Source Mechanisms of Earthquakes Theory and Practice Recent, disastrously large and tsunami-generating earthquakes, from Sumatra in 2004 to Chilein2010andJapanin2011,haveunderlinedtheneedtobetterunderstandtheorigins, dynamics and source mechanisms of earthquakes. This book presents an innovative new approach to studying source mechanisms, combining theory and observation in a unified methodology,witha key focus on themechanics governing fault failures. The authors explain source mechanisms by building from fundamental concepts, such as the equations of elasticity theory, to more advanced problems including dislocation theory, kinematic models and fracture dynamics. The theory is presented in a student- friendly form, using consistent notation throughout and with full, detailed, mathematical derivations that enable students to follow each step. The topics covered include point source models, the seismic moment tensor, the determination of point sources and source dimensions and kinematicextended sources. The later chapters present widely used practical modeling methods for source- mechanism determination, linking clearly to the theoretical foundations. These chapters highlight theprocessing ofdigital seismological data and cutting-edgecomputer methods and are fully supportedwith up-to-date seismograms and illustrative data plots. Providing a unique balance between application techniques and theory, this is an ideal guide for graduate students and researchers in seismology, tectonophysics, geodynamics and geomechanics and a valuable practical resource for professionals working in seismic hazard assessment and seismic engineering. AgustínUdíasisEmeritusProfessorattheUniversidadComplutensedeMadrid(UCM).Heis the author of many papers about seismicity, the seismotectonics of the Azores–Gibraltar regionandthephysicsofseismicsourcesandhasalsowrittenseveraltextbooksincluding PrinciplesofSeismology(Cambridge,1999).ProfessorUdíashasservedasEditor-in-Chief of Física de la Tierra and the Journal of Seismology, and earlier as the Vice-President of theEuropeanSeismologicalCommission.HeisamemberoftheAccademiaEuropeae,the Seismological Society of America and the American Geophysical Union, amongst other societies,andisaFellowoftheRoyalAstronomicalSociety. Raúl Madariaga isEmeritusProfessorofGeophysicsattheÉcoleNormaleSupérieure(ENS) in Paris. During his career he has served as Director of the Seismological Laboratory of theInstitutdePhysiqueduGlobe(IPG)andasDirectoroftheGeologyLaboratoryofENS. ProfessorMadariagaisaFellowoftheAmericanGeophysicalUnionandhasbeenawarded theStefanMullermedaloftheEuropeanGeophysicalSocietyandalsotheH.F.ReidMedal, thehighestawardoftheSeismologicalSocietyofAmerica.Hehasbeenaneditorofseveral journals as well as a member of the US science Board and he is the author of 140 papers inleadingscientificjournalsaswellasseveralarticlesinearthscienceencyclopaedias. Elisa Buforn is a Professor of Geophysics at UCM, teaching courses on geophysics, seismology, physics and numerical methods. She has published papers on topics ranging from source fracture processes and seismicity to the seismotectonics of the Ibero– MaghrebianregionandtheAzores–Gibraltarregionandistheauthorofvarioustextbooks including Solved Problems in Geophysics (Cambridge, 2012). Professor Buforn currently servesasEditor-in-ChiefofFísicadelaTierraandisontheEditorialBoardoftheJournal ofSeismology.Shehasservedonmanyinternationalscientificcommitteesandisamember of the Seismological Society of America, the American Geophysical Union, the Royal Astronomical Society and other distinguished societies. Source Mechanism of Earthquakes: Theory and Practice AGUSTÍN UDÍAS UniversidadComplutensedeMadrid RAÚL MADARIAGA ÉcoleNormaleSupérieure,Paris ELISA BUFORN UniversidadComplutensedeMadrid UniversityPrintingHouse,CambridgeCB28BS,UnitedKingdom PublishedintheUnitedStatesofAmericabyCambridgeUniversityPress,NewYork CambridgeUniversityPressispartoftheUniversityofCambridge. ItfurtherstheUniversity’smissionbydisseminatingknowledgeinthepursuitof education,learningandresearchatthehighestinternationallevelsofexcellence. www.cambridge.org Informationonthistitle:www.cambridge.org/9781107040274 ©AgustínUdías,RaúlMadariagaandElisaBuforn2014 Thispublicationisincopyright.Subjecttostatutoryexception andtotheprovisionsofrelevantcollectivelicensingagreements, noreproductionofanypartmaytakeplacewithoutthewritten permissionofCambridgeUniversityPress. Firstpublished2014 PrintedintheUnitedKingdombyXXXX AcataloguerecordforthispublicationisavailablefromtheBritishLibrary LibraryofCongressCataloguinginPublicationdata UdíasVallina,Agustín,author. Sourcemechanismsofearthquakes:theoryandpractice/AgustínUdías,UniversidadComplutensede Madrid(Spain),RaúlMadariaga,Ecolenormalesuperieure,Paris(France),ElisaBuforn,Universidad ComplutensedeMadrid(Spain). pages cm ISBN978-1-107-04027-4(Hardback) 1. Faults(Geology) 2. Seismiceventlocation. 3. Shear(Mechanics) 4. Geodynamics. 5. Seismic prospecting. I. Madariaga,Raúl,author. II. Buforn,Elisa,1954–author. III. Title. QE539.2.S37U352014 551.2201–dc23 2013049910 ISBN978-1-107-04027-4Hardback CambridgeUniversityPresshasnoresponsibilityforthepersistenceoraccuracyof URLsforexternalorthird-partyinternetwebsitesreferredtointhispublication, anddoesnotguaranteethatanycontentonsuchwebsitesis,orwillremain, accurateorappropriate. Contents Preface page ix 1. Earthquakes and fault motion 1 1.1. The originof earthquakes 1 1.2. Faultsin theEarth’scrust 2 1.3. Geometry ofa fault 5 1.4. Elasticrebound and theearthquake cycle 7 1.5. Energy,stressdrop and seismicmoment 9 1.6. Stick–slip motion ona fault 14 1.7. Seismicity and statistical properties ofearthquakes 19 2. Processing and analysis of recorded seismic signals 22 2.1. Recorded and ground motion intime and frequencydomains 22 2.2. Analogical anddigital data 25 2.3. Removing theinstrumental response 27 2.4. Processinga seismic signal 30 2.5. Displacement,velocityand acceleration 36 2.6. Continuous GPSobservations 39 3. Mathematical representation of the source 41 3.1. Fundamental equationsof motion foran elastic medium 41 3.2. Green’s and Somigliana’s tensors 44 3.3. Representation theorem 45 3.4. Somigliana’stensor foran infinite homogeneous isotropic medium 48 3.5. Green’s tensor foraninfinite homogeneous isotropic medium 52 3.6. Green’s function forlayered media 57 3.7. Focal region 59 3.8. Kinematic and dynamicmodels 61 4. Point source models 63 4.1. Point source approximation 63 4.2. Equivalent forces. Double couple 63 4.3. Shear fracture or dislocation 66 4.4. Point shear fracture inaninfinite medium 69 4.5. The geometry of a shear fracture 77 4.6. Far-field displacements referred tothegeographical axes 78 v vi Contents 4.7. Source representationusing the focal sphere 80 4.8. The source time function 81 4.9. Spectral properties ofthe source time function 86 4.10. Seismic energyradiation 88 5. The seismic moment tensor 90 5.1. Definition of themomenttensor 90 5.2. Eigenvaluesand eigenvectors 93 5.3. Components of themomenttensor 94 5.4. Moment tensor and elastic dislocations 95 5.5. Moment tensor components referredto geographical axes 98 5.6. The point source momenttensor 99 5.7. The separation ofthe moment tensor 102 5.8. Higher-ordermomenttensors 105 5.9. Moments ofthe moment-rate distribution 106 6. Determination of point source mechanisms 108 6.1. Parameters andobservations 108 6.2. The focal sphere 109 6.3. Fault-plane solutionsfrom P wave polarities 110 6.4. Inversion ofbody wave forms 116 6.5. Empirical Green’sfunctions 123 6.6. Moment tensor inversion 124 6.7. Centroid momenttensorinversion 132 7. Kinematics of extended sources 135 7.1. Source dimensions 135 7.2. Rectangular fault. Haskell’smodel 138 7.3. Bilateral rupture propagation 141 7.4. Oblique rupture propagation 142 7.5. Corner frequency 145 7.6. Directivityeffects 148 7.7. Rupture nucleation,propagation andarrest 151 7.8. Kinematic models with variable slip onthe fault plane 155 7.9. The circular kinematicmodel ofSato and Hirasawa 157 8 Determination of source dimensions 163 8.1. Parameters ofkinematic extendedsource models 163 8.2. Analysis of seismic wave spectra 163 8.3. DirectivityeffectsofRayleigh waves 167 8.4. Effectsof directivity on body wave form modeling 170 8.5. Apparent source time function 172 8.6. Far-fieldinversion of theslipdistribution on the fault plane 173 8.7. Kinematic inversion ofthe near-fieldwaves 181 vii Contents 9. Simple dynamic models 189 9.1. Kinematic and dynamicmodels 189 9.2. Static problem 190 9.3. Modes ofpropagating fractures 191 9.4. Circularfault.Staticmodel 192 9.5. Circularfault.Brune’s model 195 9.6. Scalinglaws 200 10. Dynamics of fracture. Homogeneous models 205 10.1. Griffith’sfracturemodel 205 10.2. Energyflowtowards thefracturefront for a growing crack 209 10.3. Stresssingularitiesaround therupture front moving at constant speed 212 10.4. Spontaneous shear-fracture propagation 216 10.5. Frictionmodelsoffracture 220 10.6. Rupture ofanexpanding circular fault 223 10.7 Far-field displacements of a dynamic circular fault 227 11. Fracture dynamics. Heterogeneous models 232 11.1. The cohesive zone 232 11.2. The slip-weakening friction model 233 11.3. Friction laws 237 11.4. Determinationof G from seismic data 243 c 11.5. Nucleation and arrestof rupture 246 11.6. Barriers and asperities 250 11.7. Healing and rupturepulse propagation 253 11.8. Super-shearrupture velocity 254 12. Modeling earthquakes using fracture dynamics 259 12.1. Dynamic models 259 12.2. Modelingearthquakes inthree dimensions 260 12.3. Rupture propagation on a planar uniform fault 263 12.4. Afinite circular fault ina homogeneous medium 265 12.5. Shallowstrike-slip rectangular fault 269 12.6. Spontaneous rupture on a realistic fault: theLanders1992 earthquake 270 12.7. Rupture ofa geometricallycomplex earthquake:theIzmitevent of August 1999 275 12.8. Dynamic inversion ofthe 2008 Iwate intermediate-depth earthquake 276 12.9. Conclusion andperspectives 281 References 284 Index 300 Preface A key problem in seismology is the study of the processes that give rise to earthquakes. It is well established now that earthquakes, with rare exceptions, are caused by shear fractureonpreexistingfaultsintheEarth.Thus,themodernstudyofearthquakesandtheir source mechanisms should be based on the application of dislocation theory and fracture mechanics; this is the point of view taken in this book. General textbooks of seismology have dedicated chapters to this subject, for example, Aki and Richards (1980); Ben Menahem and Singh (1981); Dahlen and Tromp (1998), Gubbins (1990), Lay and Wallace (1995) and Udías (1999). A few books have been written specifically on the subject,forexample,Kasahara(1981),KostrovandDas(1988)andScholz(1990).Eachof these three books has a different approach: Kasahara presents the state of the study of earthquakemechanismsasitwas30yearsago,beforemoderndevelopments,anddoesnot givedetailsofthemathematicaldevelopments.Scholz’approachisthatofrockmechanics, with an emphasis on qualitative descriptions and applications to Earth faulting. Finally, KostrovandDas(1988)givesanexcellentpresentationofearthquakedynamics,butitmay bedifficultforstudentstofollow.Noneofthesebooksincludesadetailedpresentationand discussion of practical methods for the determination of the earthquake mechanisms togetherwiththe theory on which they are based. At present it is very common to find, in publications and on the web, results from differentmethodsofdeterminationofthesourcemechanismsofearthquakes,forexample, centroidmomenttensor(CMT)solutions.Theuseoftheseresultsistodaystandardpractice inthestudyoftectonicsandseismichazard,oftenwithoutknowledgeofthemethodsand theory behind them. Thus the novel approach of this book is to present the theory of the source mechanisms of earthquakes and also the methods used for their determination in particular cases. The theory on which the methods are based is made clear. As nowadays digital data are used in the analysis of observations, the practical processing of seismo- grams, accelerograms and high-rate GPS data is also presented. A further novelty is the structureofthebookasatextbook,thatis,havinginminditsusebystudents.Thustopics are presented in a gradual form, starting from the basic principles and developments and progressing to more advanced and difficult problems. In many cases mathematical devel- opmentsaregivenindetail,andthroughoutthebookaunifiedterminologyandnotationare used.Thisisanimportantpointsinceinarticlesdifferentauthorsusedifferingnotation,and thismakesitdifficult,sometimes,togetaunifiedgraspofthesubject.Aboveall,consider- able efforts have been made towards clarity in the presentation and an emphasis on the fundamentals;thereforesomemoreadvancedanddifficultproblemshavebeenomitted. Thebookisdividedinto12chapters.Anintroduction,inwhichverybasicconceptsare explained(Chapter1),isfollowedbyapresentationofthebasisofdigitaldataprocessing, ix