Seismology. - GeoWeb

1 HookeHooke s Law (or )1760 MitchellRecognitionthatgroundmotiondueto earthquakesis relatedtowave propagation1821 NavierEquationofmotion1828 PoissonWave equation P&S-waves1885 RayleighTheoreticalaccountsurfacewaves Rayleigh&Love waves1892 MilneFirsthigh-qualityseismograph beginofobservationalperiod1897 WiechertPredictionofexistenceofdensecore (basedonmeteorites Fe-alloy)1900 OldhamCorrectidentificationofP,Sandsurfa cewaves1906 OldhamDemonstrationofexistenceofcorefrom seismicdata1906 GalitzinFirstfeed-backbroadbandseismogra ph1909 Mohorovi ci cCrust-mantleboundary1911 LoveLove waves(surfacewaves)1912 GutenbergDepthtocore-mantleboundary.

2 2900km1922 Turnerlocationofdeepearthquakesdownto600 km(butlocatedsomeat 2000km, )1928 WadatiAccuratelocationofdeepearthquakes Wadatai-Benioff zones1936 LehmanDiscoveryofinnercore1939 Jeffreys&BullenFirsttravel-timetables 1 DEarthmodel1948 BullenDensityprofile1977 Dziewonski&Toks ozFirst3 Dglobalmodels1996 Song&RichardsSpinninginnercore?Observati ons:1964 ISC(InternationalSeismologicalCentre) travel timesandearthquake locations1960 WWSSN(WorldwideStandardizedSeismographNe twork) (analogrecords)1978 GDSN(GlobalDigitalSeismographNetwork) (digitalrecords)1980 IRIS(IncorporatedResearchInstitutesforSe ismology) ;wecansimplynotoffera comprehensive willdiscusssomebasictheorytoshow howexpressionsforthepropagationofelastic waves,suchasPandSwaves, ,let s lookat a verybrief andincomplete overview ofthehistoricaldevelopmentofseismology.

3 Modernseismologyis characterizedbyalternationsofperiodsin whichmoreprogressismadeintheorydevelopme ntandperiodsinwhichtheemphasisseemstobem oreondatacollectionandtheapplicationofex istingtheoryonnew and often s goodtorealizethatobservationalseismology didnotkickoff untillatelastcentury( ).Priortothat seismology waseffectivelyrestrictedtothedevelopment ofthetheoryofelasticwave propagation,whichwasa ,seeattachmentabove table(thishistoricaloverview is bynomeanscompletebutit doesgive anideaofthedevelopmentsofthoughts).Lay& Wallace(1995)give theirview onthecurrentswingoftheresearchpendulumin thefollowingtables(withsourcerelatedissu eslistedontheleftandEarthstructuretopics ontheright) (latitude,longitude,depth)(crust,mantle, core) (magnitude,seismicmoment) (earthquake, explosion,other) ,structure(area,displacement) (fluid,solid) willdiscusssome classical conceptsandalsodiscusssomeofthemore current todealwithsomebasictheory.

4 Inprinciple,whatweneedis aformulationoftheseismicsource,equations todescribeelasticwave propagationoncemotionhasstartedsomewhere ,anda willconcentrateontheformertwo ,andwewillthereforehave (seismos),earthquake and (logos), , earthquakeseismology is (crust,mantle,core?) , , TERMINOLOGYF ormostofthederivationswewillusetheCartes iancoordinatesystemanddenotethepositionv ectorwitheither or . Thedisplacementofa particleatposition andtime isgivenby , thisis thevectordistancefromitspositionat someprevioustime (Lagrangiandescriptionofmotion)Theveloci tyandaccelerationoftheparticlearegivenby "!

5 And# $ "! respectively. Volumeelementsaredenotedby%'&andsurfacee lementsby(*). Body(ornon-contact) forces,suchasgravity, arewrittenas+andtractionsby,. Atractionis thestressvectorrepresentingtheforceperun itareaacrossaninternalorientedsurface( -. withina continuum,andthisis,infact,thecontactfor ce wave equationis .! .! or# /' , whichisa differentialequationdescribingthepropaga tionofa displacementdisturbance withspeed .We willseethatthefundamentaltheoryofwave propagationis primarilybasedontwo equations:Newton s secondlaw (021 354768354:9<; =?>@9<;BA) andHooke sconstitutivelaw153 DCFE =(statingthattheextensionofanelasticmate rialresultsina restoringforce1, withEtheelastic(spring)constant(notwave speedasintheboxabove!)))

6 Inonedimension,Hooke s lawcanalsobeformulatedastheproportionali tybetweenstressGandstrainH, withproportionalityfactorIisYoung s modulus:GJ3 KILH. We willseethatthislinearrelationshipbetween stressandstraindoesnotholdin2 Dor3D,inwhichcaseweneedtheso-calledgener alizedHooke s Law. For01M3D476wehave toconsiderboththenon-contactbodyforces,s uchasgravitythatworksona certainvolume, ( stressvectors ).We thereforehave tolookinsomedetailat gettheidea,let s considerthedefor-mationofa lineelementN displacedtoPQ P andPQSUPtoPQSUPQ PQ SP) andN ObecomesN;.ThestraininthePdirection,H , canthenbedefinedasH 3N;C NON O3 PRQSP ?

7 C P SP( )If weassumethatSPis smallwecanlinearizetheproblemaroundthe referencestate P byusinga Taylorexpansionon PQTSUP PRQTSUP 3 P Q 9 9<P SUPQ SP; P Q 9 9P SP( )sothatH 3 9 P 9<P 3 9 P 9PQ9 P 9<P ( )whichrepresentsthenormalstraininthex andH (etc),whichinvolve isH 3 9 P 9<P! Q9 P! " 9<P# %$3 9 9<P! Q9 9P &$3 ' 9 9P Q9 9P( )$3H ( )withnormalstrainsfor*?3,+andshearstrain sfor*.-3,+. (Inthisdiscussionofdeformationwedonotcon sidertranslationand/orrotationofthemater ialitself).Equation( )showsthatthestraintensoris symmetric,sothattherethemaximumnumberofd ifferentcoefficientsis forceperunitarea,andtheprincipleunitis Nm/ (orPascal: 1Nm/ = 1Pa).

8 Similartostrain,wecanalsodistinguishbetw eennormalstress, theforce110perunitareathatis perpendiculartothesurfaceelementS32, andtheshearstress, whichis theforce154perunitareathatisparalleltoS3 2( ).Theforce1actingonthesurfaceelementS32c anbedecomposedintothreecomponentsinthedi rectionofthecoordinateaxes:153 76O 86;86 9 . Wefurtherdefinea unitvector:;normaltothesurfaceelementS<2 . Thelengthof:;is,ofcourse,=:;=3 . vectorthatrepresentsthetotalforceperunit areaonS< , alsothetraction canbedecomposedinto 3 O 8 ;8 9 3 O POQ ;P;Q 9P9. Thetraction obtaina moreusefuldefinitionofthetraction in termsofelementsofthestresstensorconsider a is orthogonaltoP ; thefourthsurface,S32, usethefollowingnotationconvention:thecom ponentofthestressthatworksontheplane 5 POinthedirectionofP isGO , force1thatworksonS32mustbecancelledbyfor cesactingontheotherthreesurfaces:06 3 S C5GO OC5G; ;CG9 93 sothat S 3GO OQ G; ;Q G9 9.

9 We know thattheexpressionweareaftershouldnotdepe ndonourchoiceof noronS (sincetheformerwerejustchosenandthelatte risarbitrary).Thisiseasilyachievedbyreal izingthatS and arerelatedtoeachother: isnothingmorethantheorthogonalprojection ofS ontotheplaneperpendiculartotheprincipala xisP : F3 S , with theanglebetween:;, thenormaltoS , andP . But is infactsimply sothat 3 S . (We have usedthisbeforewhenwedecomposedthemomento finertia aroundanarbitraryaxis intothemomentsofinertiaaroundtheprincipa laxes).Usingthisweget: S 3GO ?O S QG; ;S QG9 9S ( )or 3GO ?OQG; ;QG9 9( )Thus:the* componentofthetractionvector is givenbya linearcombinationofstressesactinginthe* directiononthesurfaceperpendiculartoP) (orparallelto ), where+R3 8 8"!

10 ; 3G! # ( ) , anelementG! ofthestresstensorisdefinedasthe* componentofthetractionactingonthesurface perpendiculartothe+ axis(P( ):G# 3 P( & ( )The9 componentsG! ofalltractionsformtheelementsofthestress tensor:G# 3 GO OGO;GO9G;OG; ;GO;G9OG9;GO9 3 GO OGO;GO9G; ;GO;GO9 ( )Thenormalstressesarerepresentedbythedia gonalelements(i=j)andtheshearstressesare theoff diagonalelements(*-3+). It canbeshownthatinabsenceofbodyforcesthest resstensorissymmetricG# 3G sothatthereareonly6 candiagonalizethestresstensorbychangingo urcoordinatesysteminsucha waythattherearenoshearstressesonthesurfa cesperpendiculartoany 3 GO O G; ; GO9 3 GO G; GO ( )Somecasesareofspecialinterest: uni-axialstress: onlyoneoftheprincipalstressesis non-zero, ,GO-3 ,G;3G93 planestress: onlyoneoftheprincipalstressesiszero, ,GO3 ,G;8G9-3 pure shear:G93 ,GO3 JCFG; isotropic(or, hydrostatic)stress:GO32G;3G93 ( 3O9 GOQG.)))