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Seismology Lecture Notes: Reflection and Refraction of P and Sv Waves, Study notes of Geology

Lecture notes on the reflection and refraction of p and sv waves in seismology. It includes equations for potentials, boundary conditions, and the solution of zoeppritz's equations to obtain reflection and conversion coefficients. The document also discusses the dependence of coefficients on the incidence angle and frequency, as well as the concept of critical and post-critical reflection.

Typology: Study notes

2012/2013

Uploaded on 07/19/2013

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Introduction to Seismology:Lecture Notes
P-Sv CASE
A P wave incident on a free surface generates both reflected P and Sv waves (Fig. 1)
f
z
\
P
\
Sv
/
P
1
i
j
2
i
α
β
ree surface
Fig. 1
sin i
1
= sin j = sin i
2
,
α
β
j i
α β β
1
We follow the same approach as the S
H
case:
1. Potentials
2. Boundary conditions
3. Solve system of equations (often in the matrix form of Zoeppritz’s Equations)
Potentials
/ /
φ
I
= Aexp
{
i
ω
(
px
η
z t
)}
α
\ \
φ
R
= B exp
{
i
ω
(
px +
η
z t
)}
P wave
α
\ \
ϕ
R
= C exp
{
i
ω
(
px +
η
z t
)}
Sv wave
β
Boundary Conditions
The traction on the free surface must vanish
T
3
0
(
σ
,
σ
yz
,
σ
)
0
xz zz
σ
As before
ij
=
λδ
ij
+
2
µ
ε
ij
2 2
σ
=
λ
2
φ
+ 2
µ
φ
+ 2
µ
ϕ
zz
z x
z
2
2 2 2
φ
ϕ
ϕ
σ
= 2
µ
z x +
µ
x
2
z
2
xz
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Download Seismology Lecture Notes: Reflection and Refraction of P and Sv Waves and more Study notes Geology in PDF only on Docsity!

P-Sv CASE

A P wave incident on a free surface generates both reflected P and Sv waves (Fig. 1)

f

z
\ P Sv

/ P

i 1 (^) j i 2

α

β

ree surface

Fig. 1

sin i 1

sin j

sin i 2 , α 〉β ⇒ ji α β β

1

We follow the same approach as the SH case:

  1. Potentials
  2. Boundary conditions
  3. Solve system of equations (often in the matrix form of Zoeppritz’s Equations)

Potentials / /

φ I = A exp{ i ω ( px −η α z − t )}

\ \

φ R = B exp{ i ω ( px +η α z − t )} P wave

\
ϕ (^) R = C exp{ i ω ( px +η (^) β zt )} Sv wave

Boundary Conditions

The traction on the free surface must vanish

T 3 = 0 = ( σ (^) xz , σ (^) yz , σ (^) zz ) = 0

σ

As before

ij =^ λδ^ ij ∆^ +^2 με^ ij 2 2 σ =λ∇ 2 φ + 2 μ

∂ φ

  • 2 μ

∂ ϕ zzz^2 ∂ xz

∂^2 φ ⎛ ∂^2 ϕ ∂^2 ϕ ⎞ σ = 2 μ x z

  • μ ⎝

x^2

z^2 ⎟ ⎠

xz^ ⎟ ∂ ∂ ⎜

2 nd^ March 2005

Then solve

A = ⎜⎜

\

\

C

B

Which gives

( A + B ){ ( λ+ 2 )

2

p λ C μ p η β 0

( A − B ) 2 p η α − C ( p^2 −η 2 β^ )= 0

After a bit of work we can get expressions of the reflection coefficient (Rpp ) and the

conversion/refraction coefficient (Rps).

\ (^2 2 2 2 2) − 1

/ \ B (λ + 2 μ )η + p λ + 4 μ p η η α β( p −η β)

R / \ = P P = / =

α

p p − {(λ + 2 μ )η 2 + p^2 λ } + 4 μ p^2 η η^ ( p^2 −η^2 −^1

A α α β β )

\ (^2 ) / \ (^) C ⎛ (^4) p ηα ⎫ ⎟

⎧⎪ (λ + 2 μ )η α+ p λ ⎪

R / \ = P S = = ⎜ 2 2 2 2 2 2 2 − 1 ⎬

P S /^ ⎜^ p −η ⎟⎪ (λ + 2 μ )η + p λ − 4 μ p η η ( p −ηβ ) ⎪

A ⎝ β^ ⎠⎩^ α^ α^ β^ ⎭

(See figure on overhead of P and Sv reflection coefficients versus angle of incidence)

There are several things that should be noted about these equations:

  1. The coefficients are dependent on the incidence angle, through cos i 1 , η =

cos j sin i 1 ηα = α

β β

and p = α

  1. For normal incidence the case is more simple i 1 = 0 → p = 0

1 1 ηα = ; η = α

β β so R (^) pp = − 1 , a full reflection with a change in polarity

RPS = 0 , there is no P → Sv conversion /
RSP = S P = 0 /
RSS = S S = 1

  1. Notice that RPP and R (^) SS do not depend on frequency, ω. This is only true so long as :

nd (^) March 2005

R / =

ρ 1 β 1 cos i 1 − ρ 2 β 2 cos i 2 S S ρ 1 β 1 cos i 1 + ρ 2 β 2 cos i 2

2 ρ 1 β 1 cos i 1 T (^) \ \ = S S ρ 1 β 1 cos i 1 + ρ 2 β 2 cos i 2

If we increase i 1 to the critical angle, i c (Fig. 4).

S


1 S

/ 1

Figure 4

1 i 1 β 1


i 2

S 2

β (^2) π i 1 = i (^) c , i 2 = 2

p =

sin i 1

sin ic

sin i 2

⇒ sin i =

β (^1) c β 1 β 1 β 2 β 2 β (^2)

β

ηβ =

= 0 therefore there is no vertical propagation, k (^) z= 2

2 β (^2)

2

\ / R^1 1 2 \ / =^ S^ S =

ρ β cos i 1 − ρ β cos i 2 = 1 , a full reflection S S ρ 1 β 1 cos i 1 + ρ 2 β 2 cos i 2 \
T^1 \ \ =^ S^ S =^

2 ρ β cos i 1 = 2 (also known as the Head Wave) S S ρ 1 β 1 cos i 1 + ρ 2 β 2 cos i 2

When the incidence angle is increased to post-critical, i.e. i 1 〉 i (^) c , i 2 cannot increase to an angle greater than 90º.

sin i 1 sin i (^) c 1

β (^1)

β (^1)

= p = β (^2)

For the incoming wave p 〉 β (^2)

nd (^) March 2005

2 2 2

2 2 2 2

η β β

η = − p = i p − = i

η 2 is now a complex number

For an SH wave the displacement vector is U = (0,U (^) y,0)

U y , 2 = exp{ i ω ( px + η 2 z − t )}

In a postcritical situation when η 2 = i ηˆ

U y , 2 = exp{ iw ( px + i ηˆ 2 z − t )}

U y , 2 = exp{− ωηˆ^2 z }exp{ i ω( px − t )}

Where the first term on the RHS describes an exponential decay in the z- direction (there is no propagation in the z-direction). The frequency, ω, controls the rate of decay. The second term on the RHS describes a harmonic function ( x , t), therefore the wave propagation is in the x -direction.

U y , 2 = A ( z ) exp{ i ω( px − t )}

The property of decreasing wave amplitude with depth based on the frequency of the wave is known as Evanescence. 2 Π ωη = k (^) z = λ 2 Π η = λω

A wave with a short wavelength (λ), high frequency (ω), will decay more quickly. At infinite frequency the decay is instantaneous and the wave becomes a ray.

If the properties of a medium change with depth, for example there is a body which allows a wave to pass through it more quickly at depth only the low ω, long λ, waves will sample it as the high ω waves will have been stripped out. A wave with a frequency dependence is a Dispersive wave. Often a wave can be dispersive and evanescent.

(See figure on overhead of post critical reflection and transmission coefficients)