Introduction to Seismology - Eikonal Equation

Today’s class: 1) Eikonal equation (basis of ray theory) 2) Boundary conditions (Stein’s book 2.3.10) 3) Snell’s law Some remarks on what we discussed last class.: • When at surface or in a well, we measure the apparent velocity (cx, cz); but this is not a very convenient frame to work in. The examples below show that cx2 + cz2 ≠ c2. Better work with components of slowness vector s (or wave vector k); sx2 + sz2 = s2 = 1/c2, and kx2 + kz2 = k2 = (ωs)2 = ω2/c2. cx cx x x i k=(kx,kz) k=(kx,kz) cz cz z z A - Arbitrary case B- Sub-horizontal C - Subvertical case D- Limit: horizontal wave propagation x k=(kx,0) cx x i cz z k=(kx,kz) z Case B: cz large & cx decreases as the incidence angle i increases Limit: cz -> infinity, so that cx2 + cz2 ≠ c2; but sz and kz well defined Case C: cx large & cz decreases as the incidence angle i decreases Limit: cx -> infinity; but sx and kx well defined Case D: kz = ωη =0 • plane wave superposition: φ (x, y,z,t) = φ (x,t) ≈ +∞ ∫ π ∫ ∫ φ (k π − k ,ω )e i(k x−ωt )dk x dk z dω x, z −∞ We want to get rid of infinite integration boundaries to make it numerically tractable: ω o + dω 1- consider band limitation: ∫ dω ω o − dω ko + dk 2- select a certain direction (i.e., range of wave vectors) ∫ dk ko − dk 1 • The integration over frequencies is important, but for now we will consider the high frequency approximation, 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 approximation. EIKONAL EQUATION • • “eikon” means image in Greek. FUNDAMENTAL equation P wave equation: "φ=α2∇2φ If we consider the following solution, φ = A( x)e i( k.x−ωt ) = A(x)e iξ = A(x)e− iωT (x ) 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: ∇φ = u = ∇Ae − iωT ( )) ( 2 equal phase ~ equal travel time T from origin ∇ 2φ = ∇ 2 A − ω 2 A ∇T − i 2ω∇A.∇T + ωA2T e −iωT φ&& = − Aω 2e −iωT → φ&& = α 2∇ 2φ [ 2 ] ⇒ ∇ 2 A − ω 2 A ∇T − iω 2∇A.∇T + A∇ 2T = − notation : kα = ω ω ω , k = , kβ = c α β 2 ∇ 2 A − ω 2 A ∇T = − 1 2 ∇T − α2 ∇T = ∇2 A Aω 2 1 2 ∇T = = α2 1 c Aω 2 α2 Aω 2 α2 = − Akα2 Imaginary part: information about amplitude Propagation: look at the real part Take the limit: ω→∞ (at least for ω sufficiently large => ∇2A Aω 2 →0 P wave General case: c phase velocity 1 Gradient of T given by local velocity c( x) 1 ∇T (x) = k̂ = s kˆ = s s is the slowness vector (p, η) c( x) ∇T (x) = Grad(T) EIKONAL EQUATION 2 What does it mean? Gradient of a wavefront at a position x (here defined as the travel time, surface of equal phase) is equal to the local slowness. The direction of maximum 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(x) changes, the gradient of T has to change, and the direction of propagation changes at the same time. • If one knows c(x), 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. ∇2A 1 2 << ∇T − 2 for eikonal equation to be valid simplification of the wave → 2 Aω α equation. There are no fixed rules but some conditions of validity exist: • • Change in wave speed along the ray has to be small i.e. the distance over which C(x) 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 Extreme 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): 1. Kinematic BC: displacement 2. Dynamic BC: stresses, traction 3 The three principal interfaces are the following: • solid/solid (or welded interface) (example: crust/mantle) All displacements + all tractions MUST be continuous • solid/liquid (example: CMB, ocean floor) Normal displacement (i.e Uz) MUST be continuous Normal traction (Tz – σxz,σyz,σzz) MUST be continuous Tangential displacements need NOT to be continuous Tangential tractions vanish – free slip surface • free surface All tractions vanish (Tz; σxz,σyz,σzz)=0 @ z=0 Displacements are not constrained Example: 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 Ux P reflected Uz 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: 1. Describe displacement potentials 2. 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 COEFFICIENTS: Reflection coefficient R= ⊗I ⊗R ⊗I ⊗R ⊗T Transmission coefficient T = ⊗ I ⊗T SNELL’s LAW Displacement u u = ∇φ + ∇ × ψ i* ∇.ψ = 0 Upgoing P i Ti = σ ij n j Downgoing P j Downgoing SV σ ij = λδijεkk + 2µεij φ (x,t) = Ae i( k.x−ωt ) k = (k x ,0, k z ) φ (x,t) = A expi{k x x + k z z − ωt} = A expi{sin i.x + cosi.z − ωt} φ (x,t) = A expi{sini.x − cosi.z − ωt} Upgoing P u′ = ∇φ = A.(sini,0,−cosi)expi{sini.x − cosi.z − ωt} ⎛ ∂φ ∂φ ⎞ P′ = u′ = ∇φ = ⎜ ,0, ⎟ = (ux ,uy ,uz ) ⎝ ∂x ∂z ⎠ ⎛ ∂u ∂u ⎞ ∂u σ xx = λ∆ + µ⎜ x + x ⎟ = λ∆ + 2µ x ⎝ ∂x ∂x ⎠ ∂x 2 ⎛ ∂u ∂u ⎞ ∂φ σ xz = µ⎜ x + z ⎟ = 2µ ∂x∂z ⎝ ∂z ∂x ⎠ σ yz = 0 σ zz = λ∇ 2φ + 2µ Stress tensor ∂ 2φ ∂z 2 5 Downgoing P (+ sign in front of the cosine) ∂ψ ∂ψ Downgoing SV ,0, ) ∂z ∂x ⎛ ∂ 2ψ ∂ 2ψ ⎞ σ xz = µ⎜ 2 − 2 ⎟ Stress tensor ∂z ⎠ ⎝ ∂x σ yz = 0 `SV = u = (− σ zz = 2µ ∂ 2ψ ∂x∂z `SH = u = (0, uy ,0) σ xz = 0 ∂u σ yz = µ y ∂z σ zz = 0 Downgoing SH Stress tensor SV and P are coupled & SH is decoupled. Upgoing P wave potential: ⎧ ⎧ sini ⎫⎫ cosi .x − φ ′(x,t) = A exp⎨iω ⎨ .z − t⎬⎬ ⎭⎭ α ⎩ ⎩ α i* P Downgoing P wave ⎧ ⎧ sini * ⎫⎫ cosi * .x + .z − t⎬⎬ `φ (x,t) = B exp⎨iω ⎨ ⎭⎭ α ⎩ ⎩ α i P j S Downgoing SV wave ⎧ ⎧ sin j ⎫⎫ cos j .x + .z − t⎬⎬ `ψ (x,t) = C exp⎨iω ⎨ β ⎭⎭ ⎩ ⎩ β Total Displacement φ ′ + `φ = φ `ψ = ψ free surface: at z=0, τzx=τzz=0 (normal tractions) σ zx = σ zz = 0 σ zx = σ φzx + σψzx σ zz = σ φzz + σψzz 6 τzx and τzz are a sum of the following contributions: since the BC hold for all x @ z=0 ⎧ ⎧ sin j ⎫⎫ ⎧ ⎧ sini ⎧ ⎧ sini * ⎫⎫ ⎫⎫ exp⎨iω ⎨ .x − t⎬⎬;exp⎨iω ⎨ .x − t⎬⎬ .x − t⎬⎬;exp⎨iω ⎨ ⎭⎭ ⎭⎭ ⎩ ⎩ α ⎩ ⎩ α ⎭⎭ ⎩ ⎩ β ⎫ ⎧ sin i ⎫ ⎧ sini * ⎫ ⎧ sin j .x − t ⎬ = ⎨ .x − t ⎬ = ⎨ ⇒⎨ .x − t ⎬ ⎩ α ⎭ ⎩ α ⎭ ⎩ β ⎭ ⇔ sini sini α α = = 1 αx sini * α = sin j β =p = sx = p p is the ray parameter, it is constant for the entire system of rays produced by one incoming ray. 7