CWP-669
Estimation of interval shear-wave attenuation from mode-converted data
Bharath Shekar&Ilya Tsvankin
ABSTRACT
Interval attenuation measurements provide valuable information for rervoir
characterization and lithology discrimination.Here,we extend the attenuation
layer-stripping method of Behura and Tsvankin to mode-converted(PS)waves
with the goal of estimating the interval S-wave attenuation coefficient.By identi-
fying PP and PS events with shared ray gments and applying the PP+PS=SS
method,wefirst perform kinematic construction of pure shear(SS)events in the
target layer and overburden.Then,the modified spectral-ratio method is ud
to compute the effective shear-wave attenuation coefficient for the target reflec-
tion.Finally,application of the dynamic version of velocity-independent layer
stripping to the constructed SS reflections yields the interval S-wave attenuation
coefficient in the target layer.The attenuation coefficient estimated for a range
of source-receiver offts can be inverted for the interval attenuation-anisotropy
parameters.The method is tested on a multicomponent synthetic data t from
layered VTI(transverly isotropic with a vertical symmetry axis)media gen-
erated with the anisotropic reflectivity method.
Key words:attenuation,anisotropy,multicomponent data,shear waves
Introduction
Attenuation analysis provides ismic attributes nsi-
tive to the physical properties of the subsurface.Re-
liable attenuation measurements have become feasible
with acquisition of high-quality reflection and borehole
data.Attenuation is often found to be anisotropic(di-
rectionally dependent)due to a variety of factors such as
the intrinsic anisotropy of the material,the prence of
alignedfluid-fractures(Batzle et al.,2005),or interbed-
ding of thin layers with different properties(Zhu et al.,
2007).The magnitude of attenuation anisotropy can be
much higher than that of velocity anisotropy,and the
symmetry of the attenuation coefficient can be different
than that of the velocity function(Liu et al.,2007).
The quality factors Q P and Q S are widely ud as
measures of P-and S-wave intrinsic attenuation,respec-
tively(Zhu,2006).Dvorkin and Mavko(2006)obrve
that the ratio Q−1
S /Q−1
P
六年级毕业寄语can rve as an indicator of
hydrocarbons becau the values of Q P and Q S influid-saturated rocks are clo,while in dry or gas-saturated rocks Q P Q S.Adam(2008)suggests that time-lap studies of attenuation are uful in monitoring rer-voirfluids.Chichinina et al.(2009)conduct ultrasonic laboratory experiments for models with VTI symmetry. Their results show that the symmetry-axis attenuation of P-waves is muc
h greater than that of S-waves in dry samples,while for oil-saturated models,the two models have comparable attenuation.Shear-wave attenuation in heavy oils is cloly linked to temperature,and hence could be uful in ismic monitoring of thermal recov-ery process(Behura et al.,2007).
De et al.(1994)report measurements of the shear-wave quality factor from vertical ismic profiling(VSP) surveys and sonic logs.It is more difficult to study S-wave attenuation using reflection data due to such prob-lems as the high level of noi and statics problems for shear waves.Behura and Tsvankin(2009)combine the velocity-independent layer stripping(VILS)method of Dewangan and Tsvankin(2006)with the spectral-ratio method to estimate the interval attenuation of pure PP or SS reflected waves.They identify the overburden and target events that share ray gments in the overburden to compute the interval traveltime and then the interval attenuation coefficient in the target layer.Their algo-
398 B.Shekar&I.Tsvankin
rithm is data-driven and does not require information about the velocity or attenuation in the overburden.
Shear waves,however,cannot be excited offshore, and shear-wave sources are ldom ud on land.
There-fore,here we extend the technique of Behura and Tsvankin(2009)to mode-converted data by supple-menting it with the PP+PS=SS method of Grechka and Tsvankin(2002).First,we discuss how the PP+PS=SS method can be combined with VILS to construct SS-wave moveout in the target layer and overburden from PP and PS data.Then the interval S-wave attenua-tion coefficient is obtained by extending the kimenatic construction procedure to frequency-domain amplitudes procesd using the spectral-ratio method.Finally,we apply the algorithm to synthetic data generated for a layered VTI medium and investigate the accuracy of the inversion for the SV-wave attenuation-anisotropy parameters.
Methodology
For simplicity,the method is described for2D models, but it can be extended to3D wide-azimuth data.We operate with pure-mode(PP)and mode-converted(PS) reflections for a medium with an arbitrarily anisotropic, heterogeneous target layer overlaid by a laterally homo-geneous overburden with a horizontal symmetry plane in each layer.In the2D version of the method the ver-tical incidence plane is suppod to be a plane of mirror symmetry for the whole model.Therefore,both rays and the corresponding pha-velocity vectors are confined to the incidence plane,and converted waves reprent in-plane polarized PSV modes.The P-to-S conversion is assumed to occur only at the refle
ctor.We begin with a description of the algorithm designed to compute the interval shear-wave traveltimes and then discuss estima-tion of the interval shear-wave attenuation coefficient in the target layer.
Kinematic layer stripping for interval
shear-wave traveltimes
To estimate the interval shear-wave traveltimes in the target layer,we combine the PP+PS=SS method with velocity-independent layer stripping(VILS)developed by Dewangan and Tsvankin(2006).Suppo P-wave sources and receivers of both P-and S-waves are con-tinuously distributed along the acquisition line.As dis-cusd by Grechka and Tsvankin(2002),matching time slopes on common-receiver gathers at the source loca-tion A allows us to identify the PP(ARB)and PS (ARC)target events that share the downgoing gment AR and the reflection point R at the bottom of the target layer(Figure1).Likewi,for a P-wave source at B,wefind PP(BRA)and PS(BRD)target events that share the downgoing gment BR.This
procedure Figure 1.2D ray diagram illustrating the PP+PS=SS method for PP and PS reflections from the bottom of the target layer.The wavefield is excited in split-spread geome-try by P-wave sources located at points A and B.Target PP (ARB)and PS(ARC)events share the downgoing gment AR and,therefore the reflection point R at the bottom of the target layer.Another pair of PP(BRA)and PS(BRD) target events share the downgoing gment BR.The con-structed SS target event corresponds to DRC.
makes it possible to construct the SS event DRC,where C and D are the coordinates of S-wave receivers.For brevity,we denote the PP(ARB)and PS(ARC and BRD)events by PP E,PS E1,and PS E2(respectively) and the constructed SS event DRC by SS E(“E”refers to effective events reflected from the bottom of the tar-get layer).The exact shear-wave traveltime for the re-flection SS E is(Grechka and Tsvankin,2002)
t SS
E
=t P S
E1
+t P S
E2
−t P P
E
.(1) The constructed event SS E can be treated(in a kine-matic n)as a pure reflection mode excited by a shear-wave source.
Next,wefind the interval shear-wave traveltime in the target layer,which requires knowledge of the travel-time in the overburden.Since the data are assumed to be generated with a P-wave source,it is necessary to ap-ply the PP+PS=SS method repeatedly to construct SS reflections in the overburden(Figure2).To layer-strip the gment DR of the SS-wave,we need to obtain the coordinate of point I and the traveltime along the over-burden gment ID.Note that the horizontal slowness along any ray in the laterally homogeneous overburden should be prerved.
First,we form a common-receiver gather of the PS-wave at location D and identify the point(E)where the time slope(horizontal slowness)coincides with that at D.The obtained overburden PS event EID s
hares the gment ID with the target SS event CRD(Figure2b). Then we form a common-source PP gather at location E tofind the point F where the time slope(horizontal slowness)coincides with that at E,which means that
Interval shear-wave attenuation
399
(a)(b)
Figure 2.Layer stripping of the constructed SS events.(a)Application of the PP+PS=SS method is applied to kinematically construct pure SS-waves in the overburden.PP (EIF )and PS (EID )events share the downgoing gment EI and the reflection point I at the bottom of the overburden.Another pair of PP (F IE )and PS (F IG )overburden events share the downgoing gment F I .(b)The constructed overburden SS event DIG shares the gment ID with the target SS reflection.
the overburden PP event EIF shares the downgoing gment EI with the PS event EID (Figure 2a).
The moveout functions of the overburden PP,PS,and SS events are symmetric with respect to zero offt.Therefore,the receiver coordinate of the overburden PS event F IG can be found from
x G =x E +x F −x D .
(2)
The constructed event DIG (denoted by SS O1,where “O”refers to the overburden and “1”to the left gment of the target SS event in Figure 2b)shares the gment ID with the target SS event DRC (Figure 2b).The PP event EIF will be denoted by PP O1and the PS events EID and F IG by PS O1.The exact traveltime of the event SS O1is then given by
t SS O 1=2t P S O 1−t P P O 1,
(3)
and the lateral coordinate of location I is
毛肚x I =x D +x G
.(4)
Likewi,we can apply the PP+PS=SS method to con-struct the overburden SS event HJC (SS O2)that shares the gment JC with the target event SS E (Figure 3).The corresponding traveltime t SS O 2and the lateral co-ordinate of point J are obtained using the algorithm discusd above.The interval shear-wave traveltime in the target layer is given by
t SS T =t SS E −1
(t SS O 1+t SS O 2).(5)
The interval traveltime t SS T corresponds to the raypath IRJ of the target event SS T .
If the target is horizontal and laterally homoge-neous,the raypaths of the downgoing and upgoing over-burden events correspond to the same ray
parameter
Figure 3.Raypaths of the constructed SS events.The target SS event DRC shares the gments ID and JC with the overburden events DIG and HJC ,respectively.The method produces the interval traveltime along the raypath IRJ .
and,therefore,are symmetric with respect to the verti-cal.Then t SS O 1=t SS O 2,and it is sufficient to apply the PP+PS=SS method just to one of the overburden gments of the target event SS E .
Layer stripping for interval shear-wave attenuation
Behura and Tsvankin (2009)combine VILS with the spectral-ratio method and apply their attenuation layer-stripping algorithm to frequency-domain amplitudes of pure-mode reflections.This technique can be extended to the combination of PP-and PS-waves analyzed
400 B.Shekar&I.Tsvankin
above.The ray-theoretic frequency-domain amplitudes of the waves PP E,PS E1and PS E2(Figure1)can be written as
|U P P
E |=S(ω)G P P
E
e−k I P,AR l AR−k I P,RB l RB,(6)
|U P S
E1|=S(ω)G P S
E1
e−k I P,AR l AR e−k I S,RC l RC,(7)
|U P S
E2|=S(ω)G P S
E2
e−k I P,BR l BR e−k I S,RD l RD,(8)
where S(ω)is the spectrum of the source wavelet.The coefficients k I P,XY and k I S,XY are the average P-and S-wave group attenuation coefficients along the raypath XY,the length of the raypath XY is denoted by l XY.
The coefficients G P P
E ,G P S
E1
and G P S
E2
include the
source/receiver directivity,reflection/transmission coef-ficients along the raypath,and the geometrical spread-ing of the corresponding events.Equations6,7,and8 can be combined to compute the attenuation coefficient of the reflection SS E constructed by the PP+PS=SS method:
|U SS
E |=
|U P S
E1
||U P S
E2
|
|U P P
E
|
=G E S(ω)e−k I S,DR l DR−k I S,RC l RC,(9)
where the ratio G E=G P S1
E G P S2
E
/G P P
E
is as-
sumed to be independent of frequency.It should be
noted that|U SS
E |in equation9does not reprent the
actual amplitude of the primary SS reflection.While the PP+PS=SS method reproduces the kinematics of shear-wave primaries,it cannot yield the true ampli-tudes without knowledge of the velocity model(Grechka and Tsvankin,2002;Grechka and Dewangan,2003).Al-though equation9can be ud to obtain the effective S-wave attenuation coefficient by evaluating the slope
of ln|U SS新疆南山
E |,its application is hampered by the need
to evaluate the source spectrum S(ω),which is often difficult to do in practice.
However,as shown below,S(ω)is eliminated in the estimation of the interval S-wave attenuation coefficient. The ray-theoretic frequency-domain amplitudes of the waves PP O1and PS O1(Figure2)can be written as
|U P P
O1|=S(ω)G P P
O1
e−k I P,O1(l EI+l IF)
=S(ω)G P P
O1
e−2k I P,O1l EI,(10)
|U P S
O1|=S(ω)G P S
O1
e−k I P,O1l EI e−k I S,O1l ID,(11)
where k I P,O1and k I S,O1are the average P-wave and S-wave group attenuation coefficients along the raypaths PP O1and PS O1,respectively.Equations10and11can be combined to compute the attenuation of the con-structed shear-wave SS O1in the overburden:
|U SS
O1|=
|U P S
O1
|2
|U P P
O1
|
=G O1S(ω)e−2k I S,O1l ID,(12)
where G O1=G2P S
O1/G P P
O1
.Likewi,the attenuation
coefficient for the overburden event SS O2can be found
from
|U SS
O2
|=
|U P S
O2
|2
|U P P
O2
|
=G O2S(ω)e−2k I S,O2l JC,(13)
The problem is now reduced to the attenuation analy-
sis of pure modes considered by Behura and Tsvankin
早的英语怎么说
(2009).Equations9,12and13can be combined to com-
pute the interval shear-wave attenuation in the target
layer as follows:
|U SS
T
|=
|U SS
E
|2
|U SS
O1
||U SS
O2
耶稣祷告词
|
=G T e−2(k I S,DR l DR+k I S,RC l RC)
e2(k I S,O1l ID+k I S,O2l JC),(14)
where G T=G2E/(G O1G O2).Taking the logarithm of
equation14yields:
ln|U SS
T
|=ln G T−2(k I S,DR l DR+k I S,RC l RC)
+2(k I S,O1l ID+k I S,O2l JC).(15)
Since k I S,DR l DR=k I S,IR l IR+k I S,O1l ID and
k I S,RC l RC=k I S,RJ l RJ+k I S,O2l JC,equation15
can be rewritten as
ln|U SS
T
|=ln G T−2k I S,IR l IR−2k I S,RJ l RJ
=ln G T−2k I S,T(l IR+l RJ),(16)
where the coefficient k I S,T reprents the average group
attenuation coefficient along the shear-wave raypath in
the target layer.
Interval pha attenuation coefficient for a
homogeneous target layer
If the target layer is heterogeneous,equation16provides
only the offt-dependent average interval attenuation
coefficient.Interpretation of attenuation measurements
can be significantly simplified for horizontal,homoge-
neous layers with a horizontal symmetry plane.Then
the length of the raypath in the target layer is given by
l IR+l RJ=V g t SS
T
,where V g is the shear-wave group
velocity along the ray IR(Figure3),and t SS
T
is the
interval shear-wave traveltime in the target layer.As a
result,equation16reduces to
ln|U SS
T
|=ln G T−2k I S,T V g t SS
T
.(17)
Behura and Tsvankin(2009)show that equation17can
be ud to obtain the interval pha attenuation coeffi-
cient of P-or S-waves.According to their results,equa-
tion17can be rewritten as
ln|U SS
T
|=ln G T−2ωA S t SS
T
,(18)
whereωis the angular frequency and A S=k I,P h/k R,P h
比的意义教学设计is the S-wave pha attenuation coefficient(Zhu,2006)
for a zero inhomogeneity angle(the angle between the
real and imaginary parts of the wave vector);k I,P h and
Interval shear-wave attenuation401 k R,P h are the magnitudes of the imaginary and real
parts of the wave vector,respectively,for S-waves.
The shear-wave interval traveltime in the target
layer(t SS
T )is computed from equation5using the
南山塔kinematic layer stripping.Hence,the slope of the log-arithmic spectral ratio in equation18yields the pha attenuation coefficient for the pha angle correspond-ing to a given group ,to the raypath IR in Figure3).If the slope is constant,A S and the quality factor Q S≈1/(2A S)are independent of frequency.If the slope varies with frequency,A S has to be computed from the instantaneous slope,which yields a frequency-dependent attenuation coefficient and quality factor.
For VTI and orthorhombic media,the S-wave pha attenuation coefficient can be inverted for the attenuation-anisotropy parameters introduced by Zhu and Tsvankin(2006,2007).The SV-wave pha atten-uation coefficient in VTI media is approximately given by(Zhu and Tsvankin,2006):
A SV(θ)=A S0(1+σ
Q
sin2θcos2θ),(19) where A S0≈1/(2Q S0)is the symmetry-direction SV attenuation coefficient and Q S0is the vertical quality
factor.The parameterσ
Q determines the variation of
A SV away from the symmetry direction and depends
on the attenuation-anisotropy parameters
Q andδ
Q
,
as well as on the vertical velocities and quality factors P-and S-waves.
Whereas the pha attenuation coefficient is ex-presd as a function of the pha angle,our method computes A SV for a certain source-receiver offt at the top of the target layer.Estimating th
e pha angle for a given source-receiver pair generally requires knowledge of the anisotropic velocityfield in the interval of inter-est.
Synthetic example
The method was tested on synthetic multicomponent data from a horizontally stratified VTI model(Figure 4).The sources were placed on the top of the model, while the receivers were on the bottom of the water layer.Our method is applicable to this source-receiver geometry becau it utilizes events with shared ray g-ments in the overburden.
Synthetic reflection data were generated using an anisotropic reflectivity code(Schmidt and Tango,1986). PP and PS events from the top and bottom of the target were identified on the vertical and radial displacement components of the shot gather(Figure5).Kinematic layer stripping of the shear-wave traveltimes produced the interval moveout in the third(target)layer shown in Figure6.The layer-stripped interval traveltimes prac-tically coincide with the exact values computed by ray tracing.It should be noted that the maximum offt for the constructed shear-wave in the target layer is
limited Figure4.Synthetic model ud to test the algorithm.The
source is placed on the top of the model and the receivers are on the water bottom.The water is purely isotropic and elastic with the P-wave velocity V P=1500m/s and thick-ness d=2000m.The other three layers have VTI symmetry for both velocity and attenuation.For the cond layer,the vertical P-and S-wave velocities are V P0=1600m/s and V S0=800m/s,thickness d=600m,and velocity-anisotropy parameters are =0.30,andδ=0.10;the attenuation pa-rameters are Q P0=20,Q S0=50,
Q
=0.30,andδ
Q
= 0.20.In the third layer,V P0=1700m/s,V S0=900m/s, d=1000m, =0.25,δ=0.10,Q P0=100,Q S0=20,
Q
=0.20,andδ
Q
=0.10.The parameters of the bottom halfspace are V P0=2500m/s,V S0=1400m/s, =0.30,
δ=0.10,Q P0=50,Q S0=50,
Q
=0.40,andδ
Q
=0.
30.
杨幂是哪里人
(a)(b)
Figure5.Vertical(a)and horizontal(b)displacement com-ponents of a shot gather for the model from Figure4.The target PP and PS events are marked by the red arrows in(a) and(b),respectively.
by the critical angle for SP mode conversions,which is equal to32◦.
The input amplitudes were obtained by computing the vector sum of the radial and vertical displacement components.Frequency-domain amplitudes were found by windowing the arrivals and applying the Fourier transform.The target layer is horizontal,homogeneous,
