophone: g s y + (1.1) = 2 g s h − (1.2) = 2 The reason for using half the offset in the equations is to simplify and symmetrize many later equations. Offset is defined with g s rather than with s g so that positive offset means − − waves moving in the positive x direction. In the marine case, this means the ship is presumed to sail negatively along the x-axis. In reality the ship may go either way, and shot points may either increase or decrease as the survey proceeds. In some situations you can clarify matters by setting the field observer’s shot-point numbers to negative values. Data is defined experimentally in the space of (s, g). Equations (1.1) and (1.2) represent a 1 2 CHAPTER 1. FIELD RECORDING GEOMETRY change of coordinates to the space of (y, h). Midpoint-offset coordinates are especially useful for interpretation and data processing. Since the data is also a function of the travel time t, the full dataset lies in a volume. Because it is so difficult to make a satisfactory display of such a volume, what is customarily done is to display slices. The names of slices vary slightly from one company to the next. The following names seem to be well known and clearly understood: (y, h 0, t) zero-offset section = (y, h hmin, t) near-trace section = (y, h const, t) constant-offset section = (y, h hmax, t) far-trace section = (y const, h, t) common-midpoint gather = (s const, g, t) field profile (or common-shot gather) = (s, g const, t) common-geophone gather = (s, g, t const) time slice = (h, y, t const) time slice = A diagram of slice names is in Figure 1.1. Figure 1.2 shows three slices from the data volume. The first mode of display is “engineering drawing mode.” The second mode of display is on the

Load more