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PROGRAMMING GEOPHYSICS IN C Dave Nichols foundations inversion Hector Urdaneta moveout applications Hyang Im Oh missing data examples Jon Claerbout outline proofreading Lisa Laane foundations Martin Karrenbach moveout document integration Matt Schwab tutorial and convolution cAugust

Contents INTRODUCTION WHAT IS A SPACE OPERATORS SOLVERS ADJOINTS AND THE DOTPRODUCT TEST Using and understanding an operator DIRECTORY STRUCTURE A PHYSICAL OPERATOR CLASS THE TEST PROGRAM THE CONVOLUTION CLASS SKELETON STANDARD DOCUMENTATION Elementary Science Operators MATRIX MULTIPLICATION CAUSAL INTEGRATION FIRST DERIVATIVE CONVOLUTION A GENERAL MOVEOUT OPERATOR Base and Building Block Operators BASIC OPERATOR HIERARCHY COMPOSING OPERATORS UTILITY OPERATORS Solving leastsquares inverse problems i

ii CONTENTS LEAST SQUARES SOLVERS USING SOLVER OBJECTS PRECONDITIONED INVERSE PROBLEMS THE Process UTILITY Derived moveout operators LINEAR MOVEOUT NORMAL MOVEOUT ANELLIPTIC MOVEOUT KIRCHHOFF MOVEOUT Stacking operators LINEAR MOVEOUT AND STACKING NMO AND STACK KIRCHHOFF OPERATOR VELOCITY ANALYSIS SLANT STACK AND PRECONDITIONED INVERSION Missingdata restoration INTERPOLATING MISSING DATA WITH A KNOWN FILTER INTERPOLATING MISSING DATA WITH AN UNKNOWN FILTER SUMMARY Appendix Man Pages

CONTENTS PREFACE This paper document is an advertisement for an electronic document that lies behind it We plan to distribute the electronic document on CDROM late September or early October To run this version of the electronic document two commercial products are re quired First is a C compiler This books code was tested using the Sun C compiler version and the HP C compiler Both compilers are implemen tations of ATTs C compiler versions and which seems to be a unix industry standard Hopefully at a later date we will be able to include a freeware C compiler that will also do the job Secondly our code uses the a commercial matrix class library M from Dyad as a foundation for some of our C classes In the future we hope to use a public domain matrix class library to avoid this com mercial dependence We have no antipathy for commercial products in fact we like them and appreciate them but experience shows that our work will be accessible to the greatest number of people if our code depends only on noncommercial software We are collaborating with Les Dye who is aliated with Berea and Stanfords petroleum reservoir simulation group SUPRIB which is headed by Khalid Aziz These groups are involved in research on objectbased mathematical abstractions for reservoir simulation and nonlinear systems We may base our future library designs on research that results from this collaboration A recently introduced C language feature known as templates allows devel opers to write one code that can be used for multiple data types Since our current libraries only handle oat data and not complex data we hope to use templates to create a more general library later this year At mid August we have only begun to use the results of this C project for serious seismological research We hope to demonstrate its utility further by summers end To test whether your environment will build this C document press this BUTTON to destroy all the gures in this document Then view the document to see if all the gures are gone Then press this BUTTON to rebuild all the gures You could scan the document to see if all the gures have been regenerated or press this BUTTON to see a statistical analysis of the reproducibility of the illustrations in this document

CONTENTS

Chapter INTRODUCTION Every few years brings a new computer language Yet most scientic and engineering computing continues to be done in Fortran a language that has hardly changed since Here we explore the hypothesis that a new language C is a worthwhile alternative to Fortran C is based on the C language The C language has made remarkable strides in the last several years Formerly Fortran was the language that was available on most machines and it had the fewest variations from one machine to another Today C replaces Fortran in both universality and uniformity Now more young engineers and scientists learn C than any other language C overcomes the memoryallocation limitation of Fortran and oers new features such as data structures and scope C however was not designed for numerical analysis and it does not handle matrices as conveniently as does Fortran There has been little mass movement of scientic code from Fortran to C We and many others have overcome some of Fortrans most glaring decits by calling Clanguage subroutines and by using our preprocessors for memory allocation and parameter handling In the past few years C has become standardized and new developments are now centered on the C language C shares the advantages of C but it adds features that permit the addition of new more complex data types to the language and allows the author to dene all operations possible on the new data types For instance C can handle the concept of arrays at least as well as Fortran whereas C cannot We do not predict that Fortran will die and be replaced by C but there are some reasons why this might happen the growth of C has been much more rapid It already has many more users than Fortran We nd C compilers on all manner of computers also the free GNU compiler whereas our workstation manufacturers still do not oer Fortran New parallel architectures present an opportunity for a malleable language like C and a risk for static languages C was especially designed to enable concepts to be isolated leading to software that is better debugged and more reusable A particular problem in geophysical inversion problems see Processing versus inversion PVI for example Claerbout has been that the Fortran language leads to programs where the physical

CHAPTER INTRODUCTION science is closely interwoven with the numerical analysis in the conjugategradient solver One person the programmer must handle all the ingredients of both physics and numerical analysis specializations Such code is challenging to write and even more challenging to debug Further in complicated cases such as with nonlinear operations or linear operators of very high order we cannot really be sure the code is debugged which is disheartening Our ultimate goals are Write real code that is clean enough that we can include it in publications where we now use mathematics and pseudocode Reduce the overlap expertise required to merge scientic application code with numerical analysis code Write linear operators in such a way that they can be used by people other than the original author and operators written by dierent people can easily be combined to produce new operators As a rst step to meeting these goals Nichols Dunbar and Claerbout took a body of tutorial FortranRatfor code in the geophysical textbook Earth soundings analysis processing versus inversion PVI by Jon Claerbout and began converting it to C This summer more usersprogrammers were recruited The base classes were improved and many new operators were added Some examples from Claerbouts book Basic Earth Interior BEI were converted to C form The C language To ameliorate the problem of many of the readers of this document knowing little of C and C we recommend following learning pathway Begin from any of the many dozens of C tutorial books concentrating on learning particularly well the ideas of pointer structure and typedef Then switch to the book which concerns itself mainly with the class and inheritance concepts avoiding the full minutia of the C language Going beyond Fortran the rst concept needed is that of a structure For exam ple in Fortran if a single symbol could represent the idea of nxi in then that symbol would denote something like a structure Another structure might be nn xii in in A numerical analyst thinking about solving the multivariate regression y Ax might need to allocate several instances of the structure representing x Clearly the solver program written by the analyst should work for all possible structures In Fortran the application programmer ends out allocating the temporary arrays needed by the numerical analyst leading to long error prone calling sequences In Fortran the work of the applications programmer and the numerical analyst are typically brought together by subroutines C also uses subroutine libraries and additionally it makes heavy use of include les that de ne structures thus serving to better isolate science from numerical analysis C

WHAT IS A SPACE extends the structure concept to a class which is a structure containing subroutine pointers C has elaborate provisions for sharing and hiding data for defaulting and overriding functions It also uses include les to implement a new method of merging the work of the applications programmer to the numerical analyst called inheritance or derivation whose explanation is beyond the scope of this brief summary but which should become clearer as you progress through this document WHAT IS A SPACE The well known word space can lead to considerable confusion because there are many entities that are described as spaces these include continuous spaces dis crete sampled spaces abstract innite vector spaces etc In our C code the type space class describes objects that are a discrete sampling of a phys ical property in some geometric region eg pressure measured on a plane The word type denes the numerical type of the elements of the space currently floatspace is the only supported type we hope to implement a complexspace class soon An abstract space in a numerical problem is a collection of values that could be a single physical space a single instance of the type space class or an assemblage of physical spaces In our C code such assemblages are called type spacearray ie an array of type space objects In this paper we will use the word space to refer to the abstract concept and type space or type spacearray when we wish to refer to a particular representation of a space The type spacearray can be thought of as a generalization of the single type space in our code a single type space may be automatically converted to a type spacearray containing a single element The oatspace class An object of class floatspace the class to represent a regulary sampled physical space has two parts The rst part is a multidimensional array of numbers repre senting the sampled data or model This array contains oating point values The second part is a set of axes that describe the physical dimensions of region that the samples are taken from Every dimension of a floatspace has an axis associated with it which denes an origin length and sampling interval delta for the axis as well as a label Thus a floatspace is meant as a collection of real physical data rather than an abstract collection of numbers This is ideal for many scientic applications as in geophysics The publicly accessible member functions that manipulate objects of the class floatspace are relatively few as operators are the main method of transforming a space The most important member functions are

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