An Analysis of AVO Inversion for Postcritical Offsets in HTI Media
Oct 17, 2015
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An analysis of AVO inversion for postcritical offsets in HTI media
Lyubov Skopintseva1 and Tariq Alkhalifah2
ABSTRACT
Azimuthal variations of wavefield characteristics, such as tra-veltime or reflection amplitude, play an important role in theidentification of fractured media. A transversely isotropic med-ium with a horizontal symmetry axis (HTI medium) is the sim-plest azimuthally anisotropic model typically used to describeone set of vertical fractures. There exist many techniques in in-dustry to recover anisotropic parameters based on moveoutequations and linearized reflection coefficients using such amodel. However, most of the methods have limitations in defin-ing properties of the fractures due to linearizations and physicalapproximations used in their development. Thus, azimuthal ana-lysis of traveltimes based on normal moveout ellipses recovers amaximum of three medium parameters instead of the requiredfive. Linearizations made in plane-wave reflection coefficients(PWRCs) limit the amplitude-versus-offset (AVO) analysis tosmall incident angles and weak-contrast interfaces. Inversionbased on azimuthal AVO for small offsets encounters nonuni-
queness in the resolving power of the anisotropy param-eters. Extending the AVO analysis and inversion to and beyondthe critical reflection angle increases the amount of informationrecovered from the medium. However, well-accepted PWRCsare not valid in the vicinity of the critical angle and beyondit, due to frequency and spherical wave effects. Recently derivedspherical and effective reflection coefficient (ERC) methodsovercome this problem. We extended the ERCs approach toHTI media to analyze the potential of near- and postcriticalreflections in azimuthal AVO analysis. From the sensitivityanalysis, we found that ERCs are sensitive to different setsof parameters prior to and beyond the critical angle, which isuseful in enhancing our resolution of the anisotropy parameters.Additionally, the resolution of the parameters depends ona sufficient azimuthal coverage in the acquisition setup. Themost stable AVO results for the azimuthal acquisition setupwith minimum number of lines (three) are achieved when theazimuthal angle between lines is greater than 45.
INTRODUCTION
The role of anisotropy has dramatically increased over the pasttwo decades due to advances in acquisition setups, data quality, dataprocessing, and parameter estimation. It has been demonstrated thatincluding anisotropy in the data analysis considerably reduces un-certainty in interpretation (Tsvankin et al., 2010). Fracture identi-fication, direction, and density estimation have become moreattainable with the use of multiazimuth acquisition setup and multi-azimuth data analysis. However, although we see that fractured re-servoirs often adhere closely to the orthorhombic symmetry(Grechka et al., 2006), azimuthal analysis based on HTI symmetryis widely exploited for a vertical set of fractures. The HTI is thesimplest azimuthally anisotropic model (Rger, 2001).
One of the most widely exploited approaches in fracture identi-fication is the azimuthal analysis of reflection traveltimes based onthe concept of the normal moveout (NMO) ellipse (Grechka et al.,1999). Although P-wave azimuthal moveout analysis is practicallyeffective in predicting the fracture direction (Lynn et al., 1999; Todet al., 2007), the NMO ellipse constrains only three combinations ofthe medium parameters, leading naturally to a three-parameter in-version (Al-Dajani and Alkhalifah, 2000). Azimuthally dependentP-wave traveltime inversion recovers vertical P-wave velocity, ani-sotropy parameter V (or anellipticity parameter ; Alkhalifah andTsvankin, 1995), and the symmetry axis direction. The NMO el-lipse, however, is not enough to fully characterize five parametersof an HTI model.
Manuscript received by the Editor 1 August 2011; revised manuscript received 27 September 2012; published online 12 April 2013.1Norwegian University of Science and Technology, Department of Petroleum Engineering and Applied Geophysics, Trondheim, Norway. E-mail: lyus@
statoil.com.2King Abdullah University of Science and Technology, Thuwal, Saudi Arabia. E-mail: [email protected].
2013 Society of Exploration Geophysicists. All rights reserved.
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GEOPHYSICS, VOL. 78, NO. 3 (MAY-JUNE 2013); P. N11N20, 11 FIGS.10.1190/GEO2011-0288.1
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Reflection coefficients contain valuable information about the lo-cal medium properties on both sides of an interface. Therefore, ana-lysis of amplitude variations with incidence angle or offset is oftenused in reservoir characterization (Mallick and Frazer, 1991; Mal-lick et al., 1998; Avseth et al., 2001). Generally, amplitude-versus-offset (AVO) analysis has higher vertical resolution than traveltimemethods. The AVO inversion approaches used in industry are basedon the linearizations of plane-wave reflection coefficients(PWRCs), made under the assumption of weak contrast interfaces(Ostrander, 1984). As a consequence of linearizations, applicationof such AVO approaches is limited to small offsets, where a reason-ably good match with real data is achieved. The application of azi-muthally dependent approximations (Rger, 2001) in azimuthalAVO inversion, however, is hindered by nonuniqueness in param-eter estimation. Practically, azimuthal variations of AVO responseare exploited in the recovery of a fracture azimuth with a 90 un-certainty (Hall and Kendall, 2003). Despite this ambiguity, azi-muthal AVO analysis showed promise in many cases (Grayet al., 2002; Hall and Kendall, 2003; Xu and Tsvankin, 2007).Nonconventional reservoirs such as stiff-carbonate reservoirs,
heavy oil traps, salt domes, and basalts represent a challenge inthe oil industry today. These reservoirs are characterized by highcontrasts in media parameters across the interface. This results inreflections associated with the critical angle at relatively small off-sets, where conventional AVO analysis is not valid anymore. Theimportance of near- and postcritical reflections is highlighted byseveral researchers (Downton and Ursenbach, 2006; Alhussain,2007; Skopintseva et al., 2011), who show that using such reflec-tions improves inversion results. Based on PWRC analysis, Halland Kendall (2003) and Landr and Tsvankin (2007) notice thatthe critical angle is sensitive to the fracture orientation.However, PWRCs do not describe near- and postcritical reflec-
tions generated by a point source, which are functions of the wave-front radius and frequency (erven, 1961; Alhussain, 2007).Recently developed spherical and effective reflection coefficients(ERCs) for isotropic/isotropic interfaces (Ayzenberg et al., 2007;Ursenbach et al., 2007) provide better insights of the reflection be-havior observed around and beyond the critical angle. Extension ofERCs to isotropic/VTI interfaces (Ayzenberg et al., 2009) showstheir sensitivity to anisotropy parameters.In this paper, we investigate the potential of using postcritical
reflections in azimuthal AVO analysis and inversion. For this pur-pose, we extend ERCs to an isotropic/HTI interface and comparetheir azimuthal dependence prior to and beyond the critical angle.We show that the amplitude maximum observed beyond the criticalangle can be used in anisotropy parameter estimation. We providesensitivity analysis of reflection coefficient sensitivity to the under-burden parameters prior to and beyond the critical angle for multi-azimuth acquisition coverage with a minimal number of survey lines(three). Results indicate that pre- and postcritical domains are sensi-tive to different sets of parameters. In addition, we provide analysisfor the optimal acquisition setup for better stability of the inversion.
PHASE VELOCITY SURFACE VERSUS CRITICALANGLE SURFACE
Consider a two-layer model with a plane interface, in which theupper half-space is isotropic and the lower half-space represents anHTI medium. The P-wave velocity of the isotropic half-space islower than the P-wave phase velocity of the HTI medium for
any azimuthal direction. For simplicity, assume that the S-wavephase velocity in the underburden is constant and lower than theP-wave velocity in the overburden. When P-wave propagating inthe upper half-space hits the isotropic/HTI interface at the criticalangle, part of the energy reflects from the interface, part of itconverts to the transmitted SV-wave, and the rest of energy startspropagating along the interface and generates a PPP head wave(Figure 1). The velocity of the head wave corresponds to the hor-izontal P-wave phase velocity of the underburden, and it is the func-tion of angle between the incidence plane and the HTI symmetryaxis. Therefore, the critical angle cr is also a function of azimuth due to the modified Snells law (Landr and Tsvankin, 2007):
sin cr VP1
VhP2; (1)
where VP1 is the P-wave velocity in the overburden and VhP2 isthe azimuthally dependent horizontal phase P-wave velocity in theunderburden. The plot in Figure 1 defines the critical angles for twoorthogonal incidence planes. Plane k is located along the symmetryaxis of an HTI medium, and plane coincides with the isotropicplane of an HTI medium. VhP20 corresponds to the phase velocityalong the symmetry axis, and VhP290 VP2 is the phase P-wavevelocity in the isotropic plane . The horizontal velocity in plane kis less than the horizontal velocity in plane . Consequently, thecritical angle in the plane along the symmetry axis is larger thanthat for the isotropic plane. The reciprocal proportionality of thesine of the critical angle to the horizontal phase velocity (equation 1)shows that the azimuthal dependence of the critical angle mightbring an additional information about the underburden.Azimuthal dependency of the horizontal P-wave phase velocity
surface for HTI media is equivalent to the P-wave phase velocitysurface for VTI media in the vertical plane because an HTI mediumis equivalent to a VTI medium rotated by 90 in the vertical plane.According to well-known acoustic approximations (S-wave veloci-ties are assumed to be zero) for the phase velocity in VTI media(Tsvankin, 2005), the azimuthally dependent horizontal phasevelocity for an HTI medium has the following form:
Vh2P2V2P21
2Vcos2
12
14cos2V cos 22Vsin24V2cos4
q ;
(2)
where V and V are anisotropy parameters in HTI notation(Rger, 2001). Note that the horizontal P-wave velocity does notdepend on the third anisotropy parameter V, which results in non-sensitivity of the critical angle to this parameter.Substituting equation 1 into 2 and exploiting the three azimuthal
directions corresponding to 0, 45, and 90 yields the followingequations for anisotropy parameters in terms of their critical angles:
Vsin2cr90sin2cr0
2sin2cr0;
Vsin2cr90sin2cr452sin2cr0sin2cr452sin2cr90
2sin2cr0sin2cr451
2: (3)
To obtain anisotropy parameter V, we need information about thecritical angles in the incidence planes along and across the symmetry
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axis of HTI media. Anisotropy parameter V requires additionalknowledge about the critical angle for an azimuth of 45. Equation 3implies that the symmetry axis direction is known. The symmetryaxis direction can be estimated from the azimuthal distribution ofthe critical angle under two assumptions: parameter has the strongerimpact on VhP20 than and the horizontal P-wave velocity along thesymmetry axis direction VhP20 is smaller than horizontal P-wavevelocity in the isotropic plane VhP290. The second assumption im-plies that symmetry axis direction corresponds to the largest value ofthe critical angle.
EFFECTIVE REFLECTION COEFFICIENT FORTHE ISOTROPIC/HTI INTERFACE
Here, we extend the ERCs to the isotropic/HTI case. We choosethe incidence plane coinciding with the x1; x3 plane of the globalcoordinate system x1; x2; x3 and forming an angle with the sym-metry axis of an HTI medium. A point source exciting a sphericalP-wave is located in the upper half-space. The ERC represents aratio of the reflected spherical wave to the incident spherical wave(Ayzenberg et al., 2007). For the isotropic/HTI interface at the pointof a receiver, it has the form similar to ERC for isotropic/VTI inter-face (Ayzenberg et al., 2009):
PPx;;g uPPnormx;g cos x uPP tanx;g sin x
ikPrPPx
1k2Pr
2PPx
eikPr
PPx
;
(4)
where i is the imaginary part, matrix x x1; x2; : : : ; xN consistsof N source-receiver offsets in 3D space xN x1N; x2N; x3N; isthe angular frequency; g VP1; VS1; 1; VP2; VS2; V; V; V;; 2 is the vector of model parameters; VP1, VS1 are the P-and S-wave velocities of the isotropic half-space; VP2 and VS2are the P- and S-wave velocities of the HTI model in the isotropicplane; 1, 2 are the densities of the upper and lower half-spaces,respectively; V, V, V are the anisotropy parameters in HTInotation (Rger, 2001); kP VP1 is the wavenumber in the overbur-den; x is the incidence/reflection angle; and rPPx is the appar-ent radius of the wavefront at the receiver. A general form of theradius rPPx is introduced by Skopintseva et al. (2012). For a planeinterface, rPPx is the distance between the source and the receiveralong the ray (Skopintseva et al., 2011). The dimensionless normaland tangential components of the displacement vector are repre-sented by uPPnormx and uPP tanx, which have the following form:
uPPnormx;g Z 0
RPP;geix12
pJ0xd;
uPP tanx;g Z 0
RPP;gieix
12
p1 2
p J1x2d;(5)
where RPP; g is an exact PWRC for the isotropic/HTI interfacederived by Schoenberg and Protazio (1992), x kPrPPx cos x, x kPrPPx sin x, is the horizontalcomponent of the unit P-wave ray vector in the overburden, andJ0 and J1 are the Bessel functions of the zeroth and first order.Equations 4 and 5 show that reflection coefficient at the receiver
does not depend on one reflection point as it is assumed in plane
wave theory. The reflection process for spherical waves includes thecontribution of all point of the interface. The largest contribution,however, comes from reflection points of the interface included intothe Fresnel zone (Ayzenberg et al., 2007). Equations 45 can beeasily transferred from the offset domain to the angle domain.We exploit the offset domain mostly for convenient representationof the reflection coefficient at the postcritical domain.Figure 2 shows the amplitude of the ERC normalized by
the reflection amplitude at the first receiver x1 (Ax; x;x1;). It is compared with amplitudes of the normalizedPWRC ARx; RPPx;RPPx1; and reflection coefficientobtained from reflectivity modeling using the technique describedby Skopintseva et al. (2011). Reflection coefficients are calculatedfor the model water/HTI provided by Alhussain (2007), whereVP1 1.484 kms, VS1 0 kms, 1 1 gcm3, VP2 2.709 kms, VS2 1.382 kms, 2 1.2 gcm3, V 0.0019, V 0.0069, V 0.0439, 30, the frequencyis 218 Hz, and the interface depth is 240 m. The ERC and PWRCcurves coincide at precritical angles, whereas they are differentaround and beyond the critical angle. The ERC has a gradual am-plitude increase with the angle and reaches its maximum beyond thecritical angle, whereas the PWRC has an abrupt amplitude increasewith the maximum at the critical angle. Additionally, the ERC hasoscillations in the postcritical domain, whereas these are absent forthe PWRC. Effects observed for the ERC around and beyond thecritical angle occur due to the influence of the dimensionless argu-ment kPrPPx in equations 45. The perfect match of the ERC withthe reflection coefficient extracted from the synthetic data confirmsthe soundness of the observed phenomena described by the ERCs.This fact allows exploiting equations for ERC to study the influenceof different model parameters on the amplitude behavior forpre-, near-, and postcritical offsets without computing incredibleamount of synthetic seismic gathers. Moreover, these equationscan be easily exploited in the long-offset AVO inversion.
2 2
2 2
900 90
hP P
h hP P
V V
V V
II
Symmetry axis
0cr
90cr
2 90h
PV
2 0h
PV
2PV
90 0cr cr
Figure 1. Schematic plot depicting the model setting consideredhere. The ray curves correspond to a reflection at the critical anglefrom the horizontal isotropic/HTI interface.
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The argument kPrPPx represents the effect of the wavefrontcurvature and frequency on the reflection amplitude. Its value con-trols the amplitude maximum shift and amplitude oscillation inpostcritical domain associated with the interference between re-flected and head waves (Figure 3). Taking into account that the ar-gument kPrPPx is the composite of the wavefront curvature andthe frequency, the effect of this argument on the reflection coeffi-cient has a dual physical insight. Thus, the infinite value ofkPrPPx can be associated with either the infinite frequency or withthe zero-wavefront curvature, which represents the assumptions ofplane wave theory. Therefore, the ERC asymptotically tends to thePWRC with the argument kPrPPx approaching infinity. For planeinterfaces, the argument kPrPPx can be represented as the effect oftraveltime and frequency on the reflection amplitude becausekPrPPx tPPx, where tPP is the traveltime. In this situation,
the ERC tends to PWRC with the frequency or traveltime approach-ing infinity.Figures 4 and 5 show maps of azimuthal distributions of the nor-
malized reflection coefficients for different anisotropy parametersV, V, V and a frequency of 32 Hz. We used a model in whichthe isotropic half-space with parameters VP1 2 kms, VS1 1.1 kms, and 1 1.8 gcm3 overlays the HTI half-space withbackground parameters VP2 2.8 kms, VS2 1.6 kms, and2 2.1 gcm3 at a depth of 1 km. The change in velocity acrossthe interface is representative for fractured reservoirs with strongcontrast interfaces. The isotropic overburden is chosen to avoid an-isotropy effects on traveltime. This ensures that the argumentkPrPPx tPP is anisotropy independent. Therefore, we analyzethe effect of anisotropy on reflection amplitudes only. The argumentkPrPPx changes within the ranges of 201284 (Figure 4) and201635 (Figure 5) and introduces the degree of deviation of theERC from PWRC in the postcritical domain. For each row, wechange only one anisotropy parameter in the underburden whileall others are set to zero. The left column represents a weakanisotropy effect, whereas the right column corresponds to stronganisotropy. The color indicates the amplitude strength. The offsetincreases in the radial direction from the center. The circle at1 km (or, equivalently, kPrPPx 224 ) corresponds to an inci-dent/reflection angle of 30 (if anisotropy parameters are zeros),which is normally a limit for conventional AVO studies. The circleat 2 km (or, equivalently, kPrPPx 284 ) corresponds to the cri-tical angle, if anisotropy parameters are zeros. The azimuthal angle changes from 0 to 360. The offset range in Figure 4 correspondsto the precritical domain, whereas Figure 5 covers pre-, near-, andpostcritical offsets.In the case of the isotropic medium, we expect a uniform ampli-
tude response with azimuth for any offset range (circles). Anisotro-py produces deviation from such an isotropic pattern. Figure 4shows that the parameter V does not cause much azimuthal de-viation in the amplitude within the circle of 1 km. Its effect appearsat larger offsets, specifically when the incident angle reaches criticalangles. The influence of the parameter V is slightly stronger with-in the circle of 1 km, but larger offsets experience greater influenceof this parameter. The largest effect on the azimuthal distribution ofamplitudes in Figure 4 is caused by the anisotropy parameter V.The amplitude strength deviates from an isotropic pattern at alloffsets within the range 02 km.Figure 5 shows well-defined amplitude maximum contour asso-
ciated with the postcritical domain. This contour, however, doesnot coincide with the offset corresponding to the critical angledue to the influence of the finite argument kPrPPx. The shapeof these contours is somewhat proportional to the azimuthal depen-dence of the critical angle position. Sensitivity of the postcriticaldomain to the anisotropy parameters V and V is clearly ob-served, whereas sensitivity to the parameter V is not obvious.Changes in the anisotropy parameter V control the amplitudemaximum deviation from the isotropic circle in the symmetry axisdirection ( 0). Parameter V influences the amplitude maxi-mum deviation in the oblique direction ( 45), as well as am-plitude strength for 0 and 90. These observations areconsistent with equations 3, where V is a function of the criticalangles along and across the symmetry axis and V is a function ofcritical angles for three azimuths ( 0, 45, 90). Parameter Vis independent of the critical angle, which follows from equation 2.
0 20 400
1
2
Angle ()
Ampl
itude
RMERCPWRC
cr
Figure 2. Comparison between the normalized ERC, the PWRC,and the reflection response obtained from reflectivity modeling.Reflection coefficients are calculated for the model withVP1 1.484 kms, VS1 0 kms, and 1 1 gcm3 in theoverburden and VP2 2.709 kms, VS2 1.382 kms, 2 1.2 gcm3, V 0.0019, V 0.0069, and V 0.0439in the underburden. The azimuthal angle of the acquisition line re-lative to the symmetry axis is 30. The frequency of the signalis 218 Hz, and the interface depth is 240 m.
0 10 20 30 40 50 60 70
1
2
3
4
5
Ampl
itude
Angle ()
PWRCERC (Low)ERC (HIgh)
cr
Figure 3. Amplitude of normalized ERCs in the isotropic plane 90 for kPrP0 402 (high) and kPrP0 25 (low). Theargument kPrP represents the product of the raypath length and wa-venumber for the isotropic/HTI plane interface. The normalizedPWRC is given for comparison. The model parameters are asfollows: VP1 2 kms, VS1 1.1 kms, 1 1.8 gcm3, VP22.8kms, VS2 1.6 kms, 2 2.1 gcm3, V 0.1, V 0.05, and V 0.1.
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The largest anisotropic effect is observed due to the influence,whereas represents the second order effect on the postcriticaldomain. Therefore, the assumption that the largest critical anglecorresponds to the symmetry axis direction sounds reasonable.
CRITICAL OFFSET VERSUS AMPLITUDEMAXIMUM OFFSET
Figure 2 shows that the critical angle (offset) cannot be clearlydefined because it is not associated with special amplitude featuresof the ERC. However, offsets corresponding to maximum amplitudecontours as shown in Figure 5 can be exploited for anisotropyparameter detection. Assume that the deviations of the amplitudemaximum position from the critical offset are controlled only byparameter kPrPPx and are weakly dependent on the anisotropyparameters. Then the azimuthal dependence of the critical offset
is proportional to the azimuthal dependence of the amplitude max-imum offset xm:
xcr nxm; (6)
where n is an azimuthally independent proportionality coefficient,xcr is an azimuthally dependent critical offset, and xm is theazimuthally dependent amplitude maximum offset.Applying the assumption of equation 6 and the relation
sin 2cr x2cr
h2 x2cr; (7)
where h is the interface depth, we rewrite equations 3 in terms of theamplitude maximum offset xm:
0.15 V
1
2
1
2
30
210
60
240
90
270
120
300
150
330
0180
0.05 V
1
2
1
2
30
210
60
240
90
270
120
300
150
330
0180
0.05 V
1
2
1
2
30
210
60
240
90
270
120
300
150
330
0180
0.15 V
1
2
1
2
30
210
60
240
90
270
120
300
150
330
0180
0.05V
1
2
1
2
30
210
60
240
90
270
120
300
150
330
0180
0.15V
1
2
1
2
30
210
60
240
90
270
120
300
150
330
0180
0.4 0.6 0.8 1 1.2
Figure 4. Maps of the normalized azimuthal ERCs prior to thecritical offset (2 km in the isotropic plane 90) for differentanisotropy parameters. The radial direction corresponds to thesource-receiver offset from 0 to 2 km, and the angular direction cor-responds to the angle between the survey line and the symmetry axisdirection from 0 to 360. Each row represents changes in one of theanisotropy parameters, while the others are set to zero.
0.05 V
2
4
6
2
4
6
30
210
60
240
90
270
120
300
150
330
180 0
0.15 V
2
4
6
2
4
6
30
210
60
240
90
270
120
300
150
330
180 0
0.05 V
2
4
6
2
4
6
30
210
60
240
90
270
120
300
150
330
180 0
0.15 V
2
4
6
2
4
6
30
210
60
240
90
270
120
300
150
330
180 0
2
4
6
2
4
6
0.05V
2
4
6
30
210
60
240
90
270
120
300
150
330
180 0
0.15V
2
4
6
2
4
6
30
210
60
240
90
270
120
300
150
330
180 0
1 2 3 4
Figure 5. Maps of the normalized azimuthal ERCs prior to and be-yond the critical offset for different anisotropy parameters. The ra-dial direction corresponds to the source-receiver offset from 0 to6 km, and the angular direction corresponds to the angle betweenthe survey line and the symmetry axis direction from 0 to 360.Each row represents changes in one of the anisotropy parameters,while the others are set to zero.
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V cos2cr90x2m90 x2m0
2x2m0;
V M1 cos2cr90 M2 cos2cr90 cos 2cr90;(8)
where
M11x2m902x2m45
x4m90x4m45
x4m90
2x2m0x2m45;
M21
23x2m902x2m45
x4m90
x4m45x
2m90
2x2m0
x4m902x2m0x2m45
: (9)
The azimuthally independent proportionality coefficient n in equa-tions 89 is vanished. Although equations 89 are functions of theamplitude maximum offsets, the critical angle in the isotropic planeis still needed. This information can be retrieved from an AVOinversion in the isotropic plane (Skopintseva et al., 2011).Relative errors in anisotropy parameter estimates obtained from
the different types of data are shown in Figure 6. The three data setsare exploited for estimations: azimuthal dependency of the horizon-tal phase velocity VhP2 given by equation 1, azimuthal depen-dency of the critical angle cr estimated from PWRCs, andazimuthal dependency of the maximum amplitude offset xm ob-tained from ERCs. The most accurate estimates are obtained for theanisotropy parameter V. The error level is within 1% for any typeof data, and it is the smallest for horizontal P-wave velocity. Higher
inaccuracies for cr and xm are explained by PWRC and ERCdiscretization due to the receiver spacing. The error level in V
estimates increases up to 10%. The reason for such inaccuraciesis the acoustic approximation of phase velocity used in anisotropyparameter estimates. The approximation is least accurate at anazimuth of 45. This consequently results in errors for all typesof data. In general, the amplitude maximum offset xm provides asimilar level of errors as the critical angle and thus has the potentialto be used for anisotropy analysis. Nevertheless, no explicit linkbetween the critical offset and the position of the amplitudemaximum has been found yet. It is clear that the argumentkPrPPx plays the key role. The development of explicit relation-ship would dramatically simplify amplitude analysis in the post-critical domain.
SENSITIVITY ANALYSIS AND PARAMETERDEPENDENCY
The results of the previous section give the potential for aniso-tropy parameter analysis only in the case of full azimuthal coverage.In this situation, the direction of the symmetry axis is easily definedfrom the azimuthal dependence of the amplitude maximum positionas seen in Figure 5. However, quite often, data are available only forseveral azimuth directions. Then, symmetry axis direction identifi-cation is not obvious.To understand the potential for using postcritical reflections in
this situation, we provide a sensitivity analysis for multiazimuth ac-quisition coverage with a minimal number of lines, given by angles1, 2, and 3 relative to a global coordinate system (Figure 7).
The symmetry axis direction forms the angle with coordinate axis x1. For simplicity, we as-sume that the azimuthal separation between thesurvey lines is equal and, thus, satisfies 3 2 2 1. We assume that theparameters of the isotropic overburden areknown, and we investigate the sensitivity ofthe normalized reflection coefficients for an iso-tropic/HTI interface to changes in the mediumparameters of the HTI half-space. To conductthe sensitivity analysis, we exploit techniques de-scribed by Al-Dajani and Alkhalifah (2000) andbuild up the following Jacobian matrix:
JT d1; xd2; xd3; x; (10)where
dVP2VP2A
A;VS2VS2A
A;22AA
;VAA
;VAA
;VAA
;2AA
(11)
are submatrices of derivatives of the normalized reflection coeffi-cient Ax; with respect to the medium parameters for a par-ticular survey line and T indicates the transpose of a matrix. Thederivatives with respect to velocity and density are normalized toallow for direct comparison with the dimensionless anisotropicparameters.As a result, the resolution matrixM JTJ provides information
on the linear dependency of the parameters and the strength of theirresolution. Perfect resolution is given by the identity matrix, which
0.140.12
a) b)
0.10.080.060
1
2
(V )
Rel
ativ
e er
ror (
%)
V P2h
cr
xm
0.140.120.10.080.06
2
4
6
8
10
(V )
Rel
ativ
e er
ror (
%)
V P2h
cr
xm
Figure 6. Relative errors in the anisotropy parameter V (a) and V. (b) Estimatesobtained from velocity, critical angle, and offsets corresponding to the maximumamplitudes.
2x Line 2 Line 3
Line 1
Symmetry axis
32
1
1x
Figure 7. A plan view of three 2D survey lines over a horizontalHTI layer with the arbitrary symmetry axis direction from the cho-sen global coordinate system.
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indicates that all parameters are resolvable within the linear limitand do not have trade-offs between each other. However, the reso-lution matrix only allows for a linearized analysis of the sensitiv-ities, valid around a point in the model space. Depending on thelevel of nonlinearity, this analysis can also be representative ofgeneral behavior.Figure 8 represents the resolution matrix M for all underburden
model parameters for precritical (top row) and postcritical (bottomrow) offsets for the following acquisition setups: 2 30 and 60. Medium parameters in the isotropic upper layer areVP1 2.0 kms, VS1 1.1 kms, and 1 1.8 gcm3. The parameters of the underburdenare chosen to be VP2 2.8 kms, VS2 1.6 kms, 2 2.1 gcm3, V 0.1, and 0. Each column corresponds to differentcombinations of anisotropy parameters V
and V. Higher diagonal values indicate highersensitivity of the normalized reflection coeffi-cients to the particular parameter, which conse-quently results in higher resolution of thisparameter in the inversion. The nonzero off-diagonal element reveals a trade-off betweenparameters related to this element.Generally, we observe that the sensitivity of
medium parameters is model dependent, butthere are common trends for different combina-tions of V and V. Normalized ERCs are sen-sitive to different sets of parameters at pre- andpostcritical offsets. Exploiting only the precritical domain results in resolving P- and S-wave velocities with reasonable trade-off, andanisotropy parameters V and V with sometrade-off between them, as well. The sensitivityof the reflection coefficients to the symmetry axisdirection in the precritical domain is highly de-pendent on the strength of anisotropy. In thepostcritical domain, the vertical P-wave velocityremains highly resolved with less trade-off withthe shear wave velocity, whereas the resolutionof V and V decreases. Instead, the resolutionof density, anisotropy parameter V, and thesymmetry axis direction increases considerably.We observe trade-offs in the postcritical domainbetween VP2 and 2, and V, and and VP2.However, it is important to note that the symme-try axis resolution is high with a minor trade-offwith other parameters, which implies the impor-tance of the postcritical reflection coefficients inresolving the symmetry axis direction.Figure 9 gives an idea of how azimuth 2 and
separation angle between the survey lines af-fect the resolution of some parameters in the pre-critical (top row) and the postcritical (bottomrow) domains. Only diagonal values of the ma-trix M JTJ corresponding to VP2, V, V,and are exploited for this purpose. The calcu-lations are made for a model in which the param-eters of the isotropic half-space are the same as inFigure 8, and parameters of the HTI half-space
are given in the caption of Figure 9. The largest effect of the azimuth2 and angle separation on the resolution of medium param-eters is achieved in the postcritical domain rather than in theprecritical domain. Moreover, different combinations of acquisitionparameters affect the resolution of different sets of parameters.Thus, 45 < 2 < 135 and < 45 results in the best resolutionof VP2 in the postcritical domain, as this acquisition setup providesthe best coverage of the isotropic plane. The best resolution of theanisotropy parameter V is achieved when one of the survey linesis close to the symmetry axis direction (2 ). Although the
(V ) = 0.13, (V ) = 0.08VP2 VS2 2
(V ) (V ) (V ) VP2 VS2 2 (V ) (V ) (V ) VP2 VS2 2
(V ) (V ) (V )
VP2 VS2 2 (V ) (V ) (V ) VP2 VS2 2
(V ) (V ) (V ) VP2 VS2 2 (V ) (V ) (V )
VP2VS22(V )
(V )
(V )
VP2VS22(V )
(V )
(V )
VP2VS22(V )
(V )
(V )
VP2VS22(V )
(V )
(V )
VP2VS22(V )
(V )
(V )
VP2VS22(V )
(V )
(V )
(V ) = 0.07, (V ) = 0.08
Prec
ritic
al (0
1.25
km)
Post
criti
cal (3
5 km
)
(V ) = 0.13, (V ) = 0.04
1
0.5
0
0.5
1
Figure 8. Resolution matrices of parameters for the HTI layer for different values of Vand V. Other parameters remain constant: VP2 2.8 ms, VS2 1.6 kms,2 2.1 kgm3, V, and 0. Acquisition parameters are 2 300, 60.Parameters of the upper isotropic half-space are VP1 2.0 kms, VS1 1.1 kms,and 1 1.8 gcm3.
VP2
0 30 60 900
30
60
90
120
150
180(V )
0 30 60 900
30
60
90
120
150
180 (V )
0 30 60 900
30
60
90
120
150
180
() () () ()
() () () ()
VP2 (V ) (V )
0 30 60 900
30
60
90
120
150
180
2 ()
2 ()
2 ()
2 ()
2 ()
2 ()
2 ()
2 ()
0 30 60 900
30
60
90
120
150
180
0 30 60 900
30
60
90
120
150
180
0 30 60 900
30
60
90
120
150
180
Prec
ritic
al (0
0.75
km)
0 30 60 900
30
60
90
120
150
180
Post
criti
cal (3
5 km
)
0
0.2
0.4
0.6
0.8
1
Figure 9. Dependence of some diagonal values of the resolution matrix on the acquisi-tion parameters 2 and 2. The HTI medium is given by the following parametersVP2 2.8 ms, VS2 1.6 kms, 2 2.1 gcm3, V 0.13, V 0.08, V 0.1, and 0.
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normalized ERC in the postcritical domain is less sensitive to theanisotropy parameter V, its best resolution is observed when oneor more acquisition lines are close to the direction of 45 from thesymmetry axis (j2 j 45). The highest resolution of thesymmetry axis direction is achieved when survey lines deviate fromthe symmetry axis and the isotropy plane. The direction of the sym-metry axis is better resolved with an acquisition setup with > 45, where the resolution matrix shows fewer blind regionsthan other parameters.
RESOLUTION OF THE INVERSION
To gain insights on the feasibility of applying an inversionfor all or some of the parameters, we exploit the reciprocalof the condition number (Al-Dajani and Alkhalifah, 2000)1 jminjjmaxjp , where min and max are the minimum
and maximum eigenvalues of the matrix M JTJ, respectively.Larger 1 values indicate better resolution in the inversion. Thevalue of 1 depends on the number of unknown parameters usedin the inversion. To increase 1, we have to assume some of theparameters are known.Here, we focus on the assessment of the inversion stability for the
most useful HTI parameters in practice: VP2, V, and . This im-plies that the matrixM JTJ consists of derivatives of the normal-ized ERCs with respect to these three parameters only and thereforerepresents a 3 3 matrix. Figure 10 shows 1 as a function of off-set, azimuth 2, and separation angle . Cold colors are asso-ciated with poor parameter resolution and represent blind zones,whereas warm colors indicate good resolution. Horizontal slices,corresponding to pre-, near-, and postcritical offsets, show the bestresolution of all three parameters when postcritical offsets are in-volved in matrix M evaluation. However, the influence of the azi-muth angle and the separation angle in parameter resolution cannotbe disregarded, and one has to be careful with the acquisition setup.The location of the survey line around the isotropic plane(45 < 2 < 135, < 45) results in a blind zone. For optimalacquisition setup, separation angles larger than 45 ( > 45)are preferable. This observation is consistent with Figure 9.To confirm the previous observations and obtain some insights
into the changes in the accuracy of the parameter recovery processbeyond the linear limit, we monitor changes of the shape of theobjective function using postcritical reflections. For this purpose,we exploit objective functions for a single azimuth, given by
F XNn1
ADxn Axn2vuut ; (12)
where ADxn is the normalized reflection coefficient obtained fromthe data (Skopintseva et al., 2011) for a chosen azimuth and Axn isthe normalized ERC.Figure 11 shows 2D crossplots of the objective function for an
azimuth of 45. For our calculations, we exploited the modelwith parameters given in Figure 9. For each plot, we vary onlytwo parameters by 20%, whereas the rest of the parameters remain
constant corresponding to their true values (theminima of the objective function). The top rowrepresents the objective functions for only pre-critical offsets, whereas the bottom row repre-sents the objective functions with informationfrom near- and postcritical offsets included.Although Figure 11 does not represent the wholeset of possible 2D cross sections, the effect ofpostcritical offsets on the shape of the objectivefunction is obvious. We note that the shape of theobjective function is generally smooth, whichimplies that our linearized observations in Fig-ures 8, 9, and 10 can be generalized. Figure 11(top row) shows strong trade-offs between theanisotropy parameters. Specifically, the shapeof the objective function along the and V di-rections is stretched, indicating larger uncertaintycompared with parameters VP2, V, and V.This observation is consistent with the resolutionmatrix. When the postcritical offsets are involved
Figure 10. The reciprocal of the condition number (1) as a func-tion of offset, azimuth 2, and separation angle . Model param-eters are given in Figure 9.
(V
)
(V
)
(V
)
VP22.4 2.6 2.8 3 3.2
2.4 2.6 2.8 3 3.2
0.3
0.2
0.1
0
(V ) (V ) (V )
VP2 (V ) (V ) (V )
0.3 0.2 0.1 0 0.3 0.2 0.1 0
0.3 0.2 0.1 00.3 0.2 0.1 0
0.1
0
0.1
0.2
0.2
0.1
0
(V
)
(V
)
(V
)
0.2 0.1 0
0.2 0.1 0
0.7
0.8
0.9
0.3
0.2
0.1
0
0.1
0
0.1
0.2
0.2
0.1
0
0.7
0.8
0.9
0 0.2 0.4 0.6 0.8
Precritical offsets (01.75 km)
All offsets (05 km)
Figure 11. Two-dimensional crossplots of the objective function for different offsetranges for model parameters given in Figure 9. Squares denote true model values; circlesindicate a minima in the objective function.
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in the objective function, its shape significantly changes, indicatingbetter resolution of and V.
DISCUSSION
Analysis of the reflection coefficients for an isotropic/HTI inter-face shows that postcritical reflections contain information aboutthe underburden that cannot be recovered from precritical reflec-tions. Precritical reflections have higher sensitivity to P- andS-wave velocities and anisotropy parameters V and V. Sensi-tivity analysis of postcritical reflections shows that they can poten-tially be inverted to recover the P-wave velocity, the symmetry axisdirection, the anisotropy parameter V with small uncertainties,and the anisotropy parameter V with a larger uncertainty.Although the precritical reflections are sensitive to the directionof the symmetry axis, which is observed in some cases, the sensi-tivity of the postcritical reflections to this parameter is much higher.This is explained by the proportionality of the critical angle to thehorizontal velocity of the underburden.Inversion of azimuthal traveltimes (Al-Dajani and Alkhalifah,
2000) also allows the retrieval of a set of parameters, similarto inversion of postcritical reflection amplitudes. The differencebetween these two approaches is that inversion of azimuthal travel-times provides information about global properties of the overbur-den, whereas azimuthal postcritical reflections provide knowledgeabout local properties of the media just above and just below theinterface. A combination of these two approaches might improveresolution on the media parameters of the subsurface.We considered the anisotropy effect on the reflection coefficient
with one interface only, where the tuning effect due to interferenceof reflection events from neighboring interfaces is avoided. The tun-ing issue, however, should not be underestimated, especially forlong-offset data, in which a complicated interference pattern is nor-mally observed. This issue requires additional theoretical formula-tion of ERCs for a stack of layers, due to current limitations inthe apparent wavefront radius definition (Ayzenberg et al., 2009;Skopintseva et al., 2011). This additional development would bean important step in generalizing the analysis of reflection ampli-tudes for long-offset data to multilayered media.The sensitivity study is done for the ERC for single frequencies.
However, a reflection event on a seismogram is not monochromatic.It inherits a band of frequencies admitted by the source wavelet. Thepresence of multifrequencies results in oscillation smoothing in thepostcritical domain beyond the first amplitude peak (Skopintsevaet al., 2011). However, the first peak still follows the azimuthal pat-terns shown in Figure 5. Therefore, it is expected that the sensitivityof a multifrequency reflection response will be similar to a single-frequency ERC.Even though sensitivity analysis and assessment of inversion sta-
bility are performed for a multiazimuth coverage with only threesurvey lines for simplicity, some of the insights gained will helpin optimizing a more elaborate acquisition setup geared for param-eter estimation. An azimuthal distribution of survey lines in the vi-cinity of the isotropic plane results in reduced stability in the inver-sion. An increase in stability of the inversion when postcriticalreflections are included is predicted to be better than when onlyprecritical reflections are used.An analysis of the objective function for inversion for the subsur-
face parameters highlights the influence of postcritical reflectionson shape of objective function minimum. Given that the ERC shows
different sensitivities to parameters prior to and beyond the criticalangle, joint azimuthal inversion of pre- and postcritical reflectionslooks most promising. Including azimuthal traveltime informationin the inversion may also reduce uncertainty in parameter estimates.
CONCLUSIONS
Azimuthal analysis of the ERCs for a horizontal isotropic/HTIinterface shows that the reflection amplitudes beyond the criticalangle depend strongly on azimuthal variations in horizontal velocityand therefore contain additional information about the anisotropy ofthe underburden. The simple link given by Snells law between theazimuthally dependent critical angle and the azimuthally dependenthorizontal velocity provides an opportunity to use postcritical re-flections in the recovery of anisotropy parameters. However, it isdifficult in practice to retrieve the critical angle from the data.The position of the amplitude maximum beyond the critical anglecan be used instead.Sensitivity analysis of the ERCs for an isotropic/HTI interface
shows that the resolution of medium parameters is highly dependenton proximity to the critical angle. Reflections prior to the criticalangle are more sensitive to anisotropy parameters V and V,whereas reflections beyond the critical angle provide better resolu-tion of the anisotropy parameter V and the symmetry axisdirection.The acquisition setup also plays an important role in parameter
resolution, especially in the postcritical domain. Our investigation,performed for an acquisition with minimal multiazimuthal coverage(three survey lines with constant angle of separation), shows thatpoor resolution is obtained when all survey lines are located closeto the isotropic plane. The optimal acquisition setup, for best reso-lution of medium parameters, should have an azimuthal angle be-tween survey lines of more than 45 or, alternatively, a largernumber of survey lines.
ACKNOWLEDGMENTS
We acknowledge Statoil for funding the Ph.D. study of L. Sko-pintseva and KAUST for financial support. We also thank the as-sociate editor and the reviewers for their helpful suggestions andcomments.
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