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TODAY ’ S LECTURE
- Snell’s law in spherical media
- Ray equation
- Radius of curvature
- Amplitude → Geometrical spreading
- τ – p
SNELL ’ S LAW IN THE SPHERICAL MEDIA
c 1
A
B
i 1
i 2
j
r 1
r 2
c (^2)
Q
At each interface
sin i 1
=
sin j
c 1
c 2
sin j =
OQ
sin i 2
OQ
OA OB
sin j =
OB
sin i 2
r 2
sin i 2
OA r 1
r 1
sin i 1
=
r 2
sin i 2
≡ p
c 1
c 2
O
sin i r sin i
“flat earth” → p = “spherical earth” → p =
c c
r
At critical angle, p =
p
we can get depth of layer.
c ( r ) p
RAY EQUATION
dz
dx
ds
s
s
n
i
Directional cosine (3D and 2D)
dx 2
)
2
dx
dx 1
)
2
dx 3
)
2
= 1 ( )
2
dz
2
= 1
ds ds ds ds ds
∧
Direction of ray ( n )
∧ dx dz
n = ( n , 0 , n ) n = n = x z x z
ds ds
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∧ 1
Using Eikonal equation ∇ T = n ,
c
Generalized Snell ’ s law (Ray Equation)
d 1 d
1 dx i
)
dx ( ( i
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 anglei change in the direction of propagation?
d di dz di d di ( s ) dc
(sin i ) = cos i = =
ds
( pc ) ⇒
ds ds ds ds ds
= p
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
i
dx
dz
di
R
ds
R = = R =
di p dc dc
p (
dc
⇒
p ( )
dz dz
R is related to wavespeed gradient and ray parameter.
dc
If = 0 ⇒ R → ∞ Straight Ray !!
dz
2
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One layer : x = 2 h tan i
n
Multiple layers : x = 2
j 0
∑
=
h tan i j j
Continuous case
z z z z
1 dz dz
p p p p
p
2
)
−1/ 2
∫ ∫ ∫ ∫
x ( p )= 2 tan idz = 2 p ( − dz = 2 p = 2 p
c z )
2
2 2
0 0 0 /^ c^ − p 0
p
∫
dx 1 d c
z 2
dz
dp ( dc dz ) dz
2
/ 0 0
z z
dx dz d dz
p p
∫ ∫
= 2 + 2 p ⇒
dp dp
2 2 2 2
/ − (^) ⎪ / − ⎩
0 c^ p^ 0 c^ p
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.
2
d c
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
z dp
Δ
4
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(�)
(�)
(�)
(p)
(p)
(
�
)
Time
Distance
Distance
Distance
Ray parameter
Ray parameter
Intercept time
Velocity
Depth
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
16 March 2005
(�)
(�)
(�)
(p)
(p)
(
�
)
Velocity
Depth
Time
Distance
Distance
Distance
Ray parameter
Ray parameter
Intercept time
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
16 March 2005
(�)
(�)
(�)
(p)
(p)
(
�
)
Time
Distance
Distance
Distance
Ray parameter
Ray parameter
Intercept time
Velocity
Depth
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)
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