Introduction to Seismology - Anisotropy

ANISOTROPY So far we have assumed isotropy i.e. wavespeeds do not depend on the direction of wave propagation. We solved the wave equation assuming plane waves: Figure 1 x ηβ P wave S wave ηα z sin i = sin j α β α 〉 β → i〉 j Vp = α VSH = VSV = β Anisotropy however, cannot be ignored as it is the focus of increasing research in seismology. Figure 2a In anisotropic media the wavefronts are distorted (Fig. 2b). The raypaths are not perp. to the wavefronts, therefore the direction of the group velocity is not the same as the direction of the phase velocity. In a homogeneous media there are still three solutions but now they are called quasi P (q P), quasi SH (q SH) and quasi Sv (q Sv). raypath In an isotropic medium wavefronts are concentric circles with radius dependent on the velocity (Vp or Vs) as shown in Fig. 2a. The raypaths are perp. to wavefronts and the slowness vector is perp. to the wavefront. The energy goes with group velocity and the group velocity is perp. to wavefronts. Figure 2b s ~ phase velocity g group velocity Now we look at some basic theory to get insights into the problem and concepts. A full treatment is beyond the scope of this class. We have used the Generalized Hooke’s Law: τ ij = cijkl ε kl τ ij = τ ji → cijkl = c jikl ε kl = ε lk → cijkl = c jilk And from thermodynamics cijkl = cklij These relationships reduce the number of independent elements from 81 to 21 elements. In the isotropic case: cijkl = λδ ij δ kl + µ (δ ik δ jl + δ il δ jk ) Where only the two independent parameters are the Lamé constants, λ and µ. c1111 = c2222 = c3333 = λ + 2 µ c1122 = c1133 = c2233 = λ c1212 = c1313 = c2323 = µ Else = 0 An anisotropic medium is a more complex system and uses symmetries such as: Orthorhombic- (e.g. olivine) with 9 elements Hexagonal- with 5 elements Cubic- (e.g. MgO) with 3 elements Anisotropy is not due to individual minerals but the whole medium or lattice. Lattice preferred orientation (LPO) is the deformation of olivine by plate motion. Hexagonal symmetry is very useful for seismology where there is rotational symmetry around a symmetry axis (not necessarily vertical but often is). z Figure 3 x,y For example a layered system with a vertical symmetry axis which is the most useful for Earth (Fig. 3). In the transverse direction (i.e. direction perpendicular to the symmetry axis) you have isotropy. This is also known as Vertical Transverse Isotropy (VTI). The advantages are: 1. q SH and q Sv can still be treated separately 2. the velocities vary only with incidence angle and not with azimuth If the symmetry axis is horizontal you have azimuthal anisotropy or Horizontal Transverse Isotropy (HTI). For mathematical convenience we introduce a new notation convention: cijkl → C IJ 11 → 1 22 → 2 33 → 3 23 → 4 13 → 5 12 → 6 e.g .c1122 = C12 So for isotropic media: λ λ ⎛ λ + 2µ ⎜ λ + 2µ λ ⎜ λ ⎜ λ λ λ + 2µ C IJ = ⎜ 0 0 ⎜ 0 ⎜ 0 0 0 ⎜ ⎜ 0 0 0 ⎝ 0 0 0 µ 0 0 0 0 0 0 µ 0 0⎞ ⎟ 0⎟ 0⎟ ⎟ 0⎟ 0 ⎟⎟ µ ⎠⎟ The Love convention is used in global seismology (principle axis along x3): C11 = A C33 = C c1133 = C13 = F c2323 = c1313 = C55 = L c1212 = C66 = N c1122 = C12 = C 21 = A − 2N ⎛ A ⎜ ⎜ A − 2N ⎜ F C IJ = ⎜ ⎜ 0 ⎜ 0 ⎜ ⎜ 0 ⎝ A − 2N F A F 0 F C 0 0 0 0 L 0 0 0 0 0 0 0⎞ ⎟ 0⎟ 0⎟ ⎟ 0⎟ L 0 ⎟⎟ 0 L ⎟⎠ 0 0 0 0 The exploration industry convention is similar but used Thomsen parameters ε,γ,δ Looking at wave propagation: τ ij = cijkl ε kl 1 (∂ k ul + ∂ l u k ) 2 τ ij = cijkl ∂ l u k ε kl = The Equation of Motion for a homogeneous medium: ρu&&i = (∇.σ )i = ∂ jσ ij = ∂ (cijkl ∂ l u k ) = cijkl ∂ j ∂ l u k Where i is the only free index. The displacement: u = ge i ω ( t − s . x ) Where: g=polarization vector (in the direction of particle motion) 1 s=slowness vector or 1/phase velocity (direction is k̂ ) c x=position vector s is perpendicular to the wavefront since s.x is unchanged for dx perpendicular to s. If we substitute the displacement into the equation of motion: ρg i = g k cijkl s j sl or (cijkl s j sl − ρδ ik )g k = 0 Where i is the only free parameter. Now, we introduce the density- normalized elastic tensor: cijkl = Γijkl ρ And re-write the Equation of Motion as: (Γ ijkl The Christoffel Matrix: ) sˆ j sˆl − c 2δ ik g k = 0 M ik = Γijkl sˆ j sˆl The Christoffel Equation: (M ik ) − c 2δ ik g k = 0 s 1 ,s = s c The Christoffel Equation is simply an eigenvalue equation. For each direction of s there are three solutions that correspond to the three eigenvalues, c1,c2,c3, and corresponding eigenvectors, g=(g1,g2,g3), of the Christoffel matrix. Where c=phase velocity and ŝ = Now we will look at 2 specific cases. CASE 1 A plane wave in a VTI medium where propagation is in the direction of the symmetry axis. The x3 direction is the axis of symmetry and propagation direction. The displacement is: ⎧ ⎛x ⎞⎫ ui = Ai sin ⎨ω ⎜ 3 − t ⎟⎬ ⎠⎭ ⎩ ⎝ c Christoffel Matrix: ⎛L 0 0⎞ ⎟ 1⎜ M ik = ⎜ 0 L 0 ⎟ ρ⎜ ⎟ ⎝0 0 C⎠ Christoffel Equation: ⎡L 2 ⎢ρ −c ⎢ ⎢ 0 ⎢ ⎢ ⎢ 0 ⎣ 0 L ρ − c2 0 ⎤ ⎥ ⎥ ⎡ g1 ⎤ 0 ⎥ ⎢⎢ g 2 ⎥⎥ = 0 ⎥ ⎥ ⎢⎣ g 3 ⎥⎦ C − c2 ⎥ ρ ⎦ 0 The three solutions are: c1 = L ρ , c2 = g1 = (1,0,0 ) g 2 = (0,1,0 ) g 3 = (0,0,1) L ρ , c3 = c ρ Therefore the velocity in the x3 direction is different from x1 and x2 (which are the same). (In the isotropic case c1 = µ µ λ + 2µ ) ,c 2 = ,c3 = ρ ρ ρ The solutions are degenerate eigenvalues and there is no shear wave splitting. Therefore remember that if you observe no shear wave splitting the cause is either: 1. Propagation along symmetric axis 2. No anisotropy You cannot tell the difference between the two. CASE 2 Propagation in direction x1 and axis of symmetry in x3 direction. ⎧ ⎛x ⎞⎫ ui = Bi ⎨ω ⎜ 1 − t ⎟⎬ ⎠⎭ ⎩ ⎝c Christoffel Matrix: ⎛A 0 1⎜ M ij = ⎜ 0 N ρ⎜ ⎝0 0 0⎞ ⎟ 0⎟ L ⎟⎠ Christoffel Eqution: ⎡A 2 ⎢ρ −c ⎢ ⎢ 0 ⎢ ⎢ ⎢ 0 ⎣ 0 N ρ − c2 0 ⎤ ⎥ ⎥ ⎡ g1 ⎤ 0 ⎥ ⎢⎢ g 2 ⎥⎥ = 0 ⎥ ⎥ ⎢⎣ g 3 ⎥⎦ L − c2 ⎥ ρ ⎦ 0 The solutions are: c1 = A ρ , c2 = g1 = (1,0,0) g 2 = (0,1,0) g 3 = (0,0,1) N ρ , c3 = L ρ The wavespeeds in all three directions are different therefore there is shear-wave splitting(quasi P,SH, and Sv) where three polarization directions will propagate at three different speeds. quasi P α '= A ρ (α = λ + 2µ ,c1111 = λ + 2 µ = C11 A) ρ The quasi P wave is the fastest wave, polarized parallel to the direction of propagation (x1). First quasi S β '= N (C66 = c1212 = N = µ , β = ρ µ ) ρ If N>L the first quasi S wave is associated with the x2 direction. Second quasi S β ' ' = L ρ (C 44 = c2323 = λ = L, c3 = λ ) ρ The second quasi S is the slowest wave and polarized parallel to the x3 direction.