.50 Introduction to seismology /3/05 sophie michelet Today s class: ) Eikonal equation (basis of ray theory) ) Boundary conditions (Stein s book.3.0) 3) Snell s law Some remarks on what we discussed last class.: When at surface or in a well, we measure the apparent velocity (c, c z ); but this is not a very convenient frame to work in. The eamples below show that c + c z c. Better work with components of slowness vector s (or wave vector k); s + s z = s = /c, and k + k z = k = (ωs) = ω /c. c z c k=(k,k z ) c i k=(k,k z ) c z z A - Arbitrary case z B- Sub-horizontal i C - Subvertical case c D- Limit: horizontal wave propagation k=(k,0) c z z k=(k,k z ) z Case B: c z large & c decreases as the incidence angle i increases Limit: c z -> infinity, so that c + c z c ; but s z and k z well defined Case C: c large & c z decreases as the incidence angle i decreases Limit: c -> infinity; but s and k well defined Case D: k z = ωη =0 + π plane wave superposition: φ(, y,z,t) = φ(,t) π φ(k k z,ω)e ik ωt) ( dk dk dω, z We want to get rid of infinite integration boundaries to make it numerically tractable: ω o + dω - consider band limitation: dω ω o dω k o + dk - select a certain direction (i.e., range of wave vectors) dk k o dk
.50 Introduction to seismology /3/05 sophie michelet The integration over frequencies is important, but for now we will consider the high frequency approimation, which leads to ray theory. We can gain insight into the behavior of the seismic waves by considering the ray paths associated with them. This approach, studying wave propagation using ray path, is called geometric ray theory (GRT). Although it does not fully describe important aspects of wave propagation, it is widely used because it often greatly simplifies the analysis and gives a good approimation. EIKONAL EQUATION eikon means image in Greek. FUNDAMENTAL equation P wave equation: "φ= φ If we consider the following solution, i( k. ωt ) = A()e iξ iωt ( ) φ = A( )e = A()e P wave potential We choose to work at a chosen travel time T, equivalent to a phase If we look at the gradient of φ: iωt φ = u = Ae φ = ( A ω A T i(ω A. T + ω iω T A T ) e iωt & & φ = Aω e φ = φ A ω A ω A T iω[ A. T + A T ] = = Ak ω ω ω Imaginary part: information notation : k =, k =, k β = c β about amplitude Aω A ω A T = Propagation: look at the real part A T = Take the limit: ω (at least for ω Aω A sufficiently large => Aω 0 T = P wave T = General case: c phase velocity c T () = Gradient of T given by local velocity c ( ) T () = kˆ = s k ˆ = s s is the slowness vector (p, η) c ( ) equal phase ~ equal travel time T from origin EIKONAL EQUATION Grad(T)
.50 Introduction to seismology /3/05 sophie michelet What does it mean? Gradient of a wavefront at a position (here defined as the travel time, surface of equal phase) is equal to the local slowness. The direction of maimum change of the wavefront defines the direction of the wave propagation. What are the implications? Rays are perpendicular to wavefronts. The slowness gives the gradient of the travel time, and the gradient of the travel time specifies the direction of the ray. Each time c() changes, the gradient of T has to change, and the direction of propagation changes at the same time. If one knows c(), there is a way to reconstruct the direction of the ray: Æ eikonal ray tracer. Warning: only OK if ω if sufficiently large but does not have to be infinite. A << T for eikonal equation to be valid simplification of the wave Aω equation. There are no fied rules but some conditions of validity eist: Change in wave speed along the ray has to be small i.e. the distance over which C() changes has to be large compared to the wavelength. Curvature must be small (=direction of change in ray path i.e. Grad(T)) must be small compared to the wavelength. Large curvature small curvature Etreme case: reflection i.e. infinite curvature. You can study this as long as you consider really high frequencies Æ infinite frequencies Æ infinitely narrow rays. As soon as one works on a finite frequency case the task gets tough! If one works with low frequency waves (e.g. surface waves), one can only describe smooth variations when using only ray theory. BOUNDARY CONDITIONS There are two types of Boundary Conditions (BC):. Kinematic BC: displacement. Dynamic BC: stresses, traction 3
.50 Introduction to seismology /3/05 sophie michelet The three principal interfaces are the following: solid/solid (or welded interface) (eample: crust/mantle) All displacements + all tractions MUST be continuous solid/liquid (eample: CMB, ocean floor) Normal displacement (i.e Uz) MUST be continuous Normal traction (T z z, yz, zz ) MUST be continuous Tangential displacements need NOT to be continuous Tangential tractions vanish free slip surface free surface All tractions vanish (T z ; z, yz, zz )=0 @ z=0 Displacements are not constrained Eample: a welded interface An incident P wave produces a reflected and a transmitted P wave. BUT at the interface, the transmitted P wave is not sufficient to preserve the vertical component: a SV wave is needed in order to add sufficient displacement in the vertical component and to satisfy the BC P-SV coupled. P U Uz P reflected P transmitted SV needed If an SH incoming wave, the BCs can be satisfied without addition of other displacement components Æ make SH in general much easier than SV (rather, P-SV) case. Steps to follow:. Describe displacement potentials. Look at BCs 3. Write equations for displacements and stress 4. Solve these equations 5. Calculate reflection & transmission coefficient Reminder: we are still in the isotropic case. 4
.50 Introduction to seismology /3/05 sophie michelet COEFFICIENTS: Reflection coefficient R = R I R I Transmission coefficient T = T I T SNELL s LAW Displacement u u = φ + ψ.ψ = 0 Upgoing P i j i* Downgoing P T i = ij n j ij = λδ ij ε kk + µε ij i( k. ωt ) φ(,t) = Ae Downgoing SV k = (k,0, k z ) φ(,t) = A epi{ k + k z z ωt}= A epi{sin i. + cosi.z ωt} Downgoing P (+ sign in front of the cosine) φ(,t) = A epi { sini. cosi.z ωt} Upgoing P u = φ = A.(sini,0, cosi)epi{sini. cosi.z ωt} φ φ P = u = φ =,0, = (u,u y,u ) z z = λ +µ u + u u = λ + µ φ = µ u + u z z = µ z z = 0 yz φ = λ φ + µ z zz Stress tensor 5
.50 Introduction to seismology /3/05 sophie michelet ψ `S V = u = ( ψ,0, ) Downgoing SV z ψ ψ = µ Stress tensor z z = 0 yz ψ zz = µ z `S H = u = (0, u y,0) z = 0 Stress tensor Downgoing SH u = µ y yz z zz = 0 SV and P are coupled & SH is decoupled. Upgoing P wave potential: sini cosi i* φ (,t) = A ep iω..z t P i P j S Downgoing P wave cosi * `φ(,t) = B ep iω sini *. +.z t Downgoing SV wave sin j cos j `ψ(,t) = C ep iω. +.z t β β Total Displacement φ + `φ = φ `ψ =ψ = = 0 free surface: at z=0, τ z =τ zz =0 (normal tractions) z zz z = φz + ψz zz = φzz + ψzz 6
.50 Introduction to seismology /3/05 sophie michelet τ z and τ zz are a sum of the following contributions: since the BC hold for all @ z=0 j ep iω sini. t ;ep iω sin. t ;ep iω sini *. t β sin i. t = sini *. t = sin j β sini = sini * = sin j β = p. t sini = = s = p p is the ray parameter, it is constant for the entire system of rays produced by one incoming ray. 7