http://www.earthdoc.org/publication/publicationdetails/?publication=80327
1-4 June 2015 | IFEMA Madrid
Tu N102 14
Fourth-order NMO Velocity for Compressional
Waves in Layered Orthorhombic Media
Z. Koren* (Paradigm) & I. Ravve (Paradigm)
SUMMARY
We derive azimuthally dependent fourth-order effective velocity and moveout parameters for
compressional waves propagating in a layered model that consists of orthorhombic layers. The subsurface
layered medium is considered as a locally 1D model, where each layer can be characterized by
orthorhombic parameters. The orthorhombic layers have a common vertical axis but different azimuthal
orientations of horizontal axes. For a 1D vertically varying anisotropic model, the azimuth of the phase
velocity is the same for all layers, while the azimuths of the ray velocity are generally different. We extend
the existing studies on the moveout in an azimuthally anisotropic model, accounting for the azimuthal
deviation between the phase and ray velocities. We compute the lag between the azimuth of the surface
offset (source-receiver vector) and the phase velocity azimuth. An effective model that replaces the multilayer model with a single azimuthally anisotropic layer is derived, whose moveout and offset azimuth are
identical to those of the layer package up to the fourth-order terms. We verify the accuracy of the
approximation for small to moderate reflection angles vs. exact analytical ray tracing. The proposed
approximation is in particular important when analyzing residual moveouts measured along full-azimuth
common image reflection angle gathers.
77th EAGE Conference & Exhibition 2015
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1-4 June 2015 | IFEMA Madrid
Introduction
Orthorhombic models comprise subsurface anisotropy caused by vertical azimuthally-aligned
fractures and layering, or by two different orthogonal sets of fractures, with or without layering.
Fourth-order anisotropic moveout approximations have been studied for orthorhombic models by
Tsvankin (1997), Grechka and Tsvankin (1999), Bakulin et al. (2000), Pech and Tsvankin (2004),
Grechka et al. (2005). Recent advanced angle domain migrations map the seismic data into the local
phase angle/azimuth domain (e.g., Koren and Ravve, 2011). Residual moveout analysis along this
type of phase angle/azimuth common image gathers requires special moveout (and residual moveout)
formulation. We present a new derivation for the azimuthally-dependent fourth-order effective
velocity and moveout approximation in layered orthorhombic media. The moveout approximation is
valid for strong anisotropy and is formulated with respect to the phase velocity azimuth.
Wave Propagation in Orthorhombic Layered Medium
Due to Snell’s law in 1D medium, the lateral slowness and azimuth of the phase velocity do not
change along the whole package of layers. To derive the moveout approximation in the phase azimuth
domain, we present the phase velocity as a series approximation for the greatest eigenvalue of the
Christoffel matrix, and we then find the corresponding eigenvector that describes the polarization of
the compressional wave. We compute the ray velocity either through the polarization vector, or
through the gradient of the phase velocity magnitude in the directional space. The ray velocity, in
turn, yields the lateral propagation and traveltime through the layers. Next we expand the reflection
traveltime in series of a small surface offset, with hyperbolic and non-hyperbolic terms. The nonhyperbolic term yields azimuthally-dependent fourth-order effective velocity.
Effective Model
The phase velocity has a constant azimuth and a vertically varying dip. For a package of layers, the
preferred parameter of the ray pair may be the constant (depth independent) horizontal slowness
vector presented by a magnitude ph and a phase azimuth M . For nearly vertical rays studied for the
short offset moveout approximation, the lateral slowness magnitude is considered small. The lateral
propagation through all layers can be added only geometrically, like vectors, because the individual
propagations depend on the ray velocities, whose azimuths are different for various layers and all of
which generally differ from the phase velocity azimuth. We distinguish two components of the lateral
propagation: the lengthwise component hx along the phase slowness azimuth, and the transverse
component h y normal to this azimuth. For each of the two components, the contributions of layers
may be added. For the effective model, both components of the lateral propagation and the traveltime
coincide with those of the original multi-layer set, up to a specified order. In our model, the error
terms include slowness to the fifth power for the propagation components (distances between the
source and receiver on the surface) and slowness to the sixth power for the reflection traveltime. The
effective model may be described by eight independent moveout coefficients. Three of them,
U 2 ,W2 x ,W2 y belong to the second-order terms of the time series expansion, and the other five
U 4 ,W42x ,W42 y ,W44x ,W44 y are related to the fourth-order terms. The second-order parameters define
the fast and slow NMO velocities and the effective azimuth of the slow velocity,
W2 y
W2 x
U 2 W2 2
U 2 W2
V2,2fast
, V2,slow
, cos 2M 2,slow
, sin 2M 2,slow
, W2
tv
tv
W2
W2
W22x W22y ,
(1)
where tv is the two-way vertical time. The lengthwise component of the lateral propagation reads,
hx U 2 ph cos 2M W2 x sin 2M W2 y ph U 4 ph3
cos 2M W42x sin 2M W42 y ph3 cos 4M W44x sin 4M W44 y ph3 O ph5 .
The corresponding transverse component reads,
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IFEMA Madrid, Spain, 1-4 June 2015
(2)
1-4 June 2015 | IFEMA Madrid
hy
sin 2M W2 x cos 2M W2 y ph 12 sin 2M W42x cos 2M W42 y ph3
sin 4M W44x cos 4M W44 y ph3 Oph5 .
(3)
The reflection traveltime t for the package is,
1
1
3
t tv
U 2 ph2 cos 2M W2 x sin 2M W2 y ph2 U 4 ph4
2
2
4
(4)
3
4 3
4
6
cos 2M W42x sin 2M W42 y ph cos 4M W44x sin 4M W44 y ph O ph ,
4
4
The above global effective moveout parameters can be either obtained from azimuthally-dependent
moveouts, or computed by summing the corresponding local parameters of the individual layers:
u2 , w2 x , w2 y and u4 , w42x , w42 y , w44x , w44 y , which depend on the thickness, azimuthal orientation
and elastic properties. We have derived the explicit expressions for all eight moveout coefficients for
an orthorhombic layered model, but the formulae are too lengthy for this abstract.
Effective Fourth-Order Moveout Velocity
The fourth-order effective velocity V4 is related to the normalized coefficient A4 in the moveout
series and to the effective anellipticity Keff . According to Tsvankin and Thomsen (1994),
t 2 t v2
h2
t v2
t v2 V22
A4
h4
˜
t v2 V22 t v2 V22 E h 2
, E
A4 Vh2
Vh2 V22
,
V 4 V24
4
, A4
4V24
A4
2Keff
,
(5)
where Vh is the horizontal velocity and V2 is the NMO velocity. Although this approximation was
originally derived for a VTI medium, it is suitable for azimuthal anisotropy as well. Alkhalifah and
Tsvankin (1995) suggested a simplified form of approximation 5 for compressional waves, where
E 1 2Keff . We present the fourth-order velocity for a layered orthorhombic structure as,
KV M V44 M t v3
˜
>
V2,4slow cos2 M M 2,slow V2,4fast sin 2 M M 2,slow
V2,2slow cos2
M
M 2,slow V2,2fast
2
sin M M 2,slow
@
4
.
(6)
Kernel KV , in turn, may be presented as a bilinear form of row vector of length 5 that includes the
high-order parameters, 5u 7 matrix whose components depend on the low-order parameters, and
column vector of length 7 that depends on the phase velocity azimuth alone.
Numerical Tests
We performed numerical tests to assess the accuracy of the fourth-order approximation. We compared
it with the second-order approximation and with an exact ray tracing. We studied a layered structure
and a single orthorhombic layer. The layer properties are presented in Tables 1 and 2, respectively.
Table 1 Layer properties in multi-layer structure.
G1
.05
.12
.04
.03
.06
#
1
2
3
4
5
G 2 G 3 H1 H 2
.05
0
.10 .10
-.08 .03 .05 -.03
-.07 -.01 .05 -.02
-.08 -.02 .05 -.04
-.07 .01 .05 -.04
J1
.03
.03
.02
.03
.03
J 2 VP
.03 2.0
-.03 2.5
-.02 2.75
-.02 3.0
-.02 3.25
Max
0
20
40
65
110
Table 2 Properties of single layer.
#
1
G1
.1
G2
-.1
G 3 H1 H 2
J1 J 2
.05 .12 -.08 .08 -.07
VP
3.0
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Max
110
1-4 June 2015 | IFEMA Madrid
Two-way time vs. half-opening angle
2.35
3.5
2.30
3.0
2.25
2.5
2.0
Tracing
Order 1
Order 3
1.5
1.0
Two-way time (s)
Two-way lateral propagation (km)
Lateral propagation vs. half-opening angle
4.0
0.5
2.20
2.15
Tracing
Order 2
Order 4
2.10
2.05
2.00
1.95
1.90
0.0
0
5
10 15 20 25 30 35
Half-opening angle (deg)
40
0
45
Two-way time vs. full offset
40
45
Two-way time vs. phase velocity azimuth
2.30
2.11
2.25
2.10
2.20
2.15
2.10
Tracing
Order 2
Order 4
2.05
2.00
Two-way time (s)
Two-way time (s)
10 15 20 25 30 35
Half-opening angle (deg)
Figure 2.Two-way traveltime vs. half-opening
angle for multi-layer orthorhombic structure.
Figure 1 Lateral propagation vs. half-opening
angle for multi-layer orthorhombic structure.
2.09
2.08
Tracing
Order 2
Order 4
2.07
2.06
1.95
1.90
2.05
0.0
0.5
1.0
1.5 2.0 2.5
Offset (km)
3.0
3.5
0
Lengthwise propagation vs. half-opening angle
1.6
Tracing
Order 1
Order 3
0.4
0.0
0
5
10 15 20 25 30 35
Half-opening angle (deg)
40
45
Figure 5 Lengthwise component of propagation vs.
half-opening angle for a single orthorhombic layer.
Two-way transverse propagation (km)
2.0
0.8
60
90
120
Azimuth (deg)
150
180
Transverse propagation vs. half-opening angle
2.4
1.2
30
Figure 4 Two-way time vs. phase velocity
azimuth for multi-layer orthorhombic structure.
Figure 3 Two-way time vs. full offset for
multi-layer orthorhombic structure.
Two-way lengthwise propagation (km)
5
0.35
0.30
0.25
0.20
Tracing
Order 1
Order 3
0.15
0.10
0.05
0.00
0
5
10 15 20 25 30 35
Half-opening angle (deg)
40
45
FFiigure 6 Transverse component of propagation vs.
hahalf-opening angle for a single orthorhombic layer.
For all layers, we assumed VP 2VS , where VP and VS are vertical compressional and x1 -polarized
shear velocities, respectively. The layer thickness is 500 m for the multi-layer model, and 1 km for the
single layer. The velocities are in km/s, and the azimuths Max of the local orthorhombic axes x1 are
in degrees. The layers are numerated from top to bottom, where only the upper layer is VTI. Figures 1
and 2 present the lateral propagation and the traveltime vs. half-opening angle for the multi-layer
structure. In Figure 3 we present the two-way time vs. offset. In all these cases, the azimuth of the
phase velocity was zero (in the global frame), and the half opening angle was varying up to 45
degrees. In Figure 4 we present the traveltime for a fixed half-opening angle 30o and all azimuths of
77th EAGE Conference & Exhibition 2015
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the phase velocity. In Figures 5 and 6 we plot the
lengthwise and transverse components of the offset
(i.e., along and across the phase velocity azimuth)
for the single layer. In Figures 7 and 8 we plot the
diagrams for the NMO velocity V2 and the fourthorder effective velocity V4 for the multi-layer
structure and the single layer, respectively. Finally,
in Figure 9 we present the azimuthally-dependent
effective anellipticity Keff .
V2 and V4 vs. phase velocity azimuth
3
2
Vy (km/s)
1
0
V2
V4
-1
-2
-3
-3
-2
-1
0
1
2
Vx (km/s)
3
4
Conclusions
5
Figure 7 NMO velocity and fourth-order
effective velocity vs. phase velocity azimuth
for multi-layer orthorhombic structure.
V2 and V4 vs. phase velocity azimuth
3
Vy (km/s)
2
1
0
V2
V4
-1
-2
-3
-4
-3
-2
-1
0
1
Vx (km/s)
2
3
4
Figure 8 NMO velocity and fourth-order
effective velocity vs. phase velocity
azimuth for a single orthorhombic layer.
Effective anellipticity
0.04
0.03
Eta
0.02
0.01
0.00
Multi-layered
Single layer
We derived new relationships for moveout
coefficients that constitute the fourth-order
effective velocity for an azimuthally anisotropic
layered model. The effective model represents a
multi-layer anisotropic structure with orthorhombic
layers with different azimuthal orientations by a
single effective layer. It yields the same
components of the lateral propagation and the same
traveltime as the original structure for any azimuth
of the phase velocity, with fourth-order accuracy. It
is clearly shown that even for moderate reflection
angles (less than 30o) the accuracy of the secondorder approximation is insufficient, and the
additional fourth-order terms are essential to match
the exact ray tracing solution.
References
Alkhalifah, T. and Tsvankin, I. [1995] Velocity
analysis
for
transversely
isotropic
media. Geophysics, 60, 1550-1566.
Bakulin, A., Grechka, V. and Tsvankin, I. [2000]
Estimation of fracture parameters from reflection
seismic data – Part II: Fractured models with orthorhombic symmetry. Geophysics 65, 1803-1817.
Grechka, V., Pech, A. and Tsvankin, I. [2005]
Parameter estimation in orthorhombic media using
multicomponent wide-azimuth reflection data.
Geophysics, 70, D1-D8.
-0.0
Grechka, V., and Tsvankin, I. [1999] 3-D moveout
0
30
60
90
120
150
180
Phase velocity azimuth (deg)
velocity analysis and parameter estimation for
orthorhombic
media. Geophysics, 64, 820-837.
Figure 9 Effective anellipticity vs. phase
Koren, Z. and Ravve, I. [2011] Full azimuth
velocity azimuth for the two structures.
subsurface angle domain wavefield decomposi-tion
and imaging, Part1: Directional and reflection image gathers. Geophysics, 76, S1-S13.
Pech, A. and Tsvankin, I. [2004] Quartic moveout coefficient for a dipping azimuthally anisotropic
layer. Geophysics, 69, 699-707.
Tsvankin, I. [1997] Reflection moveout and parameter estimation for horizontal transverse
isotropy. Geophysics, 62, 614-629.
Tsvankin, I., and Thomsen, L. [1994] Nonhyperbolic reflection moveout in anisotropic
media. Geophysics, 59, 1290-1304.
-0.01
77th EAGE Conference & Exhibition 2015
IFEMA Madrid, Spain, 1-4 June 2015