TB, PM, JGE/205491, 27/10/2005
NANJING INSTITUTE OF GEOPHYSICAL PROSPECTING AND INSTITUTE OF PHYSICS PUBLISHING
JOURNAL OF GEOPHYSICS AND ENGINEERING
doi:10.1088/1742-2132/2/0/000
J. Geophys. Eng. 2 (2005) 1–11
Layer stripping of azimuthal anisotropy
from P-wave reflection moveout in
orthogonal survey lines
Shangxu Wang1 and Xiang-Yang Li2
1
CNPC Geophysical Key Lab, China University of Petroleum, Changping, Beijing 102200, China
Edinburgh Anisotropy Project, British Geological Survey, Murchison House, West Mains Road,
Edinburgh EH9 3LA, UK
2
Received 2 August 2005
Accepted for publication 13 October 2005
Published DD MMM 2005
Online at stacks.iop.org/JGE/2/1
Abstract
This paper presents a layer-stripping procedure to determine interval measurements of fracture
parameters in multi-layered fractured media with vertically varying strike directions. The
procedure is based on the P-wave travel time difference between two orthogonal seismic survey
lines, and this difference is referred to as the P-wave azimuthal moveout response (AMR). The
interval AMR of a fracture target for a fixed offset is a function of cos 2(α − ϕi ) with respect to
the line-azimuth α and the fracture-strike azimuth ϕi . Consequently two pairs of orthogonal
survey lines can be used to determine the local fracture strike ϕi if the interval AMR of the
target is known. In the case of a weakly fractured overburden underlain by a fractured target,
layer stripping can be achieved through the alignment of the top-target event by performing
NMO correction separately for all survey lines. The interval AMR of the target layer may then
be calculated from the residual moveout of the bottom-target event, if any. In the general case,
a ray-tracing procedure, similar to that used in AVO analysis, is required to perform effective
layer stripping. Full-wave modelling is used to verify and illustrate these procedures.
Keywords: layer stripping, azimuthal anisotropy, fractures, reflection moveout
1. Introduction
Traditionally, azimuthal anisotropy has been largely associated
with shear-wave splitting (birefringence). In the last century
and early 1990s, much effort has been focused on the analysis
of shear-wave splitting by recording multicomponent shearwave data (Crampin 1985). These studies revealed that depth
changes of the principal direction of azimuthal anisotropy are
common in the Earth’s crust, and a layer-stripping process
is then necessary to obtain the interval measurements of
azimuthal anisotropy. For shear-wave data analysis, various
processing techniques have been developed to perform layer
stripping (Winterstein and Meadows 1991, MacBeth et al
1992, Thomsen et al 1995, Li and MacBeth 1997, Dai and
Li 1998).
In recent years, the use of azimuthally varying information
in P-waves for studying azimuthal anisotropy has become a
common practice (Tsvankin 1995, Lynn et al 1996, MacBeth
1742-2132/05/000001+11$30.00
and Li 1999, Smith and McGarrity 2001, Hall and Kendall
2003, and amongst others). However, most of these studies
assume either an azimuthally isotropic overburden, or a
depth-invariant principal direction of azimuthal anisotropy.
To overcome this restriction, Grechka and Tsvankin (1998)
extend the NMO approach of Tsvankin (1995) to vertically
inhomogeneous anisotropic media. In practice, the NMO
approach requires careful data processing to minimize the error
propagation and magnification through various processing
steps (Al-Dajani and Tsvankin 1998), and this can be a
challenge in some cases.
Li (1997, 1999) presented an alternative approach for
determining the fracture orientation from P-wave seismic data,
intended to overcome some of the practical difficulties often
encountered by the use of azimuthal P-wave AVO and NMO
velocity analysis (Al-Dajani and Tsvankin 1998) in marine
streamer data. The new approach is based on the P-wave
travel time (moveout) difference between two orthogonal
© 2005 Nanjing Institute of Geophysical Prospecting
Printed in the UK
1
S Wang and X-Y Li
survey lines, and this difference is referred to as the P-wave
azimuthal moveout response (AMR). This approach requires
a configuration of four intersecting survey lines which form
two orthogonal pairs. The fracture orientation can be obtained
by analysing the cross plot of the two corresponding AMRs.
This technique is straightforward and is particularly useful in
marine exploration with repeated surveys of various vintages
where continuous azimuthal coverage is often not available.
However, this technique, as originally formulated, is restricted
to an azimuthally isotropic overburden.
Changes of fracture strike with depth are in fact very
common, which resulted in an azimuthally anisotropic
overburden. For examples, the studies cited in this paper
(e.g. Winterstein and Meadows (1991), Thomsen et al (1995),
Lynn et al (1996) and Hall and Kendall (2003), amongst others)
showed changes of the fracture orientation with depth. The
only exception is in the case of Smith and McGarrity (2001)
where the overburden appears to be azimuthally isotropic.
Therefore, layer-stripping procedures for evaluating azimuthal
anisotropy are widely applicable.
Here, we extend the approach of Li (1999) to media
with vertically varying fracture orientation. Assuming multilayered azimuthally anisotropic media with depth-change of
fracture orientation, we develop analytical expressions for
quantifying the interval AMR of any given layer in the
multi-layered media. Layer-stripping procedures are then
formulated to obtain the interval AMR. Cross-plotting analysis
may then be performed on these interval AMRs to obtain the
local fracture orientation. Full-wave modelling and synthetic
data examples are used to illustrate the methodology. Note
that real data examples are not presented here for two simple
reasons: first, it would be too expensive to conduct a field trial
for this purpose, and second the result from any real data may
not be conclusive unless confirmed by drilling activities.
2. Fracture-induced azimuthal anisotropy
This section introduces the Thomsen parameters and the
reflection moveout equation for fracture-induced TIH media,
and defines the AMR, which are needed for establishing the
analysis procedures for determining the fracture orientation.
2.1. Thomsen parameters
A medium containing aligned vertical penny-shaped fractures
gives rise to transverse isotropy with a horizontal axis of
symmetry (TIH). Use vp0 and vs0 to denote the vertical
velocities of the P-wave and the fast split shear wave of the TIH
medium, respectively. Under the natural coordinate system
determined by the fracture normal (x1 ), strike (x2 ) and the
vertical axis (x3 ), with the stiffness tensor Cij and density ρ,
the Thomsen parameters may be defined as
C33
C44
,
vs0 =
,
vp0 =
ρ
ρ
C33 − C11
C44 − C66
(1)
=
,
γ =
,
2C11
2C66
δ=
2
(C13 + C66 )2 − (C11 − C66 )2
.
2C11 (C11 − C66 )
These are generic Thomsen parameters defined with
respect to the symmetry axis of the TIH medium. An
alternative way is to define effective Thomsen parameters for
the TIH medium, as described by Tsvankin (1997).
2.2. P-wave moveout equation in a single TIH layer
For a survey line at the azimuthal angle φ to the fracture strike
of a single-layered TIH medium, the reflection moveout can
be written as, following Sayers and Ebrom (1997),
t 2 (φ, x) = t02 +
x2
Ax 4
−
,
2
2
vnmo
x 2 + t02 vp0
(2)
where t (x, φ) is the reflection travel time at offset x, t0 is the
two-way zero-offset travel time, vnmo is the NMO velocity and
A is a moveout coefficient. Equation (2) is obtained for weak
anisotropy (see also Sena (1991) and Li and Crampin (1993)),
and for general anisotropy, an empirical but more accurate
equation is given in Al-Dajani and Tsvankin (1998). From
Al-Dajani and Tsvankin (1998), vnmo and coefficient A can be
written, to the first order in the anisotropy parameters, as
1
1
= 2 [1 − 2(δ − 2) sin2 φ]
2
vnmo
vp0
A=
2( − δ) 4
sin φ.
2
vp0
(3)
(4)
2
Note the coefficient A satisfying A = −t02 vp0
A4 , where A4
is the quartic moveout coefficient defined by Al-Dajani and
Tsvankin (1998). Also, δ − 2 ≈ δ (V ) in the weak-anisotropy
approximation, where δ (V ) is the effective Thomsen parameter
from Tsvankin (1997).
Substituting equations (3) and (4) into equation (2), taking
the square root and linearizing with respect to the anisotropic
parameters and δ gives
x2
t (φ, x) = t02 + 2 [1 − (δ − 2) sin2 θ sin2 φ
vp0
− ( − δ) sin4 θ sin4 φ],
(5)
where θ is the incidence (ray) angle at the reflector measured
from vertical. And the square-root term in equation (5) is a
standard normal moveout term.
Introducing t and t⊥ as the reflection moveouts at offset
x for the survey lines parallel and perpendicular to the fracture
strike, respectively, yields
x2
t (x) = t (φ = 0, x) = t02 + 2 ,
(6)
vp0
t⊥ (x) = t (φ = π/2, x)
= t (x) − t (x)(δ − 2) sin2 θ − t (x)( − δ) sin4 θ.
(7)
Substituting equations (6) and (7) into equation (5) yields
t (φ, x) = t (x) cos2 φ + t⊥ (x) sin2 φ
+ t (x)( − δ) sin4 θ sin2 φ cos2 φ.
(8)
Layer stripping of azimuthal anisotropy
3.1. Cross-plotting analysis for fracture strike
(a)
(b)
Figure 1. Seismic surveys with orthogonal lines: (a) two
intersecting orthogonal lines, and (b) four intersecting lines forming
two orthogonal pairs.
A special four-line configuration can be used to determine
the fracture orientation, utilizing the cos 2φ variation of the
AMR. The four lines form two orthogonal pairs separated by
an arbitrary angle ϕ0 (figure 1(b)). Denote the AMR for the
first set (lines 1 and 3) as t 31 , and the second set (lines 2
and 4) as t 42 . Note that we have used the following angle
definition in figure 1(b): a positive φ representing an anticlockwise rotation from line 1 to the fracture strike. This
is consistent with the 2D rotation convention under a righthanded coordinate system with the third axis pointing to the
reader. It follows that
2.3. Azimuthal moveout response (AMR)
Let us assume two orthogonal CMP (common-mid point) lines
at azimuths φ and π/2−φ measured from the fracture strike in
a single-layered medium (figure 1(a)). The AMR of a fracture
target is defined as the travel time difference (
t) between the
two orthogonal lines from the bottom of the target:
t (φ, x) = t (π/2 − φ, x) − t (φ, x).
t 42
t 31 = t (φ, x) = B cos 2φ;
= t (ϕ0 − φ, x) = B cos 2(ϕ0 − φ),
which leads to
tc42 = B sin 2φ = (
t 42 − cos 2ϕ0 t 31 )/sin 2ϕ0 , (13)
tan 2φ = sin 2φ/cos 2φ = tc42 t 31 .
(14)
(9)
As shown in equation (5), the travel time equation explicitly
contains the square-root moveout term. Sometimes, it may be
convenient to apply a common hyperbolic moveout correction
to both azimuthal lines before the calculation of t. This
implies to re-arrange equation (9) as
x2
2
t (φ, x) = t (π/2 − φ, x) − t0 + 2
vmo
x2
− t (φ, x) − t02 + 2
,
(10)
vmo
Thus, for the four-line configuration in figure 1(b), after
correcting t 42 using equation (13), the cross plot of t 31
versus tc42 shows a linear trend indicating the direction of
2φ with respect to the axis of t 31 . This axis represents the
direction of line 1 in figure 1(b). Thus, a special four-line
configuration allows a simple way for the determination of the
fracture strike using cross-plot analysis.
Since only the moveout difference between two
orthogonal seismic lines shows the cos 2φ variation, a
minimum of two pairs of orthogonal lines is required to
estimate the fracture strike.
3.2. Angle definition and mapping
where vmo is the choosing moveout velocity.
2.4. AMR of a single TIH layer
From equation (8), t can be written as
t (φ, x) = (t⊥ − t ) cos 2φ = B0 (x, , δ) cos 2φ,
(11)
where, to the first order in the anisotropy parameters,
B0 (x, , δ) =
x
sin θ [2 − δ − ( − δ) sin2 θ ].
vp0
(12)
As B0 (x, , δ) is independent of azimuth, equation (11) shows
that in the weak-anisotropy approximation, the AMR is a
function of cos 2φ for a fixed offset. This feature allows us
to determine the fracture strike without the need to know t⊥
and t .
3. AMR analysis: single fractured TIH layer
This section reviews the analysis procedure for a single-layered
TIH medium.
Use φ0 to denote the direction of the linear trend to the t 31
axis. Noting equation (14) gives

t 42


if t 31 > 0,
tan−1 c31


t


42
φ0 = 2φ = tan−1 tc31 + π if t 31 < 0 and tc42 > 0

t



42


tan−1 tc31 − π if t 31 < 0 and tc42 < 0,
t
(15)
where the main ranges of φ0 are (−π, π ), and the axes of t 31
and tc42 form a right-handed coordinate system with the third
axis pointing to thereader. φ0
is positive if the linear trend is in
if the linear trend
quadrants I and II tc42 > 0 , and negative
is in quadrants III and IV tc42 < 0 . Once φ0 is determined,
the fracture strike φ is determined by φ = φ0 /2.
With the above angle definition, the determined fracture
strike from the cross plot can be mapped into the survey lines
in figure 1(b) by analogy of the axis of t 31 in the cross plot
to the direction of line 1 in the acquisition system. A positive
angle indicates an anti-clockwise rotation and a negative angle
indicates a clockwise rotation (figure 1(b)) in a right-handed
3
S Wang and X-Y Li
coordinate system with the third axis pointing to the reader.
Note that if the third axis points away from the reader in
the right-handed system, a positive angle then indicates a
clockwise rotation, whilst a negative angle indicates an anticlockwise rotation.
3.3. Least-square cross-plot analysis
There are several different schemes of least-square analysis
which can be applied to the cross plot. Care should be taken
to resolve the non-uniqueness of the inverse tangent function.
Here we give two examples. The first one is linear regression,
which yields
31
42 x t tc
−1
± π.
(16)
φ0 = tan
31
31
x t t
The summation is over all offsets, and the non-uniqueness can
be resolved in the same manner as in equation (15). The second
scheme is to minimize one of the coordinates by rotating the
axes of t 31 and tc42 , which yields
31
42
2
t
t
1
nπ
c
±
,
φ0 = tan−1 31 x 31
42
42
2
2
− tc tc
x t t
n = 1, 2.
(17)
For this scheme, the non-uniqueness can only be resolved by
interactively checking the cross plot to see which quadrant the
linear trend is located in.
Synthetic and real data examples of the above analysis
procedure can be found in Li (1999).
3.4. Inversion for TIH parameters
Once the fracture strike is determined, equations (6), (7) and
(8) can be used to invert the TIH parameters.
Firstly estimate t (x) and t⊥ (x). For near to middle offset
ranges (the same offset range for AVO analysis), equation (8)
can be further approximated by ignoring the sin4 θ term,
t (φ, x) = t (x) cos2 φ + t⊥ (x) sin2 φ.
(18)
This indicates that for the near-to-mid offset range, the sum
of travel time of any two orthogonal lines is a constant and
equals the sum of the travel time of the two lines parallel and
perpendicular to the fracture strike,
t (φ, x) = t (φ, x) + t (π/2 − φ, x) = t (x) + t⊥ (x). (19)
Combining with equation (11), one can determine t (x) and
t⊥ (x) from any orthogonal line pair,
t (x) = 12 [t (φ, x) − t (φ, x)/cos 2φ]
(20)
t⊥ (x) = 12 [t (φ, x) + t (φ, x)/cos 2φ].
(21)
Secondly analyse the variation of the normalized travel
time with offset. Using equations (6) and (7) gives
t⊥ (x)
= 1 − (δ − 2) sin2 θ − ( − δ) sin4 θ.
(22)
t (x)
Thus, a procedure similar to AVO analysis can be used
to estimate the gradient of the normalized travel time curve.
This gradient equals δ − 2 which is a good estimation of
4
(a)
(b)
Figure 2. (a) A cross-section of multi-layered azimuthally
anisotropic media. Lnk (x) marks the ray path at offset x from the
bottom of the nth layer for the kth line azimuth in a multi-azimuthal
survey. (b) The down-going ray segment components for the ith
layer.
the intensity of the fracturing inside the medium (Li 1997).
We call this procedure normalized travel time versus offset
(NTVO).
In theory, the higher order term sin4 θ may also be used to
yield an estimation of ( − δ). In this way, both and δ may
be determined.
4. AMR analysis: multi-layered media
Here we consider the more general case of multi-layered
fractured TIH media with arbitrary fracture orientations, and
the case with uniform fracture orientation is treated as a special
case. We first introduce the concepts of total and interval
AMRs and then derive the expressions for these AMRs which
are needed to establish the layer-stripping procedure.
4.1. Total AMR
Consider multi-azimuth seismic surveys over a stack of n TIH
layers with arbitrary fracture orientations. Assume a reflection
ray from the bottom of the nth layer with offset x at the kth line
azimuth with angle α from North. This ray is referred to as
Lnk (x) (figure 2). For the ith layer, we introduce the following
azimuthally invariant interval properties: vp0i , vertical
P-wave velocity; t0i , one-way zero-offset travel time; i and
δi , Thomsen parameters and ϕi , fracture-strike azimuth from
North. For the ray segment in the ith layer corresponding to the
kth line-azimuth, I use xki and θki as the horizontal component
and incidence angle of the ray segment, respectively, and tki
as the travel time along the ray segment (figure 2(b)).
The total AMR for the stack of n layers associated with
ray Lnk (x) is defined as
t1n (α, x) = t (α + π/2, x) − t (α, x),
(23)
where t (α, x) is the travel time for the kth line azimuth and
t (α + π/2, x) is the travel time for the line perpendicular to
the kth line azimuth. Subscript 1n denotes that the AMR is
defined from layer ‘1’ to layer n.
Similarly to the single layer case, introduce ti and t⊥i
as the travel time inside the ith layer for the survey lines
parallel and perpendicular to the fracture strike of the ith layer,
respectively, and xi , θi , x⊥i and θ⊥i as the corresponding raysegment components with the same total offset x. As shown
Layer stripping of azimuthal anisotropy
in the appendix, to the first order of the anisotropy parameters,
we have
2
xi
ti (xi ) = t0i2 + 2 ,
(24)
vp0i
t⊥i (xi ) = ti − ti (δi − 2i ) sin2 θi − ti (i − δi ) sin4 θi ,
(25)
tki (xi ) = ti cos2 (α − ϕi ) + t⊥i sin2 (α − ϕi )
+ ti (i − δi ) sin4 θi sin2 (α − ϕi ) cos2 (α − ϕi ),
4.3. Special case: multi-layered media with uniform fracture
orientation
Assume the target fractured layer is the nth layer in the media.
As shown in equation (A.15) in the appendix, the total AMR
is reduced to
t1n (φ, x) = (t⊥ − t ) cos 2φ = B(x, , δ) cos 2φ.
If the target layer is embedded into an azimuthally isotropic
background, B(x, , δ) is evaluated locally at the fractured
target,
(26)
B(x, , δ) = Bn =
and
t1n (α, x) =
n
Bi (i , δi , xi ) cos 2(α − ϕi ),
(32)
(27)
2xn
sin θn [2 − δ − ( − δ) sin2 θn ].
vp0n
(33)
i=1
where
Bi (i , δi , xi ) =
2xi
sin θi [2i − δi − (i − δi ) sin2 θi ].
vp0i
(28)
Note that the ray-segment components xi and θi are
used in all the above equations, and should be evaluated at the
azimuthal direction parallel to the local fracture strike.
Equation (32) has exactly the same form as the singlelayer medium. Thus, the same analysis procedure derived for
a single layer can also be used for multi-layered media with
uniform fracture orientation. In this case, layer stripping is not
required for the fracture orientation, but is required to invert
for the anisotropy parameters.
5. Layer-stripping procedures and results
5.1. Layer stripping by isotropic ray tracing
4.2. Interval AMR
Since equation (28) has the same form as the single-layer case
but all in terms of the local interval quantities in the direction
parallel to the fracture strike. Thus one can define
ti (α − ϕi , xi ) = Bi (i , δi , xi ) cos 2(α − ϕi ),
(29)
as the interval AMR for the ith layer. Comparing the definition
of (29) with the single-layer response (11) reveals that the
interval AMR is the travel time difference between two
orthogonal lines within the ith layer with offset xi .
Thus, the total AMR for the stack of n layers is the sum
of the interval AMR for each individual layer,
t1n (α, x) =
n
ti (α − ϕi , xi ),
(30)
i=1
where offset x satisfies,
x=2
n
xi .
(31)
i=1
Note that equation (31) implies that the ray is confined to
the incidence plane and is valid only for weak azimuthal
anisotropy.
As the interval AMR ti also shows cos 2φi variations,
where φi = α − ϕi , the same four-line configuration and
cross-plotting procedure as in the single-layer case can be
used to determine the local fracture orientation φi , if ti can
be extracted from the total AMR (30) by some forms of layer
stripping. Also equations (24), (25) and (26) have the same
form as the single-layer case; the same procedure as in the
single-layer case can also be used to estimate ti and t⊥i , and
to perform NTVO analysis to obtain the fracture intensity.
From equations (29) and (30), one can see that layer stripping
requires knowing the ray-segment components xi and
θi . Accurately determining these ray-segment components
requires anisotropic ray tracing.
Since the anisotropic
parameters are unknown at this stage of data processing,
anisotropic ray tracing is thus not feasible. Fortunately, as
shown in equation (A.17) in the appendix, the error introduced
to the AMRs by performing isotropic ray tracing is of second
order in terms of the anisotropy parameters. Thus, for
multi-layered weakly TIH media, one may use the isotropic
ray-tracing procedure, similar to AVO analysis, to perform
layer stripping. The full layer-stripping procedure may be
summarized as follows:
Data preparation
(1) Locate the four CDP gathers at the intersecting point of the
four lines (figure 1(b)), and perform velocity analysis for
each CDP gather separately to build an optimum velocity
model.
(2) Pick travel times for the four CDP gathers without NMO
correction and build the travel time table for all layers.
(3) Perform isotropic ray tracing and build an offset/raysegment-component table for each layer.
(4) For layer 1, calculate
t131 (x) = line 3 (layer 1) − line 1 (layer 1),
t142 (x) = line 4 (layer 1) − line 2 (layer 1),
perform cross-plot analysis using the single-layer method
to determine ϕ1 , and store t131 (x) and t142 (x) in a table
for future use.
5
S Wang and X-Y Li
Receiver
Source
Fr
ac
tu
re
St
rik
e(
lay
22
1)
o
er
Ttop
Tbottom
3
Line
(a)
Line
o
1
e2
Lin
Fracture Layer
Fracture Strike (Layer 2)
75
e4
Lin
(b)
Figure 3. (a) A cross section of a two-layer model for illustrating overburden correction. The model consists of a fractured target overlain
by an azimuthally anisotropic overburden. (b) A plan view of a multi-azimuthal survey. The survey consists of four intersecting lines
forming two orthogonal pairs.
Table 1. The elastic parameters for the model in figure 3(a). Layer 1 is a fractured overburden, layer 2 is the fractured target and layer 3 is
an isotropic basement.
Layer 1: fracture
ρ = 2.3 g cm−3 , vp = 3048 m s−1 , vs = 1574 m s−1 . Aspect ratio: 0.01,
Crack density: 10% . Thickness = 1.5 km
Layer 2: fracture
ρ = 2.19 g cm−3 , vp = 2183 m s−1 , vs = 1502 m s−1 . Aspect ratio: 0.01,
Crack density: 10%. Thickness = 0.3 km
Layer 3: isotropic
ρ = 2.3 g cm−3 , vp = 3048 m s−1 , vs = 1574 m s−1 . Half space
Main stripping loop
greater than x2 . Thus one may approximate t1 (x1 ) by
(1) For layer 2, select one of the orthogonal pairs (for example
lines 1 and 3) and calculate
31
t12
(x) = line 3 (layer 2) − line 1 (layer 2)
= t131 (x1 ) + t231 (x2 ),
where x1 and x2 are the ray-segment components in layers
1 and 2 respectively, calculated in step 3.
(2) From the offset-
t1 (x) table stored in step 4, estimate
t131 (x1 ), calculate
31
t231 (x2 ) = t12
(x) − t131 (x1 )
(34)
and store the results in a table for future use.
(3) Repeat the above procedures for lines 2 and 4, and obtain
t242 (x2 ).
(4) Cross plot the two interval AMRs, t231 and t242 to
determine ϕ2 .
(5) Repeat the above four steps for the remaining layers.
t1 (x1 ) ≈ t1 (x),
(35)
whilst t1 (x) can be compensated for by NMO correction
alone. The procedure can be summarized as follows:
(1) Locate the four CDP gathers at the intersecting point, and
carefully select the overburden and target horizons.
(2) Perform velocity analysis and NMO correction to the
overburden horizon for each CDP gather separately, so
that the overburden horizons in all four gathers are
aligned properly. In this way, the azimuthal AMR in
the overburden is completely removed.
(3) Apply NMO correction to the target horizon (the bottom
of the target) using the same velocity as the overburden.
So that, the amount of moveout removed from the target
horizon is almost the same as t1 (x).
(4) Pick the residual moveout for the target horizon. Using
equation (35) gives
t231 (x) = Residual moveout (line 3)
5.2. Layer stripping by NMO correction
Most often, one may approximate the multi-layered TIH media
in terms of an overburden underlain by a target layer with
respect to the vertical variation of the azimuthal anisotropy
(figure 3). In this way, layer stripping is reduced to some forms
of overburden correction. For a weakly anisotropic overburden
with its thickness far greater than the thickness of the target
layer and with weak impedance contrast, this overburden
correction of azimuthal anisotropy may be accomplished
simply by NMO correction within the conventional near-tomid offset ranges. This is because in such case, x1 is often far
6
− Residual moveout (line 1)
t242 (x)
(36)
= Residual (line 4) − Residual (line 2).
(5) Perform cross-plotting analysis of t231 (x) and t242 (x)
to quantify the fracture strike of the target.
5.3. Testing with full-wave synthetic data
A three-layer model is constructed to illustrate the layerstripping procedure using full-wave synthetics calculated by
the reflectivity method (Taylor 1996). As shown in table 1
and figure 3(a), the first layer is 1500 m thick, representing a
Layer stripping of azimuthal anisotropy
Figure 4. The four CDP gathers at the intersecting point in figure 3(b), calculated for the two-layer model in figure 3(a) with the parameters
listed in table 1. The red line marks the travel time picks along the top-target event and the blue line marks the bottom-target event. The
lines are: CDP 1001-line 1, 1002-line 2, 1003-line 3 and 1004-line 4.
weakly anisotropic overburden with 3% azimuthal anisotropy
and a fracture strike of 22◦ from line 1. The second layer is
300 m thick, representing a fractured target with 10%
azimuthal anisotropy and a fracture strike of 75◦ from line 1.
The four lines are separated by 45◦ (ϕ0 = 45◦ , figure 3(b)).
The calculated CDP gathers at the intersecting point of the
four lines are show in figure 4(b), and travel time picks of
the top- and bottom-target events are marked with the red and
the blue lines, respectively.
Firstly, we apply the single-layer method for each
individual event. From the travel time picks, we calculate
t131 = line 3–line 1 and t142 = line 4–line 2 for the top
31
42
= line 3–line 1 and t12
= line 4–line 2 for
event, and t12
the bottom event. Noting ϕ0 = π/4 here, we cross plot t131
31
42
versus t142 , t12
versus t12
(figures 5(a) and (c)). We also
calculate the fracture orientation using equation (15) for each
individual offset (figures 5(b) and (d)).
Figure 5(a) reveals a perfect linear trend, confirming the
prediction of equation (14), and the trend is at a direction of 44◦
(2 × 22◦ ) and the estimations for each individual offset yield
22◦ on average. All this agrees with the model parameter
(table 1). However, analysis of the bottom event shows a
deviated trend (figure 5(c)) and individual offset estimation
gives an average angle of 30◦ , quite close to the fracture
orientation in the overburden. This is not surprising since the
total AMR for the two layers is largely due to the overburden
because of its thickness.
Secondly, we perform layer stripping using the ray-tracing
procedure in order to estimate the fracture orientation in the
target. From the picked travel times and the ray-segment
components, the interval AMR is calculated: t231 (x2 ) =
31
42
(x) − t131 (x1 ) and t242 (x2 ) = t12
(x) − t142 (x1 ). The
t12
cross-plot and the individual offset estimation are shown in
figures 7(a) and (b). Now the linear trend for the target event is
improved at the angle of 150◦ (2 × 75◦ ) and the average angle
from the individual offset estimation is 75◦ . This confirms the
validity of the ray-tracing layer-stripping procedure.
Thirdly we perform the overburden correction procedure
based on NMO correction. Figure 6 shows the NMO-corrected
CDP gathers. The top events in all four lines are reasonably
flat and show almost no azimuthal variation, that is, the
AMR of the overburden is fully compensated. However,
there are significant residual moveouts in the bottom-target
events, which also clearly display azimuthal variations. The
picked residual moveout of the bottom target was then input to
equation (36) to calculate the interval AMR. Figures 7(c) and
(d) show the cross plot and the estimation for each offset. The
cross plot shows a slightly degraded linear trend. However,
the overall trend direction is 150◦ and the average angle from
each offset is 75◦ . These results are consistent with the model
and the ray-tracing result.
Mathematically speaking, cross plotting represents a
least-square linear-regression process. Thus, it is very robust
in the presence of noise. The method is indeed able to account
for more than 50% of synthetic random noise, as demonstrated
in Li (1999) and Liu (2003).
To sum up, the analysis of AMRs is a straightforward way
of quantifying the local fracture orientation. In the presence
of azimuthal anisotropy in the overburden, both a ray-tracing
procedure and an effective NMO procedure can be used to
perform overburden correction. The results after overburden
correction are in agreement with the model.
An intrigue question that may be asked is: how many
layers could be reasonably removed using this method? In
theory, there is no limit to the number of layers with ray
tracing and high quality data. However, in practice, due to data
uncertainties and error propagation, up to two to three layers
may be reliably removed, which turns out to be sufficient
in most cases. This is because the subsurface can often
be considered as a four-layer model, consisting of the near
surface, overburden, target and basement, for data processing
purposes. In some cases, the near-surface layer may even be
optional and a single overburden correction will suffice.
7
S Wang and X-Y Li
Results - Layer 1
Crossplot - Layer 1
120
Detected Angles (degree)
100
∆t142(x)
80
60
40
20
0
0
20
40
60
∆t31
(x)
1
80
100
120
90
85
80
75
70
65
60
55
50
45
40
35
30
25
20
15
10
5
0
1000
2000
(a)
Detected Angles (degree)
100
12
∆t42(x)
80
60
40
20
20
40
60
31(x)
∆t12
(c)
6000
5000
6000
Results - Layer 2
Crossplot - Layer 2
0
5000
(b)
120
0
3000 4000
Offset (m)
80
100
120
90
85
80
75
70
65
60
55
50
45
40
35
30
25
20
15
10
5
0
1000
2000
3000 4000
Offset (m)
(d)
Figure 5. AMR analysis using the single-layer method directly without layer stripping. (a) Cross plotting of t131 versus t142 from the
top-target event, and (b) estimating the fracture orientation of the overburden for each individual offset using equation (15). Parts (c)
and (d) are the same as (a) and (b) but for the bottom-target event.
Figure 6. The same CDP gathers as in figure 4 but after NMO correction which aligned the top events in all four lines.
8
Layer stripping of azimuthal anisotropy
Results - Layer 2
Crossplot - Layer 2
250
Detected Angles (degree)
200
2
∆t42(x)
150
100
50
0
-250
-200
-150
-100
-50
0
90
85
80
75
70
65
60
55
50
45
40
35
30
25
20
15
10
5
0
1000
2000
∆t31(x)
2
(a)
Detected Angles (degree)
40
2
∆t42(x)
30
20
10
-30
∆t 31
(x)
2
5000
6000
5000
6000
Results - Layer 2
Crossplot - Layer 2
-40
4000
(b)
50
0
-50
3000
Offset (m)
-20
-10
0
(c)
90
85
80
75
70
65
60
55
50
45
40
35
30
25
20
15
10
5
0
1000
2000
3000 4000
Offset (m)
(d)
Figure 7. Analysis results of the fractured target after layer stripping. Parts (a) and (b) are the results with the ray-tracing procedure;
parts (c) and (d) are those with NMO-correction procedure.
6. Conclusions
In this paper, we have presented an alternative approach on the
basis of the P-wave AMR (Li 1999) to achieve layer stripping
of azimuthal anisotropy. Assuming multi-layered azimuthal
anisotropic media containing vertical fractures with arbitrary
strike direction, we have developed analytical expressions for
quantifying the interval AMR of any given layer. The approach
requires a special four-line configuration with two orthogonal
pairs and the interval AMR for a fixed offset is a function of
cos(α − 2ϕi ) with respect to line-azimuth α and fracture-strike
azimuth ϕi in the multi-layered media. Thus, the cross plot
of two corresponding AMRs from two pairs of orthogonal
lines can be used to determine the local fracture orientation
φi = α − 2ϕi , if the interval AMR can be extracted from the
moveout data.
For this, a layer-stripping procedure is required. In
the case of a weakly fractured overburden underlain by a
heavily fractured target, layer stripping can be effectively
achieved using normal moveout correction. Each line should
be processed separately so that the top event of the target is
properly aligned. The residual moveout of the bottom-target
event, if any, after the moveout correction, can then be used
to calculate the interval AMR. In the general case, similarly
to AVO analysis, ray tracing can be used to calculate the raysegment components in each layer using the velocity model
built from stacking velocity analysis, and layer stripping can
then be performed using these ray-segment components. In
practice, two passes of the layer-stripping procedure are often
sufficient to correct for the effects of the near surface and the
overburden.
Acknowledgments
We thank John Lovell for commenting on the manuscript. This
work is partly supported by the internation collaboration fund
of the CNPC through its Geophysical Key Lab in Beijing, and
by the Edinburgh Anisotropy Project (EAP). We thank for their
permission to present the result. We also thank Liu Zhenwu
and Fang Chaoliang of CNPC for supporting the collaboration.
The work is published with the approval of the Director of the
British Geological Survey (NERC), and the EAP sponsors:
BG plc, BGP-CNPC, BP, Chevron, ConocoPhillips, ENIAgip, ExxonMobil, GX Technology, KerrMcGee, Landmark,
Marathon, Norsk Hydro, PDVSA, Schlumberger, Shell, Total
and Veritas DGC.
9
S Wang and X-Y Li
and
Appendix. Azimuthal moveout response in
multi-layered weakly anisotropic TIH media
Assuming multi-azimuth seismic surveys over a stack of n
TIH layers with arbitrary fracture orientation, we compute the
travel time, t (α, x), from the bottom of the nth layer for offset x
and for the kth line azimuth at angle α from North (ray Lnk (x),
figure 2), and then use this expression to derive the total AMR
of equation (30).
The total travel time t (α, x) for the given ray Lnk (x) is, as
shown in figure 2,
n
t (α, x) = 2
tki ,
(A.1)
i=1
xki .
(A.2)
i=1
Equation (A.2) implies that the ray is confined to the incidence
plane and is valid only for weak azimuthal anisotropy.
From equation (5), tki can be written as
x2
tki = t0i2 + 2ki [1 − (δi − 2i ) sin2 θki sin2 (α − ϕi )
vp0i
− (i − δi ) sin4 θki sin4 (α − ϕi )],
(A.3)
and the corresponding ti and t⊥i inside the ith layer are
2
xi
ti = t0i2 + 2 ,
(A.4)
vp0i
x2
t⊥i = t0i2 + 2⊥i [1 − (δi − 2i ) sin2 θ⊥i
vp0i
− (i − δi ) sin4 θ⊥i ],
(A.5)
where xi , θi , x⊥i and θ⊥i are the corresponding ray-segment
components with the same total offset x at the directions
parallel and perpendicular to the fracture strike of the ith layer,
respectively.
Let
xki = xi (1 + νki ),
x⊥i = xi (1 + ν⊥i ),
θki = θi + θki ,
θ⊥i = θi + θ⊥i ,
and νki and ν⊥i satisfy
n
n
xi νki =
xi ν⊥i ≡ 0,
i=1
(A.6)
(A.7)
−2
n
sin θi
sin θi
xi ν⊥i sin2 (α − ϕi ) + 2
xi νki .
vp0i
vp0i
i=1
(A.11)
A.2. Total AMR
Assume an orthogonal line at azimuth α + π/2.
In
equation (A.11), νki is the only variable that still implicitly
to denote the corresponding
varies with azimuth. Use νki
value for the orthogonal line at azimuth α + π/2. From
equation (A.11) the total AMR for the stack of n layers t1n
can then be written as
t1n (α, x) = t (α + π/2, x) − t (α, x)
n sin θi
t⊥i − ti −
xi ν⊥i cos 2(α − ϕi )
=2
vp0i
i=1
+2
n
sin θi
i=1
vp0i
n
sin θi
i=1
vp0i
(A.12)
xi (νki
− νki ) = p
n
xi (νki
− νki ) ≡ 0,
i=1
where p is the horizontal slowness (ray parameter). Using
equations (A.4) and (A.9) yields
t1n (α, x) =
n
Bi (i , δi , xi ) cos 2(α − ϕi ),
(A.13)
i=1
where
(A.8)
2xi
sin θi [2i − δi − (i − δi ) sin2 θi ].
vp0i
(A.14)
For the special case of multi-layered media with a uniform
fracture-strike azimuth ϕi ≡ ϕ, equation (A.12) becomes
t (α, x) = 2 cos 2(α − ϕ)
n
(t⊥i − ti ) − 2p cos 2(α − ϕ)
i=1
×
(A.9)
xi (νki
− νki ).
To the first order of the small anisotropy quantities,
i=1
t⊥i = ti − ti (δi − 2i ) sin2 θi
sin θi
− ti (i − δi ) sin4 θi +
xi ν⊥i ,
vp0i
ti (i − δi ) sin4 θi sin2 (α − ϕi ) cos2 (α − ϕi )
i=1
n
i=1
i=1
Bi (i , δi , xi ) =
where νki , θki , ν⊥i and θ⊥i are small quantities of the
same order as the anisotropy parameters i and δi . For weak
anisotropy, higher orders of these terms can also be neglected
in searching for linearized solutions. Substituting (A.7) into
(A.5), equation (A.6) into (A.3), and linearizing over the small
quantities gives
10
i=1
n
Note that evaluating equations (A.9) and (A.10) at offset
xi yields equations (25) and (26).
where offset x satisfies,
n
(A.10)
These lead to
n
n
t (α, x) = 2
tki = 2 [ti cos2 (α − ϕi ) + t⊥i sin2 (α − ϕi )]
+2
A.1. Travel time equation
x=2
tki = ti − ti (δi − 2i ) sin2 θi sin2 (α − ϕi )
sin θi
− ti (i − δi ) sin4 θi sin4 (α − ϕi ) +
xi νki .
vp0i
n
xi ν⊥i = (t⊥ − t ) cos 2(α − ϕ),
i=1
which reduces to equation (32).
(A.15)
Layer stripping of azimuthal anisotropy
A.3. Error analysis of isotropic ray tracing
For the ith layer (figure 2), introduce xi and θi as the raysegment components obtained by isotropic ray tracing. For
weak anisotropy, xi and θi satisfy
xi = xi (1 + νi ),
θi = θi + θi ,
(A.16)
where νi and θi are small quantities of the same order as the
anisotropy parameters i and δi . Substituting equation (A.16)
into (A.14) and linearizing over the small quantities gives
Bi (i , δi , xi ) =
2xi
sin θi [2i − δi − (i − δi ) sin2 θi ].
vp0i
(A.17)
Thus, isotropic ray tracing can replace anisotropic ray
tracing to obtain the ray-segment components, and the error
introduced to the AMR is of second order of the anisotropic
parameters.
References
Al-Dajani A and Tsvankin I 1998 Nonhyperbolic reflection moveout
for horizontal transverse isotropy Geophysics 63 1738–53
Crampin S 1985 Evaluation of anisotropy by shear-wave splitting
Geophysics 50 142–52
Dai H and Li X-Y 1998 Interpreting the residual wavefield for
polarization change in 4-C shear-wave data 68th Int. Mtg., Soc.
Explor. Geophys. Expanded Abstracts pp 726–9
Grechka V and Tsvankin I 1998 3-D description of normal moveout
in anisotropic in homogeneous media Geophysics 63 1079–92
Hall S A and Kendall J-M 2003 Fracture characterization at Valhall:
application of P-wave amplitude variation with offset and
amplitude (AVOA) analysis to a 3D ocean-bottom data set
Geophysics 68 1150–60
Li X-Y 1997 Viability of azimuthal variation in P-wave moveout for
fracture detection 67th Int. Mtg., Soc. Explor. Geophys.
Expanded Abstracts pp 1555–8
Li X-Y 1999 Fracture detection using azimuthal variation of P-wave
moveout from orthogonal seismic survey lines Geophysics 64
1193–201
Li X-Y and Crampin S 1993 Approximations to shear-wave velocity
and moveout equations in anisotropic media Geophys.
Prospect. 41 833–57
Li X-Y and MacBeth C 1997 Data-matrix asymmetry and
polarization changes from multicomponent surface seismic
Geophysics 62 630–43
Liu Y-Q 2003 Analysis of P-wave seismic reflection data for
azimuthal anisotropy PhD Thesis University of Edinburgh
Lynn H B, Simon K M, Bates C R and Van Dok R 1996 Naturally
fractured gas reservoir’s seismic characterization 66th Int.
Mtg., Soc. Explor. Geophys. Expanded Abstracts pp 1360–3
MacBeth C and Li X-Y 1999 AVD—an emerging new marine
technology for reservoir characterization acquisition and
application Geophysics 64 1153–9
MacBeth C, Li X-Y, Crampin S and Mueller M C 1992 Detecting
lateral viability in crack parameters from surface data 62nd Int.
Mtg., Soc. Explor. Geophys. Expanded Abstracts pp 816–9
Sayers C M and Ebrom D A 1997 Seismic traveltime analysis for
azimuthally anisotropic media: theory and experiment
Geophysics 62 1570–82
Sena A G 1991 Seismic travel time equations for azimuthally
anisotropic and isotropic media: estimation of interval elastic
properties Geophysics 56 2090–101
Smith R L and McGarrity J P 2001 Cracking the fractures from
seismic anisotropy in an offshore reservoir The Leading Edge
20 19–26
Taylor D B 1996 ANISEIS V manual: Applied Geophysical Inc
Thomsen L A 1986 Weak elastic anisotropy Geophysics 51 1954–66
Thomsen L A, Tsvankin I and Mueller M 1995 Layer-stripping of
azimuthal anisotropy from reflection shear-wave data 65th Int.
Mtg., Soc. Explor. Geophys. Expanded Abstracts pp 289–92
Tsvankin I 1995 Inversion of moveout velocities for horizontal
transverse isotropy 65th Int. Mtg., Soc. Explor. Geophys.
Expanded Abstracts pp 735–8
Tsvankin I 1997 Reflection moveout and parameter estimation for
horizontal transverse isotropy Geophysics 62 614–29
Winterstein D F and Meadows M A 1991 Shear-wave polarization
and substructure stress directions at lost hill field Geophysics
56 1331–48
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