Snell's Law in Spherical Media

TODAY’S LECTURE 1. Snell’s law in spherical media 2. Ray equation 3. Radius of curvature 4. Amplitude → Geometrical spreading 5. τ – p SNELL’S LAW IN THE SPHERICAL MEDIA i1 A c1 At each interface c2 sin i1 sin j = c2 c1 i2 B j OQ OQ sin i2 = OA OB r OB sin j = sin i2 = 2 sin i2 OA r1 sin j = Q r2 r1 r1 sin i1 r2 sin i2 = ≡p c2 c1 O p= “flat earth” → At critical angle, sin i c p= “spherical earth” → rp c( rp ) p= r sin i c we can get depth of layer. RAY EQUATION Directional cosine (3D and 2D) s1 s2 dz i ds ( dx dx1 2 dx ) + ( 2 )2 + ( 3 )2 = 1 ds ds ds ( dz dx 2 ) + ( )2 = 1 ds ds ∧ Direction of ray ( n ) dx ∧ n n = (n x ,0, n z ) 1 nx = dx ds nz = dz ds 1∧ n, c Generalized Snell’s law (Ray Equation) Using Eikonal equation ∇T = d 1 d 1 dxi ) ( )= ( ds c( x) ds dxi c( x) This equation means that the change of wavespeed is related to change of ray geometry. If there is no change in x direction, the derivative of x direction should be zero. d 1 dx ( )=0 ds c(x) ds ⇒ 1 dx = Const. c ds ⇒ sin i = Const. C ⇒ Snell’s law !! How does this angle i change in the direction of propagation? d di dz di d (sin i) = cos i = = ( pc) ds ds ds ds ds di ( s ) dc =p ds dz ⇒ Therefore, the change of angle is related to the change of velocity. If dc is large dz If dc is zero (c = const.) dz di is large ds ⇒ di is zero (i = const.) ds ⇒ Straight Ray !! RADIUS OF CURVATURE R : the radius of curvature ds = Rdi R= di dz i ds dx R ds 1 dz = = di p dc 1 dc p( ) dz ⇒ R= 1 dc p( ) dz R is related to wavespeed gradient and ray parameter. If dc =0 dz 2 ⇒R → ∞ Straight Ray !! If dc large ⇒ rapid change in c Strong Gradient dz from p = r sin i , c small i → small p → large R i AMPLITUDE-GEOMETRICAL SPREADING Focusing-defocusing Shadow Zone Defocusing effect Focusing effect We examine the property of dp / dx dp d dT d 2T ( )= 2 = dx dx dx dx Small dx and large dp → dp / dx goes to infinity → large amplitude (focusing) Large dx and small dp → dp / dx goes to zero We also examine x( p ) x T =2 i c ds → small amplitude (Shadow zone) h 3 ds c tan i = x/2 h One layer : x = 2h tan i Multiple layers : x=2 n ∑h j =0 j tan i j Continuous case zp zp zp zp 1 dz dz x( p ) = 2 ∫ tan idz = 2 p ∫ ( − p 2 ) −1/ 2 dz = 2 p ∫ = 2p∫ 2 η 1/ c 2 − p 2 0 0 c( z ) 0 0 z zp zp ⎫⎪ ⎫ ⎧⎪ p d 2 c ⎫⎪ dx ⎧ 1 dx dz d ⎧⎪ dz ≈ ⎨− = 2∫ + 2p ⎨∫ ⎬ + ⎨+ ∫ 2 dz ⎬ ⎬⇒ 2 2 2 2 dp ( dz dc / ) dp dp ⎪⎭ c p c p 1 / − 1 / − ⎪ ⎪ 0 ⎭ ⎪⎩ 0 dz ⎩ 0 ⎩0 ⎭ The change of distance in terms of ray parameter is related to gradient of wave speed at surface and gradient of the change in wavespeed between surface and turning point. Changes of velocity gradient, parameter p, d 2c , are small → large distance x for smaller ray dz 2 dx < 0 → “Normal” or Prograde behavior dp T c(z) dx <0 dp z Δ 4 Depth Intercept time (�) Distance (�) Velocity Time Ray parameter (p) Ray parameter (p) Distance (�) Distance (�) Figure by MIT OCW. This figure represents ray paths, T ( ∆ ) , p( ∆ ) , and velocity increasing slowly with depth. ( Adapted from S. Stein and M. Wysession (2003), and Earth Sturcture, Blackwell Publishing, p160) 5 τ ( p) relationships for An Introduction to Seismology, Earthquakes, Changes of v elocity gradient, parameter p, If dp d 2c , are large → samll distance x for smaller ray dz 2 dx > 0 → Retrograde behavior dp ≠ 0 and dx = 0 → dx = 0 → “Caustic” or focusing effect dp c(z) z dx >0 dp dx Caustic, dp = 0 large amplitude dx <0 dp dx <0 dp 6 Depth Intercept time (�) Distance (�) Velocity Time Ray parameter (p) Ray parameter (p) Distance (�) Distance (�) This figure represents ray paths, T ( ∆ ) , p( ∆ ) , and τ ( p) relationships for velocity increasing rapidly with depth. In this case we can see the triplication and retrograde behavior. ( Adapted from S. Stein and M. Wysession (2003), An Introduction to Seismology, Earthquakes, and Earth Sturcture, Blackwell Publishing, p160) 7 Depth Intercept time (�) Distance (�) Velocity Time Ray parameter (p) Ray parameter (p) Distance (�) Distance (�) This figure represents ray paths, T ( ∆ ) , p( ∆ ) , and τ ( p) relationships for velocity decreasing slowly within a low-velocity zone. In this case we can see the shadow zone where no direct geometric arrivals appear, and hence discontinuous T ( ∆ ) , p( ∆ ) , and τ ( p) curves. ( Adapted from Earthquakes, S. Stein and M. Wysession (2003), An Introduction to Seismology, and Earth Sturcture, Blackwell Publishing, p161) 8 τ – p dT x = τ ( p) + px dx τ ( p) = T ( p ) − px T ( p) = τ ( p) + T ⇒ dτ = − x( p) dp ⇒ τ2 The function τ(p) is called the intercept- τ1 slowness representation of the travel time curve. Just as p is the slope of the travel x1 x2 Δ time curve, T(x), the distance x is minus the slope of the τ(p) curve. 9