Introduction to Theoretical Seismology - fyzikazeme.sk

1 Comenius UniversityFaculty of Mathematics, Physics and InformaticsDepartment of Astronomy, Physics of the Earth, and Meteorology( KAFZM FMFI UK )Peter MoczoIntroductionto Theoretical SeismologyLecture notes for students of 2006c Peter Moczo 2006 PrefaceThe lecture notes are just transcription of what I originally hand-wrote on transparencies forstudents of the courseTheory of Seismic Wavesat Universit at Wien in 2001. In other words, thematerial was not and is not intended as a standard introductory text on Theoretical material is based on several textbooks, monographs and journal articles. The mainsources are: Aki and Richards (1980, 2002) - chapters 1, 2, and 5; Cerven y and Hron (1980), Cerven y (1985) - chapter 8, and Novotn y (1999) - chapter 6. Though the material is clearlyfar from well elaborated it can be useful for students who want to learn basics of theory want to acknowledge help from Peter Pa z ak as well as technical assistance of Martin Minkaand Lenka Moln arov of ContentsPreface.

2 I1. BASIC RELATIONS OF CONTINUUM MECHANICS.. Introduction .. Body forces.. Stress, traction.. Displacement, strain.. Stress tensor, equation of motion.. Stress - strain relation. Strain - energy .. Uniqueness theorem.. Reciprocity theorem.. Green s function.. Representation theorem..202. SEISMIC SOURCE.. Representation theorems for an internal surface.. Body-force equivalents.. Effective point source.. Moment density tensor.. Effective point source and scalar seismic moment.. Volume Source..333. METHODS OF SOLUTION OF THE EQUATION OF MOTION.. Equations of motion 3D problem.. 1D Problems.. 2D Problems.. Solving equations of motion in the time and frequency domains.. Methods of solving the equation of motion..404. ELASTIC WAVES IN UNBOUNDED HOMOGENEOUSISOTROPIC MEDIUM.. Wave potentials and separation of the equation of equations for P and S waves.. Plane waves.. Harmonic plane wave.. Spherical waves.

3 495. REFLECTION AND TRANSMISSION OF PLANE WAVES AT A PLANEINTERFACE.. Conditions at interface.. Reflection of the plane P and S waves at a free surface.. Reflection and transmission of the plane SH waves at a solid-solid interface.. The case of the critical incidence..576. SURFACE WAVES.. Love waves in a layered halfspace.. Love waves in a single layer over halfspace..637. SEISMIC RESPONSE OF A SYSTEM OF HORIZONTAL LAYERS OVERA HALFSPACE TO A VERTICALLY INCIDENT PLANE SH WAVE.. The case ofnlayers over halfspace.. The case of a single layer over halfspace..678. THE RAY METHOD.. The ray series in the frequency domain.. The ray series in the time domain.. The basic system of equations of the ray method.. The first equations in the basic system.. Rays and ray fields.. Ray parameters.. Ray coordinates.. FunctionJ.. The ray tube.. Relation betweenJand .. Determination of functionJ.. The ray-centered coordinate system.

4 Transport equations.. Solution of transport equations.. Medium with interfaces.. Ray tracing across an interface.. Amplitudes in a medium with interfaces.. Elementary seismogram.. Ray synthetic seismogram.. Elementary and synthetic seismograms - computation in the frequency domain.. Rays in a radially symmetric medium.. Benndorf s equation..90 Appendix..91 Convolution..91 References..95 Index..961. BASIC RELATIONS OF CONTINUUM IntroductionAn application of a force to real object causes some deformation of the object, , change of itsshape. If the deformation is negligibly small, we can work with a concept of a rigid body. Therigid body retains a fixed shape under all conditions of applied forces. If the deformations arenot negligible, we have to consider the ability of an object to undergo the deformation, , itselasticity, viscosity or , we will restrict ourselves to the elastic behavior. For the purpose of the macroscopicdescription both the rigid and elastic bodies can be defined as a system of material particles( not atoms or molecules!)

5 At the same time we assume a continuous distribution of mass a continuum. In a continuum we assign values of material parameters to geometric , we can make use of the theory of continuous value of a material parameter assigned to a geometric point represents an average value forsuch a volume of the material in which the real discontinuous ( atomic or molecular ) structureneed not be a rigid body, relative coordinates connecting all of the constituent particles remain constant, , the particles do not undergo any relative an elastic body, the particles can undergo relative displacements if forces are of continuum usually is used for description of elastic bodies and fluids. The elasticbehavior or objects is a subject of the continuum BASIC Body forcesNon-contact forces proportional to mass contained in a considered volume of a between particles that are not adjacent; , mutual gravitational forces-forces due to the application of physical processes external to the considered volume; ,forces acting on buried particles of iron when a magnet is moving outside the consideredvolumeLet~f(~x, t) be a body force acting per unit volume on the particle that was at position~xat somereference time.

6 An important case of a body force a force applied impulsively to one particleat~x=~ andt= in the direction of thexn-axisfi(~x, t) =A (~x ~ ) (t ) in( )[fi]U=Nm 3,[ (~x ~ )]U=m 3[A]U=Ns,[ (t )]U=s Stress, tractionIf forces are applied at a surfaceSsurrounding some volume of continuum, that volume of con-tinuum is in a condition of stress. This is due to internal contact forces acting mutually betweenadjacent particles within a continuum. Consider an internal surfaceSdividing a continuum intopartAand ~n unit normal vector toS ~F an infinitesimal force acting across an infinitesimal area S force due to materialAacting upon materialB~T(~n) = lim S 0 ~F S( )[~T]U= N m 2~T(~n) traction vector (stress vector) force per unit area exerted by the material in the direction of~nacross the surfaceThe part of~T that is normal to the surface normal stress that is parallel to the surface shear stressTraction depends on the orientation of the surface element Sacross which contract force :VnHnVT(n ) = T(n ) Displacement, strain3 The state of stress at a point has to be described by a Displacement, strainLagrangian description follows a particular particle that is specified by its original position atsome reference time.

7 Eulerian description follows a particular spatial position and thus whateverparticle that happens to occupy that a real seismogram is a record of Lagrangian motion, we will use the Lagrangian of the particleat time torigin0originuxxPosition of the particleat time tDisplacement~u=~u(~x, t) is the vector distance of a particle at timetfrom the position~xof theparticle at some reference timet0.~X=~x+~uis the new position.[~u]U=m ~u t particle velocity, 2~u t2 particle acceleration41. BASIC RELATIONS~ucan generally include both the deformation and rigid body translation and rotation. To analyzethe deformation, we compare displacements of two neighboring particles.~D=~u(~x+~d) ~u(~x)( )Di=ui(xj+dj) ui(xj)( )ui(xj+dj).=ui(xj) +ui,jdj( )(ui,j= ui xj)~u(~x+~d).=~u(~x) + (~d )~u(~x)( )~u(~x+~d) =~u(~x) + u1,1u1,2u1,3u2,1u2,2u2,3u3,1u3,2u3,3 d1d2d3 ( )Di=ui,jdj( )ui,j=12(ui,j+uj,i) +12(ui,j uj,i)( )eij=12(ui,j+uj,i)symmetric tensor( ) ij=12(ui,j uj,i)antisymmetric tensor( )eij= u1,112(u1,2+u2,1)12(u1,3+u3,1)12(u2,1+u1 ,2)u2,212(u2,3+u3,2)12(u3,1+u1,3)12(u3,2 +u2,3)u3,3 ( ) ij= 012(u1,2 u2,1)12(u1,3 u3,1)12(u2,1 u1,2)012(u2,3 u3,2)12(u3,1 u1,3)12(u3,2 u2,3)0 ( )Di=eijdj+ ijdj( )zxConsider 2D case a square in 0.

8 Thenu1,3= u3,1and ij=[0u1,3 u1,30](u2,j=ui,2= 0) Stress tensor, equation of motion5u1,3u = - u1,33,1zxpure rotation (no deformation)Let ij= 0 and assume no volume change. Thenu1,3=u3,1andeij=[0u1,3u1,30]u1,3u = u1,33,1zxshear deformation (no rotation)Generally,12(ui,j uj,i)dj=12 ijk jlmum,ldk=12(rot~u ~d)i( )12rot~urepresents a rigid body rotation if|ui,j| ,eijrepresents deformation. Therefore,eijis called the strain tensor. [eij]U= [ui,j]U= can be also shown by investigating a change of distance between two particles since thechange can be only due to deformation|~d|2=didi( )|~d+~D|2= (di+Di)(di+Di)= (di+ui,jdj)(di+ui,jdj)=didi+ 2ui,jdidj+ui,jui,kdjdk=|~d|2+ui,jdidj+uj ,idjdi+uk,juk,idjdi=|~d|2+ (ui,j+uj,i+uk,iuk,j )didj( )we can neglect since|uk,i| 1|~d+~D|2=|~d|2+ 2eijdidj( )Displacement is a local measure of an absolute change in is a local measure of relative change in position and displacement field due to Stress tensor, equation of motionConsider a volume V with surface rate of change of momentum of particles = forces acting on particles61.

9 BASIC RELATIONS t V ~u tdV= V ~f dV+ S ~T(~n)dS( )Since V and S move with the particles (Lagrangian description), dVdoes not change with timeand t V ~u tdV= V 2~u t2dV( )Consider a particle P inside the volumeVfor which none of acceleration, body force and tractionhave singular value. ShrinkVdown onto P and compare relative magnitudes of the terms inequation ( ). Both the volume integrals are of orderVwhile the surface integral is of orderV23. This means that the surface integral approaches zero more slowly than the volume integraldoes. Then ( )/ S dSleads tolimV 0| S ~T dS| S dS= limV 0O(V13) = 0( )Apply equation ( ) to two caseLetVbe a disc with a negligibly small area of the edgenT(n )s-nT(-n )Equation ( ) limV 0[~T(~n) +~T( ~n)]S2S= 0 ~T( ~n) = ~T(~n)( )2nd caseLetVbe a tetrahedronEquation ( ) limV 0~T(~n).ABC+~T( x1).OBC+~T( x2).OCA+~T( x3).OABABC+OBC+OCA+OAB= 0( ) Stress tensor, equation of motion7 Since~n= (n1, n2, n3);n1=OBC/ABC,n2=OCA/ABC,n3=OAB/ABC( )and~T( xi) = ~T( xi);i= 1,2,3we get from ( ) after dividing it by ABClimV 0~T(~n) ~T( x1)n1 ~T( x2)n2 ~T( x3)n3 ABC+OBC+OCA+OAB= 0and consequently~T(~n) =~T( xj)nj( )Ti(~n) =Ti( xj)njBoth properties ( ) and ( ) are important since they are valid in a dynamic case.

10 (Theirvalidity in a static case is trivial.)Equation ( ) can be written as[T1(~n), T2(~n), T3(~n)] = [n1, n2, n3] T1( x1)T2( x1)T3( x1)T1( x2)T2( x2)T3( x2)T1( x3)T2( x3)T3( x3) ( )Define stress tensor ji ji=Ti( xj)( )[ ji]U=Nm 2 Then ( ) and ( ) can be rewritten asTi(~n) = jinj Cauchy s stress formula( )and[T1(~n), T2(~n), T3(~n)] = [n1, n2, n3] 11 12 13 21 22 23 31 32 33 ( ) jiis thei-th component of the traction exerted by a material with greaterxjacross the planenormal to thej-th axis on material with BASIC RELATIONSE xample:Stress tensor fully describes a state of stress at a given we can apply eq. ( ) to eq. ( ). Eq. ( ) in the index notation is V ui,ttdV= V fidV+ S Ti(~n)dS( )Using eq. ( ) the surface integral becomes S jinjdS= S jidSj( )The surface integral can be transformed into a volume integral using Gauss s divergence theorem S ~a~dS= V div ~a dV S ajdSj= V aj jdV(~ )In our problem, the particles constituting S have moved from their original positions~xat thereference time to position~X=~x+~uat time t.