Post on 16-Jan-2016
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Long Offset Moveout Approximation in Layered Elastic Orthorhombic Media
Zvi Koren and Igor Ravve
© 2013, PARADIGM. ALL RIGHTS RESERVED.
Locally 1D Orthorhombic Layered Model
Multi-layer orthorhombic structure Different azimuthal orientation at each layer Common vertical axis Wave type: Compressional
In 1D model, the magnitude and the azimuth of horizontal slowness are the same for all layers
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Layer Parameters
Thickness and Vertical compressional velocity:
Vertical shear-to-compression velocity ratio:
Thomsen-like parameters:
Azimuthal orientation:
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ii vz ,P,
iiiiiii ,2,1,2,1,3,2,1 ,,,,,,
2,P
2,S /1
1 iixi vvf
ix ,1
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Effective Model
Effective model is presented by a single layer with azimuthal anisotropy that• yields the same moveout as the original
layered medium, for any magnitude and azimuth of the horizontal slowness
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Direct & Parametric Presentation of Azimuthally Dependent Normal Moveout
Direct NMO vs. surface azimuth/offset: Direct NMO vs. phase velocity azimuth/offset: NMO vs. phase velocity azimuth/horizontal slowness:
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h
hT
hLpt
ph
ph,,
,
,phs
phs
phs
ht
ht
,
,
phs
off
Lengthwise – offset along phase azimuthTransverse – offset normal to Lengthwise 90phs
phs
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Classical Moveout Approximation for ORT
Approximation for VTI, Tsvankin & Thomsen (1994)
Choices for effective horizontal velocity:
Alkhalifah and Tsvankin, 1995:
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2
22
h
2h4
eff42
42
44
4
22o
22
2o
22
44
2o
22
2
2o
2o
2
,24 VV
VAA
V
VVA
hAtVtV
hA
tV
h
t
tt
iii VVVVVV ,hh,hh,hh RMQ,RMS,MAX
eff21 A
works for azimuthal anisotropy as well
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Why Parametric Approximation?
Fractional moveout approximation has the asymptotic correction factor in the denominator of the nonhyperbolic term, A
While it works perfectly for compressional waves in VTI layered medium, it may lead to negative correction factor A for ORT medium
It never happens in ORT planes of symmetry, but may occur for some azimuths in between
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Effective Model for Short Offsets
Short-offset effective model has 8 coefficients:• 3 low-order coefficients: fast and slow NMO
velocities and slow azimuth• 5 high order coefficients: three effective
anellipticities and two additional azimuths
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Short-Offset Model for Fixed Phase Azimuth
Generally (for any azimuth), there are eight short-offset moveout coefficients
For a fixed phase velocity azimuth, there are only one low-order and one high-order short-offset coefficients
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Effective Model for Fixed Azimuth
The short-offset coefficients are related to power series expansion for infinitesimal horizontal slowness
To accurately describe the moveout, expansion coefficients are needed also in the proximity of the critical slowness (long-offset parameters)
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Long-Offset Parameters: Per Azimuth
Short-offset parameters are computed in the proximity of the vertical direction, it has no azimuth
Long-offset parameters are computed for a proximity of a horizontal direction, characterized by a fixed azimuth. We compute them per azimuth
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Separation of Long-Offset Parameters Two long-offset coefficients enforce convergence of
moveout to correct asymptote for infinite offsets Unlike short-offset parameters related to all layers, the
two long-offset parameters are separated The first is related to the “fast” layer (with fastest
horizontal velocity for given phase azimuth) and controls the tilt of the asymptote
The second is related to all other (“slow”) layers and controls the elevation of the asymptote
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Gluing (Combining) the Coefficients
With the given short-offset and long-offset coefficients, we can “glue” them into a unique continuous function for the whole feasible range of the horizontal slowness
Expansions of the synthetized function into a power series for the infinitesimal horizontal slowness and in the proximity of the critical slowness yield the required computed coefficients
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Short-Offset Expansions
Expansions of three moveout components for infinitesimal horizontal slowness
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crit
442o
53
53
)Transverse(
)Lengthwise(
p
pp
pOpbpatt
pOpbpah
pOpbpah
hh
hhtht
hhThTT
hhLhLL
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Short-Offset Coefficients
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4critphs44phs44phs42phs424phs
2critphs2phs22phs
3critphs44phs44phs42phs42phs
critphs2phs2phs
3critphs44phs44phs42phs424phs
critphs2phs22phs
4sin4cos2sin2cos4
3
,2sin2cos2
1
Traveltime
4cos4sin2cos2
12sin
2
1
,2cos2sin
componentoffset Transverse
4sin4cos2sin2cos
,2sin2cos
componentoffset Lengthwise
pWWWWUb
pWWUa
pWWWWb
pWWa
pWWWWUb
pWWUa
yxyxt
yxt
yxyxT
yxT
yxyxL
yxL
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Long-Offset Expansions
Expansions of three moveout components in the proximity of critical slowness
Unbounded term includes the small value in the denominator
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t
hh
h
t
TT
T
LL
L
p
pp
p
Ottc
t
Ohzc
h
Ohzc
h
cri
2
slowfasto,
slow,fast
slow,fast
1,1
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How We Compute Contribution of Slow Layers
Contribution of “slow” layers in the lengthwise and transverse offset components and traveltime is computed per slowness azimuth:• Assume in the “fast” layer propagation occurs in the
horizontal plane, zenith angle 90 deg.• Applying Snell’s law, compute zenith angle of the
phase velocity for each “slow” layers• Given phase velocity direction, compute for each
“slow” layer
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slowslow,slow, ,, thh TL
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Contribution of Fast Layer into Moveout
We assume that the phase velocity direction in the local orthorhombic frame of the “fast” layer is , where the vertical component is infinitesimal, and the horizontal components are
is the local ORT axis of the “fast” layer
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32,1 ,nnn
23phs2
23phs1 1sin,1cos
11nnnn xx
1x
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Contribution of Fast Layer (Continued)
The horizontal slowness in the “fast” layer is
Performing the infinitesimal analysis, we obtain the phase velocity, the polarization vector, the ray velocity components, the components of the lateral propagation and the traveltime vs. the infinitesimal parameter
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321phs
23
phsphsphs
phs2
phsphsphs
phs
,,
1
,
cos1
,
sin
nnnV
n
VVph
2crit
2 /1 pph
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From Coefficients at Two Ends of Slowness Interval to Combined Continuous Moveout
The offset components and the traveltime are approximated for the whole feasible range of horizontal slowness with continuous functions
Coefficients of continuous functions are obtained by combining short- and long-offset coefficients
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Combined Moveout Functions
The moveout approximation functions are
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2
crit
4242o
33
33
1
10
h
h
hh
hthththt
hThThThTT
hLhLhLhLL
p
pp
pp
pDpCpBpAtt
pDpCpBpAh
pDpCpBpAh
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Intercept Time
With the parametric functions, we obtain the moveout approximation in domain in a straightforward way
The horizontal slowness has no transverse component while the offset has both components
The intercept time simplifies to
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hhhhh ppptp hp
p
hLhhh phpptp
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Test for Multi-Layer Structure
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# δ1 δ2 δ3 ε1 ε2 γ1 γ2 V, km/s f Δz, km φaxo
1 0.15 0.15 0 0.20 0.20 0.06 0.06 2.0 0.72 0.5 VTI
2 0.12 -0.08 0.07 0.18 -0.15 0.03 -0.03 3.0 0.74 0.5 20
3 0.09 -0.10 -0.06 0.16 -0.12 0.10 -0.08 4.8 0.75 0.5 110
4 0.13 -0.08 -0.07 0.15 -0.14 0.08 -0.09 3.0 0.76 0.5 60
5 0.16 -0.17 0.05 0.12 -0.17 0.05 -0.06 3.5 0.78 0.5 140
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Lengthwise and Transverse Offset Components vs. Slowness for Constant Phase Azimuth
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0.0
0.5
1.0
1.5
2.0
2.5
0.0 0.2 0.4 0.6 0.8 1.0
Nor
mal
ized
off
set c
ompo
nent
s
Normalized horizontal slowness
Lengthwise & transverse offset componentsvs. horizontal slowness, ψphs=0
Exact hxExact hyApprox hxApprox hy
7.6θphs , deg
15.3 23.2 31.3 40.00
-8.E-03
-6.E-03
-4.E-03
-2.E-03
0.E+00
2.E-03
0.0 0.2 0.4 0.6 0.8 1.0
Rel
ativ
e er
ror
Normalized horizontal slowness
Relative error of lengthwise & transverse offset components vs. horizontal slowness, ψphs=0
Error hxError hy
7.6θphs , deg
15.3 23.2 31.3 40.00
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Lag between Acquisition Azimuthand Phase Velocity Azimuth
31Const. reflection angle, varying azimuthConst. phase azm, varying reflection angle
2
4
6
8
10
12
0.0 0.2 0.4 0.6 0.8 1.0
Azi
mut
hal l
ag, d
eg
Normalized horizontal slowness
Lag between acquisition and phasevelocity azimuths, ψphs=0
ExactApprox
7.6 θphs , deg.15.3 23.2 31.3 40.00
-6
-4
-2
0
2
4
6
8
10
0 30 60 90 120 150 180
Azi
mut
hal l
ag, d
eg
Phase velocity azimuth, deg.
Lag between acquisition and phase velocityazimuths vs. phase velocity azimuth, θphs=38o
ExactApprox
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Traveltime vs. Slowness & its Error for Constant Phase Velocity Azimuth
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traveltime vs. slowness error of traveltime vs. slowness
1.00
1.25
1.50
1.75
2.00
2.25
2.50
0.0 0.2 0.4 0.6 0.8 1.0
Nor
mal
ized
tim
e
Normalized horizontal slowness
Traveltime vs. slowness, ψphs=0
ExactApprox
7.6θphs , deg.
15.3 23.2 31.3 40.00
0.0E+00
2.0E-03
4.0E-03
6.0E-03
8.0E-03
1.0E-02
1.2E-02
1.4E-02
0.0 0.2 0.4 0.6 0.8 1.0
Rel
ativ
e ti
me
erro
r
Normalized horizontal slowness
Relative error of traveltime vs. slowness, ψphs=0
Error
7.6θphs , deg.
15.3 23.2 31.3 40.00
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Traveltime vs. Offset for Constant Phase Azimuth: Parametric Model and Alkhalifah Strong & Weak
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traveltime vs. offset error of traveltime vs. offset
1.0
1.5
2.0
2.5
3.0
3.5
4.0
0.0 1.0 2.0 3.0 4.0
Nor
mal
ized
trav
elti
me
Normalized offset
Traveltime vs. offset, ψphs=0
ExactA StrongA WeakApprox
θphs , deg.
36.6 38.80 39.5 39.7
-0.01
0.00
0.01
0.02
0.03
0.04
0 1 2 3 4
Rel
ativ
e er
ror
of tr
avel
tim
e
Normalized offset
Relative error of traveltime vs. offset, ψphs=0
A StrongA WeakApprox
36.6
θphs , deg.
38.8 39.50 39.7
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Lengthwise and Transverse Offset Components vs. Phase Azimuth for Constant Reflection Angle
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lengthwise & transverse offset components error of offset components
-0.5
0.0
0.5
1.0
1.5
0 30 60 90 120 150 180
Nor
mal
ized
off
set c
ompo
nent
s
Phase velocity azimuth, deg.
Lengthwise & transverse offset componentsvs. phase velocity azimuth, θphs=38o
Exact hxExact hyApprox hxApprox hy
-1.6E-02
-1.2E-02
-8.0E-03
-4.0E-03
0.0E+00
4.0E-03
0 30 60 90 120 150 180
Abs
olur
e er
ror
of o
ffse
t com
pone
nts
Phase velocity azimuth, deg.
Absolute error of lengthwise & transverse offset components vs. phase velocity azimuth, θphs=38o
Error hxError hy
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Traveltime vs. Phase Azimuth for Constant Refl. Angle: Parametric and Alkhalifah Strong & Weak
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traveltime vs. phase velocity azimuth error of traveltime vs. phase azimuth
1.15
1.20
1.25
1.30
1.35
1.40
1.45
1.50
1.55
0 30 60 90 120 150 180
Nor
mal
ized
tim
e
Phase velocity azimuth, deg.
Traveltime vs. phase velocity azimuth, θphs=38o
ExactA StrongA WeakApprox
0.00
0.01
0.02
0.03
0.04
0 30 60 90 120 150 180
Rel
ativ
e tr
avel
tim
e er
ror
Phase velocity azimuth, deg.
Error of traveltime vs. phase velocity azimuth, θphs=38o
A StrongA WeakApprox
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Conclusions
We derived new asymptotic correction of the moveout approximation for ORT layered media
The approximation has the same power series expansion of the moveout components as the moveout of the original multi-layer package for• infinitesimal slowness and• nearly critical slowness
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Conclusions (continued)
For infinitesimal slowness, we keep two terms of the moveout series per azimuth (and vertical time)
For nearly critical slowness, we keep two terms of the moveout series per azimuth as well
One long-offset term characterizes the propagation through the layer with the fastest horizontal velocity, while the other term describes the propagation through the “slower” layers
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Conclusions (continued)
The approximation is parametric: lengthwise and transverse offset components and traveltime are functions of horizontal slowness and its azimuth
Parametric functions allow approximating the moveout in both t-x and tau-p domains
For wide opening angles the asymptotic correction terms are essential to match the exact ray tracing
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