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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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Free-surface reflection
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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Free-surface reflection
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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Free-surface reflection
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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Free-surface reflection
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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Free-surface reflection
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