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- 1. Introduction to Seismology: Lecture Notes 16 March 2005 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 c1 At each interface c2 sin i1 sin j i1 A = c1 c2 i2 B j OQ OQ sin j = sin i2 = OA OB Q OB r r2 sin j = sin i2 = 2 sin i2 r1 OA r1 r1 sin i1 r2 sin i2 = ≡p c1 c2 O sin i r sin i “flat earth” → p= “spherical earth” → p= c c rp At critical angle, p= we can get depth of layer. c(rp ) RAY EQUATION Directional cosine (3D and 2D) s1 dx1 2 dx dx dx 2 dz ( ) + ( 2 )2 + ( 3 )2 = 1 ( ) + ( )2 = 1 s2 ds ds ds ds ds dz i ds ∧ Direction of ray ( n ) dx ∧ dx dz n n = (n x ,0, n z ) nx = nz = ds ds 1
- 2. Introduction to Seismology: Lecture Notes 16 March 2005 1∧ Using Eikonal equation ∇T = n, c Generalized Snell’s law (Ray Equation) d 1 d 1 dxi ( )= ( ) dxi c( x) ds c( x) ds 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 1 dx sin i ( )=0 ⇒ = Const. ⇒ = Const. ⇒ Snell’s law !! ds c(x) ds c ds C How does this angle i change in the direction of propagation? d di dz di d di ( s ) dc (sin i) = cos i = = ( pc) ⇒ =p ds ds ds ds ds ds dz Therefore, the change of angle is related to the change of velocity. dc di If is large ⇒ is large dz ds dc di If is zero (c = const.) ⇒ is zero (i = const.) Straight dz ds Ray !! RADIUS OF CURVATURE R : the radius of curvature ds = Rdi ds 1 dz 1 1 R= = = ⇒ R= di di p dc dc dc p( ) p( ) dz dz R R is related to wavespeed gradient and ray parameter. dz i ds dc dx If =0 ⇒R → ∞ Straight Ray !! dz 2
- 3. Introduction to Seismology: Lecture Notes 16 March 2005 dc If large ⇒ rapid change in c Strong Gradient dz r sin i from p = , c small i → small p → large R i AMPLITUDE-GEOMETRICAL SPREADING Focusing-defocusing Shadow Zone Focusing effect Defocusing 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 → small amplitude (Shadow zone) We also examine x( p ) x ds x/2 T =2 tan i = i c h c ds h 3
- 4. Introduction to Seismology: Lecture Notes 16 March 2005 One layer : x = 2h tan i n Multiple layers : x = 2 ∑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 ∫ = 2 p ∫ 0 0 c( z ) 2 0 1 / c 2 − p 2 0 η d ⎧ ⎫ dx ⎧ ⎫ ⎧ p d 2c ⎫ zp zp z dx dz ⎪ dz ⎪ 1 ⎪ ⎪ = 2 ∫ + 2 p ⎨ ∫ ⎬ ⇒ ≈ ⎨− ⎬ + ⎨+ ∫ 2 dz ⎬ dp 2 1 / c − p 2 2 2 dp ⎪ 0 1 / c − p ⎪ dp ⎩ (dc / dz ) 0 ⎭ ⎪ 0 dz ⎩ ⎪ ⎭ 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. d 2c Changes of velocity gradient, , are small → large distance x for smaller ray dz 2 dx parameter p, < 0 → “Normal” or Prograde behavior dp T c(z) dx <0 z dp Δ 4
- 5. Introduction to Seismology: Lecture Notes 16 March 2005 Distance (�) Intercept time (�) Depth Velocity Ray parameter (p) Ray parameter (p) Time Distance (�) Distance (�) Figure by MIT OCW. This figure represents ray paths, T ( ∆ ) , p( ∆ ) , and τ ( p) relationships for velocity increasing slowly with depth. ( Adapted from S. Stein and M. Wysession (2003), An Introduction to Seismology, Earthquakes, and Earth Sturcture, Blackwell Publishing, p160) 5
- 6. Introduction to Seismology: Lecture Notes 16 March 2005 d 2c Changes of v elocity gradient, , are large → samll distance x for smaller ray dz 2 dx parameter p, > 0 → Retrograde behavior dp dx If dp ≠ 0 and dx = 0 → = 0 → “Caustic” or focusing effect dp c(z) z dx Caustic, dp = 0 large amplitude dx >0 dp dx <0 dp dx <0 dp 6
- 7. Introduction to Seismology: Lecture Notes 16 March 2005 Distance (�) Intercept time (�) Depth Velocity Ray parameter (p) Ray parameter (p) Time Distance (�) Distance (�) Figure by MIT OCW. 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
- 8. Introduction to Seismology: Lecture Notes 16 March 2005 Distance (�) Intercept time (�) Depth Velocity Ray parameter (p) Ray parameter (p) Time Distance (�) Distance (�) Figure by MIT OCW. 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 S. Stein and M. Wysession (2003), An Introduction to Seismology, Earthquakes, and Earth Sturcture, Blackwell Publishing, p161) 8
- 9. Introduction to Seismology: Lecture Notes 16 March 2005 τ – p dT T ( p) = τ ( p) + x = τ ( p) + px T dx ⇒ τ ( p) = T ( p ) − px dτ ⇒ = − x( p) τ2 dp 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

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