Long Offset Moveout Approximation in
Layered Elastic Orthorhombic Media
Zvi Koren and Igor Ravve
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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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2
Layer Parameters
 Thickness and Vertical compressional velocity:
z i , v P ,i
 Vertical shear-to-compression velocity ratio:
2
f i  1  vSx1 ,i
2
/ vP,i
 Thomsen-like parameters:
 1,i ,  2,i ,  3,i ,  1,i ,  2,i ,  1,i ,  2,i
 Azimuthal orientation:  x1 ,i
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3
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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4
Direct & Parametric Presentation of
Azimuthally Dependent Normal Moveout
t  off , h 
 Direct NMO vs. surface azimuth/offset:
 Direct NMO vs. phase velocity azimuth/offset: t  phs , h
 NMO vs. phase velocity azimuth/horizontal slowness:
hL  phs , ph
, t  phs , ph
hT  phs , ph

phs
Lengthwise – offset along phase azimuth

Transverse – offset normal to Lengthwise  phs  90



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 



5
Classical Moveout Approximation for ORT
 Approximation for VTI, Tsvankin & Thomsen (1994)
2
2
2
4
works
for
t  to
h
A4 h


2
2 2
to
V2 to V 2t 2 V 2t 2  A h 2 
azimuthal
2 o
2 o
4
4
2
anisotropy
V4  V2
A4 Vh
A4  


2

,
A


eff
4
2
2
4V2
Vh  V2
as well
 Choices for effective horizontal velocity:
Vh  MAX Vh ,i , Vh  RMS Vh ,i , Vh  RMQ Vh ,i 
 Alkhalifah and Tsvankin, 1995:
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A  1  2 eff
6
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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7
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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8
Short-Offset Model for Fixed Phase Azimuth
 Generally (for any azimuth), there are eight shortoffset 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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9
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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10
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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11
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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12
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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13
Short-Offset Expansions
 Expansions of three moveout components for
infinitesimal horizontal slowness
hL 
hT 
3
a L p h  bL p h
3
aT p h  bT p h
t  t o  at
2
ph
4
 bt p h
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 
 Op 
 Op 
O
5
ph
5
h
4
h
(Lengthwise)
(Transverse)
ph
ph 
p crit
14
Short-Offset Coefficients
Lengthwise offset component
  

3
bL  phs   U 4  W42 x cos 2 phs  W42 y sin 2 phs  W44 x cos 4 phs  W44 y sin 4 phs  pcrit
a L  phs  U 2  W2 x cos 2 phs  W2 y sin 2 phs pcrit ,
Transverse offset component
  

1
1
 3
bT  phs    W42 x sin 2 phs  W42 y cos 2 phs  W44 x sin 4 phs  W44 y cos 4 phs  pcrit
2
2
aT  phs  W2 x sin 2 phs  W2 y cos 2 phs pcrit ,


Traveltime




at  phs
bt  phs


1
2
 U 2  W2 x cos 2 phs  W2 y sin 2 phs pcrit ,
2
3
4
 U 4  W42 x cos 2 phs  W42 y sin 2 phs  W44 x cos 4 phs  W44 y sin 4 phs pcrit
4

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
15
Long-Offset Expansions
 Expansions of three moveout components in the
proximity of critical slowness
c L z fast
hL 
 h L, slow  O 

cT z fast
hT 
 hT , slow  O 

c t t o, fast
t
 t slow  O 

  1
ph
ph 
p crit
2
ph
,
  1
 Unbounded term includes the small value in the
denominator
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21
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 h L, slow , hT , slow , t slow
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22
Contribution of Fast Layer into Moveout
 We assume that the phase velocity direction in the
local orthorhombic frame of the “fast” layer is
n1, n2 , n3 , where the vertical component is
infinitesimal, and the horizontal components are

n1  cos  phs  x1

2
1  n3

, n2  sin  phs  x1

2
1  n3
 x1 is the local ORT axis of the “fast” layer
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23
Contribution of Fast Layer (Continued)
 The horizontal slowness in the “fast” layer is
ph 
sin  phs

Vphs  phs ,  phs
1  cos  phs
2



Vphs  phs ,  phs


2
1  n3
Vphs n1 , n2 , n3 
 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
2
2
infinitesimal parameter   1  ph / pcrit
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24
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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25
Combined Moveout Functions
 The moveout approximation functions are
AL p h 
 C L ph  
hL 

3
3
AT p h  BT p h  CT p h   DT p h 
hT 

2
4
2
4
t o  At p h  Bt p h  C t p h   Dt p h 
t

3
BL ph
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3
DL ph 
ph
ph 
p crit
0  ph  1
  1
2
ph
26
Intercept Time
 With the parametric functions, we obtain the
moveout approximation in   p domain in a
straightforward way
  ph   t  ph   p h  ph   h  ph 
 The horizontal slowness has no transverse
component while the offset has both components
 The intercept time simplifies to
  p h   t  p h   p h hL  p h 
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28
Test for Multi-Layer Structure
#
δ1
δ2
δ3
ε1
ε2
γ1
γ2
V, km/s
f
Δz, km
φax
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
3.5
0.78
0.5
140
0.05
0.12 -0.17 0.05 -0.06
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o
29
2.5
0
Lengthwise & transverse offset components
vs. horizontal slowness, ψphs=0
23.2
31.3
40.0
15.3
7.6
Relative error of lengthwise & transverse offset
components vs. horizontal slowness, ψphs=0
40.0
23.2
31.3
15.3
0
7.6
2.E-03
θphs , deg
θphs , deg
2.0
0.E+00
1.5
Exact hx
Exact hy
Approx hx
Approx hy
1.0
0.5
0.0
Relative error
Normalized offset components
Lengthwise and Transverse Offset Components
vs. Slowness for Constant Phase Azimuth
-2.E-03
Error hx
Error hy
-4.E-03
-6.E-03
-8.E-03
0.0
0.2
0.4
0.6
0.8
Normalized horizontal slowness
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1.0
0.0
0.2
0.4
0.6
0.8
Normalized horizontal slowness
1.0
30
Lag between Acquisition Azimuth
and Phase Velocity Azimuth
12
0
Lag between acquisition and phase
velocity azimuths, ψphs=0
31.3
40.0
23.2
15.3
7.6
Lag between acquisition and phase velocity
azimuths vs. phase velocity azimuth, θphs=38o
10
θphs , deg.
10
8
Exact
Approx
6
4
Azimuthal lag, deg
Azimuthal lag, deg
8
6
4
2
Exact
Approx
0
-2
-4
2
-6
0.0
0.2
0.4
0.6
0.8
Normalized horizontal slowness
1.0
Const. phase azm, varying reflection angle
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0
30
60
90
120
150
Phase velocity azimuth, deg.
180
Const. reflection angle, varying azimuth
31
Traveltime vs. Slowness & its Error for
Constant Phase Velocity Azimuth
Traveltime vs. slowness, ψphs=0
2.50
0
7.6
15.3
23.2
31.3
Relative error of traveltime vs. slowness, ψphs=0
40.0
1.4E-02
θphs , deg.
7.6
15.3
23.2
31.3
40.0
θphs , deg.
1.2E-02
2.00
1.75
Exact
Approx
1.50
Relative time error
2.25
Normalized time
0
1.0E-02
8.0E-03
4.0E-03
1.25
2.0E-03
1.00
0.0E+00
0.0
0.2
0.4
0.6
0.8
Normalized horizontal slowness
traveltime vs. slowness
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1.0
Error
6.0E-03
0.0
0.2
0.4
0.6
0.8
Normalized horizontal slowness
1.0
error of traveltime vs. slowness
32
Traveltime vs. Offset for Constant Phase Azimuth:
Parametric Model and Alkhalifah Strong & Weak
Traveltime vs. offset, ψphs=0
36.6
38.8
39.5
39.7
0.04
θphs , deg.
Normalized traveltime
3.5
3.0
Exact
A Strong
A Weak
Approx
2.5
2.0
1.5
1.0
Relative error of traveltime
0
4.0
Relative error of traveltime vs. offset, ψphs=0
0
36.6
38.8
39.5
39.7
θphs , deg.
0.03
0.02
A Strong
A Weak
Approx
0.01
0.00
-0.01
0.0
1.0
2.0
3.0
Normalized offset
traveltime vs. offset
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4.0
0
1
2
3
Normalized offset
4
error of traveltime vs. offset
33
Lengthwise and Transverse Offset Components
vs. Phase Azimuth for Constant Reflection Angle
Normalized offset components
1.5
1.0
Exact hx
Exact hy
Approx hx
Approx hy
0.5
0.0
-0.5
0
30
60
90
120
150
Phase velocity azimuth, deg.
180
lengthwise & transverse offset components
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Absolure error of offset components
Lengthwise & transverse offset components
vs. phase velocity azimuth, θphs=38o
Absolute error of lengthwise & transverse offset
components vs. phase velocity azimuth, θphs=38o
4.0E-03
0.0E+00
-4.0E-03
Error hx
Error hy
-8.0E-03
-1.2E-02
-1.6E-02
0
30
60
90
120
Phase velocity azimuth, deg.
150
180
error of offset components
34
Traveltime vs. Phase Azimuth for Constant Refl.
Angle: Parametric and Alkhalifah Strong & Weak
Traveltime vs. phase velocity azimuth, θphs=38o
Error of traveltime vs. phase velocity azimuth, θphs=38o
0.04
Normalized time
1.50
1.45
1.40
Exact
A Strong
A Weak
Approx
1.35
1.30
1.25
Relative traveltime error
1.55
0.03
0.02
A Strong
A Weak
Approx
0.01
1.20
0.00
1.15
0
30
60
90
120
150
Phase velocity azimuth, deg.
180
traveltime vs. phase velocity azimuth
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0
30
60
90
120 150
Phase velocity azimuth, deg.
180
error of traveltime vs. phase azimuth
35
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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40
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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41
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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42