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GEOPHYSICAL DATA ANALYSIS Discrete Inverse Theory GEOPHYSICAL DATA ANALYSIS Discrete Inverse Theory FOURTH EDITION W M ILLIAM ENKE AcademicPressisanimprintofElsevier 125LondonWall,LondonEC2Y5AS,UnitedKingdom 525BStreet,Suite1800,SanDiego,CA92101-4495,UnitedStates 50HampshireStreet,5thFloor,Cambridge,MA02139,UnitedStates TheBoulevard,LangfordLane,Kidlington,OxfordOX51GB,UnitedKingdom #2018ElsevierInc.Allrightsreserved. Nopartofthispublicationmaybereproducedortransmittedinanyformorbyanymeans,electronicor mechanical,includingphotocopying,recording,oranyinformationstorageandretrievalsystem,withoutpermission inwritingfromthepublisher.Detailsonhowtoseekpermission,furtherinformationaboutthePublisher’s permissionspoliciesandourarrangementswithorganizationssuchastheCopyrightClearanceCenterandthe CopyrightLicensingAgency,canbefoundatourwebsite:www.elsevier.com/permissions. ThisbookandtheindividualcontributionscontainedinitareprotectedundercopyrightbythePublisher(otherthan asmaybenotedherein). Notices Knowledgeandbestpracticeinthisfieldareconstantlychanging.Asnewresearchandexperiencebroadenour understanding,changesinresearchmethods,professionalpractices,ormedicaltreatmentmaybecomenecessary. Practitionersandresearchersmustalwaysrelyontheirownexperienceandknowledgeinevaluatingandusing anyinformation,methods,compounds,orexperimentsdescribedherein.Inusingsuchinformationormethodsthey shouldbemindfuloftheirownsafetyandthesafetyofothers,includingpartiesforwhomtheyhavea professionalresponsibility. Tothefullestextentofthelaw,neitherthePublishernortheauthors,contributors,oreditors,assumeanyliability foranyinjuryand/ordamagetopersonsorpropertyasamatterofproductsliability,negligenceorotherwise,or fromanyuseoroperationofanymethods,products,instructions,orideascontainedinthematerialherein. LibraryofCongressCataloging-in-PublicationData AcatalogrecordforthisbookisavailablefromtheLibraryofCongress BritishLibraryCataloguing-in-PublicationData AcataloguerecordforthisbookisavailablefromtheBritishLibrary ISBN978-0-12-813555-6 ForinformationonallAcademicPresspublications visitourwebsiteathttps://www.elsevier.com/books-and-journals Publisher:CandiceJanco AcquisitionEditor:MarisaLaFleur EditorialProjectManager:KaterinaZaliva ProductionProjectManager:PremKumarKaliamoorthi CoverDesigner:ChristianJ.Bilbow TypesetbySPiGlobal,India Preface For now we see through a glass, darkly, but parameters that either are truly discrete or then… Paul of Tarsus canbeadequatelyapproximatedasdiscrete. Byadheringtotheselimitations,inversethe- Every researcher in the applied sciences orycanbepresentedonalevelthatisacces- whohasanalyzeddatahaspracticedinverse sibletomostfirst-yeargraduatestudentsand theory. Inverse theory is simply the set of manycollegeseniorsintheappliedsciences. methods used to extract useful inferences The only mathematics that is presumed is a about the world from physical measure- working knowledge of the calculus and lin- ments.Thefittingofastraightlinetodatain- ear algebra and some familiarity with gen- volvesasimpleapplicationofinversetheory. eral concepts from probability theory and Tomography,popularizedbythephysician’s statistics. CT and MRI scanners, uses it on a more Nevertheless,thetreatmentinthisbookis sophisticated level. in no sense simplified. Realistic examples, The study of inverse theory, however, is drawnfromthescientificliterature,areused morethanthecatalogingofmethodsofdata to illustrate the various techniques. Since in analysis. It is an attempt to organize these practice the solutions to most inverse prob- techniques, to bring out their underlying lems require substantial computational ef- similarities and pin down their differences, fort, attention is given to how realistic and to deal with the fundamental question problemscan be solved. of the limits of information that can be The treatment of inverse theory in this gleanedfrom any given dataset. book is divided into four parts. Chapters 1 Physical properties fall into two general and 2 provide a general background, classes: those that can be described by dis- explaining what inverse problems are and crete parameters (e.g., the mass of the earth what constitutes their solution as well as orthepositionoftheatomsinaproteinmol- reviewing some of the basic concepts from ecule) and those that must be described by linear algebra and probability theory that continuousfunctions(e.g.,temperatureover willbeappliedthroughoutthetext.Chapters thefaceoftheearthorelectricfieldintensity 3–7 discuss the solution of the canonical in- in a capacitor). Inverse theory employs dif- verse problem: the linear problem with ferent mathematical techniques for these Gaussian statistics. This is the best under- twoclassesofparameters:thetheoryofma- stood of all inverse problems, and it is here trix equations for discrete parameters and thatthefundamentalnotionsofuncertainty, the theory of integral equationsfor continu- uniqueness, and resolution can be most ous functions. clearly developed. Chapters 8–11 extend Being introductory in nature, this book the discussion to problems that are non- dealsmainlywith“discreteinversetheory,” Gaussian, nonlinear, and continuous. thatis,thepartofthetheoryconcernedwith Chapter 11 devotes special attention to the ix x PREFACE so-calledadjointmethod,amathematicaltech- Manypeoplehelpedmewritethisbook.I niquethat,overthepasttwodecades,hasbe- amverygratefultomystudentsatColumbia come an increasingly important tool for University and at Oregon State University solving inverse problems in seismology forthehelpfulcommentstheygavemedur- andclimatescience.Chapters12–13provide ingthecoursesIhavetaughtoninversethe- examples of the use of inverse theory and a ory. Mike West, of the Alaska Volcano discussion of the steps that must be taken Observatory,didmuchtoinspirerecentrevi- to solve inverse problemson a computer. sions of the book, by inviting me to teach a MatLab scripts are used throughout the mini-course on the subject in the fall of book as a means of communicating how 2009.TheuseofMatLabinthisbookparallels the formulas of inverse theory can be used theusageinEnvironmentalDataAnalysiswith in computer-based data processing scenar- MatLab (Menke and Menke, 2011), a data ios.MatLabisacommercialsoftwareproduct analysis textbook that I wrote with my son ofTheMathWorks,Inc.andiswidelyusedin Joshua Menke in 2011. The many hours we university settings as an environment for spent working together on its tutorials scientificcomputing.Allofthebook’sexam- taught us both a tremendous amount about ples,itsrecommendedhomeworkproblems, howtousethatsoftwareinapedagogicalset- and the case studies of Chapter 12 use ting. Finally, I thank the many hundreds of MatLab extensively. Further, all the MatLab scientists and mathematicians whose ideas scripts used in the book are made available I drew upon in writing this book. to readers through the book’s website. The bookisself-contained;itcanbereadstraight Reference through, and profitably, even by someone with no access to MatLab. But it is meant to Menke,W.,Menke,J.,2011.EnvironmentalDataAnaly- be used in a setting where students are ac- sis With MatLab. Academic Press, Elsevier Inc., Oxford.263pp. tivelyusing MatLab bothas anaidto study- ing(thatis,byreproducingtheexamplesand case studies described in the book) and as a Companion Web Site toolforcompletingtherecommendedhome- https://www.elsevier.com/books-and-journals/book- work problems. companion/9780128135556. Introduction I.1 FORWARD AND INVERSE THEORIES Inversetheoryisanorganizedsetofmathematicaltechniquesforreducingdatatoobtain knowledge about the physical world on the basisof inferences drawn from observations. In- versetheory,asweshallconsideritinthisbook,islimitedtoobservationsandquestionsthat canberepresentednumerically.Theobservationsoftheworldwillconsistofatabulationof measurements,ordata.Thequestionswewanttoanswerwillbestatedintermsofthenumer- ical values(andstatistics) of specific (but notnecessarily directly measurable)propertiesof theworld.Thesepropertieswillbecalledmodelparametersforreasonsthatwillbecomeap- parent.Weshallassumethatthereissomespecificmethod(usuallyamathematicaltheoryor model) for relatingthe modelparametersto the data. Thequestion,“whatcausesthemotionoftheplanets?,”forexample,isnotonetowhich inversetheorycanbeapplied.Eventhoughitisperfectlyscientificandhistoricallyimportant, itsanswerisnotnumericalinnature.Ontheotherhand,inversetheorycanbeappliedtothe question,assumingthatNewtonianmechanicsapplies,determinethenumberandorbitsof theplanetsonthebasisoftheobservedorbitofHalley’scomet.Thenumberofplanetsand their orbital ephemerides are numerical in nature. Another important difference between these two problems is that the first asks us to determine the reason for the orbital motions, andthesecondpresupposesthereasonandasksusonlytodeterminecertaindetails.Inverse theoryrarelysuppliesthekindofinsightdemandedbythefirstquestion;italwaysrequires that the physical model or theorybespecified beforehand. Theterminversetheoryisusedincontrasttoforwardtheory,whichisdefinedastheprocess ofpredictingtheresultsofmeasurements(predictingdata)onthebasisofsomegeneralprin- cipleormodelandasetofspecificconditionsrelevanttotheproblemathand.Inversetheory, roughlyspeaking,addressesthereverseproblem:startingwithdataandageneralprinciple, theory,orquantitativemodel,itdeterminesestimatesofthemodelparameters.Intheearlier example,predictingtheorbitofHalley’scometfromthepresumablywell-knownorbitaleph- emeridesof the planets isa problem for forward theory. Anothercomparisonofforwardandinverseproblemsisprovidedbythephenomenonof temperaturevariationasafunctionofdepthbeneaththeearth’ssurface.Letusassumethat thetemperatureincreaseslinearlywithdepthintheearth,thatis,temperatureTisrelatedto depthzbytheruleT(z)¼az+b,whereaandbarenumericalconstantsthatwewillrefertoas modelparameters.Ifoneknowsthata¼0.1andb¼25,thenonecansolvetheforwardproblem simply by evaluating the formula for any desired depth. The inverse problem would be to determine a and b on the basis of a suite of temperature measurements made at different xi xii INTRODUCTION depthsin,say,aborehole.Onemayrecognizethatthisistheproblemoffittingastraightline to data, which is a substantially harder problem than the forward problem of evaluating a first-degree polynomial. Thisbrings out a property of most inverse problems: that they are substantiallyharder to solve than theircorresponding forward problems. Forwardproblem: estimatesofmodelparameters!quantitativemodel!predictionsofdata Inverseproblem: observationsofdata!quantitativemodel!estimatesofmodelparameters Note that the role of inverse theory is to provide information about unknown numerical parametersthatgointothemodel,nottoprovidethemodelitself.Nevertheless,inversethe- orycan oftenprovide ameansfor assessing thecorrectnessofagivenmodelorofdiscrim- inatingbetween several possible models. The model parameters one encounters in inverse theory vary from discrete numerical quantities to continuous functions of one or more variables. The intercept and slope of the straight line mentioned earlier are examples of discrete parameters. Temperature, which varies continuously with position, is an example of a continuous function. This book deals mainly with discrete inverse theory, in which the model parameters are represented as a set of a finite number of numerical values. This limitation does not, in practice, exclude the study of continuous functions, since they can usually be adequately approximated by a finite number of discrete parameters. Temperature, for example, might be represented byitsvalueatafinitenumberofcloselyspacedpointsorbyasetofsplineswithafinitenum- berofcoefficients.Thisapproachdoes,however,limittherigorwithwhichcontinuousfunc- tionscanbestudied.Parameterizationsofcontinuousfunctionsarealwaysbothapproximate and,tosomedegree,arbitraryproperties,whichcastacertainamountofimprecisionintothe theory.Nevertheless,discreteinversetheoryisagoodstartingplaceforthestudyofinverse theory,ingeneral,sinceitreliesmainlyonthetheoryofvectorsandmatricesratherthanon thesomewhatmorecomplicatedtheoryofcontinuousfunctionsandoperators.Furthermore, carefulapplicationofdiscreteinversetheorycanoftenyieldconsiderableinsight,evenwhen appliedto problems involving continuous parameters. Althoughthemainpurposeofinversetheoryistoprovideestimatesofmodelparameters, thetheoryhasaconsiderablylargerscope.Evenincasesinwhichthemodelparametersare theonlydesiredresults,thereisaplethoraofrelatedinformationthatcanbeextractedtohelp determine the “goodness” of the solution to the inverse problem. The actual values of the modelparametersareindeedirrelevantincaseswhenwearemainlyinterestedinusingin- versetheoryasatoolinexperimentaldesignorinsummarizingthedata.Someoftheques- tionsinverse theory can helpanswer are the following: 1. What arethe underlying similarities among inverse problems? 2. How are estimates of modelparametersmade? 3. Howmuchoftheerrorinthemeasurementsshowsupaserrorintheestimatesofthemodel parameters? 4. Givenaparticularexperimentaldesign,canacertainsetofmodelparametersreallybe determined? These questions emphasize that there are many different kinds of answers to inverse problemsandmanydifferentcriteriabywhichthegoodnessofthoseanswerscanbejudged. xiii INTRODUCTION Muchofthesubjectofinversetheoryisconcernedwithrecognizingwhencertaincriteriaare moreapplicablethanothers,aswellasdetectingandavoiding(ifpossible)thevariouspitfalls that can arise. Inverseproblemsariseinmanybranchesofthephysicalsciences.Anincompletelistmight includesuch entries as 1. medical andseismictomography, 2. image enhancement, 3. curve fitting, 4. earthquakelocation, 5. oceanographic andmeteorological dataassimilation, 6. factor analysis, 7. determination of earth structure from geophysical data, 8. satellite navigation, 9. mappingof celestial radio sources with interferometry, and 10. analysisofmolecular structure by X-ray diffraction. Inverse theory was developed by scientists and mathematicians having various back- grounds and goals. Thus, although the resulting versions of the theory possess strong and fundamental similarities, they have tended to look, superficially, very different. One of the goals of this book is to present the variousaspects of discrete inverse theory in such a way that boththe individual viewpoints and the “bigpicture” can beclearly understood. Thereareperhapsthreemajorviewpointsfromwhichinversetheorycanbeapproached. Thefirstandoldestsprangfromprobabilitytheory—anaturalstartingplaceforsuch“noisy” quantities as observations of the real world. In this version of inverse theory, the data and model parameters are treated as random variables, and a great deal of emphasis is placed ondeterminingtheprobabilitydensityfunctionsthattheyfollow.Thisviewpointleadsvery naturallyto the analysisoferrorand to tests of thesignificanceofanswers. Thesecondviewpointdevelopedfromthatpartofthephysicalsciencesthatretainsadeter- ministicstanceandavoidstheexplicituseofprobabilitytheory.Thisapproachhastendedto dealonlywithestimatesofmodelparameters(andperhapswiththeirerrorbars)ratherthan withprobabilitydensityfunctionsperse.Yetwhatonemeansbyanestimateisoftennothing more than the expected value of a probability density function; the difference is only one of emphasis. The third viewpoint arose from a consideration of model parameters that are inherently continuousfunctions.Whereas theothertwoviewpointshandledthisproblem byapproxi- matingcontinuousfunctionswithafinitenumberofdiscreteparameters,thethirddeveloped methodsforhandlingcontinuousfunctionsexplicitly.Althoughcontinuousinversetheoryis not the primary focus of this book, many of the concepts originally developed for it have applicationtodiscreteinversetheory,especiallywhenitisusedwithdiscretizedcontinuous functions. I.2 MATLAB AS A TOOL FOR LEARNING INVERSE THEORY Thepracticeofinversetheoryrequirescomputer-basedcomputation.Apersoncanlearn manyoftheconceptsofinversetheorybyworkingthroughshortpencil-and-paperexamples andbyexaminingprecomputedfiguresandgraphs.Butheorshecannotbecomeproficientin xiv INTRODUCTION the practice of inverse theory that way because it requires skills that can only be obtained through the experience of working with large data sets. Three goals are paramount: to develop the judgment needed to select the best solution method among many alternatives; to build confidence that the solution can be obtained even though it requires many steps; and to strengthenthe critical faculties needed to assess thequality of theresults. This book devotesconsiderablespacetocasestudiesandhomeworkproblemsthatprovidethepractical problem-solving experience needed to gain proficiency ininverse theory. Computer-based computation requires software. Many different software environments are available for the type of scientific computation that underpins data analysis. Some are moreapplicableandotherslessapplicabletoinversetheoryproblems,butamongtheappli- cableones,nonehasadecisiveadvantage.Nevertheless,wehavechosenMatLab,acommer- cialsoftwareproductofTheMathWorks,Inc.asthebook’ssoftwareenvironmentforseveral reasons, somehaving to do with its designs and other more practical. The most persuasive designreasonisthatitssyntaxfullysupportslinearalgebra,whichisneededbyalmostevery inverse theory method. Furthermore, it supports scripts, that is, sequences of data analysis commands that are communicated in written form and which serve to document the data analysisprocess.Practicalconsiderationsincludethefollowing:itisalong-livedandstable product, available since the mid-1980s; implementations are available for most commonly used typesof computers; its price, especiallyfor students, is fairly modest; andit is widely used,at least, in university settings. In MatLab’s scripting language, dataarepresented as one ormorenamedvariables (inthe same sense that c and d in the formula, c¼πd, are named variables). Data are manipulated by typing formula that create new variables from old ones and by running scripts, that is, sequences of formulas stored in a file. Much of inverse theory is simply the application of well-known formulas to novel data, so the great advantage of this approach is that the formulas that are typed usually have a strong similarity to those printed in a textbook. Furthermore, scripts provide both a way of documenting the sequence of a formula used toanalyzeaparticulardatasetandawaytotransfertheoveralldataanalysisprocedurefrom one data set to another. However, one disadvantage is that the parallel between the syntax ofthescriptinglanguageandthesyntaxofstandardmathematicalnotationisnowherenear perfect. A person needsto learnto translate one into the other. I.3 A VERY QUICK MATLAB TUTORIAL Unfortunately, this book must avoid discussion of the installation of MatLab and the appearanceofMatLabonyourcomputerscreen,forproceduresandappearancesvaryfromcom- putertocomputerandquicklybecomeoutdated,anyway.Wewillassumethatyouhavesuccess- fully installed it and that you can identify the Command Window, the place where MatLab formulaandcommandsaretyped.OnceyouhaveidentifiedtheCommandWindow,trytyping: date MatLabshouldrespondbydisplayingtoday’sdate.AlltheMatLabcommandsthatareused in this book are in freely available MatLab scripts. This one is named gda01_01 and is in a