Reflection+ Transmission Coefficients for 2-Layer Problem.

Reflection+ Transmission Reflection+ Transmission Coefficients for 2-Layer Coefficients for 2-Layer

ProblemProblem

OutlineOutline1.1. Two-Layer Problem: Reflection+Transmission Two-Layer Problem: Reflection+Transmission

Coefficients, Snell’s Law, Refractions, Critical Coefficients, Snell’s Law, Refractions, Critical angle, Post-critical Waves.angle, Post-critical Waves.

D = e D = e i(k x - k z - wt) i(k x - k z - wt) x x z z

(2)

U = R e U = R e i(k x + k z - wt) i(k x + k z - wt) x x z z

x x

z z

(2) increase x, z {- increase t

{

D = TD = Te e i(k i(k xx - k’ - k’ zz - - wt)wt) x x z z

k = w – k k = w – k x x z z 2 2

2 2 2 2

cc

k’ = k’ = ww – k – k x x z z 2 2

2 2 2 2

c’c’

c’

c

’

2-Layer Medium: Find R & 2-Layer Medium: Find R & TT

(2)

x x

z z

c’

c

’

2-Layer Medium: Find R & 2-Layer Medium: Find R & TT

TTe e i(k i(k xx - k’ - k’ zz - - wt)wt) x x z z

P = P = -

e e i(k x - k z - wt) i(k x - k z - wt)

x x z z

P = U + D = P = U + D = + e e

i(k x + k z - wt) i(k x + k z - wt) x x z z

+ R+ R

Two unknowns R & Two unknowns R & TT, need, need

two equations of constraint two equations of constraint

B.C. #1: Pressure Must Match at z=0B.C. #1: Pressure Must Match at z=0

ik ik xx TTe e

x x

e e ik x ik x

x x e e

ik x ik x x x + R+ R = =

+ -P = P = PP

@z=0@z=0

TTe e i(k i(k xx - - wt)wt) x x

e e i(k x - wt) i(k x - wt)

x x e e

i(k x - wt) i(k x - wt) x x + R+ R = = @z=0@z=0

Divide by e i( wt) i( wt)

Implies k = Implies k = kkxx xxsin O = sin sin O = sin O’O’

cc c’c’

ik x ik x x x

O O

O O

c’

c

’

Snell’s Law

Pressure must match at z=0Pressure must match at z=0

+ -P = P = PP

@z=0@z=0

O O

O O

ik ik xx TTe e

x x

e e ik x ik x

x x e e


ik x ik x x x @z=0@z=0

+ -w = w = ww.. .. .. ..

Recall w = -1 dP/dzRecall w = -1 dP/dz

.. ..

ik xik x -ik T-ik Te e

x x

-ik e -ik e ik x ik x

x x e e

ik x ik x x x + ik R+ ik R = = z z z z z z

@z=0@z=0

z=0z=0

c’

c

’

B.C. #2: Particle Accel. Must Match at z=0B.C. #2: Particle Accel. Must Match at z=0

Vertical acceleration matches @ z=0 Vertical acceleration matches @ z=0

so w+ = w- @ z=0so w+ = w- @ z=0

Pressure & accelration must match at z=0Pressure & accelration must match at z=0

O O

O O

ik ik xx TTe e

x x

e e ik x ik x

x x e e


ik x ik x x x @z=0@z=0


x x


x x e e


@z=0@z=0

c’c’ cos O - c cos cos O - c cos OOc cos c cos OO + + c’c’ cos O cos O

,, ,,

R =

c’

c

’

c’c’ cos O cos Oc cos c cos OO + + c’c’ cos O cos O

,, ,,

T =22

Special Cases: Bright SpotsSpecial Cases: Bright Spots


,, ,,

R =

O = 0:O = 0: c’c’ - c - c ,,

R =c’c’ + c + c ,,

Positive Refle. Coeff whenPositive Refle. Coeff when

Model is low over hi velocityModel is low over hi velocity

Negative Refle. Coeff whenNegative Refle. Coeff when

Model is hi over low velocityModel is hi over low velocity

BRIGHT SPOTBRIGHT SPOT

Gas sand has lower impedanceGas sand has lower impedance

than wet sand. Works well in GOMthan wet sand. Works well in GOM

and young sedimentary basinsand young sedimentary basins

Special Cases: Critical AngleSpecial Cases: Critical Angle


,, ,,

R =

O = critical angle:O = critical angle:

O O

O O

c’

c

’

sin O = sin sin O = sin OO’ ccc’c’

1 < ccc’c’

Special Cases: Post Critical ReflectionsSpecial Cases: Post Critical Reflections


,, ,,

R =


O O

O O

c’

c

’


1 < ccc’c’


,, ,,

R =


O O

O O

c’

c

’


1 < ccc’c’



,, ,,

R =


O O

O O

c’

c

’

sin O = 1 sin O = 1 ccc’c’

1 < ccc’c’



,, ,,

R =

O = post critical angle:O = post critical angle:

O O

O O

c’

c

’

sin O > 1 sin O > 1 ccc’c’

1 < ccc’c’

Total Energy Reflection Post CriticalTotal Energy Reflection Post Critical

Phase ShiftPhase Shift


Special Cases: Post Critical Reflection Coeff.Special Cases: Post Critical Reflection Coeff.


,, ,,

R =

sin = sin = c’c’sinOsinO OOcc

= 1 – = 1 – 22 c’c’sinOsinO

cc{

{

c’c’sinOsinO

cc> 1 > 1

- 1 - 1 22 c’c’sinOsinO

cc{

{

= i= i - 1 - 1 22 c’c’sinOsinO

cc{

{

= e = e ii/2 /2

e e ii/2 /2

OOcoscos = 1 – sin= 1 – sin OO22

AA

BB e e ii/2 /2

Special Cases: Post Critical Reflection Coeff.Special Cases: Post Critical Reflection Coeff.

c’c’ cos O - c cos cos O - c cos OO

,,

,,

R =

R =c’c’ A - c A - c e e

ii/2 /2

,, BB

c’c’ A + c A + c e e ii/2 /2

,,

BB

Reflection coefficient is nowReflection coefficient is now

a complex numbera complex number

U = R e U = R e i(k x + k z - wt) i(k x + k z - wt) x x z z

e e ii

c’c’ cos O cos O c cos c cos OO ++

- 1 - 1 22 c’sinOc’sinO

cc{

{

= e = e ii/2 /2

BB e e ii/2 /2

cos cos OO = =

AA

Phase change in upgoing post critical reflectionsPhase change in upgoing post critical reflections

Havoc AVO and stacking!Havoc AVO and stacking!

Layered Medium & Critical AngleLayered Medium & Critical Angle

3 km/s3 km/s

6600X (km)X (km)

6600X (km)X (km)

00

3.53.5

Z (

km

)Z

(k

m)

Tim

e (s

)T

ime

(s)

00

Post-critical reflection rayPost-critical reflections

Sea floor

4.04.0 1.5 km/s1.5 km/s

CSGCSG ModelModel

Special Cases: Pressure+Marine Free SurfaceSpecial Cases: Pressure+Marine Free Surface


,, ,,

R =

Free Surface Free Surface ’c’=’c’=0:0:

R =- c cos - c cos OO

c cos c cos OO

= -1

P = 1 + R = 1 – 1 = 0

1 -1

ghostsghosts

c’c’ cos O + c cos cos O + c cos OOc cos c cos OO + + c’c’ cos O cos O

,, ,,

R =

Free Surface Free Surface ’c’=’c’=0:0:

R =+ c cos + c cos OO

c cos c cos OO

= 1

w = 1 + R = 1 + 1 = 2

1 +1

ghostsghosts

Special Cases: Part. Vel. + Land Free SurfaceSpecial Cases: Part. Vel. + Land Free Surface

Part.

Part.

Part.

Special Cases: Part. Vel. + Land Free SurfaceSpecial Cases: Part. Vel. + Land Free Surface

Why is particle velocity R opposite polarity to pressure R?Why is particle velocity R opposite polarity to pressure R?


x x


x x e e


@z=0@z=0

so w+ = w- @ z=0so w+ = w- @ z=0Recall w = -1 dP/dzRecall w = -1 dP/dz

.. ..

- DDDD - U R- U R = = z z z z z z

Downgoing particle-z velocityDown particle-z vel. Up particle-z vel.

R =-RR =-Rpart.part. press.press.

Special Cases: ImpedanceSpecial Cases: Impedance

P = e P = e ik z ik z

z z

Recall w = -1 dP/dzRecall w = -1 dP/dz

.. .. ik z ik z z z

ik ik zzdP/dz = e dP/dz = e wherewhere

ik ik zzdP/dz = P = P

Therefore wTherefore w

.. .. ik ik zzdP/dz = - P = - P But w = -iBut w = -iw w

.. .. . .

. . Therefore wTherefore w

ik ik zzdP/dz = P = P

iior P/w = or P/w = cc.ImpedanceImpedance

Special Cases: Conservation of Energy FlowSpecial Cases: Conservation of Energy Flow

cc

Previously we found that the Energy Density in deformingPreviously we found that the Energy Density in deforming

a cube was given by a cube was given by PP

22

22

Therefore the rate at which energy is flowing acrossTherefore the rate at which energy is flowing across

a flat interface by a propagating plane wave is bya flat interface by a propagating plane wave is by

PPcc

22

Energy/area/time = = Energy/area/time = = cc22

PPcc

22

Conservation of energy demands energy flow of incidentConservation of energy demands energy flow of incident

wave is same as the transmitted and reflected wave:wave is same as the transmitted and reflected wave:

cc

22

11 = R + T = R + T cc c’c’

22 22

’’Note: > 1> 1

+ + cc

c’c’22 ’’

c’c’’’

Special CasesSpecial Cases

P = sinO1/v1 = sinO2/v2=sinO3/v3=sinO4/v4P = sinO1/v1 = sinO2/v2=sinO3/v3=sinO4/v4

v1v1

v2v2

v3v3

v4v4

Slowness = inverse apparent Vx

Tim

e T

ime

v1v1

v2v2

v3v3

v4v4

Slope = v5

v5v5

Date post:	22-Dec-2015
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Reflection+ Transmission Coefficients for 2-Layer Problem.

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