8 reflectivity

7/30/2019 8 Reflectivity

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GEOL882.3GEOL483.3

Reflection

coefficientsReflection and conversion ofplane waves

Snell's law

P/SV wave conversion

Scattering matrix

Zoeppritz equations

Amplitude vs. Angle and Offsetrelations

Reading: Telford et al., Section 4.2. Shearer, 6.3, 6.5

Sheriff and Geldart, Chapter 3


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Surface reflection

transmission, and

conversion

Consider waves incident on a weldedhorizontal interface of two uniform half-

spaces:Because of their vertical motion, PandSVwaves couple to each other on theinterface,

therefore, there are 8 possible wavesinteracting with each other at the

boundary.

1, VP1, VS1

2, V

P2, V

S2

...what about SHwaves?


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Free-surface reflection

and conversion

Consider a Pwave incident on a free surface:

IncidentP ReflectedP

Reflected S

Boundary condition: xz

= yz

= zz

= 0 on z=0 x

z

Each of the P- or S-waves is described bypotentials:

u=

uPx , z=

x,0,

z,

uSx , z=

z, 0,

x,

P-

waves

SV-wave

=inc

refl

=refl

inc refl

refl


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and conversion (2)

Traction (force acting on the surface):

FPx , z=

2

2

x z, 0, 2

2

2

z2

, P-wave

SV-wave

inc=APinc

exp[ ix ninc PVP t]

FSx , z= 2

x2

2

z2

, 0, 22

2

x z, Considerplane harmonicwaves:

refl=ASrefl

exp[i x nrefl SVS t]

refl=APrefl

exp

[i

x nrefl PV

P

t

]

incidentP

reflectedP

reflected SV

Q: What are the dependencies of

and above on coordinatex?


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and conversion (3)

The boundary condition is: Force(x,t)=0

Note that functional dependencies of

and on (x,t) are:

exp[i sin iVP xt],exp

[i

sin i*

VP

xt

],

exp[i sin jVS xt],

IncidentP ReflectedP

Reflected S

Boundary condition: xz

= yz

= zz

= 0 on z=0 x

z

These must satisfyfor anyx,consequently, the

Snell's law:

sin i

VP=sin i

*

VP= sin j

VS=p


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and conversion (4)

Displacement in plane waves is thus:

uPx , z=i p , 0,i

cos j

VP , P-waves

SV-wave

...and traction:

uSx , z=icos j

VP, 0, i p ,

FPx ,z =2 VS2p,0,12V2p2i2 VS,

FSx ,z =12V2p

2i

2VS,0,2VS

2p,0.


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and conversion (5)

Traction vector at the surface must vanish

Fx

=Fz

=0

Therefore, we have two equations toconstrain the amplitudes of the tworeflected waves;

Their solution:

APrefl

APinc

=

4V S4

p2 cos i

VP

cos jVS

12VS2

p22

4VS4

pcos i

VP

cos j

VS12VS

2p

22,

ASrefl

APinc

=

4VS2

p cos iVP

12VS2

p2

4V S4

pcos i

V P

cos j

VS12VS

2p

22.


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and conversion (5)

Normal

incidenceGrazing

incidence

VP

= 5 km/s

VS= 3 km/s


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Completereflection/transmission

problem

There are 16 possiblereflection/transmission coefficients on awelded contact of two half-spaces


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Scattering matrix

All 16 possible reflection coefficients can besummarized in the scattering matrix:

S=P P S P P P S P

PS S S P S S S

P P S P P P S P

PS

SS

PS

SS

1, V

P1, V

S1

2, V

P2, V

S2

Incident Scattered

P1S1P2

S2=S

P1S1P2

S2.


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All reflection and refraction

amplitudes at an interface(Derivation of the Scattering Matrix)

The scattering matrix can be used to easilyderive all possible reflection and refraction

amplitudes at once:consider matrix N that is givingdisplacement and traction at the interfacefor the incident field, and a similar matrixM for the scattered field:

This is a general (matrix) form ofZoeppritz' equations (relating theincident, reflected, and converted waveamplitudes).

Their general solution:

uxuyxzzz

=MP1S1P2S2=N

P1S1P2S2.

S=M1N


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M and N

The matrices M and N consist of thecoefficients of plane-wave amplitudes and

tractions for P- and SV-waves:

M=VP1 p cos j1 VP2 p cos j2

cosi1 VS1 p cos i2 VS2 p

2 1VS12

p cosi1 1VS112 VS12

p2 2 2 VS2

2p cosi2 2VS212VS2

2p

2

1VP112 VS12

p2 2 1 VS1

2p cos j1 2VP212VS2

2p

2 2 2 VS1

2p cos j2

,N=

VP1p cos j1 VP2 p cos j2cosi1 VS1 p cosi2 VS2 p

2 1 VS12

p cos i1 1 VS112 VS12

p2 2 2VS2

2p cosi 2 2 VS212 VS2

2p

2

1 VP112 VS12

p2 2 1VS1

2p cos j 1 2VP212 VS2

2p

2 2 2 VS1

2p cos j 2 ,

SP P S P P P S P

PS S S P S S S

P P S P P P S P

P S S S P S S S=M 1 N .

This is matrix form ofKnott' equations (solutions forreflected and refracted amplitudes)


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Partitioning at

normal incidence

At normal incidence, i1=i

2=j

1=j

2=0, andp=0:

M=0 1 0 11 0 1 0

0 1VS1 0 2VS21 VP1 0 2VP2 0 , N=

0 1 0 1

1 0 1 0

0 1 VS1 0 2 VS21 VP1 0 2VP2 0 ,

P S PP PSS S

The P- and S-waves do not interact at normalincidence, and so we can look, e.g., at P-waves

only (extract the odd-numbered columns):

N=0 0

1 1

0 0

1 VP1 2 VP2,M=

0 0

1 1

0 0

1VP1 2VP2,

Note that

these two

constraints

are satisfied

automatically

Drop the two trivial equations (#1and 3) and obtain:

P P P PP P P P=M 1 N= 1 1Z1 Z21

1 1Z1 Z2=1

Z1Z2 Z2Z1 2Z2

2Z1 Z1Z2.

Impedance,

V=Z

Reflection and transmission coefficients


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Reflection andTransmission at

normal incidence

Thus, at normal incidence (in practice, for anglesup to ~15)

Reflection coefficient:

Transmission coefficient:

Energy Reflection coefficient:

Energy Transmission coefficient:

Note that the energy coefficients do not dependon the direction of wave propagation, but Rchanges its sign.

R < 0 leads tophase reversal in reflectionrecords.

R=Z2Z1Z1Z2

Z2Z

1

2lnZ

1

2

VPVP

T=2Z1

Z1Z

2

ER=R2

ET=1ER=2Z

1Z

2

Z1Z2.


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Typical impedance contrasts

and reflectivities


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Oblique incidenceAmplitude versus Angle

(AVA)variation

At oblique incidence, we have to use the full M-1N expression for S

Amplitudes andpolarities of the reflections varywith incidence angles.

Fast to slow:

VP2

/VP1

= 0.5,2/

1= 0.8;

2= 0.25

Fast to slow:

VP2

/VP1

= 0.5,2/

1= 0.8;

2= 0.25

Slow to Fast:

VP2

/VP1

= 2.0,2/

1= 0.5;

2= 0.3

Slow to Fast:

VP2

/VP1

= 2.0,2/

1= 0.5;

2= 0.3

Fraction ofP-wave reflection

energy,for various V

P2/V

P1

2/

1= 1.0;

1=

2= 0.25

Fraction ofP-wave reflection

energy,for various V

P2/V

P1

2/

1= 1.0;

1=

2= 0.25

Fraction ofP-wave reflection energy,

for various2/1V

P2/V

P1= 1.5;

1=

2= 0.25

Fraction ofP-wave reflection energy,

for various2/1

VP2

/VP1

= 1.5; 1

=2

= 0.25


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Oblique incidence

Small-contrast AVA approximation

VP, V

S,

, and therefore, ray angle variations

are considered small

Shuey's (1985) formula gives the variation of Rfrom the case on normal incidence in terms ofV

P

and (Poisson's ratio):

where:

R

R0

1Psin2 Q tan 2sin 2

R01

2 VPVP

,P=[Q

21121 ] R012 ,

Q=

VPVP

VPVP

=1

1/

VP/VP

.

Important at >~30

Important at typical

reflection angles


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Amplitude Variation

with Offset (AVO)

AVO is a group of interpretationtechniques designed to detect reflection

AVA effects:Records processed with true amplitudes(preserving proportionality to the actualrecorded amplitudes);

Source-receiver offsets converted to the

incidence angles;

From pre-stack (variable-offset) datagathers, parameters R(0), Pand Q areestimated:

Thus, additional attributes are extractedto distinguish between materials withvarying .

RR0[1Psin

2

Q tan

2

sin

2

].


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Three practical AVA cases

Three typical AVA behaviours:

1)Amplitude decreases with angle without

crossing 0;2)Amplitude increases;

3)Amplitude decreases and crosses 0(reflection polarity changes)

2

= 1

=0.3 (solid)

=0.2 (dashed)

2

= 1

=0.3 (solid)

=0.2 (dashed)

1)

2)1)

2)3)

3)

2

< 1

: 0.4 to 0.1 (solid)

: 0.3 to 0.1 (dashed)

2

< 1

: 0.4 to 0.1 (solid)

: 0.3 to 0.1 (dashed)

2

> 1

: 0.1 to 0.4 (solid)

: 0.1 to 0.2 (dashed)

2

> 1

: 0.1 to 0.4 (solid)

: 0.1 to 0.2 (dashed)

From Ostrander, 1984

Normal caseNormal case AVO (AVA) anomalies AVO (AVA) anomalies

(Above: VP2

/VP1

= 2/

1=1.25; 1.11; 1.0; 0.9, and 0.8)


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AVA (AVO) anomalies

2)

2)

3)

3)

From Ostrander, 1984

Gas/water contact Base of gas sand embedded in shale

Top of gas sand embedded in shale Base of

high-impedance reservoir

Top of

high-impedance reservoir


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Amplitude Variation withOffset (AVO)Gas sand vs. wet sand

Gas-filled pores tend to reduce VP

more than VS, and as

a result, the Poisson's ratio () is reduced.

Negative VPandthus cause negative-polarity bright

reflection (bright spot) andan AVO effect (increase inreflection amplitude with offset) that are regarded ashydrocarbon indicators.

However, not every AVO anomaly is related to acommercial reservoir...

Fr

om

Yu

,1

985


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AVO cross-plotting

From Young et al, 200


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Cross-plotting

From Young et al, 200


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Rock-physics

Indicators

Rock-physics parameters can bederived from the shapes of AVO(AVA) responses:

(fluid incompressibility) isconsidered the most sensitive fluidindicator

(rigidity) is insensitive to fluid butsensitive to the matrix.

increases with increasing quartz

content (e.g., in sand vs. clay). is sensitive to gas content.


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Colours correspondto identified (lr,mr) zone

-- cross-plotting

Colours correspondto identified (,) zon

8 reflectivity

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