Directory UMM :Data Elmu:jurnal:J-a:Journal Of Applied Geophysics:Vol46.Issue1.2001:
Journal of Applied Geophysics 46 Ž2001. 1–29
www.elsevier.nlrlocaterjappgeo
AVO-A response of an anisotropic half-space bounded by a
dipping surface for P–P, P–SV and P–SH data
Luc T. Ikelle a,) , Lasse Amundsen b
a
CASP Project, Department of Geology and Geophysics, Texas A & M UniÕersity, College Station, TX 77843-3115, USA
b
Statoil Exploration and Petroleum Technology Research Centre, USA
Received 20 October 1999; accepted 29 August 2000
Abstract
We analyze amplitude variations with offsets and azimuths ŽAVO-A. of an anisotropic half-space bounded by a dipping
surface. By analyzing the response of a dipping reflector instead of a horizontal one, we integrate the fundamental problem
of lateral heterogeneity vs. anisotropy into our study. This analysis is limited to the three scattering modes that dominate
ocean bottom seismic ŽOBS. data: P–P, P–SV and P–SH. When the overburden is assumed isotropic, the AVO-A of each of
these three scattering modes can be cast in terms of a Fourier series of azimuths, f , in general form,
4
R avoa Ž f . s F0 q
Ý w Fn cos Ž n f . q Gn sin Ž n f . x ,
ns 1
where F0 , Fn and Gn are the functions that describe the seismic amplitude variations with offsets ŽAVO. for a given
azimuth. The forms of AVO functions are similar to those of classical AVO formulae; for instance, the AVO functions
corresponding to the P–P scattering mode can be interpreted in terms of the intercept and gradient, although the resulting
numerical values can differ significantly from those of isotropic cases or horizontal reflectors.
One of the benefits of describing the AVO-A as a Fourier series is that the contribution of amplitude variations with
azimuths ŽAVAZ. is distinguishable from that of AVO. The AVAZ is characterized by the functions 1, cos f , sin f , cos2 f ,
sin2 f , cos3f , sin3f , cos4f , sin4f 4, that are mutually orthogonal. Thus, the AVO–A inversion can be formulated as a
series of AVO inversions where the AVO behaviors are represented by the functions F0 , Fn and Gn .
When the coordinate system of seismic acquisition geometry coincides with the symmetry planes of the rock formations,
the series corresponding to P–P and P–SV simplify even further; they reduce to F0 for azimuthally isotropic symmetry and
to F0 , F2 , F4 , G 2 and G4 for orthorhombic symmetry. The series corresponding to P–SH scattering is reduced to G 2 and G4
for these two symmetries.
Unfortunately, the coordinate system of seismic acqusition geometry rarely coincides with the symmetry planes of the
rock formations; therefore, the other terms are rarely zero. In particular, the functions F1, F3 , G 1 and G 3 become important
for large dips and are actually largely dependent on the angle of the dipping reflector. For P–P scattering, these functions are
zero if the reflector is horizontal, irrespective of the anisotropic behavior. For P–SV and P–SH scattering, these functions
)
Corresponding author. Fax: q1-979-458-1513.
E-mail address: ikelle@nutmeg.tamu.edu ŽL.T. Ikelle..
0926-9851r01r$ - see front matter q 2001 Elsevier Science B.V. All rights reserved.
PII: S 0 9 2 6 - 9 8 5 1 Ž 0 0 . 0 0 0 3 1 - 8
2
L.T. Ikelle, L. Amundsenr Journal of Applied Geophysics 46 (2001) 1–29
are not necessarily zero for horizontal reflectors because they are affected by the asymmetry of the P–S reflection in addition
to the effect of dip. q 2001 Elsevier Science B.V. All rights reserved.
Keywords: AVO-A; Ocean bottom seismic; Dipping surface
1. Introduction
Ocean bottom seismic ŽOBS. data acquisition and
processing system is a new oil and gas exploration
and production tool that not only aims at remotely
identifying fluid-saturated rock formations but also
at determining their characteristics as well as any
changes in their characteristics over time. These
determinations are possible because OBS surveys
have overcome three key limitations of towedstreamer surveys: Ži. azimuthal coverage, Žii. multicomponent recordings and Žiii. repeatability of surveys. The 3D OBS can be designed with small
enough spacing in both source and receiver coordinates to allow a full azimuthal coverage. Also, since
it is carried out at the sea floor, three component
particle velocity Žgeophone. data can be recorded
along with pressure Žhydrophone. data. This recording system gives us direct access to the shear-wave
in addition to the P-wave. By considering a scenario
where sensors are permanently placed at the sea
floor, the repeatability of OBS is a realistic possibility although some technological issues, such as the
coupling of geophones to the sea floor, still remain
to be overcome. To use these three new features of
marine seismic measurements for retrieving fluid
properties of rock formations, we need to gain more
insight on how the physical properties which characterize fluid-saturated rocks are related to seismic
data.
Fluid-saturated rocks contain small scale inhomogeneities such as fractures that control the fluid flow
in these rocks. At the scale of seismic wavelength,
rocks containing these types of small scale inhomogeneities behave as azimuthally anisotropic materials. Therefore, changes in the physical properties of
such rock formations can produce amplitude variations with offsets ŽAVO. as well as amplitude variations with azimuths ŽAVAZ.. In short, we will call
such amplitude variations AVO-A.
To identify and characterize fluid-saturated rock
formations, we need to understand the correlations
between anisotropic parameters and AVO-A on one
hand and the correlations between anisotropic parameters and properties of fluid-saturated rock formations on the other. Such correlations make it
possible to reduce the number of parameters in our
inversion algorithms, which are generally ill-posed
due partially to the large number of unknowns.
These types of correlations can also help distinguish
between different petrological models. Here we present an analysis of an AVO-A response to an
anisotropic half-space bounded by a dipping surface
as one way of improving our understanding of these
correlations. The dipping surface can be planed or
curved. By considering the response of a dipping
reflector instead of a horizontal one, we have tried to
integrate the fundamental problem of lateral heterogeneity vs. anisotropy into our study. Our analysis
will be limited to the three scattering modes which
dominate OBS data: P–P, P–SV and P–SH.
We will use the weak-contrast assumption in our
derivations Ži.e., Born approximation., but no weakly
anisotropic assumptions are made of this material.
We will consider transversely isotropic ŽTI. and
orthorhombic symmetries as most experimental and
theoretical studies suggest that the majority of rock
formations possess these symmetries Že.g., Backus,
1962; White, 1975; Thomsen, 1986; Ikelle et al.,
1993; Sayers, 1998.. However, these symmetries are
of interest in the derivations of AVO-A only if the
axis of these symmetries coincides with the coordinate system of seismic acquisition geometries. Unfortunately, the coordinate system of seismic acquisition geometries sometimes does not coincide with
the symmetry plane of rock formations. Even if they
do coincide, such information is generally unknown.
Our AVO-A analysis will show that the total
anisotropic effect of reflection data can be described
by a small number of simple linear combinations of
the elastic stiffness tensor coefficients.
There are many publications on reflection coefficients for weakly anisotropic media Že.g., Daley and
Horn, 1977; Gibson and Ben-Menahem, 1991; Ikelle,
L.T. Ikelle, L. Amundsenr Journal of Applied Geophysics 46 (2001) 1–29
1996; Zillmer et al., 1997, 1998; Ruger, 1998; Ursin
and Haugen, 1996; Vavrycuk
˘ and Psencik,
˘ ˘ 1998..
The novelty here with respect to the previous works
is that our derivations include the effect of a dipping
reflector. We also produce mathematical constructs
of AVO-A, which can be useful for AVO-A analysis
and inversions by seeking the combinations of elastic
parameters that best constrained P–P, P–SV and
P–SH data. These mathematical constructs are similar to those derived by Ikelle Ž1996. and corroborated by real data experiments ŽLeaney et al., 1999.
for P–P data when the acquisition plane is parallel to
the dip direction.
A summary of the rest of this paper is as follows.
In the next section, we will describe the basic elastodynamic properties of anisotropic media needed to
introduce the general formulae of the AVO-A. We
will also recall these formulae for the three scattering
modes considered here. In the third, fourth and fifth
sections, we will derive and analyze AVO-A formulae for each of the three scattering modes. Finally, in
the sixth section, we will indicate the key preprocessing requirements for AVO-A inversion.
2. Introductory definitions
Consider two media separated by a dipping interface Žsee Fig. 1.. The top medium, in which the
sources and receivers are located, is homogeneous
and isotropic; the bottom medium is also homogeneous but anisotropic. Our objective in this paper is
3
to derive the AVO-A response of this configuration
under the weak-contrast approximation ŽBorn approximation.. In this particular section, we introduce
the basic definitions that we need for the AVO-A
formulation.
The AVO-A formulation requires the slowness
and polarization vectors which characterize the Pwave propagation from the source point to the scattered point on the dipping interface and those which
characterize P- and S-wave upward propagation from
the scattered point to the receiver point. As we have
assumed that the top medium is homogeneous and
isotropic, we are simply going to recall the well
known analytic formulae of the slowness and polarization vectors for this particular case. We will also
recall the compact definition of AVO-A on which
our derivation will be based. To clarify the presentation of these definitions, we will start by introducing
our notation convention.
2.1. Notations
Position in the configuration in Fig. 1 is specified
by the coordinates x, y, z 4 with respect to a fixed
Cartesian reference frame with the origin O and the
three mutually perpendicular base vectors i x , i y , i z 4
each of unit length. i z points vertically downwards.
To accommodate anisotropy, the subscript notation
for vectors and tensors and the Einstein summation
convention are adopted. Lower case Latin subscripts
are employed for this purpose; they are to be assigned the values 1, 2 and 3 if not specified other-
Fig. 1. Isotropicranisotropic configuration with sources and receivers in the isotropic media.
L.T. Ikelle, L. Amundsenr Journal of Applied Geophysics 46 (2001) 1–29
4
wise. The lower case Latin subscripts s and r are
reserved symbols for indicating source and receiver,
respectively.
As we will see later, the quantities used to describe AVO-A formulae depend on several variables.
We will sometimes write these quantities without
some of the variables when the context unambiguously indicates the quantity currently under consideration.
2.2. Vertical waÕenumbers, polarization Õectors and
slowness Õectors in an isotropic homogeneous
medium
We start with the vertical wavenumbers corresponding to an isotropic, homogeneous, elastic
medium. They are defined as follows for downgoing
P-waves from a source point, Ž x s , ys , 0.,
qsŽP . s qsŽP . Ž k s ,kXs , v . s
v
(
1y
ÕP
Õ P2
k s2 q kXs2
v
,
2
Let us now define the polarization vectors. The
polarization vector b˜ ŽP. s b ŽP. Ž k s , kXs , v . of the
downgoing P-waves is
b̃ ŽP . s b ŽP . k s ,kXs , v
Ž
qrŽP . s qrŽP . Ž k r ,kXr , v . s
v
(
1y
ÕP
Õ P2 k r2 q kXr2
v
,
2
Ž 2.
qrŽS. s qrŽS.
Ž
k r ,kXr , v
.s
v
Õs
(
1y
Õ S2 k r2 q kXr2
v2
,
Ž 3.
k s , kXs , k r , and kXr are the horizontal wavenumbers
corresponding to x s , ys , x r , and yr , respectively; v
is the temporal frequency corresponding to time t;
Õ P and Õ S are the P- and S-wave velocities of the top
medium in Fig. 1, respectively. The vertical
wavenumber qsŽP. characterizes downgoing P-waves
propagating through the top medium Žsee Fig. 1.
from the source to the scattered point, and qrŽP. and
qrŽS. characterize upgoing P- and S-waves propagating through the top medium from the scattered point
to the receiver.
0
v
Ž 4.
,
qsŽ P .
and those of the upgoing P-, SV- and SH-waves are
defined as follows:
b̂
ŽP .
sb
ŽP .
Ž
k r ,kXr , v
.s
0
kr
kXr
ÕP
v
Ž 5.
,
qrŽ P .
bˆ ŽSV . s b ŽSV . Ž k r ,kXr , v .
s
1
ÕS
v
(
k r2 q kXr2
Ž 1.
and for upgoing P- and S-waves to a receiver location, Ž x r , yr , 0.,
.s
ks
kXs
ÕP
k r qrŽ S .
kXr qrŽ S .
yŽ
b̂ ŽSH . s b ŽSH . Ž k r ,kXr , v . s
k r2 q kXr2
.
0
1
(k q k
2
r
X2
r
Ž 6.
,
ykXr
k r , Ž 7.
0
0
respectively.
Another set of vectors that we are going to need
in the definition of AVO-A are slowness vectors.
They are denoted by g˜ ŽP. s g ŽP. Ž k s , kXs , v . for a
downgoing P-wave and defined as follows:
g˜ ŽP . s g ŽP . Ž k s ,kXs , v . s
1
v
k s ,kXs ,qsŽP .
T
Ž 8.
.
For upgoing P-, SV- and SH-waves, they are defined
as follows:
gˆ ŽP . s g ŽP . Ž k r ,kXr , v . s
1
v
gˆ ŽSV . s g ŽSV . Ž k r ,kXr , v . s
gˆ ŽSH . s gˆ ŽSV . ,
k r ,kXr ,qrŽP .
1
v
T
k r ,kXr ,qrŽS.
Ž 9.
,
T
,
Ž 10 .
Ž 11 .
respectively, where upper-case T indicates transpose
vectors.
L.T. Ikelle, L. Amundsenr Journal of Applied Geophysics 46 (2001) 1–29
l
2.3. AVO-A for a weak-contrast interface
We can describe the materials in Fig. 1 by their
density of mass, r , and their stiffness tensor, c i jk l :
r Ž0. ; c iŽ0.jk l 4 characterizes the isotropic top medium
with the following relationship:
c iŽ0.jk l
r Ž0.
s Õ P2 y 2 Õ S2 d i j d k l q Õ S2 Ž d i k d jl q d i l d jk . ,
Ž 12 .
while r Ž1., c iŽ1.jk l 4 characterizes the anisotropic bottom medium. For an incident P-wave, the reflection
at the interface between the two media produces
three scattering fields: P–P, P–SV and P–SH. Under
the weak-contrast approximation, the reflection coefficients characterizing the amplitude variations with
offsets and azimuths can be written Žsee de Hoop et
al., 1994; Ikelle, 1996.:
R Ž1.
avoa s
˜
ˆ
biŽP .D rbiŽP .
q
˜ ˜
biŽP .g jŽP .D c i jk l
for P–P scattering
ˆ ˆ
b kŽP .g lŽP .
Ž 13 .
q b˜iŽP .g˜ jŽP .D c i jk l bˆkŽSV .gˆ lŽSV . ,
Ž 14 .
˜ ŽP . ˆ ŽSH .
R Ž3.
avoa s b i D rb i
q b˜iŽP .g˜ jŽP .D c i jk l bˆkŽSH .gˆ lŽSH . ,
for P–SH scattering
r Ž1.
r Ž0.
D c i jk l s
,
c iŽ1.jk l
r Ž0. Õ P2
.
3. AVO-A derivation and analysis for P–P data
3.1. Dip and azimuthal angles
The amplitude variations with offsets and azimuths R Ž1.
avoa for P–P scattering are given in Eq.
Ž13.. To gain physical insight into R Ž1.
avoa , it is useful
to express it in terms of dip and azimuthal angles.
This can be done by relating the wavenumbers k s ,
kXs , k r and kXr to incident and reflected angles as
follows:
ks s
Ž 15 .
where
D rs
™
scripts I and J run from 1 to 6, with ij I according to 11, 22, 33, 23, 31, 12 1, 2, 3, 4, 5, 6.
The Eqs. Ž13., Ž14. and Ž15. are just alternative
forms of small angle approximation of the reflection
coefficients Žsee Ikelle et al., 1992; Ikelle, 1995..
These formulae are derived from scattering theory
assuming that density and stiffnesses are spatial
step-functions. They do not use plane wave approximation nor weak anisotropic approximation. However, they assume a weak-contrast interface Ži.e.,
Born approximation..
The effect of dip in Eqs. Ž13., Ž14. and Ž15. is
taken into account by the upgoing polarization and
slowness vectors. As we will see later, the direction
of the upgoing rays is modified if the interface is a
dipping one instead of a horizontal one.
,
˜ ŽP . ˆ ŽSV .
R Ž2.
avoa s b i D rb i
for P–SV scattering
5
kXs s
Ž 16 .
kr s
Ž 17 .
kXr s
Alternatively, the tensorial stiffness D c i jk l will also
be denoted by DCI J , respectively, where the sub-
v
ÕP
v
ÕP
v
ÕP
v
ÕP
sin us cos fs ,
Ž 18 .
sin us sin fs ,
Ž 19 .
sin ur cos fr ,
Ž 20 .
sin ur sin fr ,
Ž 21 .
where us is the angle between the incident ray and
the vertical axis; ur is the angle between the re-
L.T. Ikelle, L. Amundsenr Journal of Applied Geophysics 46 (2001) 1–29
6
flected ray and the vertical axis; fs is the azimuthal
angle between the x-axis and the line connecting the
projection of the shot point in the plane Ž x–y . with
that of the scattering point; fr is the angle between
the x-axis and the line connecting the projection of
the receiver point in the plane Ž x–y . with that of the
scattering point. These four angles are shown in
Figs. 2 and 3. The angles us and ur vary over w0,
pr2x; the azimuthal angles fs and fr vary over w0,
2p x.
Alternatively, the angles u , u X , f and f X will be
used instead of us , ur , fs and fr . They are introduced as follows:
u s us y ur ,
Ž 22 .
u X s us q ur ,
Ž 23 .
fs
fs q fr
2
Ž 24 .
,
gle between the plane containing the scattering point
and shot point and that containing the scattering
point and receiver points. These four new angles are
also shown in Figs. 2 and 3. As we can see in Fig. 2,
u X is zero for the particular case where the reflector
is horizontal.
3.2. Decoupling of AVAZ and AVO
By substituting Eqs. Ž18. – Ž25. in Eq. Ž13. and
regrouping the different elements as a linear combination of 1, cos f , sin f , cos2 f , sin2 f , cos3f ,
sin3f , cos4f , sin4f 4 , the amplitude variations with
offsets and azimuths ŽAVO-A. can be cast in terms
of a Fourier series of the azimuthal angle f as
follows:
4
Ž1.
R Ž1.
avoa s F0 q
Ý
FnŽ1. cos Ž n f . q GnŽ1. sin Ž n f . .
ns1
fX s
fs y fr
2
Ž 25 .
,
X
where u is the total reflection angle; u is the angle
of the dipping reflector; f is the azimuthal angle
between the x-axis and the plane containing the
scattering point and midpoint; f X the azimuthal an-
Ž 26 .
Expressions of functions F0Ž1. s F0Ž1. Ž u , u X , f X ., FnŽ1.
s FnŽ1. Ž u , u X , f X . and GnŽ1. s GnŽ1. Ž u , u X , f X . can be
obtained from Table 1 Žthe subscript n runs from 1
to 4.. Furthermore, the different elements of the
series Ž26. can also be regrouped as a linear combi-
Fig. 2. P–P reflection at an isotropicranisotropic interface. Ža. corresponds to a horizontal interface with, us , the incident angle equal to, ur ,
the reflected angle. Žb. corresponds to a dipping interface where the incident angle no longer equals the reflected angle.
L.T. Ikelle, L. Amundsenr Journal of Applied Geophysics 46 (2001) 1–29
7
Fig. 3. Azimuthal angles used in the radiation patterns for a case where the receivers are outside the acquisition plane.
nation of cos f X , sin f X , cos2 f X , sin2 f X , cos3f X ,
sin3f X , cos4f X , sin4f X 4 , leading to
X
X
Ž1.
Ž1.
Ž1.
R Ž1.
avoa s f 00 q f 02 cos2 f q f 04 cos4f
Ž1c .
sin f X
q f 11Ž1c . cos f X q g 11
Ž1c .
sin3f X cos f
qf 13Ž1c . cos3f X q g 13
Ž1s.
sin f X
q f 11Ž1s. cos f X q g 11
Ž1s.
qf 13Ž1s. cos3f X q g 13
sin3f X sin f
Ž1c .
Ž1c .
Ž1c .
q f 20
q f 22
cos2 f X q g 22
sin2 f X cos2 f
Ž1s.
Ž1s.
Ž1s.
sin2 f X sin2 f
q f 20
q f 22
cos2 f X q g 22
Ž1c .
Ž1c .
sin f X cos3f
q f 31
cos f X q g 31
Ž1s.
Ž1s.
sin f X sin3f
q f 31
cos f X q g 31
Ž1c .
q f 40Ž1c . cos4f q g 40
sin4f .
Ž 27 .
X
Ž1. Ž
. Ž1c.
Expressions of functions f 0Ž1.
m s f 0 m u , u , fn m s
X
X
Ž1c. Ž
Ž1s.
Ž1s. Ž
Ž1c.
Ž1c. Ž
.
.
fn m u , u , fn m s fn m u , u , g n m s g n m u , u X .
X
Ž1s. Ž
. Žthe subscripts n and m run
and g nŽ1s.
m s gnm u , u
from 0 to 4. can be deduced from Table 1.
X
The dependence of R Ž1.
avoa on f and f describes
amplitude variations with azimuths ŽAVAZ. while
its dependence on u and u X describes amplitude
variations with offsets ŽAVO.. Thus, the AVO effect
in Eq. Ž27. of R Ž1.
avoa is represented by the functions
Ž1c.
Ž1s.
Ž1s.
f 0Ž1.
,
f
,
f
,
g nŽ1c.
m
nm
nm
m , and g n m , all independent of
the azimuthal angles f and f X .
Ž .
To simplify our analysis of R Ž1.
avoa in Eqs. 26 and
Ž27., we will limit our discussion to one azimuthal
angle by taking f X s 0 for the rest of this section
Ži.e., the source and receivers lie on the same line
similar to when seismic data are arranged in common azimuthal sections, as described in Ikelle, 1996.
and we will use the series Ž26. instead of Eq. Ž27..
So, the AVO functions are F0Ž1. s F0Ž1. Ž u , u X ., FnŽ1. s
FnŽ1. Ž u , u X . and GnŽ1. s GnŽ1. Ž u , u X .. They can be
obtained from Table 1 by taking f X s 0 Žthe subscript n runs from 1 to 4..
Before we discuss the implication of these conŽ .
structs of R Ž1.
avoa in Eq. 26 for an inversion algorithm, let us examine the AVO-A formula in Eq.
Ž26. for three symmetries regularly observed in ultrasonic laboratory measurements Že.g., Jones and
Wang, 1981; Lo et al., 1981; Cheadle et al., 1991;
L.T. Ikelle, L. Amundsenr Journal of Applied Geophysics 46 (2001) 1–29
8
Table 1
AVO functions which describe amplitude variations with offsets and azimuths for P–P
AVO formulae
X
F0Ž1. s D r cos us cos ur q D r sin us sin ur
AVO parameters
Elastic parameters
R 00 s D r q G 33
D r s D r cos2 f
X
X
X
1
G 11 s Ž DC11 q DC22 . = Ž2 q cos4f X .
8
qG 11 sin2us sin2ur q G 33 cos 2us cos 2ur
R 01 s yD r y D r y 2 G 33 y 4 G44 q 2 G 13
qG 13 Žcos 2us sin2ur q sin2us cos 2ur .
R 02 s D r y D r y 2 G 33 q 4 G44 q 2 G 13
1
X
q DC12 Ž2 y cos4f .
4
qG44 sin2 us sin2 ur
R 03 s y2 G 11 q 2 G 33
1
X
q DC66 cos4f
2
F0Ž1. s R 00 q R 01 sin2
qR 03 sin2
qR 04 sin4
u
2
u
2
u
2
q R 02 sin2
sin2
X
uX
2
uX
2
q R 05 sin4
R 04 s G 11 q G 33 q 4 G44 y 2 G 13
G 33 s DC33
R 05 s G 11 q G 33 y 4 G44 y 2 G 13
1
G 13 s Ž DC13 q DC23 .
2
uX
1
G44 s Ž DC44 q DC55 .cos2 f X
2
2
F1Ž1. s F Bc Žsin2us sin2 ur q sin2 us sin2ur .
R10 s 2FGc
1
F Bc s Ž DC15 q DC25 .cos f X
2
qFGc Žcos 2us sin2 ur q sin2 us cos 2ur .
R11 s y2F Bc y 2FGc
1
X
q Ž DC15 y DC25 .cos3f
4
qF Bx Žsin2us sin2 ur y sin2 us sin2ur .
R12 s 2F Bc y 2FGc
1
X
q DC46 cos3f
2
qFGx Žcos 2us sin2 ur y sin2 us cos 2ur .
Rf 0 s y2FGx
ž
F1Ž1. s R10 q R11 sin2
ž
u
2
q R12 sin2
q Rf 0 q Rf 1 sin2
u
2
uX
2
/
q Rf 2 sin2
sin u
uX
2
/
X
sin u
1
F Bx s y Ž DC24 q DC14 .sin f X
2
Rf 1 s y2F Bx q 2FGx
1
X
y Ž DC14 y DC24 .sin3f
4
Rf 2 s 2F Bx q 2FGx
1
X
q DC56 sin3f
2
FGc s DC35 cos f X
FGx s yDC34 sin f X
G1Ž1. s F BsŽsin2us sin2 ur q sin2 us sin2ur .
R10 s 2FGs
1
F Bs s Ž DC14 q DC24 .cos f X
2
qFGs Žcos 2us sin2 ur q sin2 us cos 2ur .
R11 s y2F Bs y 2FGs
1
X
y Ž DC14 y DC24 .cos3f
4
qF By Žsin2us sin2 ur y sin2 us sin2ur .
R12 s 2F Bs y 2FGs
1
X
q DC56 cos3f
2
qFGy Žcos 2us sin2 ur y sin2 us cos 2ur .
Rf 0 s y2FGy
Rf 1 s y2F By q 2FGy
Rf 2 s 2F By q 2FGy
1
F By s Ž DC15 q DC25 .sin f X
2
1
X
y Ž DC15 y DC25 .sin3f
4
1
X
y DC46 sin3f
2
L.T. Ikelle, L. Amundsenr Journal of Applied Geophysics 46 (2001) 1–29
9
Table 1 Ž continued .
AVO formulae
AVO parameters
ž
G1Ž1. s R10 q R11 sin2
u
q R12 sin2
2
ž
q Rf 0 q Rf 1 sin2
u
2
u
X
/
2
q Rf 2 sin2
sin u
uX
2
/
X
Elastic parameters
FGs s DC34 cos f X
FGy s DC35 sin f X
sin u
R 21 s y4 GGc q 2 G Hc
1
G Bc s Ž DC11 y DC22 .cos2 f X
2
qG Hc Žcos 2us sin2ur q sin2us cos 2ur .
R 22 s 4 GGc q 2 G Hc
1
GGc s Ž DC55 y DC44 .
2
yG Kc Žcos 2us sin2ur y sin2us cos 2ur .
R 23 s y2 G Bc
1
G Hc s Ž DC13 y DC23 .cos2 f X
2
R 24 s G Bc q 4 GGc y 2 G Hc
G Kc s DC36 sin2 f X
F2Ž1. s G Bc sin2us sin2ur q GGc sin2 us sin2 ur
F2Ž1. s R 21 sin2
u
q R 22 sin2
2
qR 23 sin2
qR 25 sin4
u
sin2
2
uX
uX
2
uX
2
q R 24 sin4
X
q Rf sin u sin u
2
u
2
X
Rf s G Kc
2
2
G Ž1.
2 s G Bs sin us sin ur q GGs sin2 us sin2 ur
qG Hs Žcos 2us sin2ur q sin2us cos 2ur .
2
2
2
2
yG Ks Žcos us sin ur y sin us cos ur .
G 2Ž1. s R 21 sin2
u
q R 22 sin2
2
qR 23 sin2
qR 25 sin4
u
sin2
2
uX
2
uX
2
uX
2
q R 24 sin4
X
q Rf sin u sin u
u
2
X
qF Ex Žsin2us sin2 ur y sin2 us sin2ur .
ž
F3Ž1. s R 31 sin2
ž
u
2
q R 32 sin2
q Rf 1 sin2
u
2
u
2
q Rf 2 sin2
/
sin u
uX
2
/
R 21 s y4 GGs q 2 G Hs
G Bs s Ž DC16 q DC26 .cos2 f X
R 22 s 4 GGs q 2 G Hs
GGs s DC45
R 23 s y2 G Bs
G Hs s DC36 cos2 f X
R 24 s G Bs q 4 GGs y 2 G Hs
1
G Ks s Ž DC23 y DC13 .sin2 f X
2
R 25 s G Bs y 4 GGs y 2 G Hs
Rf s G Ks
F3Ž1. s F Ec Žsin2us sin2 ur q sin2 us sin2ur .
X
R 25 s G Bc y 4 GGc y 2 G Hc
R 31 s yR 32 s y2F Ec
Rf 1 s yRf 2 s y2F Ex
q2DC56 xsin f
sin u
qF Ey Žsin2us sin2 ur y sin2 us sin2ur .
y2DC46 xcos f
X
1
F Ex s wŽ DC14 y DC24 .
4
X
2 .
Ž 2
G Ž1.
3 s F Es sin us sin2 ur q sin2 us sin ur
1
F Ec s wŽ DC15 y DC25 .
4
R 31 s yR 32 s y2F Es
Rf 1 s yRf 2 s y2F Ey
X
1
F Es s wŽ DC14 y DC24 .
4
q2DC56 xcos f
X
(continued on next page)
L.T. Ikelle, L. Amundsenr Journal of Applied Geophysics 46 (2001) 1–29
10
Table 1 Ž continued .
AVO formulae
AVO parameters
ž
G 3Ž1. s R 31 sin2
u
2
q R 32 sin2
ž
q Rf 1 sin2
u
2
u
X
2
q Rf 2 sin2
F4Ž1. s G Ec sin2us sin2ur
F4Ž1. s R 43 sin2
u
2
qR 44 sin4
sin2
u
u
2
qR 44 sin4
u
2
uX
2
/
1
F Ey s wŽyDC15 q DC25 .
4
X
q2DC46 xsin f
sin u
u
X
G Ec s Ž1r8.Ž DC11 q DC22 .
R 43 s y2 R 44 s y2 R 45
uX
2
G Es s Ž1r2.Ž DC16 y DC26 .
R 43 s y2 R 44 s y2 R 45
uX
s y2 GEs
2
q R 45 sin4
yŽ1r4. DC12 y Ž1r2. DC66
s y2 GEc
2
q R 45 sin4
sin2
sin u
X
2
G4Ž1. s G Es sin2 sin2us sin2ur
G4Ž1. s R 43 sin2
/
Elastic parameters
uX
2
Hornby, 1996.: transversely isotropic ŽTI. symmetry,
orthorhombic symmetry and monoclinic symmetry.
Using conditions D c i jk l Žsee Auld Ž1990.., the
AVO-A formula in Eq. Ž26. for TI media with
respect to the vertical axis Žalso known as TIV.
reduces to F01 only:
Ž1.
R Ž1.
avoa s F0 ,
Ž 28 .
which means that the amplitudes are invariant with
azimuths. This result is consistent with the fact that
the medium is azimuthally isotropic.
Let us now look at an example of azimuthally
anisotropic media. For that, we consider an orthorhombic medium. Using the conditions on D c i jk l
associated with orthorhombic symmetry Žsee Auld
Ž1990.., the AVO-A formula in Eq. Ž26. is reduced
to
Ž1.
Ž1.
Ž1.
R Ž1.
avoa s F0 q F2 cos Ž 2 f . q F4 cos Ž 4f . .
Ž 29 .
We can see that the amplitudes are no longer invariant with azimuths when medium perturbations are
azimuthally anisotropic.
For the third example, we consider monoclinic
symmetry. The conditions on D c i jk l associated with
monoclinic symmetry can be found in Auld Ž1990..
With these conditions, the AVO-A formula in Eq.
Ž26. is reduced to
4
Ž1.
R Ž1.
avoa s F0 q
Ý FnŽ1. cos Ž nf . .
Ž 30 .
ns1
3.3. Effect of dip
A widespread concern in the interpretation of
AVO-A response is the ambiguity between heterogeneity and anisotropy. By deriving our AVO-A
formula in Eq. Ž26. for a dipping reflector instead of
the usual horizontal one, we can provide some insight on how these two physical properties affect
amplitude variations with offsets and azimuths. Let
us start our discussion by assuming that the interface
between the isotropic and anisotropic media is horizontal Ži.e., u X s 0, in the formulae in Table 1 assuming that f X s 0.. The AVO-A formula in Eq. Ž26. is
then reduced to
Ž1.
Ž1.
Ž1.
Ž1.
R Ž1.
avoa s F0 q F2 cos2 f q G 2 sin2 f q F4 cos4f
q G4Ž1. sin4f .
Ž 31 .
L.T. Ikelle, L. Amundsenr Journal of Applied Geophysics 46 (2001) 1–29
Basically, the f- and 3 f-terms of the Fourier
series are zero irrespective of anisotropic symmetry.
Therefore, the AVO functions F1Ž1., G 1Ž1., F3Ž1. and
G 3Ž1. are direct indicators of the dipping effect.
For some particular anisotropic symmetries like
azimuthally isotropic symmetry and orthorhombic
symmetry, the AVO functions F1Ž1., G 1Ž1., F3Ž1. and
G 3Ž1. are zero, as we can see in Eqs. Ž28. and Ž29..
The question then becomes: how to identify the
effect of dip in these cases? For orthorhombic symmetry, we can use the function F2Ž1. to identify the
effect of dip. In fact, the AVO-A for orthorhombic
symmetry is described by three AVO functions: F0Ž1.,
F2Ž1. and F4Ž1.. We can notice from their expressions
in Table 1 that they can be written in the form of
classical AVO formulae. For instance, F0Ž1. and F2Ž1.
can be written:
Ž1.
2
F0Ž1. s AŽ1.
0 f q B0 f sin
Ž1.
2
F2Ž1. s AŽ1.
2 f q B2 f sin
u
2
u
2
u
q C0Ž1.f sin4
2
u
q C2Ž1.f sin4
2
,
Ž 32 .
,
Ž 33 .
with
Ž1.
AŽ1.
0 f s A0 f
X
Ž u . s R 00 q R 02 sin
2
uX
2
q R 05 sin
4
uX
2
,
Ž 34 .
B0Ž1.f s B0Ž1.f Ž u X . s R 01 q R 03 sin2
u
X
Ž u . s R 22 sin
2
uX
2
2
,
q R 25 sin
B2Ž1.f s B2Ž1.f Ž u X . s R 21 q R 23 sin2
C2Ž1.f s R 24 .
cept of F2Ž1., which is A 2f , is zero for the particular
case where the interface is horizontal Ži.e., uX s 0..
Therefore, the value of the intercept of F2Ž1. is a
direct indicator of the dipping effect for orthorhombic symmetry.
The latest indicator of the dipping effect is based
on the AVO function F2Ž1. which is zero for azimuthally isotropic symmetry as shown in Eq. Ž28..
Hence, how can we recognize the dipping effect for
this commonly used model of anisotropy? For this
case, the series in Eq. Ž26. is reduced to AVO
variations only, and it is described by F0Ž1. only. As
expressed in Eq. Ž32., the intercept AŽ1.
0f and gradient
B0fŽ1. of F0Ž1. vary with the angle of the dipping
reflector in such a way that other equations with
similar parameters are needed to distinguish the dipping effect. By combining the AVO F0Ž1. of P–P
with that of P–SV, we will later see that it is
possible to identify the dipping effect even for azimuthally isotropic symmetry.
Let us make another important remark about the
AVO function F0Ž1.. If we assume that the bottom
medium in Fig. 1 is also isotropic, the intercept and
gradient of F0Ž1., defined in Eqs. Ž34. and Ž35., are
independent of u X , irrespective of the shape of the
interface between the isotropic and anisotropic media, i.e.,
AŽ1.
0 f s R 00 ,
Ž 40 .
Ž 35 .
B0Ž1.f s R 01 .
Ž 41 .
Ž 36 .
Therefore, in isotropic cases, the AVO of a dipping
interface has the same form as that of a horizontal
interface. This conclusion is generally translated by a
statement such as: Athe AVO is a 1D effectB. Let us
emphasize that this statement is only correct if the
media under consideration are isotropic.
X
C0Ž1.f s R 04 ,
Ž1.
AŽ1.
2 f s A2 f
11
uX
2
,
4
uX
2
,
Ž 37 .
Ž 38 .
Ž 39 .
The definitions of the other parameters are given in
Table 1. Using AVO terminology, we remark that
Ž1.
AŽ1.
0f and A 2f are the intercepts for the AVO funcŽ1.
Ž1.
Ž1.
tions F0 and F2Ž1., respectively; and B0f
and B2f
Ž1.
are the gradients for the AVO functions F0 and
F2Ž1., respectively. We can also notice that the inter-
3.4. AVO-A analysis for inÕersion purposes
Basically, we would like to organize the parameters D r and D c i jk l into combinations that are as
independent as possible, in terms of the information
contained in R avoa . Also, we would like to select
only the parameters which have a significant effect
12
L.T. Ikelle, L. Amundsenr Journal of Applied Geophysics 46 (2001) 1–29
Fig. 4. The AVO-A of P–P. The anisotropic materials used here are given in Table 2. Ža. and Žb. correspond to the orthorhombic material in
Table 2a for a horizontal reflector and a 308 dipping reflector, respectively. Žc. and Žd. correspond to the arbitrarily anisotropic material in
Table 2b for a horizontal reflector and a 308 dipping reflector, respectively.
on R Ž1.
avoa . These two important issues are discussed
in this section.
The dependence of R Ž1.
avoa on f describes amplitude variations with azimuths ŽAVAZ. while its dependence on u and u X describes amplitude variations
with offsets ŽAVO.. Thus, the AVO effect in Eq.
Ž1.
Ž26. of R Ž1.
avoa is represented by the functions F0 ,
Ž1.
Ž1.
Fn and Gn , which are all independent of the
azimuthal angle f . The AVAZ effect in this equation is represented by the trigonometric basic functions 1, cos f , sin f , cos2 f , sin2 f , cos3f , sin3f ,
cos4f , sin4f 4 . We can see that the AVAZ effect on
seismic amplitudes can be decoupled from the AVO
effect. Furthermore, we remark that the trigonometric functions describing the AVAZ effect are mutually orthogonal. This AVAZ property suggests that
the AVO functions F0Ž1., FnŽ1. and GnŽ1. can be extracted and processed separately based on the following equations:
2p
F0Ž1. Ž u , u X . s
H
0
X
R Ž1.
avoa Ž u , u , f . d f ,
Ž 42 .
2p
FnŽ1. Ž u , u X . s
H
for n s 1,2,3,4,
0
X
R Ž1.
avoa Ž u , u , f . cos n f d f ,
Ž 43.
L.T. Ikelle, L. Amundsenr Journal of Applied Geophysics 46 (2001) 1–29
13
Table 2
Normalized stiffness tensor D c i jk l . Ža. The anisotropic material used here is an orthorhombic rock as described by Cheadle et al. Ž1991.. Žb.
We have rotated it to simulate the case where the axes of the symmetries do coincide with the coordinate system of the acquisition geometry
Ža.
Parameter
Value
DC11
0.4211
DC22
0.2578
DC33
y0.0488
DC44
y0.0655
DC55
y0.0344
DC66
y0.0044
DC12
0.2500
DC13
0.2133
DC23
0.1844
DC11
DC14
DC45
0.2546
y0.0251
y0.0138
DC22
DC15
DC46
0.3045
0.0354
0.0191
DC33
DC16
DC56
y0.0025
y0.0045
y0.0228
DC44
DC24
Dr
y0.0338
y0.0697
0.1
DC55
DC25
DC66
DC26
DC12
DC34
DC13
DC35
DC23
DC36
y0.0498
0.0184
y0.0213
y0.0369
0.27037
y0.0565
0.1995
0.0323
0.2144
y0.01291
Dr
0.1
Žb.
Parameter
Value
2p
GnŽ1. Ž u , u X . s
H
for n s 1,2,3,4.
0
functions can be expressed in those terms; for instance, G 1Ž1. in Table 1 can be rewritten:
X
R Ž1.
avoa Ž u , u , f . sin n f d f ,
Ž 44.
Another advantage of inverting each of these functions separately is that we significantly reduce the
number of parameters to be estimated in each case.
Before we discuss this point further, let us examine
the contribution of these AVO functions to the
AVO-A, R Ž1.
avoa . We have plotted in Fig. 4 the AVO-A
for P–P scattering corresponding to the two models
described in Table 2. We can see that small offsets
behave as azimuthally isotropic media when the
interface is horizontal Ži.e., F0Ž1. is the dominant
.
function in R Ž1.
avoa . The presence of dip, which introduces the contributions of the functions F1Ž1., G 1Ž1.,
F3Ž1. and G 3Ž1., completely changes this pattern.
Ikelle Ž1996. has suggested that the effect of F3Ž1.,
Ž1.
G 3 , F4Ž1. and G4Ž1. might be small. The AVO-A in
Fig. 5 computed without these terms shows that they
are negligible, especially at small incident angles.
This is an unfortunate outcome because the 3f- and
4f-terms involve single parameter inversion Žsee
Table 1.; therefore, they are easy to perform. However, their extractions from R Ž1.
avoa will be unreliable
for noisy data.
The inversion for parameters contained in the
AVO functions F0Ž1., F1Ž1., F2Ž1., G 1Ž1., and G Ž1.
2 can be
performed using classical AVO techniques. In fact,
we have shown in Eqs. Ž32. and Ž33. that F0Ž1. and
F2Ž1. can be expressed in terms of the intercept and
gradient just as in classical AVO. Actually, all these
Ž1.
2
G 1Ž1. s AŽ1.
1g q B1g sin
u
2
Ž 45 .
,
with
Ž1.
AŽ1.
1g s A1g
X
Žu . s
RX10 q RX12 sin2
uX
2
sin u X ,
Ž1.
Ž1.
B1g
s B1g
Ž u X . s RX11 sin u X .
Ž 46 .
Ž 47 .
The definitions of the other parameters are given in
Table 1. Using AVO terminology, we remark that
Ž1.
AŽ1.
1g and B1g are the intercept and gradient of the
AVO function G 1Ž1., respectively. With the AVO
constructs in Eqs. Ž32., Ž33. and Ž45., we reduce the
number of parameters to be inverted to two or three
only.
The AVO-A without the 3f- and 4f-terms can be
finally regrouped into the classical AVO form as
follows:
Ž1.
Ž1.
Ž1.
Ž1.
R Ž1.
avoa s Ž A 0 f q A1f cos f q A1g sin f q A 2 f cos2 f
Ž1.
Ž1.
qAŽ1.
2 g sin2 f . q Ž B0 f q B1f cos f
L.T. Ikelle, L. Amundsenr Journal of Applied Geophysics 46 (2001) 1–29
14
Ž1.
qB1g
sin f q B2Ž1.f cos2 f q B2Ž1.g sin2 f . sin2
q Ž C0Ž1.f q C2Ž1.f cos2 f q C2Ž1.g sin2 f . sin
u
4
2
u
2
4.1. Dip and azimuthal angles
,
Ž 48 .
and reduces to
Ž1.
Ž1.
Ž1.
Ž1.
2
R Ž1.
avoa s A 0 f q Ž B0 f q B2 f cos2 f q B2 g sin2 f . sin
q Ž C0Ž1.f q C2Ž1.f cos2 f q C2Ž1.g sin2 f . sin
u
4
2
u
2
,
Ž 49 .
when the interface is horizontal.
4. AVO-A derivation and analysis for P–SV data
Our task in this section is to derive and analyze
the AVO-A for P–SV scattering. We will seek to
utilize the AVO-A of P–SV to resolve some of the
elastic parameters which cannot be recovered from
P–P scattering alone.
As in the previous section, we will begin by
relating the wavenumbers k s , kXs , k r and kXr to
incidence and reflecton angles as follows:
v
k s s sin us cos fs ,
Ž 50 .
ÕP
Fig. 5. The AVO-A of P–P where the 3f- and 4f-terms are dropped. The anisotropic materials used here are given in Table 2. Ža. and Žb.
correspond to the orthorhombic material in Table 2a for a horizontal reflector and a 308 dipping reflector, respectively. Žc. and Žd.
correspond to the arbitrarily anisotropic material in Table 2b for a horizontal reflector and a 308 dipping reflector, respectively.
L.T. Ikelle, L. Amundsenr Journal of Applied Geophysics 46 (2001) 1–29
kXs s
kr s
kXr s
v
ÕP
v
ÕS
v
ÕS
sin us sin fs ,
Ž 51 .
sin ur cos fr ,
Ž 52 .
sin ur sin fr .
Ž 53 .
The angles fs and fr are the same as those introduced in Fig. 3. The angles us and ur are shown in
Fig. 6. Notice that, although these angles have the
same meaning as those introduced in Fig. 2 for P–P,
their physical behaviors are quite different. For instance, us is not equal to ur even when the interface
is horizontal, due to the asymmetry between P-wave
and SV-wave reflections. However, they are related
through Snell’s law:
sin us s
ÕP
ÕS
sin ur .
15
Alternatively, the angles u, uX , f and fX will be
used. They are introduced as follows:
u s us y ur ,
Ž 55 .
u X s us q ur ,
Ž 56 .
fs
fX s
fs q fr
2
fs y fr
2
Ž 57 .
,
,
Ž 58 .
where u is the total reflection angle, u X is the angle
due to the asymmetry between the P- and SV-wave
reflection plus the dip angle of the reflector. The
angles f and f X have the same meaning as those
introduced in Fig. 3 for P–P scattering. The two new
angles, u and u X , are also shown in Fig. 6. As we
can see in Fig. 6, u X is non-zero even when the
reflector is horizontal, contrary to the P–P case.
Ž 54 .
This relationship is only valid when the interface is
horizontal because our angles, us and ur , are defined
with respect to the vertical axis and not with respect
to the normal vector of the reflector.
4.2. Decoupling of AVAZ and AVO
As we did for P–P scattering, By substituting
Eqs. Ž50. – Ž58. in Eq. Ž14. and regrouping the different elements as a linear combination of 1, cos f ,
Fig. 6. P–SV reflection at an isotropicranisotropic interface. Ža. corresponds to a horizontal interface. Contrary to the P–P reflection case,
the incident angle us is no longer equal to ur , the reflected angle, due to the asymmetry of the P–SV reflection. Žb. corresponds to a dipping
interface.
L.T. Ikelle, L. Amundsenr Journal of Applied Geophysics 46 (2001) 1–29
16
Table 3
AVO functions that describe amplitude variations with offsets and azimuths for P–SV scattering
AVO formulae
AVO parameters
F0Ž1. s D r sin us cos ur y D r cos us sin ur
R 00 s
1
2
Elastic parameters
Ž D r q D r q G 33 y G 13 q 2 G44 .
1
q G 11 sin2us sin2 ur
2
1
R 01 s y Ž G 33 q G 11 . q G 13 y 2 G44
2
1
y G 33 cos 2us sin2 ur
2
R 02 s yR 04 s
1
q G 13 cos2 us sin2 ur
2
R 03 s
1
q G44 sin2 us cos2 ur
2
R 05 s
ž
F0Ž1. s R 00 q R 01 sin2
u
2
q R 02 sin2
u
ž
q R 03 q R 04 sin2
2
uX
/
2
q R 05 sin2
1
2
1
2
1
2
Ž G 11 y G 33 .
2
/
X
1
G 11 s Ž DC11 q DC22 .
8
X
=Ž2 q cos4f .
Ž D r y D r y G 33 q G 13 q 2 G44 .
1
X
q DC12 Ž2 y cos4f .
4
Ž G 33 q G 11 . y G 13 y 2 G44
1
X
q DC66 Žcos4f .
2
sin u
uX
D r s D r cos2 f
G 33 s DC33
sin u
1
G 13 s Ž DC13 q DC23 .
2
X
1
G44 s Ž DC44 q DC55 .cos2 f X
2
ž
F1Ž1. s F Bc sin2us cos2 ur q
1
2
ž
ž
ž
sin2 us sin2 ur
qFGc cos 2us cos2 ur y
qF Bx sin2us cos2 ur y
qFGx cos 2us cos2 ur q
F1Ž1. s R10 q R11 sin2
qR13 sin2
qR15 sin4
u
2
2
1
2
1
2
sin2 us sin2 ur
sin2 us sin2 ur
sin2 us sin2 ur
q R12 sin2
sin2
2
u
u
1
u
uX
2
X
2
/
q R14 sin4
u
2
R10 s FGc q FGx
/
/
/
1
F Bc s Ž DC15 q DC25 .cos f X
2
R13 s 4 Rf 1 s 2ŽF Bc q FGc q F Bx q FGx .
1
X
q Ž DC15 y DC25 .cos3f
4
R14 s 4ŽyF Bx q FGx .
1
X
q DC46 cos3f
2
R15 s y4ŽF Bc y FGc .
1
F Bx s y Ž DC24 q DC14 .sin f X
2
1
R11 s y Ž2 R14 q R 13 .
2
1
X
y Ž DC14 y DC24 .sin3f
4
1
R12 s y Ž2 R15 q R13 .
2
1
X
q DC56 sin3f
2
X
2
q Rf 1 sin u sin u
X
FGc s DC35 cos f X
FGx s yDC34 sin f X
ž
G1Ž1. s F Bs sin2us cos2 ur q
1
2
sin2 us sin2 ur
/
R10 s FGs q FGy
1
F Bs s Ž DC14 q DC24 .cos f X
2
L.T. Ikelle, L. Amundsenr Journal of Applied Geophysics 46 (2001) 1–29
17
Table 3 Ž continued .
AVO formulae
AVO parameters
ž
ž
ž
qFGs cos 2us cos2 ur y
qF By sin2us cos2 ur y
qFGy cos 2us cos2 ur q
G1Ž1. s R10 q R11 sin2
qR13 sin2
qR15 sin4
u
2
uX
2
u
2
1
2
1
2
1
2
sin2 us sin2 ur
sin2 us sin2 ur
sin2 us sin2 ur
q R12 sin2
sin2
u
uX
2
q R14 sin4
q Rf 1 sin u sin u
R13 s 4 Rf 1 s 2ŽF Bs q FGs q F By q FGy .
1
X
y Ž DC14 y DC24 .cos3f
4
R14 s 4ŽyF By q FGy .
1
X
q DC56 cos3f
2
R15 s y4ŽF Bs y FGs .
1
F By s Ž DC15 q DC25 .sin f X
2
1
R 11 s y Ž2 R14 q R 13 .
2
X
2
/
/
/
Elastic parameters
u
R12 s
2
1
2
1
X
y Ž DC15 y DC25 .sin3f
4
1
X
y DC46 sin3f
2
Ž2 R15 q R13 .
X
FGs s DC34 cos f X
FGy s DC35 sin f X
F2Ž1. s
1
2
G Bc sin2us sin2 ur
1
2
G Hc q
1
2
G Kc
1
q GGc sin2 us cos2 ur
2
1
R 21 s y G Bc q G Hc y 2 GGc
2
1
q G Hc cos2 us sin2 ur
2
R 22 s yR 24 s
1
y G Kc sin2 ur
2
R 23 s GGc q
ž
F2Ž1. s R 20 q R 21 sin2
u
2
q R 22 sin2
ž
q R 23 q R 24 sin2
q R 25 sin2
G Ž1.
2 s
R 20 s GGc y
1
2
uX
2
/
uX
2
/
sin u
R 25 s
1
2
1
2
1
2
G Bc y G Kc
G Hc y
1
2
G Kc
1
G Bc s Ž DC11 y DC22 .cos2 f X
2
1
GGc s Ž DC55 y DC44 .
2
1
G Hc s Ž DC13 y DC23 .cos2 f X
2
G Kc s DC36 sin2 f X
G Bc y 2 GGc y G Hc
u
2
sin u
X
G Bs sin2us sin2 ur
R 20 s GGs y
1
2
G Hs q
1
2
G Ks
1
q GGs sin2 us cos2 ur
2
1
R 21 s y G Bs q G Hs y 2 GGs
2
1
q G Hs cos2 us sin2 ur
2
R 22 s yR 24 s
1
y G Ks sin2 ur
2
R 23 s GGc q
1
2
1
2
G Bs y G Ks
G Hs y
1
2
G Ks
G Bs s Ž DC16 q DC26 .cos2 f X
GGs s DC45
G Hs s DC36 cos2 f X
1
G Ks s Ž DC23 y DC13 .sin2 f X
2
(continued on next page .
L.T. Ikelle, L. Amundsenr Journal of Applied Geophysics 46 (2001) 1–29
18
Table 3 Ž continued .
AVO formulae
AVO parameters
ž
2
G Ž1.
2 s R 20 q R 21 sin
u
q R 22 sin2
2
ž
q R 23 q R 24 sin2
F3Ž1. s
1
2
u
2
ž
ž
u
q R 32 sin2
2
qR 33 sin2
qR 35 sin4
uX
/
2
u
sin2
2
uX
2
uX
2
sin u
sin2 us sin2 ur
2
u
q R 34 sin4
q Rf 3 sin u sin u
q R 32 sin2
2
qR 33 sin2
qR 35 sin4
1
2
u
u
2
uX
2
ž
u
2
2
sin2 us sin2 ur
ž
2
q R 34 sin4
u
ž
ž
u
2
q R 44 sin2
u
2
/
/
R 33 s 4 Rf 3 s 2ŽF Es q F Ey .
R 34 s y4F Ey
uX
2
/
1
s yR 45 s y G Ec
2
sin u
uX
2
/
sin u
2
q R 45 sin2
/
1
s yR 45 s y G Es
2
sin u
uX
2
/
y2DC46 xcos f
X
1
F Ex s wŽ DC14 y DC24 .
4
q2DC56 xsin f
X
sin u
1
F Es s wŽ DC14 y DC24 .
4
q2DC56 xcos f
X
1
F Ey s wŽyDC15 q DC25 .
4
q2DC46 xsin f
X
G Ec s Ž1r8.Ž DC11 q DC22 .
yŽ1r4. DC12 y Ž1r2. DC66
X
R 41 s yR 42 s R 44
uX
1
F Ec s wŽ DC15 y DC25 .
4
1
R 32 s y Ž2 R 35 q R 33 .
2
X
R 41 s yR 42 s R 44
q R 45 sin2
q R 42 sin2
R 34 s y4F Ex
1
R 31 s y Ž2 R 34 q R 33 .
2
2
G Es sin2us sin2 ur
G4Ž1. s R 41 sin2
R 33 s 4 Rf 3 s 2ŽF Ec q F Ex .
R 35 s y4F Es
2
q Rf 3 sin u sin u
u
X
1
R 32 s y Ž2 R 35 q R 33 .
2
X
uX
q R 42 sin2
q R 44 sin2
2
uX
1
2
Elastic parameters
G Bs y 2 GGs y G Hs
1
R 31 s y Ž2 R 34 q R 33 .
2
2
G Ec sin2us sin2 ur
F4Ž1. s R 41 sin2
1
sin2
1
2
R 35 s y4F Ec
1
F Es sin2us cos2 ur q sin2 us sin2 ur
2
2
G 3Ž1. s R 31 sin2
/
/
2
ž
ž
1
1
R 25 s
uX
qF Ey sin2us cos2 ur y
G4Ž1. s
sin u
1
F Ec sin2us cos2 ur q sin2 us sin2 ur
2
F3Ž1. s R 31 sin2
F4Ž1. s
/
2
q R 25 sin2
qF Ex sin2us cos2 ur y
G 3Ž1. s
u
X
X
G Es s Ž1r2.Ž DC16 y DC26 .
L.T. Ikelle, L. Amundsenr Journal of Applied Geophysics 46 (2001) 1–29
sin f , cos2 f , sin2 f , cos3f , sin3f , cos4f , sin4f 4 ,
the amplitude variations with offsets and azimuths
ŽAVO-A. can be cast in terms of a Fourier series of
the azimuthal angle f as follows:
4
Ž2.
R Ž2.
avoa s F0 q
Ý
FnŽ2. cos Ž n f . q GnŽ2. sin Ž n f . .
ns1
Ž 59 .
Expressions of functions F0Ž2. s F0Ž2. Ž u , u X , f X ., FnŽ2.
s FnŽ2. Ž u , u X , f X . and GnŽ2. s GnŽ2. Ž u , u X , f X . can be
obtained from Table 3 Žthe subscript n runs from 1
to 4.. Furthermore, the different elements of the
series Ž59. can also be regrouped as a linear combination of cos f X , sin f X , cos2 f X , sin2 f X , cos3f X ,
sin3f X , cos4f X , sin4f X 4 , leading to
X
X
Ž2.
Ž2.
Ž2.
R Ž2.
avoa s f 00 q f 02 cos2 f q f 04 cos4f
Ž2.
R Ž2.
avoa s F0 ,
Ž 61 .
which means that the amplitudes are invariant with
azimuths. For orthorhombic symmetry, we must add
the functions F2Ž2. and F4Ž2. to Eq. Ž59.:
Ž 62 .
For monoclinic symmetry, the AVO-A formula in
Eq. Ž59. becomes
Ž2 c .
qf 13Ž2 c . cos3f X q g 13
sin3f X cos f
Ž2 s.
q f 11Ž2 s. cos f X q g 11
sin f X
4
Ž2.
R Ž2.
avoa s F0 q
Ž2 s.
qf 13Ž2 s. cos3f X q g 13
sin3f X sin f
Ý FnŽ2. cos Ž nf . .
Ž 63 .
ns1
Ž2 c .
Ž2 c .
Ž2 c .
q f 20
q f 22
cos2 f X q g 22
sin2 f X
Ž2 s.
Ž2 s.
=cos2 f q f 20
q f 22
cos2 f X
Ž2 s.
Ž2 c .
cos f X
qg 22
sin2 f X sin2 f q f 31
Ž2 c .
Ž2 s.
cos f X
qg 31
sin f X cos3f q f 31
Ž2 s.
Ž2 c .
cos4f
qg 31
sin f X sin3f q f 40
Ž2 c .
q g 40
sin4f .
u X . and GnŽ2. s GnŽ2. Ž u , u X .. They can be obtained
from Table 3 by taking f X s 0 Žthe subscript n runs
from 1 to 4..
By comparing Eqs. Ž26. and Ž59., we can remark
that the AVAZ behavior of P–SV scattering has
exactly the same form as that of P–P. This similarity
is preserved for transversely isotropic ŽTI., orthorhombic and monoclinic symmetries. In fact, for
TI symmetry with respect to the vertical axis ŽTIV.,
the AVO-A formula in Eq. Ž59. is reduced to F02
only:
Ž2.
Ž2.
Ž2.
R Ž2.
avoa s F0 q F2 cos Ž 2 f . q F4 cos Ž 4f . .
Ž2 c .
q f 11Ž2 c . cos f X q g 11
sin f X
19
Ž 60 .
X
Ž2. Ž
. Ž2c.
Expressions of functions f 0Ž2.
m s f 0 m u , u , fn m s
X
X
Ž2c. Ž
Ž2s.
Ž2s.
Ž2c.
Ž2c.
f n m u , u ., f n m s f n m Ž u , u ., g n m s g n m Ž u , u X .
X
Ž2s. Ž
. Žthe subscripts n and m run
and g nŽ2s.
m s gnm u , u
from 0 to 4. can be deduced from Table 3.
Just like for the P–P case, the dependence of
X
R Ž2.
avoa on f and f describes amplitude variations
with azimuths ŽAVAZ. while its dependence on u
and u X describes amplitude variations with offsets
ŽAVO.. Thus, the AVO effect in Eq. Ž26. of R Ž1.
avoa is
Ž2c.
Ž2s.
Ž2c.
represented by the functions f 0Ž2.
m , fn m , fn m , g n m .
We will limit the rest of our discussion in this
section to one azimuthal angle by taking f X s 0,
using the series Ž59. instead of Eq. Ž60.. Hence, the
AVO functions are F0Ž2. s F0Ž2. Ž u , u X ., FnŽ2. s FnŽ2. Ž u ,
Notice that 61., Ž62. and Ž63. are similar to Eqs.
Ž28. – Ž30., respectively.
We have established that the structure of AVAZ
of P–SV is similar to that of P–P. By comparing the
third columns of Tables 1 and 3, we can also observe
that the combinations of elastic parameters invoked
in P–SV are exactly the same as those in P–P. The
differences between P–P and P–SV are in their
AVO behaviors. We will analyze these differences in
more detail below.
4.3. Effect of dip
As discussed in the previous sections, the dipping
and anisotropic effects on AVO-A of P–P scattering
are distinguishable. However, the case where the
bottom medium Žsee Fig. 1. is azimuthally isotropic
has not yet been resolved. The AVO function F0Ž2. of
P–SV provides the answer to this case. As described
in Table 3, F0Ž2. is reduced to
F0Ž2. s yD r sin ur y
1
2
Ž G 33 y gG13 . sin2 ur ,
Ž 64 .
20
L.T. Ikelle, L. Amundsenr Journal of Applied Geophysics 46 (2001) 1–29
Fig. 7. The AVO-A of P–SV. The anisotropic materials used here are given in Table 2. Ža. and Žb. correspond to the orthorhombic material
in Table 2a for a horizontal reflector and a 308 dipping reflector, respectively. Žc. and Žd. correspond to the arbitrarily anisotropic material in
Table 2b for a horizontal reflector and a 308 dipping reflector, respectively.
at normal incidence Ži.e., us s 0.. The definitions of
G33 and G13 are given in Table 3. If the interface is
assumed horizontal, the reflected angle ur is zero
Ž ur s 0. whenever us s 0; thus, F0Ž2. is zero. However, if the interface is a dipping one as described in
Fig. 1, ur / 0 even when us s 0; hence, F0Ž2. is
non-zero. This remark can be used as a dip indicator
for azimuthally isotropic symmetry.
Contrary to what we have seen for P–P scattering,
notice that the functions F1Ž2., F3Ž2., G 1Ž2. and G 3Ž2. are
non-zero even if the interface is horizontal because
u X / 0 due to asymmetry between the P–SV reflection.
4.4. AVO-A analysis for inÕersion purposes
The inversion procedure of AVO-A of P–SV is
similar to that outlined earlier for P–P. We first
extract the AVO functions F0Ž2., FnŽ2. and G
www.elsevier.nlrlocaterjappgeo
AVO-A response of an anisotropic half-space bounded by a
dipping surface for P–P, P–SV and P–SH data
Luc T. Ikelle a,) , Lasse Amundsen b
a
CASP Project, Department of Geology and Geophysics, Texas A & M UniÕersity, College Station, TX 77843-3115, USA
b
Statoil Exploration and Petroleum Technology Research Centre, USA
Received 20 October 1999; accepted 29 August 2000
Abstract
We analyze amplitude variations with offsets and azimuths ŽAVO-A. of an anisotropic half-space bounded by a dipping
surface. By analyzing the response of a dipping reflector instead of a horizontal one, we integrate the fundamental problem
of lateral heterogeneity vs. anisotropy into our study. This analysis is limited to the three scattering modes that dominate
ocean bottom seismic ŽOBS. data: P–P, P–SV and P–SH. When the overburden is assumed isotropic, the AVO-A of each of
these three scattering modes can be cast in terms of a Fourier series of azimuths, f , in general form,
4
R avoa Ž f . s F0 q
Ý w Fn cos Ž n f . q Gn sin Ž n f . x ,
ns 1
where F0 , Fn and Gn are the functions that describe the seismic amplitude variations with offsets ŽAVO. for a given
azimuth. The forms of AVO functions are similar to those of classical AVO formulae; for instance, the AVO functions
corresponding to the P–P scattering mode can be interpreted in terms of the intercept and gradient, although the resulting
numerical values can differ significantly from those of isotropic cases or horizontal reflectors.
One of the benefits of describing the AVO-A as a Fourier series is that the contribution of amplitude variations with
azimuths ŽAVAZ. is distinguishable from that of AVO. The AVAZ is characterized by the functions 1, cos f , sin f , cos2 f ,
sin2 f , cos3f , sin3f , cos4f , sin4f 4, that are mutually orthogonal. Thus, the AVO–A inversion can be formulated as a
series of AVO inversions where the AVO behaviors are represented by the functions F0 , Fn and Gn .
When the coordinate system of seismic acquisition geometry coincides with the symmetry planes of the rock formations,
the series corresponding to P–P and P–SV simplify even further; they reduce to F0 for azimuthally isotropic symmetry and
to F0 , F2 , F4 , G 2 and G4 for orthorhombic symmetry. The series corresponding to P–SH scattering is reduced to G 2 and G4
for these two symmetries.
Unfortunately, the coordinate system of seismic acqusition geometry rarely coincides with the symmetry planes of the
rock formations; therefore, the other terms are rarely zero. In particular, the functions F1, F3 , G 1 and G 3 become important
for large dips and are actually largely dependent on the angle of the dipping reflector. For P–P scattering, these functions are
zero if the reflector is horizontal, irrespective of the anisotropic behavior. For P–SV and P–SH scattering, these functions
)
Corresponding author. Fax: q1-979-458-1513.
E-mail address: ikelle@nutmeg.tamu.edu ŽL.T. Ikelle..
0926-9851r01r$ - see front matter q 2001 Elsevier Science B.V. All rights reserved.
PII: S 0 9 2 6 - 9 8 5 1 Ž 0 0 . 0 0 0 3 1 - 8
2
L.T. Ikelle, L. Amundsenr Journal of Applied Geophysics 46 (2001) 1–29
are not necessarily zero for horizontal reflectors because they are affected by the asymmetry of the P–S reflection in addition
to the effect of dip. q 2001 Elsevier Science B.V. All rights reserved.
Keywords: AVO-A; Ocean bottom seismic; Dipping surface
1. Introduction
Ocean bottom seismic ŽOBS. data acquisition and
processing system is a new oil and gas exploration
and production tool that not only aims at remotely
identifying fluid-saturated rock formations but also
at determining their characteristics as well as any
changes in their characteristics over time. These
determinations are possible because OBS surveys
have overcome three key limitations of towedstreamer surveys: Ži. azimuthal coverage, Žii. multicomponent recordings and Žiii. repeatability of surveys. The 3D OBS can be designed with small
enough spacing in both source and receiver coordinates to allow a full azimuthal coverage. Also, since
it is carried out at the sea floor, three component
particle velocity Žgeophone. data can be recorded
along with pressure Žhydrophone. data. This recording system gives us direct access to the shear-wave
in addition to the P-wave. By considering a scenario
where sensors are permanently placed at the sea
floor, the repeatability of OBS is a realistic possibility although some technological issues, such as the
coupling of geophones to the sea floor, still remain
to be overcome. To use these three new features of
marine seismic measurements for retrieving fluid
properties of rock formations, we need to gain more
insight on how the physical properties which characterize fluid-saturated rocks are related to seismic
data.
Fluid-saturated rocks contain small scale inhomogeneities such as fractures that control the fluid flow
in these rocks. At the scale of seismic wavelength,
rocks containing these types of small scale inhomogeneities behave as azimuthally anisotropic materials. Therefore, changes in the physical properties of
such rock formations can produce amplitude variations with offsets ŽAVO. as well as amplitude variations with azimuths ŽAVAZ.. In short, we will call
such amplitude variations AVO-A.
To identify and characterize fluid-saturated rock
formations, we need to understand the correlations
between anisotropic parameters and AVO-A on one
hand and the correlations between anisotropic parameters and properties of fluid-saturated rock formations on the other. Such correlations make it
possible to reduce the number of parameters in our
inversion algorithms, which are generally ill-posed
due partially to the large number of unknowns.
These types of correlations can also help distinguish
between different petrological models. Here we present an analysis of an AVO-A response to an
anisotropic half-space bounded by a dipping surface
as one way of improving our understanding of these
correlations. The dipping surface can be planed or
curved. By considering the response of a dipping
reflector instead of a horizontal one, we have tried to
integrate the fundamental problem of lateral heterogeneity vs. anisotropy into our study. Our analysis
will be limited to the three scattering modes which
dominate OBS data: P–P, P–SV and P–SH.
We will use the weak-contrast assumption in our
derivations Ži.e., Born approximation., but no weakly
anisotropic assumptions are made of this material.
We will consider transversely isotropic ŽTI. and
orthorhombic symmetries as most experimental and
theoretical studies suggest that the majority of rock
formations possess these symmetries Že.g., Backus,
1962; White, 1975; Thomsen, 1986; Ikelle et al.,
1993; Sayers, 1998.. However, these symmetries are
of interest in the derivations of AVO-A only if the
axis of these symmetries coincides with the coordinate system of seismic acquisition geometries. Unfortunately, the coordinate system of seismic acquisition geometries sometimes does not coincide with
the symmetry plane of rock formations. Even if they
do coincide, such information is generally unknown.
Our AVO-A analysis will show that the total
anisotropic effect of reflection data can be described
by a small number of simple linear combinations of
the elastic stiffness tensor coefficients.
There are many publications on reflection coefficients for weakly anisotropic media Že.g., Daley and
Horn, 1977; Gibson and Ben-Menahem, 1991; Ikelle,
L.T. Ikelle, L. Amundsenr Journal of Applied Geophysics 46 (2001) 1–29
1996; Zillmer et al., 1997, 1998; Ruger, 1998; Ursin
and Haugen, 1996; Vavrycuk
˘ and Psencik,
˘ ˘ 1998..
The novelty here with respect to the previous works
is that our derivations include the effect of a dipping
reflector. We also produce mathematical constructs
of AVO-A, which can be useful for AVO-A analysis
and inversions by seeking the combinations of elastic
parameters that best constrained P–P, P–SV and
P–SH data. These mathematical constructs are similar to those derived by Ikelle Ž1996. and corroborated by real data experiments ŽLeaney et al., 1999.
for P–P data when the acquisition plane is parallel to
the dip direction.
A summary of the rest of this paper is as follows.
In the next section, we will describe the basic elastodynamic properties of anisotropic media needed to
introduce the general formulae of the AVO-A. We
will also recall these formulae for the three scattering
modes considered here. In the third, fourth and fifth
sections, we will derive and analyze AVO-A formulae for each of the three scattering modes. Finally, in
the sixth section, we will indicate the key preprocessing requirements for AVO-A inversion.
2. Introductory definitions
Consider two media separated by a dipping interface Žsee Fig. 1.. The top medium, in which the
sources and receivers are located, is homogeneous
and isotropic; the bottom medium is also homogeneous but anisotropic. Our objective in this paper is
3
to derive the AVO-A response of this configuration
under the weak-contrast approximation ŽBorn approximation.. In this particular section, we introduce
the basic definitions that we need for the AVO-A
formulation.
The AVO-A formulation requires the slowness
and polarization vectors which characterize the Pwave propagation from the source point to the scattered point on the dipping interface and those which
characterize P- and S-wave upward propagation from
the scattered point to the receiver point. As we have
assumed that the top medium is homogeneous and
isotropic, we are simply going to recall the well
known analytic formulae of the slowness and polarization vectors for this particular case. We will also
recall the compact definition of AVO-A on which
our derivation will be based. To clarify the presentation of these definitions, we will start by introducing
our notation convention.
2.1. Notations
Position in the configuration in Fig. 1 is specified
by the coordinates x, y, z 4 with respect to a fixed
Cartesian reference frame with the origin O and the
three mutually perpendicular base vectors i x , i y , i z 4
each of unit length. i z points vertically downwards.
To accommodate anisotropy, the subscript notation
for vectors and tensors and the Einstein summation
convention are adopted. Lower case Latin subscripts
are employed for this purpose; they are to be assigned the values 1, 2 and 3 if not specified other-
Fig. 1. Isotropicranisotropic configuration with sources and receivers in the isotropic media.
L.T. Ikelle, L. Amundsenr Journal of Applied Geophysics 46 (2001) 1–29
4
wise. The lower case Latin subscripts s and r are
reserved symbols for indicating source and receiver,
respectively.
As we will see later, the quantities used to describe AVO-A formulae depend on several variables.
We will sometimes write these quantities without
some of the variables when the context unambiguously indicates the quantity currently under consideration.
2.2. Vertical waÕenumbers, polarization Õectors and
slowness Õectors in an isotropic homogeneous
medium
We start with the vertical wavenumbers corresponding to an isotropic, homogeneous, elastic
medium. They are defined as follows for downgoing
P-waves from a source point, Ž x s , ys , 0.,
qsŽP . s qsŽP . Ž k s ,kXs , v . s
v
(
1y
ÕP
Õ P2
k s2 q kXs2
v
,
2
Let us now define the polarization vectors. The
polarization vector b˜ ŽP. s b ŽP. Ž k s , kXs , v . of the
downgoing P-waves is
b̃ ŽP . s b ŽP . k s ,kXs , v
Ž
qrŽP . s qrŽP . Ž k r ,kXr , v . s
v
(
1y
ÕP
Õ P2 k r2 q kXr2
v
,
2
Ž 2.
qrŽS. s qrŽS.
Ž
k r ,kXr , v
.s
v
Õs
(
1y
Õ S2 k r2 q kXr2
v2
,
Ž 3.
k s , kXs , k r , and kXr are the horizontal wavenumbers
corresponding to x s , ys , x r , and yr , respectively; v
is the temporal frequency corresponding to time t;
Õ P and Õ S are the P- and S-wave velocities of the top
medium in Fig. 1, respectively. The vertical
wavenumber qsŽP. characterizes downgoing P-waves
propagating through the top medium Žsee Fig. 1.
from the source to the scattered point, and qrŽP. and
qrŽS. characterize upgoing P- and S-waves propagating through the top medium from the scattered point
to the receiver.
0
v
Ž 4.
,
qsŽ P .
and those of the upgoing P-, SV- and SH-waves are
defined as follows:
b̂
ŽP .
sb
ŽP .
Ž
k r ,kXr , v
.s
0
kr
kXr
ÕP
v
Ž 5.
,
qrŽ P .
bˆ ŽSV . s b ŽSV . Ž k r ,kXr , v .
s
1
ÕS
v
(
k r2 q kXr2
Ž 1.
and for upgoing P- and S-waves to a receiver location, Ž x r , yr , 0.,
.s
ks
kXs
ÕP
k r qrŽ S .
kXr qrŽ S .
yŽ
b̂ ŽSH . s b ŽSH . Ž k r ,kXr , v . s
k r2 q kXr2
.
0
1
(k q k
2
r
X2
r
Ž 6.
,
ykXr
k r , Ž 7.
0
0
respectively.
Another set of vectors that we are going to need
in the definition of AVO-A are slowness vectors.
They are denoted by g˜ ŽP. s g ŽP. Ž k s , kXs , v . for a
downgoing P-wave and defined as follows:
g˜ ŽP . s g ŽP . Ž k s ,kXs , v . s
1
v
k s ,kXs ,qsŽP .
T
Ž 8.
.
For upgoing P-, SV- and SH-waves, they are defined
as follows:
gˆ ŽP . s g ŽP . Ž k r ,kXr , v . s
1
v
gˆ ŽSV . s g ŽSV . Ž k r ,kXr , v . s
gˆ ŽSH . s gˆ ŽSV . ,
k r ,kXr ,qrŽP .
1
v
T
k r ,kXr ,qrŽS.
Ž 9.
,
T
,
Ž 10 .
Ž 11 .
respectively, where upper-case T indicates transpose
vectors.
L.T. Ikelle, L. Amundsenr Journal of Applied Geophysics 46 (2001) 1–29
l
2.3. AVO-A for a weak-contrast interface
We can describe the materials in Fig. 1 by their
density of mass, r , and their stiffness tensor, c i jk l :
r Ž0. ; c iŽ0.jk l 4 characterizes the isotropic top medium
with the following relationship:
c iŽ0.jk l
r Ž0.
s Õ P2 y 2 Õ S2 d i j d k l q Õ S2 Ž d i k d jl q d i l d jk . ,
Ž 12 .
while r Ž1., c iŽ1.jk l 4 characterizes the anisotropic bottom medium. For an incident P-wave, the reflection
at the interface between the two media produces
three scattering fields: P–P, P–SV and P–SH. Under
the weak-contrast approximation, the reflection coefficients characterizing the amplitude variations with
offsets and azimuths can be written Žsee de Hoop et
al., 1994; Ikelle, 1996.:
R Ž1.
avoa s
˜
ˆ
biŽP .D rbiŽP .
q
˜ ˜
biŽP .g jŽP .D c i jk l
for P–P scattering
ˆ ˆ
b kŽP .g lŽP .
Ž 13 .
q b˜iŽP .g˜ jŽP .D c i jk l bˆkŽSV .gˆ lŽSV . ,
Ž 14 .
˜ ŽP . ˆ ŽSH .
R Ž3.
avoa s b i D rb i
q b˜iŽP .g˜ jŽP .D c i jk l bˆkŽSH .gˆ lŽSH . ,
for P–SH scattering
r Ž1.
r Ž0.
D c i jk l s
,
c iŽ1.jk l
r Ž0. Õ P2
.
3. AVO-A derivation and analysis for P–P data
3.1. Dip and azimuthal angles
The amplitude variations with offsets and azimuths R Ž1.
avoa for P–P scattering are given in Eq.
Ž13.. To gain physical insight into R Ž1.
avoa , it is useful
to express it in terms of dip and azimuthal angles.
This can be done by relating the wavenumbers k s ,
kXs , k r and kXr to incident and reflected angles as
follows:
ks s
Ž 15 .
where
D rs
™
scripts I and J run from 1 to 6, with ij I according to 11, 22, 33, 23, 31, 12 1, 2, 3, 4, 5, 6.
The Eqs. Ž13., Ž14. and Ž15. are just alternative
forms of small angle approximation of the reflection
coefficients Žsee Ikelle et al., 1992; Ikelle, 1995..
These formulae are derived from scattering theory
assuming that density and stiffnesses are spatial
step-functions. They do not use plane wave approximation nor weak anisotropic approximation. However, they assume a weak-contrast interface Ži.e.,
Born approximation..
The effect of dip in Eqs. Ž13., Ž14. and Ž15. is
taken into account by the upgoing polarization and
slowness vectors. As we will see later, the direction
of the upgoing rays is modified if the interface is a
dipping one instead of a horizontal one.
,
˜ ŽP . ˆ ŽSV .
R Ž2.
avoa s b i D rb i
for P–SV scattering
5
kXs s
Ž 16 .
kr s
Ž 17 .
kXr s
Alternatively, the tensorial stiffness D c i jk l will also
be denoted by DCI J , respectively, where the sub-
v
ÕP
v
ÕP
v
ÕP
v
ÕP
sin us cos fs ,
Ž 18 .
sin us sin fs ,
Ž 19 .
sin ur cos fr ,
Ž 20 .
sin ur sin fr ,
Ž 21 .
where us is the angle between the incident ray and
the vertical axis; ur is the angle between the re-
L.T. Ikelle, L. Amundsenr Journal of Applied Geophysics 46 (2001) 1–29
6
flected ray and the vertical axis; fs is the azimuthal
angle between the x-axis and the line connecting the
projection of the shot point in the plane Ž x–y . with
that of the scattering point; fr is the angle between
the x-axis and the line connecting the projection of
the receiver point in the plane Ž x–y . with that of the
scattering point. These four angles are shown in
Figs. 2 and 3. The angles us and ur vary over w0,
pr2x; the azimuthal angles fs and fr vary over w0,
2p x.
Alternatively, the angles u , u X , f and f X will be
used instead of us , ur , fs and fr . They are introduced as follows:
u s us y ur ,
Ž 22 .
u X s us q ur ,
Ž 23 .
fs
fs q fr
2
Ž 24 .
,
gle between the plane containing the scattering point
and shot point and that containing the scattering
point and receiver points. These four new angles are
also shown in Figs. 2 and 3. As we can see in Fig. 2,
u X is zero for the particular case where the reflector
is horizontal.
3.2. Decoupling of AVAZ and AVO
By substituting Eqs. Ž18. – Ž25. in Eq. Ž13. and
regrouping the different elements as a linear combination of 1, cos f , sin f , cos2 f , sin2 f , cos3f ,
sin3f , cos4f , sin4f 4 , the amplitude variations with
offsets and azimuths ŽAVO-A. can be cast in terms
of a Fourier series of the azimuthal angle f as
follows:
4
Ž1.
R Ž1.
avoa s F0 q
Ý
FnŽ1. cos Ž n f . q GnŽ1. sin Ž n f . .
ns1
fX s
fs y fr
2
Ž 25 .
,
X
where u is the total reflection angle; u is the angle
of the dipping reflector; f is the azimuthal angle
between the x-axis and the plane containing the
scattering point and midpoint; f X the azimuthal an-
Ž 26 .
Expressions of functions F0Ž1. s F0Ž1. Ž u , u X , f X ., FnŽ1.
s FnŽ1. Ž u , u X , f X . and GnŽ1. s GnŽ1. Ž u , u X , f X . can be
obtained from Table 1 Žthe subscript n runs from 1
to 4.. Furthermore, the different elements of the
series Ž26. can also be regrouped as a linear combi-
Fig. 2. P–P reflection at an isotropicranisotropic interface. Ža. corresponds to a horizontal interface with, us , the incident angle equal to, ur ,
the reflected angle. Žb. corresponds to a dipping interface where the incident angle no longer equals the reflected angle.
L.T. Ikelle, L. Amundsenr Journal of Applied Geophysics 46 (2001) 1–29
7
Fig. 3. Azimuthal angles used in the radiation patterns for a case where the receivers are outside the acquisition plane.
nation of cos f X , sin f X , cos2 f X , sin2 f X , cos3f X ,
sin3f X , cos4f X , sin4f X 4 , leading to
X
X
Ž1.
Ž1.
Ž1.
R Ž1.
avoa s f 00 q f 02 cos2 f q f 04 cos4f
Ž1c .
sin f X
q f 11Ž1c . cos f X q g 11
Ž1c .
sin3f X cos f
qf 13Ž1c . cos3f X q g 13
Ž1s.
sin f X
q f 11Ž1s. cos f X q g 11
Ž1s.
qf 13Ž1s. cos3f X q g 13
sin3f X sin f
Ž1c .
Ž1c .
Ž1c .
q f 20
q f 22
cos2 f X q g 22
sin2 f X cos2 f
Ž1s.
Ž1s.
Ž1s.
sin2 f X sin2 f
q f 20
q f 22
cos2 f X q g 22
Ž1c .
Ž1c .
sin f X cos3f
q f 31
cos f X q g 31
Ž1s.
Ž1s.
sin f X sin3f
q f 31
cos f X q g 31
Ž1c .
q f 40Ž1c . cos4f q g 40
sin4f .
Ž 27 .
X
Ž1. Ž
. Ž1c.
Expressions of functions f 0Ž1.
m s f 0 m u , u , fn m s
X
X
Ž1c. Ž
Ž1s.
Ž1s. Ž
Ž1c.
Ž1c. Ž
.
.
fn m u , u , fn m s fn m u , u , g n m s g n m u , u X .
X
Ž1s. Ž
. Žthe subscripts n and m run
and g nŽ1s.
m s gnm u , u
from 0 to 4. can be deduced from Table 1.
X
The dependence of R Ž1.
avoa on f and f describes
amplitude variations with azimuths ŽAVAZ. while
its dependence on u and u X describes amplitude
variations with offsets ŽAVO.. Thus, the AVO effect
in Eq. Ž27. of R Ž1.
avoa is represented by the functions
Ž1c.
Ž1s.
Ž1s.
f 0Ž1.
,
f
,
f
,
g nŽ1c.
m
nm
nm
m , and g n m , all independent of
the azimuthal angles f and f X .
Ž .
To simplify our analysis of R Ž1.
avoa in Eqs. 26 and
Ž27., we will limit our discussion to one azimuthal
angle by taking f X s 0 for the rest of this section
Ži.e., the source and receivers lie on the same line
similar to when seismic data are arranged in common azimuthal sections, as described in Ikelle, 1996.
and we will use the series Ž26. instead of Eq. Ž27..
So, the AVO functions are F0Ž1. s F0Ž1. Ž u , u X ., FnŽ1. s
FnŽ1. Ž u , u X . and GnŽ1. s GnŽ1. Ž u , u X .. They can be
obtained from Table 1 by taking f X s 0 Žthe subscript n runs from 1 to 4..
Before we discuss the implication of these conŽ .
structs of R Ž1.
avoa in Eq. 26 for an inversion algorithm, let us examine the AVO-A formula in Eq.
Ž26. for three symmetries regularly observed in ultrasonic laboratory measurements Že.g., Jones and
Wang, 1981; Lo et al., 1981; Cheadle et al., 1991;
L.T. Ikelle, L. Amundsenr Journal of Applied Geophysics 46 (2001) 1–29
8
Table 1
AVO functions which describe amplitude variations with offsets and azimuths for P–P
AVO formulae
X
F0Ž1. s D r cos us cos ur q D r sin us sin ur
AVO parameters
Elastic parameters
R 00 s D r q G 33
D r s D r cos2 f
X
X
X
1
G 11 s Ž DC11 q DC22 . = Ž2 q cos4f X .
8
qG 11 sin2us sin2ur q G 33 cos 2us cos 2ur
R 01 s yD r y D r y 2 G 33 y 4 G44 q 2 G 13
qG 13 Žcos 2us sin2ur q sin2us cos 2ur .
R 02 s D r y D r y 2 G 33 q 4 G44 q 2 G 13
1
X
q DC12 Ž2 y cos4f .
4
qG44 sin2 us sin2 ur
R 03 s y2 G 11 q 2 G 33
1
X
q DC66 cos4f
2
F0Ž1. s R 00 q R 01 sin2
qR 03 sin2
qR 04 sin4
u
2
u
2
u
2
q R 02 sin2
sin2
X
uX
2
uX
2
q R 05 sin4
R 04 s G 11 q G 33 q 4 G44 y 2 G 13
G 33 s DC33
R 05 s G 11 q G 33 y 4 G44 y 2 G 13
1
G 13 s Ž DC13 q DC23 .
2
uX
1
G44 s Ž DC44 q DC55 .cos2 f X
2
2
F1Ž1. s F Bc Žsin2us sin2 ur q sin2 us sin2ur .
R10 s 2FGc
1
F Bc s Ž DC15 q DC25 .cos f X
2
qFGc Žcos 2us sin2 ur q sin2 us cos 2ur .
R11 s y2F Bc y 2FGc
1
X
q Ž DC15 y DC25 .cos3f
4
qF Bx Žsin2us sin2 ur y sin2 us sin2ur .
R12 s 2F Bc y 2FGc
1
X
q DC46 cos3f
2
qFGx Žcos 2us sin2 ur y sin2 us cos 2ur .
Rf 0 s y2FGx
ž
F1Ž1. s R10 q R11 sin2
ž
u
2
q R12 sin2
q Rf 0 q Rf 1 sin2
u
2
uX
2
/
q Rf 2 sin2
sin u
uX
2
/
X
sin u
1
F Bx s y Ž DC24 q DC14 .sin f X
2
Rf 1 s y2F Bx q 2FGx
1
X
y Ž DC14 y DC24 .sin3f
4
Rf 2 s 2F Bx q 2FGx
1
X
q DC56 sin3f
2
FGc s DC35 cos f X
FGx s yDC34 sin f X
G1Ž1. s F BsŽsin2us sin2 ur q sin2 us sin2ur .
R10 s 2FGs
1
F Bs s Ž DC14 q DC24 .cos f X
2
qFGs Žcos 2us sin2 ur q sin2 us cos 2ur .
R11 s y2F Bs y 2FGs
1
X
y Ž DC14 y DC24 .cos3f
4
qF By Žsin2us sin2 ur y sin2 us sin2ur .
R12 s 2F Bs y 2FGs
1
X
q DC56 cos3f
2
qFGy Žcos 2us sin2 ur y sin2 us cos 2ur .
Rf 0 s y2FGy
Rf 1 s y2F By q 2FGy
Rf 2 s 2F By q 2FGy
1
F By s Ž DC15 q DC25 .sin f X
2
1
X
y Ž DC15 y DC25 .sin3f
4
1
X
y DC46 sin3f
2
L.T. Ikelle, L. Amundsenr Journal of Applied Geophysics 46 (2001) 1–29
9
Table 1 Ž continued .
AVO formulae
AVO parameters
ž
G1Ž1. s R10 q R11 sin2
u
q R12 sin2
2
ž
q Rf 0 q Rf 1 sin2
u
2
u
X
/
2
q Rf 2 sin2
sin u
uX
2
/
X
Elastic parameters
FGs s DC34 cos f X
FGy s DC35 sin f X
sin u
R 21 s y4 GGc q 2 G Hc
1
G Bc s Ž DC11 y DC22 .cos2 f X
2
qG Hc Žcos 2us sin2ur q sin2us cos 2ur .
R 22 s 4 GGc q 2 G Hc
1
GGc s Ž DC55 y DC44 .
2
yG Kc Žcos 2us sin2ur y sin2us cos 2ur .
R 23 s y2 G Bc
1
G Hc s Ž DC13 y DC23 .cos2 f X
2
R 24 s G Bc q 4 GGc y 2 G Hc
G Kc s DC36 sin2 f X
F2Ž1. s G Bc sin2us sin2ur q GGc sin2 us sin2 ur
F2Ž1. s R 21 sin2
u
q R 22 sin2
2
qR 23 sin2
qR 25 sin4
u
sin2
2
uX
uX
2
uX
2
q R 24 sin4
X
q Rf sin u sin u
2
u
2
X
Rf s G Kc
2
2
G Ž1.
2 s G Bs sin us sin ur q GGs sin2 us sin2 ur
qG Hs Žcos 2us sin2ur q sin2us cos 2ur .
2
2
2
2
yG Ks Žcos us sin ur y sin us cos ur .
G 2Ž1. s R 21 sin2
u
q R 22 sin2
2
qR 23 sin2
qR 25 sin4
u
sin2
2
uX
2
uX
2
uX
2
q R 24 sin4
X
q Rf sin u sin u
u
2
X
qF Ex Žsin2us sin2 ur y sin2 us sin2ur .
ž
F3Ž1. s R 31 sin2
ž
u
2
q R 32 sin2
q Rf 1 sin2
u
2
u
2
q Rf 2 sin2
/
sin u
uX
2
/
R 21 s y4 GGs q 2 G Hs
G Bs s Ž DC16 q DC26 .cos2 f X
R 22 s 4 GGs q 2 G Hs
GGs s DC45
R 23 s y2 G Bs
G Hs s DC36 cos2 f X
R 24 s G Bs q 4 GGs y 2 G Hs
1
G Ks s Ž DC23 y DC13 .sin2 f X
2
R 25 s G Bs y 4 GGs y 2 G Hs
Rf s G Ks
F3Ž1. s F Ec Žsin2us sin2 ur q sin2 us sin2ur .
X
R 25 s G Bc y 4 GGc y 2 G Hc
R 31 s yR 32 s y2F Ec
Rf 1 s yRf 2 s y2F Ex
q2DC56 xsin f
sin u
qF Ey Žsin2us sin2 ur y sin2 us sin2ur .
y2DC46 xcos f
X
1
F Ex s wŽ DC14 y DC24 .
4
X
2 .
Ž 2
G Ž1.
3 s F Es sin us sin2 ur q sin2 us sin ur
1
F Ec s wŽ DC15 y DC25 .
4
R 31 s yR 32 s y2F Es
Rf 1 s yRf 2 s y2F Ey
X
1
F Es s wŽ DC14 y DC24 .
4
q2DC56 xcos f
X
(continued on next page)
L.T. Ikelle, L. Amundsenr Journal of Applied Geophysics 46 (2001) 1–29
10
Table 1 Ž continued .
AVO formulae
AVO parameters
ž
G 3Ž1. s R 31 sin2
u
2
q R 32 sin2
ž
q Rf 1 sin2
u
2
u
X
2
q Rf 2 sin2
F4Ž1. s G Ec sin2us sin2ur
F4Ž1. s R 43 sin2
u
2
qR 44 sin4
sin2
u
u
2
qR 44 sin4
u
2
uX
2
/
1
F Ey s wŽyDC15 q DC25 .
4
X
q2DC46 xsin f
sin u
u
X
G Ec s Ž1r8.Ž DC11 q DC22 .
R 43 s y2 R 44 s y2 R 45
uX
2
G Es s Ž1r2.Ž DC16 y DC26 .
R 43 s y2 R 44 s y2 R 45
uX
s y2 GEs
2
q R 45 sin4
yŽ1r4. DC12 y Ž1r2. DC66
s y2 GEc
2
q R 45 sin4
sin2
sin u
X
2
G4Ž1. s G Es sin2 sin2us sin2ur
G4Ž1. s R 43 sin2
/
Elastic parameters
uX
2
Hornby, 1996.: transversely isotropic ŽTI. symmetry,
orthorhombic symmetry and monoclinic symmetry.
Using conditions D c i jk l Žsee Auld Ž1990.., the
AVO-A formula in Eq. Ž26. for TI media with
respect to the vertical axis Žalso known as TIV.
reduces to F01 only:
Ž1.
R Ž1.
avoa s F0 ,
Ž 28 .
which means that the amplitudes are invariant with
azimuths. This result is consistent with the fact that
the medium is azimuthally isotropic.
Let us now look at an example of azimuthally
anisotropic media. For that, we consider an orthorhombic medium. Using the conditions on D c i jk l
associated with orthorhombic symmetry Žsee Auld
Ž1990.., the AVO-A formula in Eq. Ž26. is reduced
to
Ž1.
Ž1.
Ž1.
R Ž1.
avoa s F0 q F2 cos Ž 2 f . q F4 cos Ž 4f . .
Ž 29 .
We can see that the amplitudes are no longer invariant with azimuths when medium perturbations are
azimuthally anisotropic.
For the third example, we consider monoclinic
symmetry. The conditions on D c i jk l associated with
monoclinic symmetry can be found in Auld Ž1990..
With these conditions, the AVO-A formula in Eq.
Ž26. is reduced to
4
Ž1.
R Ž1.
avoa s F0 q
Ý FnŽ1. cos Ž nf . .
Ž 30 .
ns1
3.3. Effect of dip
A widespread concern in the interpretation of
AVO-A response is the ambiguity between heterogeneity and anisotropy. By deriving our AVO-A
formula in Eq. Ž26. for a dipping reflector instead of
the usual horizontal one, we can provide some insight on how these two physical properties affect
amplitude variations with offsets and azimuths. Let
us start our discussion by assuming that the interface
between the isotropic and anisotropic media is horizontal Ži.e., u X s 0, in the formulae in Table 1 assuming that f X s 0.. The AVO-A formula in Eq. Ž26. is
then reduced to
Ž1.
Ž1.
Ž1.
Ž1.
R Ž1.
avoa s F0 q F2 cos2 f q G 2 sin2 f q F4 cos4f
q G4Ž1. sin4f .
Ž 31 .
L.T. Ikelle, L. Amundsenr Journal of Applied Geophysics 46 (2001) 1–29
Basically, the f- and 3 f-terms of the Fourier
series are zero irrespective of anisotropic symmetry.
Therefore, the AVO functions F1Ž1., G 1Ž1., F3Ž1. and
G 3Ž1. are direct indicators of the dipping effect.
For some particular anisotropic symmetries like
azimuthally isotropic symmetry and orthorhombic
symmetry, the AVO functions F1Ž1., G 1Ž1., F3Ž1. and
G 3Ž1. are zero, as we can see in Eqs. Ž28. and Ž29..
The question then becomes: how to identify the
effect of dip in these cases? For orthorhombic symmetry, we can use the function F2Ž1. to identify the
effect of dip. In fact, the AVO-A for orthorhombic
symmetry is described by three AVO functions: F0Ž1.,
F2Ž1. and F4Ž1.. We can notice from their expressions
in Table 1 that they can be written in the form of
classical AVO formulae. For instance, F0Ž1. and F2Ž1.
can be written:
Ž1.
2
F0Ž1. s AŽ1.
0 f q B0 f sin
Ž1.
2
F2Ž1. s AŽ1.
2 f q B2 f sin
u
2
u
2
u
q C0Ž1.f sin4
2
u
q C2Ž1.f sin4
2
,
Ž 32 .
,
Ž 33 .
with
Ž1.
AŽ1.
0 f s A0 f
X
Ž u . s R 00 q R 02 sin
2
uX
2
q R 05 sin
4
uX
2
,
Ž 34 .
B0Ž1.f s B0Ž1.f Ž u X . s R 01 q R 03 sin2
u
X
Ž u . s R 22 sin
2
uX
2
2
,
q R 25 sin
B2Ž1.f s B2Ž1.f Ž u X . s R 21 q R 23 sin2
C2Ž1.f s R 24 .
cept of F2Ž1., which is A 2f , is zero for the particular
case where the interface is horizontal Ži.e., uX s 0..
Therefore, the value of the intercept of F2Ž1. is a
direct indicator of the dipping effect for orthorhombic symmetry.
The latest indicator of the dipping effect is based
on the AVO function F2Ž1. which is zero for azimuthally isotropic symmetry as shown in Eq. Ž28..
Hence, how can we recognize the dipping effect for
this commonly used model of anisotropy? For this
case, the series in Eq. Ž26. is reduced to AVO
variations only, and it is described by F0Ž1. only. As
expressed in Eq. Ž32., the intercept AŽ1.
0f and gradient
B0fŽ1. of F0Ž1. vary with the angle of the dipping
reflector in such a way that other equations with
similar parameters are needed to distinguish the dipping effect. By combining the AVO F0Ž1. of P–P
with that of P–SV, we will later see that it is
possible to identify the dipping effect even for azimuthally isotropic symmetry.
Let us make another important remark about the
AVO function F0Ž1.. If we assume that the bottom
medium in Fig. 1 is also isotropic, the intercept and
gradient of F0Ž1., defined in Eqs. Ž34. and Ž35., are
independent of u X , irrespective of the shape of the
interface between the isotropic and anisotropic media, i.e.,
AŽ1.
0 f s R 00 ,
Ž 40 .
Ž 35 .
B0Ž1.f s R 01 .
Ž 41 .
Ž 36 .
Therefore, in isotropic cases, the AVO of a dipping
interface has the same form as that of a horizontal
interface. This conclusion is generally translated by a
statement such as: Athe AVO is a 1D effectB. Let us
emphasize that this statement is only correct if the
media under consideration are isotropic.
X
C0Ž1.f s R 04 ,
Ž1.
AŽ1.
2 f s A2 f
11
uX
2
,
4
uX
2
,
Ž 37 .
Ž 38 .
Ž 39 .
The definitions of the other parameters are given in
Table 1. Using AVO terminology, we remark that
Ž1.
AŽ1.
0f and A 2f are the intercepts for the AVO funcŽ1.
Ž1.
Ž1.
tions F0 and F2Ž1., respectively; and B0f
and B2f
Ž1.
are the gradients for the AVO functions F0 and
F2Ž1., respectively. We can also notice that the inter-
3.4. AVO-A analysis for inÕersion purposes
Basically, we would like to organize the parameters D r and D c i jk l into combinations that are as
independent as possible, in terms of the information
contained in R avoa . Also, we would like to select
only the parameters which have a significant effect
12
L.T. Ikelle, L. Amundsenr Journal of Applied Geophysics 46 (2001) 1–29
Fig. 4. The AVO-A of P–P. The anisotropic materials used here are given in Table 2. Ža. and Žb. correspond to the orthorhombic material in
Table 2a for a horizontal reflector and a 308 dipping reflector, respectively. Žc. and Žd. correspond to the arbitrarily anisotropic material in
Table 2b for a horizontal reflector and a 308 dipping reflector, respectively.
on R Ž1.
avoa . These two important issues are discussed
in this section.
The dependence of R Ž1.
avoa on f describes amplitude variations with azimuths ŽAVAZ. while its dependence on u and u X describes amplitude variations
with offsets ŽAVO.. Thus, the AVO effect in Eq.
Ž1.
Ž26. of R Ž1.
avoa is represented by the functions F0 ,
Ž1.
Ž1.
Fn and Gn , which are all independent of the
azimuthal angle f . The AVAZ effect in this equation is represented by the trigonometric basic functions 1, cos f , sin f , cos2 f , sin2 f , cos3f , sin3f ,
cos4f , sin4f 4 . We can see that the AVAZ effect on
seismic amplitudes can be decoupled from the AVO
effect. Furthermore, we remark that the trigonometric functions describing the AVAZ effect are mutually orthogonal. This AVAZ property suggests that
the AVO functions F0Ž1., FnŽ1. and GnŽ1. can be extracted and processed separately based on the following equations:
2p
F0Ž1. Ž u , u X . s
H
0
X
R Ž1.
avoa Ž u , u , f . d f ,
Ž 42 .
2p
FnŽ1. Ž u , u X . s
H
for n s 1,2,3,4,
0
X
R Ž1.
avoa Ž u , u , f . cos n f d f ,
Ž 43.
L.T. Ikelle, L. Amundsenr Journal of Applied Geophysics 46 (2001) 1–29
13
Table 2
Normalized stiffness tensor D c i jk l . Ža. The anisotropic material used here is an orthorhombic rock as described by Cheadle et al. Ž1991.. Žb.
We have rotated it to simulate the case where the axes of the symmetries do coincide with the coordinate system of the acquisition geometry
Ža.
Parameter
Value
DC11
0.4211
DC22
0.2578
DC33
y0.0488
DC44
y0.0655
DC55
y0.0344
DC66
y0.0044
DC12
0.2500
DC13
0.2133
DC23
0.1844
DC11
DC14
DC45
0.2546
y0.0251
y0.0138
DC22
DC15
DC46
0.3045
0.0354
0.0191
DC33
DC16
DC56
y0.0025
y0.0045
y0.0228
DC44
DC24
Dr
y0.0338
y0.0697
0.1
DC55
DC25
DC66
DC26
DC12
DC34
DC13
DC35
DC23
DC36
y0.0498
0.0184
y0.0213
y0.0369
0.27037
y0.0565
0.1995
0.0323
0.2144
y0.01291
Dr
0.1
Žb.
Parameter
Value
2p
GnŽ1. Ž u , u X . s
H
for n s 1,2,3,4.
0
functions can be expressed in those terms; for instance, G 1Ž1. in Table 1 can be rewritten:
X
R Ž1.
avoa Ž u , u , f . sin n f d f ,
Ž 44.
Another advantage of inverting each of these functions separately is that we significantly reduce the
number of parameters to be estimated in each case.
Before we discuss this point further, let us examine
the contribution of these AVO functions to the
AVO-A, R Ž1.
avoa . We have plotted in Fig. 4 the AVO-A
for P–P scattering corresponding to the two models
described in Table 2. We can see that small offsets
behave as azimuthally isotropic media when the
interface is horizontal Ži.e., F0Ž1. is the dominant
.
function in R Ž1.
avoa . The presence of dip, which introduces the contributions of the functions F1Ž1., G 1Ž1.,
F3Ž1. and G 3Ž1., completely changes this pattern.
Ikelle Ž1996. has suggested that the effect of F3Ž1.,
Ž1.
G 3 , F4Ž1. and G4Ž1. might be small. The AVO-A in
Fig. 5 computed without these terms shows that they
are negligible, especially at small incident angles.
This is an unfortunate outcome because the 3f- and
4f-terms involve single parameter inversion Žsee
Table 1.; therefore, they are easy to perform. However, their extractions from R Ž1.
avoa will be unreliable
for noisy data.
The inversion for parameters contained in the
AVO functions F0Ž1., F1Ž1., F2Ž1., G 1Ž1., and G Ž1.
2 can be
performed using classical AVO techniques. In fact,
we have shown in Eqs. Ž32. and Ž33. that F0Ž1. and
F2Ž1. can be expressed in terms of the intercept and
gradient just as in classical AVO. Actually, all these
Ž1.
2
G 1Ž1. s AŽ1.
1g q B1g sin
u
2
Ž 45 .
,
with
Ž1.
AŽ1.
1g s A1g
X
Žu . s
RX10 q RX12 sin2
uX
2
sin u X ,
Ž1.
Ž1.
B1g
s B1g
Ž u X . s RX11 sin u X .
Ž 46 .
Ž 47 .
The definitions of the other parameters are given in
Table 1. Using AVO terminology, we remark that
Ž1.
AŽ1.
1g and B1g are the intercept and gradient of the
AVO function G 1Ž1., respectively. With the AVO
constructs in Eqs. Ž32., Ž33. and Ž45., we reduce the
number of parameters to be inverted to two or three
only.
The AVO-A without the 3f- and 4f-terms can be
finally regrouped into the classical AVO form as
follows:
Ž1.
Ž1.
Ž1.
Ž1.
R Ž1.
avoa s Ž A 0 f q A1f cos f q A1g sin f q A 2 f cos2 f
Ž1.
Ž1.
qAŽ1.
2 g sin2 f . q Ž B0 f q B1f cos f
L.T. Ikelle, L. Amundsenr Journal of Applied Geophysics 46 (2001) 1–29
14
Ž1.
qB1g
sin f q B2Ž1.f cos2 f q B2Ž1.g sin2 f . sin2
q Ž C0Ž1.f q C2Ž1.f cos2 f q C2Ž1.g sin2 f . sin
u
4
2
u
2
4.1. Dip and azimuthal angles
,
Ž 48 .
and reduces to
Ž1.
Ž1.
Ž1.
Ž1.
2
R Ž1.
avoa s A 0 f q Ž B0 f q B2 f cos2 f q B2 g sin2 f . sin
q Ž C0Ž1.f q C2Ž1.f cos2 f q C2Ž1.g sin2 f . sin
u
4
2
u
2
,
Ž 49 .
when the interface is horizontal.
4. AVO-A derivation and analysis for P–SV data
Our task in this section is to derive and analyze
the AVO-A for P–SV scattering. We will seek to
utilize the AVO-A of P–SV to resolve some of the
elastic parameters which cannot be recovered from
P–P scattering alone.
As in the previous section, we will begin by
relating the wavenumbers k s , kXs , k r and kXr to
incidence and reflecton angles as follows:
v
k s s sin us cos fs ,
Ž 50 .
ÕP
Fig. 5. The AVO-A of P–P where the 3f- and 4f-terms are dropped. The anisotropic materials used here are given in Table 2. Ža. and Žb.
correspond to the orthorhombic material in Table 2a for a horizontal reflector and a 308 dipping reflector, respectively. Žc. and Žd.
correspond to the arbitrarily anisotropic material in Table 2b for a horizontal reflector and a 308 dipping reflector, respectively.
L.T. Ikelle, L. Amundsenr Journal of Applied Geophysics 46 (2001) 1–29
kXs s
kr s
kXr s
v
ÕP
v
ÕS
v
ÕS
sin us sin fs ,
Ž 51 .
sin ur cos fr ,
Ž 52 .
sin ur sin fr .
Ž 53 .
The angles fs and fr are the same as those introduced in Fig. 3. The angles us and ur are shown in
Fig. 6. Notice that, although these angles have the
same meaning as those introduced in Fig. 2 for P–P,
their physical behaviors are quite different. For instance, us is not equal to ur even when the interface
is horizontal, due to the asymmetry between P-wave
and SV-wave reflections. However, they are related
through Snell’s law:
sin us s
ÕP
ÕS
sin ur .
15
Alternatively, the angles u, uX , f and fX will be
used. They are introduced as follows:
u s us y ur ,
Ž 55 .
u X s us q ur ,
Ž 56 .
fs
fX s
fs q fr
2
fs y fr
2
Ž 57 .
,
,
Ž 58 .
where u is the total reflection angle, u X is the angle
due to the asymmetry between the P- and SV-wave
reflection plus the dip angle of the reflector. The
angles f and f X have the same meaning as those
introduced in Fig. 3 for P–P scattering. The two new
angles, u and u X , are also shown in Fig. 6. As we
can see in Fig. 6, u X is non-zero even when the
reflector is horizontal, contrary to the P–P case.
Ž 54 .
This relationship is only valid when the interface is
horizontal because our angles, us and ur , are defined
with respect to the vertical axis and not with respect
to the normal vector of the reflector.
4.2. Decoupling of AVAZ and AVO
As we did for P–P scattering, By substituting
Eqs. Ž50. – Ž58. in Eq. Ž14. and regrouping the different elements as a linear combination of 1, cos f ,
Fig. 6. P–SV reflection at an isotropicranisotropic interface. Ža. corresponds to a horizontal interface. Contrary to the P–P reflection case,
the incident angle us is no longer equal to ur , the reflected angle, due to the asymmetry of the P–SV reflection. Žb. corresponds to a dipping
interface.
L.T. Ikelle, L. Amundsenr Journal of Applied Geophysics 46 (2001) 1–29
16
Table 3
AVO functions that describe amplitude variations with offsets and azimuths for P–SV scattering
AVO formulae
AVO parameters
F0Ž1. s D r sin us cos ur y D r cos us sin ur
R 00 s
1
2
Elastic parameters
Ž D r q D r q G 33 y G 13 q 2 G44 .
1
q G 11 sin2us sin2 ur
2
1
R 01 s y Ž G 33 q G 11 . q G 13 y 2 G44
2
1
y G 33 cos 2us sin2 ur
2
R 02 s yR 04 s
1
q G 13 cos2 us sin2 ur
2
R 03 s
1
q G44 sin2 us cos2 ur
2
R 05 s
ž
F0Ž1. s R 00 q R 01 sin2
u
2
q R 02 sin2
u
ž
q R 03 q R 04 sin2
2
uX
/
2
q R 05 sin2
1
2
1
2
1
2
Ž G 11 y G 33 .
2
/
X
1
G 11 s Ž DC11 q DC22 .
8
X
=Ž2 q cos4f .
Ž D r y D r y G 33 q G 13 q 2 G44 .
1
X
q DC12 Ž2 y cos4f .
4
Ž G 33 q G 11 . y G 13 y 2 G44
1
X
q DC66 Žcos4f .
2
sin u
uX
D r s D r cos2 f
G 33 s DC33
sin u
1
G 13 s Ž DC13 q DC23 .
2
X
1
G44 s Ž DC44 q DC55 .cos2 f X
2
ž
F1Ž1. s F Bc sin2us cos2 ur q
1
2
ž
ž
ž
sin2 us sin2 ur
qFGc cos 2us cos2 ur y
qF Bx sin2us cos2 ur y
qFGx cos 2us cos2 ur q
F1Ž1. s R10 q R11 sin2
qR13 sin2
qR15 sin4
u
2
2
1
2
1
2
sin2 us sin2 ur
sin2 us sin2 ur
sin2 us sin2 ur
q R12 sin2
sin2
2
u
u
1
u
uX
2
X
2
/
q R14 sin4
u
2
R10 s FGc q FGx
/
/
/
1
F Bc s Ž DC15 q DC25 .cos f X
2
R13 s 4 Rf 1 s 2ŽF Bc q FGc q F Bx q FGx .
1
X
q Ž DC15 y DC25 .cos3f
4
R14 s 4ŽyF Bx q FGx .
1
X
q DC46 cos3f
2
R15 s y4ŽF Bc y FGc .
1
F Bx s y Ž DC24 q DC14 .sin f X
2
1
R11 s y Ž2 R14 q R 13 .
2
1
X
y Ž DC14 y DC24 .sin3f
4
1
R12 s y Ž2 R15 q R13 .
2
1
X
q DC56 sin3f
2
X
2
q Rf 1 sin u sin u
X
FGc s DC35 cos f X
FGx s yDC34 sin f X
ž
G1Ž1. s F Bs sin2us cos2 ur q
1
2
sin2 us sin2 ur
/
R10 s FGs q FGy
1
F Bs s Ž DC14 q DC24 .cos f X
2
L.T. Ikelle, L. Amundsenr Journal of Applied Geophysics 46 (2001) 1–29
17
Table 3 Ž continued .
AVO formulae
AVO parameters
ž
ž
ž
qFGs cos 2us cos2 ur y
qF By sin2us cos2 ur y
qFGy cos 2us cos2 ur q
G1Ž1. s R10 q R11 sin2
qR13 sin2
qR15 sin4
u
2
uX
2
u
2
1
2
1
2
1
2
sin2 us sin2 ur
sin2 us sin2 ur
sin2 us sin2 ur
q R12 sin2
sin2
u
uX
2
q R14 sin4
q Rf 1 sin u sin u
R13 s 4 Rf 1 s 2ŽF Bs q FGs q F By q FGy .
1
X
y Ž DC14 y DC24 .cos3f
4
R14 s 4ŽyF By q FGy .
1
X
q DC56 cos3f
2
R15 s y4ŽF Bs y FGs .
1
F By s Ž DC15 q DC25 .sin f X
2
1
R 11 s y Ž2 R14 q R 13 .
2
X
2
/
/
/
Elastic parameters
u
R12 s
2
1
2
1
X
y Ž DC15 y DC25 .sin3f
4
1
X
y DC46 sin3f
2
Ž2 R15 q R13 .
X
FGs s DC34 cos f X
FGy s DC35 sin f X
F2Ž1. s
1
2
G Bc sin2us sin2 ur
1
2
G Hc q
1
2
G Kc
1
q GGc sin2 us cos2 ur
2
1
R 21 s y G Bc q G Hc y 2 GGc
2
1
q G Hc cos2 us sin2 ur
2
R 22 s yR 24 s
1
y G Kc sin2 ur
2
R 23 s GGc q
ž
F2Ž1. s R 20 q R 21 sin2
u
2
q R 22 sin2
ž
q R 23 q R 24 sin2
q R 25 sin2
G Ž1.
2 s
R 20 s GGc y
1
2
uX
2
/
uX
2
/
sin u
R 25 s
1
2
1
2
1
2
G Bc y G Kc
G Hc y
1
2
G Kc
1
G Bc s Ž DC11 y DC22 .cos2 f X
2
1
GGc s Ž DC55 y DC44 .
2
1
G Hc s Ž DC13 y DC23 .cos2 f X
2
G Kc s DC36 sin2 f X
G Bc y 2 GGc y G Hc
u
2
sin u
X
G Bs sin2us sin2 ur
R 20 s GGs y
1
2
G Hs q
1
2
G Ks
1
q GGs sin2 us cos2 ur
2
1
R 21 s y G Bs q G Hs y 2 GGs
2
1
q G Hs cos2 us sin2 ur
2
R 22 s yR 24 s
1
y G Ks sin2 ur
2
R 23 s GGc q
1
2
1
2
G Bs y G Ks
G Hs y
1
2
G Ks
G Bs s Ž DC16 q DC26 .cos2 f X
GGs s DC45
G Hs s DC36 cos2 f X
1
G Ks s Ž DC23 y DC13 .sin2 f X
2
(continued on next page .
L.T. Ikelle, L. Amundsenr Journal of Applied Geophysics 46 (2001) 1–29
18
Table 3 Ž continued .
AVO formulae
AVO parameters
ž
2
G Ž1.
2 s R 20 q R 21 sin
u
q R 22 sin2
2
ž
q R 23 q R 24 sin2
F3Ž1. s
1
2
u
2
ž
ž
u
q R 32 sin2
2
qR 33 sin2
qR 35 sin4
uX
/
2
u
sin2
2
uX
2
uX
2
sin u
sin2 us sin2 ur
2
u
q R 34 sin4
q Rf 3 sin u sin u
q R 32 sin2
2
qR 33 sin2
qR 35 sin4
1
2
u
u
2
uX
2
ž
u
2
2
sin2 us sin2 ur
ž
2
q R 34 sin4
u
ž
ž
u
2
q R 44 sin2
u
2
/
/
R 33 s 4 Rf 3 s 2ŽF Es q F Ey .
R 34 s y4F Ey
uX
2
/
1
s yR 45 s y G Ec
2
sin u
uX
2
/
sin u
2
q R 45 sin2
/
1
s yR 45 s y G Es
2
sin u
uX
2
/
y2DC46 xcos f
X
1
F Ex s wŽ DC14 y DC24 .
4
q2DC56 xsin f
X
sin u
1
F Es s wŽ DC14 y DC24 .
4
q2DC56 xcos f
X
1
F Ey s wŽyDC15 q DC25 .
4
q2DC46 xsin f
X
G Ec s Ž1r8.Ž DC11 q DC22 .
yŽ1r4. DC12 y Ž1r2. DC66
X
R 41 s yR 42 s R 44
uX
1
F Ec s wŽ DC15 y DC25 .
4
1
R 32 s y Ž2 R 35 q R 33 .
2
X
R 41 s yR 42 s R 44
q R 45 sin2
q R 42 sin2
R 34 s y4F Ex
1
R 31 s y Ž2 R 34 q R 33 .
2
2
G Es sin2us sin2 ur
G4Ž1. s R 41 sin2
R 33 s 4 Rf 3 s 2ŽF Ec q F Ex .
R 35 s y4F Es
2
q Rf 3 sin u sin u
u
X
1
R 32 s y Ž2 R 35 q R 33 .
2
X
uX
q R 42 sin2
q R 44 sin2
2
uX
1
2
Elastic parameters
G Bs y 2 GGs y G Hs
1
R 31 s y Ž2 R 34 q R 33 .
2
2
G Ec sin2us sin2 ur
F4Ž1. s R 41 sin2
1
sin2
1
2
R 35 s y4F Ec
1
F Es sin2us cos2 ur q sin2 us sin2 ur
2
2
G 3Ž1. s R 31 sin2
/
/
2
ž
ž
1
1
R 25 s
uX
qF Ey sin2us cos2 ur y
G4Ž1. s
sin u
1
F Ec sin2us cos2 ur q sin2 us sin2 ur
2
F3Ž1. s R 31 sin2
F4Ž1. s
/
2
q R 25 sin2
qF Ex sin2us cos2 ur y
G 3Ž1. s
u
X
X
G Es s Ž1r2.Ž DC16 y DC26 .
L.T. Ikelle, L. Amundsenr Journal of Applied Geophysics 46 (2001) 1–29
sin f , cos2 f , sin2 f , cos3f , sin3f , cos4f , sin4f 4 ,
the amplitude variations with offsets and azimuths
ŽAVO-A. can be cast in terms of a Fourier series of
the azimuthal angle f as follows:
4
Ž2.
R Ž2.
avoa s F0 q
Ý
FnŽ2. cos Ž n f . q GnŽ2. sin Ž n f . .
ns1
Ž 59 .
Expressions of functions F0Ž2. s F0Ž2. Ž u , u X , f X ., FnŽ2.
s FnŽ2. Ž u , u X , f X . and GnŽ2. s GnŽ2. Ž u , u X , f X . can be
obtained from Table 3 Žthe subscript n runs from 1
to 4.. Furthermore, the different elements of the
series Ž59. can also be regrouped as a linear combination of cos f X , sin f X , cos2 f X , sin2 f X , cos3f X ,
sin3f X , cos4f X , sin4f X 4 , leading to
X
X
Ž2.
Ž2.
Ž2.
R Ž2.
avoa s f 00 q f 02 cos2 f q f 04 cos4f
Ž2.
R Ž2.
avoa s F0 ,
Ž 61 .
which means that the amplitudes are invariant with
azimuths. For orthorhombic symmetry, we must add
the functions F2Ž2. and F4Ž2. to Eq. Ž59.:
Ž 62 .
For monoclinic symmetry, the AVO-A formula in
Eq. Ž59. becomes
Ž2 c .
qf 13Ž2 c . cos3f X q g 13
sin3f X cos f
Ž2 s.
q f 11Ž2 s. cos f X q g 11
sin f X
4
Ž2.
R Ž2.
avoa s F0 q
Ž2 s.
qf 13Ž2 s. cos3f X q g 13
sin3f X sin f
Ý FnŽ2. cos Ž nf . .
Ž 63 .
ns1
Ž2 c .
Ž2 c .
Ž2 c .
q f 20
q f 22
cos2 f X q g 22
sin2 f X
Ž2 s.
Ž2 s.
=cos2 f q f 20
q f 22
cos2 f X
Ž2 s.
Ž2 c .
cos f X
qg 22
sin2 f X sin2 f q f 31
Ž2 c .
Ž2 s.
cos f X
qg 31
sin f X cos3f q f 31
Ž2 s.
Ž2 c .
cos4f
qg 31
sin f X sin3f q f 40
Ž2 c .
q g 40
sin4f .
u X . and GnŽ2. s GnŽ2. Ž u , u X .. They can be obtained
from Table 3 by taking f X s 0 Žthe subscript n runs
from 1 to 4..
By comparing Eqs. Ž26. and Ž59., we can remark
that the AVAZ behavior of P–SV scattering has
exactly the same form as that of P–P. This similarity
is preserved for transversely isotropic ŽTI., orthorhombic and monoclinic symmetries. In fact, for
TI symmetry with respect to the vertical axis ŽTIV.,
the AVO-A formula in Eq. Ž59. is reduced to F02
only:
Ž2.
Ž2.
Ž2.
R Ž2.
avoa s F0 q F2 cos Ž 2 f . q F4 cos Ž 4f . .
Ž2 c .
q f 11Ž2 c . cos f X q g 11
sin f X
19
Ž 60 .
X
Ž2. Ž
. Ž2c.
Expressions of functions f 0Ž2.
m s f 0 m u , u , fn m s
X
X
Ž2c. Ž
Ž2s.
Ž2s.
Ž2c.
Ž2c.
f n m u , u ., f n m s f n m Ž u , u ., g n m s g n m Ž u , u X .
X
Ž2s. Ž
. Žthe subscripts n and m run
and g nŽ2s.
m s gnm u , u
from 0 to 4. can be deduced from Table 3.
Just like for the P–P case, the dependence of
X
R Ž2.
avoa on f and f describes amplitude variations
with azimuths ŽAVAZ. while its dependence on u
and u X describes amplitude variations with offsets
ŽAVO.. Thus, the AVO effect in Eq. Ž26. of R Ž1.
avoa is
Ž2c.
Ž2s.
Ž2c.
represented by the functions f 0Ž2.
m , fn m , fn m , g n m .
We will limit the rest of our discussion in this
section to one azimuthal angle by taking f X s 0,
using the series Ž59. instead of Eq. Ž60.. Hence, the
AVO functions are F0Ž2. s F0Ž2. Ž u , u X ., FnŽ2. s FnŽ2. Ž u ,
Notice that 61., Ž62. and Ž63. are similar to Eqs.
Ž28. – Ž30., respectively.
We have established that the structure of AVAZ
of P–SV is similar to that of P–P. By comparing the
third columns of Tables 1 and 3, we can also observe
that the combinations of elastic parameters invoked
in P–SV are exactly the same as those in P–P. The
differences between P–P and P–SV are in their
AVO behaviors. We will analyze these differences in
more detail below.
4.3. Effect of dip
As discussed in the previous sections, the dipping
and anisotropic effects on AVO-A of P–P scattering
are distinguishable. However, the case where the
bottom medium Žsee Fig. 1. is azimuthally isotropic
has not yet been resolved. The AVO function F0Ž2. of
P–SV provides the answer to this case. As described
in Table 3, F0Ž2. is reduced to
F0Ž2. s yD r sin ur y
1
2
Ž G 33 y gG13 . sin2 ur ,
Ž 64 .
20
L.T. Ikelle, L. Amundsenr Journal of Applied Geophysics 46 (2001) 1–29
Fig. 7. The AVO-A of P–SV. The anisotropic materials used here are given in Table 2. Ža. and Žb. correspond to the orthorhombic material
in Table 2a for a horizontal reflector and a 308 dipping reflector, respectively. Žc. and Žd. correspond to the arbitrarily anisotropic material in
Table 2b for a horizontal reflector and a 308 dipping reflector, respectively.
at normal incidence Ži.e., us s 0.. The definitions of
G33 and G13 are given in Table 3. If the interface is
assumed horizontal, the reflected angle ur is zero
Ž ur s 0. whenever us s 0; thus, F0Ž2. is zero. However, if the interface is a dipping one as described in
Fig. 1, ur / 0 even when us s 0; hence, F0Ž2. is
non-zero. This remark can be used as a dip indicator
for azimuthally isotropic symmetry.
Contrary to what we have seen for P–P scattering,
notice that the functions F1Ž2., F3Ž2., G 1Ž2. and G 3Ž2. are
non-zero even if the interface is horizontal because
u X / 0 due to asymmetry between the P–SV reflection.
4.4. AVO-A analysis for inÕersion purposes
The inversion procedure of AVO-A of P–SV is
similar to that outlined earlier for P–P. We first
extract the AVO functions F0Ž2., FnŽ2. and G