REFLECTION SEISMOLOGY: THEORY, DATA PROCESSING AND INTERPRETATION W Y ENCAI ANG AMSTERDAM(cid:129)BOSTON(cid:129)HEIDELBERG(cid:129)LONDON NEWYORK(cid:129)OXFORD(cid:129)PARIS(cid:129)SANDIEGO SANFRANCISCO(cid:129)SINGAPORE(cid:129)SYDNEY(cid:129)TOKYO Elsevier 225,WymanStreet,Waltham,MA02451,USA TheBoulevard,LangfordLane,Kidlington,OxfordOX51GB,UK Radarweg29,POBox211,1000AEAmsterdam,TheNetherlands (cid:1)2014PetroleumIndustryPress.PublishedbyElsevierInc.Allrightsreserved. Nopartofthispublicationmaybereproduced,storedinaretrievalsystemortransmitted inanyformorbyanymeanselectronic,mechanical,photocopying,recordingor otherwisewithoutthepriorwrittenpermissionofthepublisher PermissionsmaybesoughtdirectlyfromElsevier’sScience&TechnologyRights DepartmentinOxford,UK:phone(+44)(0)1865843830;fax(+44)(0)1865853333;email: permissions@elsevier.com.Alternativelyyoucansubmityourrequestonlinebyvisiting theElsevierwebsiteathttp://elsevier.com/locate/permissions,andselectingObtaining permissiontouseElseviermaterial Notice Noresponsibilityisassumedbythepublisherforanyinjuryand/ordamagetopersons orpropertyasamatterofproductsliability,negligenceorotherwise,orfromanyuseor operationofanymethods,products,instructionsorideascontainedinthematerial herein.Becauseofrapidadvancesinthemedicalsciences,inparticular,independent verificationofdiagnosesanddrugdosagesshouldbemade LibraryofCongressCataloging-in-PublicationData WencaiYang,1942- Reflectionseismology:theory,dataprocessing,andinterpretation/WencaiYang. pagescm Includesbibliographicalreferences. ISBN978-0-12-409538-0(hardback) 1.Seismicreflectionmethod.I.Title. TN269.84.Y262013 551.22–dc23 2013029456 BritishLibraryCataloguinginPublicationData AcataloguerecordforthisbookisavailablefromtheBritishLibrary ISBN:978-0-12-409538-0 ForinformationonallElsevierpublications visitourwebsiteatstore.elsevier.com PrintedandboundinChina 1314151617 10987654321 Preface ReflectionseismologybelongstoasmallbranchofsolidEarthphysics. Doesithaveitsowntheoreticalsystems?ThisisthequestionthatIkeep considering.Ifonecopieswordbywordtheformulafromelasticorcon- tinuum mechanics to form the theory of reflection seismology, one looks upon reflection seismology as a branch of applied technology, but not asabranchofappliedscience.Eventoday,manygeoscientistsstillregard reflection seismology as engineering technology that surveys mineral exploration.Asreflectionseismologists,wecannotblametheirignorance, but can only blame ourselves for not having built the theoretical system perfectly andcompletely. It is true that reflection seismology was an oil/gas exploration tech- nique at the beginning. An oil field was found according to the seismic reflection data in the continent of Oklahoma in the USA in 1926, which provesthatseismicreflectionisreallyagoodpracticaltechniquehaving enormous economic worth. After the 1930s, based on decades of effort, especiallyafter the invention of computers, thetheory ofreflection seis- mology has been developed rapidly. Some monographs on reflection seismology have been published in the 1970s. However, because the renewal speed of seismic exploration methods has been too fast, these monographs cannot include enough newly developed content on the theory of reflection seismology. On the other hand, textbooks used in universities mostly quote elastic mechanics formulas, and have not absorbed enough newly developed theories on reflection seismology. From 1983 to 2013, I have written a few monographs on geophysical inversion theory and methods that have been my main research area. As the foundation of performing seismic inversion is built on solving the forward problems in reflection seismology, I have devoted my time for summarizing the theory of reflection seismology, merging for- ward and inverse problems into a textbook, and bringing out the best in each other. This book is designed as a textbook for professional graduate stu- dentsmajoringinappliedgeophysics.Itisimpossibletocoveralldevel- opments in reflection seismology theories. I have outlined just the equationsystemsofreflectionseismologybasedonthetheoryofcontin- uum mechanics. The equation family constructs the theoretical frame- work of reflection seismology. I do not want to emphasize the abstract beautyoftheseequations,butinthisbook,Itrytoputupsystematically ix x PREFACE a skeleton that is formed with equations and mathematical formulae, showing the spirit and values of classical physics. Equations and mathematical formulae are the essential language for communication between human beings and computers, applicable to all. One can be an excellent geologist who may not be good at mastering this kind of language, but one cannot be an excellent seismologist who is not good at mastering this kind of language. The corresponding teaching of how to use this textbook takes about 50h.Asteachingmaterials,theauthorsmustconsiderthefollowingthree points and guard against three misleading factors, namely first, write teachingmaterialsasresearchreports,discussalotofdetails,andmislead studentstoignorethebackbone.Second,writethetextbookasacollection of practical techniques that will be updated soon, and do not probe into their theoretical bases deeply. Students tend to listen excitedly, think thatalotofusefulknowledgewouldbeobtained,butignoretheacademic ability and marrow. Third, write the teaching materials as an encyclope- diathatincludeseveryaspectofthesubject,butnotfullysystematically. Books become very voluminous when they provide too much of knowl- edgeforstudents,butbecomeunfavorablefortraininggraduatestudents whohave asystematic thinkingability.Toavoid misleadingthereaders, thisbookadherestothemainidea,anddoesdiscussthemainbranchesin detail; it only explains the theory and not already existing market-based technology. I am not afraid of writing little, but only afraid of writing too much. The backbone must explain clearly, the minor branches may only be clicked. It not only lets students assimilate information but also confers the ability to combine bits of knowledge to solve practical prob- lems in application. This book is divided into 7 chapters as follows: The first chapter reviews the basic wave theory of classical physics, and the second sum- marizeselasticwaveequationsthatbuildupthefoundationofreflection seismology. Particle dynamics used to be improperly applied to explain vibration and wave propagation in the past, and I have tried my best to correctimproperconceptsbyusingcontinuummechanics.Chapterthree summarizesseismicwavepropagationinthesolidEarth,especiallysome special characteristics different from elastic waves in common solid media. Chapter four deals with wave equation changes along with seis- micdataprocessing,togetherwithchangesontherelevantboundarycon- ditions.Chapterfivediscussestheintegralsolutionsofwavepropagation problems and Green’s function methods. Chapter six discusses the decomposition and continuation of seismic wave fields, with emphasis on the properties of wave equations with variable coefficients and the operatorexpansionmethodfortheproblemsunderstudy.Thelastchap- ter briefly introduces inverse problems involved in seismic exploration and typical solving numericalmethods. xi PREFACE IwouldliketothankprofessorsFuChengyiandW.Telfordforguid- ingmeonsolidEarthgeophysics.IalsoappreciatethehelpofDrs.Yang Wuyang, Li Lin, Xie Chunhui, Wang Enli, Wang Wanli, and Zhu Xiaoshan in the translation of this book to English. The author is much indebtedtoMohanapriyanRajendranforimprovingtheEnglishpresen- tationofthisbookandgratefultoallcolleaguesandreaderswhoinform errors left in this book. Wencai Yang National Lab on Tectonics and Dynamics, Institute of Geology, CAGS.PRC C H A P T E R 1 Introduction to the Wave Theory O U T L I N E 1.1. Wave Motion inContinuous Media 2 1.2. Vibration 5 1.3. Propagation and Diffusion 6 1.4. AcousticWave Equation 8 1.5. AcousticWave Equationwith ComplexCoefficients 11 1.5.1. ComplexElastic Modulus and the Complex Wave Velocity 11 1.5.2. DampingWave Equations inViscoelastic Media 13 1.5.3. Viscoelastic Models 14 1.6. AcousticWave Equationwith VariantDensity orVelocity 16 1.7. Summary 18 Physics is the study of the motion of matter. It originates from New- tonian mechanics theory. In classical mechanics, which is established by Newtonintheseventeenthcentury,particledynamicswasfirstemployed to describe the motion of macroscopic objects. However, we must deal withcontinuouslyconstructedmediasuchastheearth;theapplicationof particle dynamics has some limitations. Thus, scientists developed continuum mechanics in the early twentieth century, including fluid mechanicsandsolidmechanics.Quantummechanicshasbeendeveloped in the same period to describe the motion of microscopic particles. The theoryinthisbookisbasedoncontinuummechanicsandthemethodsof mathematical physics. ReflectionSeismology 1 (cid:1)2014PetroleumIndustryPress.Publishedby http://dx.doi.org/10.1016/B978-0-12-409538-0.00001-4 ElsevierInc.Allrightsreserved. 2 1. INTRODUCTIONTOTHEWAVETHEORY Themotionofmacroscopicobjectsingeneralcanbedividedintothree types:Thefirstisdisplacement,suchaslinearmovement,rotation,flight, and flow. The second is vibration and wave motion, such as periodic motion,waterwave,acousticwave,andseismicwave.Thethirdischaotic movement,suchasintermittentmotion,turbulence,andnonlinearwave motion.Thisbookonlydiscussesclassicalvibrationandwavetheory.Asa kind of physical movement, vibration can be described using an initial value problem of ordinary differential equations for a closed system, in whichenergyandinformationdonotgetexchangedbetweenthesystem and the outside world. For an open system, in which energy and infor- mation get exchanged between the system and the outside world, the initialandboundaryvalueproblemsofpartialdifferentialequationsmust beapplied.Thischapterdiscussesthebasicwavetheory,focusingonthe acoustic wave equation andrelatedwave behaviors. To make mathematical formulas clear, we use bold English letters for vectorsor matrices, and normal Greek letters for scalars in thisbook. 1.1. WAVE MOTION IN CONTINUOUS MEDIA Wave motion refers to the propagation of vibration in continuous media, which can bedescribed by using the following formulations: Wave motion¼vibrationþpropagation (in continuousmedia); Vibration¼periodicmotion that anobjectmovesaround its equilibrium point; Propagation¼interaction between thevibration andadjoiningmass grains and diffusionof the vibrationenergy. The above concepts are limited to the situation that the vibration equilibrium point of mass grains is fixed and is stationary during wave propagation; theyareaccuratefor theelastic waves propagatinginsolid media. Generalized waves involve situations in which the vibration equilibriumpointisnotfixedandmovable.Forexample,waterwavesare akindofgeneralizedwaveswithmovingequilibriumpoints.Wedonot discuss generalized waves in thisbook. Three assumptions are usually accepted in the study of continuum mechanics and are as follows (Fung,1977; Spencer, 1980; Du Xun, 1985): 1. Mass motion follows the law of conservation of mass, that isthe first derivative of mass M with respectto time tiszero: dM ¼0 (1.1) dt 2. Theequilibriumpointofmassegrainvibrationinawavefieldisfixed and stable. 3 1.1. WAVEMOTIONINCONTINUOUSMEDIA 3. Vibrationoccurs in continuous media,and the interaction between adjacentmassgrainsfollowsthecontinuityequation,whichdefinesthe continuous media. What kind of media can be defined as continuous media? In other words,whatkindofmechanicallawsarefollowedbycontinuousmedia? Definition: for a small deformation, continuous media are defined by the continuity equation as follows: r¼r0ð1(cid:2)divuÞ (1.2) Inthisequation,rdenotesthedensityofamassgraininthemediawith respecttotimet;r0denotesthedensityofthemassgrainwithrespecttoan initial time t0; u denotes the displacement vector of the mass grain; divu indicatesthedivergenceofthedisplacementvector.Equation(1.2)couldbe derived from Eqn (1.1), which describes the conservation of mass. The productofthegrainvolumeanddensityiscalledthemasselement. SetVasthevolumeofcontinuousmediaandthemassinEqn(1.1)can be expressedas follows: Z M¼ rdV V In Cartesian coordinates, according to the law of the conservation of mass,masscan beexpressedas follows: Z Z M¼ rdV ¼ r0dV (1.3) V V wherer0representsthedensityattheinitialtimet0.Becausedv¼dxdydz anddvo ¼dxo dyo dzo, denotinga determinant as (cid:2) (cid:2) (cid:2) (cid:2) (cid:2) vx vx vx(cid:2) (cid:2)vx0 vy0 vz0(cid:2) (cid:2) (cid:2) J ¼(cid:2)(cid:2) vy vy vy(cid:2)(cid:2) (1.4) (cid:2)vx0 vy0 vz0(cid:2) (cid:2) (cid:2) (cid:2) vz vz vz (cid:2) vx0 vy0 vz0 Wehave dV ¼JdV0 Substituting Eqn (1.4) into Eqn (1.3) yields Z Z rJdV ¼ r dV (1.5) o o o Vo Vo 4 1. INTRODUCTIONTOTHEWAVETHEORY or Z ðrJ(cid:2)r ÞdV ¼0 (1.6) o o Vo Hence,theintegral kernel shouldbeequal to zeroas r0 ¼rJ (1.7) Suppose a small deformation is being generated during the wave motion.Then, (cid:2) (cid:2) (cid:2) (cid:2) (cid:2) (cid:2) (cid:2) (cid:2) (cid:2)(cid:2)vux(cid:2)(cid:2)(cid:3)1 and (cid:2)(cid:2)vux(cid:2)(cid:2)(cid:3)1; etc: vy vz Ifoneignoresthesecond-orderterms,Eqn(1.3)canbetransformedas J ¼1þvuxþvuyþvuz ¼1þdivu (1.8) vx vy vz IfwesubstituteEqn(1.8)intoEqn(1.7),thecontinuousEqn(1.2)canbe obtained. It is true that wave motion involves only small deformations, but ex- plosions involve large deformations. In seismic explorations, explosions areusedasthevibrationsourcestoproduceseismicwaves.Bothvibration and movement of mass grains occur around the shots, where one has to use the theory of explosion that we do not discuss in this book. We will explainthemovementinthefarfieldarea,wheretheequilibriumpoints arefixed andstableand small deformationsareaccepted. ThecontinuityEqn(1.2)impliesthatthedensityofthemasselements may vary during wave motion and that the variation amplitude is pro- portional to the divergence ofthe displacement vector. The movements of the mass grains in nature can be classified into severalkinds,andvibrationandwavemotionaretwoofthem.Thereare manyphysicalbranchesthatdescribethemovementsofthemassgrains, including classical mechanics, continuum mechanics, and nonlinear dy- namics. Classical mechanics is the study of mass motion in free space, usuallywithoutadditionalconstraints,andishelpfulforthestudyofgas motion in a vacuum and the Brownian motion of molecules. Mass movementfollowsNewton’slawsofmotionanduniversalgravitationin an enclosed dynamic system and can be expressed by some ordinary differential equations with initial values. Continuum mechanics studies constrainedmotionincontinuumspace,itisbasedonNewton’sequation of motion and specific constitutive equation, and it can be described by partial differential equations with initial and boundary values. The constitutive equation in elastic mechanics is called the generalized Hooke’s law. Continuity Eqn (1.2) is the starting point of continuum 5 1.2. VIBRATION mechanics, which used to be divided into statics and dynamics. Wave motion isthe resultof some forces, andbelongsto dynamics. 1.2. VIBRATION Vibration can be described as the motion of a mass constraint to an equilibrium point with a limited distance. No matter how the mass oscillates, it will always return to the equilibrium point, due to the di- rection of the force acting on the oscillator always pointing to the equi- libriumpoint.Onlywhenthedirectionofforce(i.e.elasticforceinasolid) isoppositethatofthemovement,thevibrationcanbealwaysaroundthe equilibrium point. Therefore, vibration is a kind of mass motion whose workingforceand displacement arein reversedirections. Inthecaseofone-dimensional(1D)motion,wedenotetheelasticforce as F, andthe displacement of movement isindicated as u; then, Hooke’s law is F¼(cid:2)ku (1.9) where k indicates the elastic coefficient, the negative sign indicates that the direction of the force is opposite to the displacement. Denoting the mass of the oscillator as m, we can apply the equation using Newton’s second law as (Budak et al., 1964) 2 d u m ¼(cid:2)ku¼F (1.10) 2 dt Settingthecircularfrequencyasu,wesubstituteu2 ¼ k intoEqn(1.10) m and obtain 2 d uþu2u¼0 (1.11) 2 dt Theaboveequationisthevibrationequationwithoutdamping,andits general solutionis u¼AcosðutþfÞ (1.12) whereAindicates the amplitudeand findicatesthevibration phase. Any movement will be affected by resistance, and a vibration with resistance is called the damping vibration. Damping comes from the constraints of surrounding media and creates a resistance f , which is r proportionaltothespeedoftheoscillationwhenthespeedisnottoohigh and their directions are opposite. If we denote g as the proportional coefficient, then du f ¼(cid:2)g (1.13) r dt