TODAY S LECTURE SNELL S LAW IN THE SPHERICAL MEDIA RAY EQUATION. sin i 1 = sin j. sin j = OQ sin i 2 = OQ OA OB. sin i 2 OA r 1. sin j = OB sin i 2

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1 TODAY S LECTURE. Snell s law in spherial media 2. Ray equation 3. Radius of urvature 4. Amplitude Geometrial spreading 5. τ p SNELL S LAW IN THE SPHERICAL MEDIA i A i 2 j B 2 At eah interfae sin i = sin j 2 sin j = OQ sin i 2 = OQ OA OB Q r r 2 sin j = OB sin i 2 = r 2 sin i 2 OA r r sin i = r sin i p O sin i flat earth p = spherial earth p = r sin i At ritial angle, p = r p r ) ( p we an get depth of layer. RAY EQUATION Diretional osine (3D and 2D) s i s2 2 ( ) 2 + ( ) 2 + ( 3 ) 2 = ( ) 2 + ( ) 2 = Diretion of ray ( n ) n n = (n x, 0, n z ) n x = n z =

2 Using Eikonal equation T = n, Generalized Snell s law (Ray Equation) d d ( ) = ( i ) i ( x ) ( x ) This equation means that the hange of wavespeed is related to hange of ray geometry. If there is no hange in x diretion, the derivative of x diretion should be zero. d sin i ( ) = 0 = Const. = Const. Snell s law!! (x) C How does this angle i hange in the diretion of propagation? d di di d di ( s ) (sin i) = os i = = ( p) = p Therefore, the hange of angle is related to the hange of veloity. di If is large is large di If is zero ( = onst.) is zero (i = onst.) Straight Ray!! RADIUS OF CURVATURE R : the radius of urvature = Rdi i di R R = = R = di p = p( ) p( ) R is related to wavespeed gradient and ray parameter. If = 0 R Straight Ray!! 2

3 If large rapid hange in Strong Gradient i r sin i from p =, small i small p large R AMPLITUDE-GEOMETRICAL SPREADING Fousing-defousing Shadow Zone Fousing effet Defousing effet We examine the property of / 2 d dt d T ( = ) = 2 Small and large / goes to infinity large amplitude (fousing) Large and small / goes to zero small amplitude (Shadow zone) We also examine x ( p ) x x / 2 T = 2 tan i = i h h 3

4 One layer : x = 2h tan i Multiple layers : x =2 Continuous ase n j = 0 h j tan i j z p z p z p z p tan i =2p p 2 ) / 2 = p z ) x ( p ) =2 ( 2 =2 p ( / p η z p z p z p 2 d d = p ( / ) 2 0 / p 0 / p 0 0 The hange of distane in terms of ray parameter is related to gradient of wave speed at surfae and gradient of the hange in wavespeed between surfae and turning point. Changes of veloity gradient, d 2 2, are small large distane x for smaller ray parameter p, < 0 Normal or Prograde behavior (z) T z < 0 Δ 4

5 Ray parameter (p) Ray parameter Time (p) Depth Interept time (τ) Veloity Figure by MIT OCW. This figure represents ray paths, T ( ), p( ), and τ ( p) relationships for veloity inreasing slowly with depth. ( Adapted from S. Stein and M. Wysession (2003), An Introdution to Seismology, Earthquakes, and Earth Sturture, Blakwell Publishing, p60) 5

6 d 2 Changes of v eloity gradient,, are large samll distane x for smaller ray 2 parameter p, > 0 Retrograde behavior If 0 and = 0 = 0 Causti or fousing effet (z) z > 0 Causti, = 0 < 0 large amplitude < 0 6

7 Ray parameter (p) (p) Depth (τ) Veloity Ray parameter Time Interept time Figure by MIT OCW. This figure represents ray paths, T ( ), p( ), and τ ( p) relationships for veloity inreasing rapidly with depth. In this ase we an see the tripliation and retrograde behavior. ( Adapted from S. Stein and M. Wysession (2003), An Introdution to Seismology, Earthquakes, and Earth Sturture, Blakwell Publishing, p60) 7

8 Ray parameter (p) (p) Depth Interept time (τ) Veloity Ray parameter Time Figure by MIT OCW. This figure represents ray paths, T ( ), p( ), and τ ( p) relationships for veloity dereasing slowly within a low-veloity zone. In this ase we an see the shadow zone where no diret geometri arrivals appear, and hene disontinuous T ( ), p( ), and τ ( p) urves. ( Adapted from S. Stein and M. Wysession (2003), An Introdution to Seismology, Earthquakes, and Earth Sturture, Blakwell Publishing, p6) 8

9 τ p T τ 2 τ dt T ( p ) = τ ( p) + x = τ ( p) + px τ ( p) = T ( p ) px dτ = x ( p ) The funtion τ(p) is alled the interept- slowness representation of the travel time urve. Just as p is the slope of the travel x x 2 Δ time urve, T(x), the distane x is minus the slope of the τ(p) urve. 9