Practical Aspects of AVO Modeling.li.Downton.xu.Leadingedge

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AVO (amplitude variation with offset)modeling contributes significantly in

seismic data acquisition design, andprestack seismic data processing andinterpretation. It has become a commonpractice in prestack seismic analysis. Itis also used for verifying and develop-ing AVO theories. Conducting AVOmodeling is, by nature, an exercise ofmultidiscipline integration. It thusenhances reservoir characterization andreduces the risk in hydrocarbon explo-ration.

This paper presents practicalaspects of AVO modeling with the goalof better understanding and more effec-

tively using AVO modeling in bothprestack data processing and interpre-tation. Numerous examples demon-strate the applications of AVO modeling.We present approaches for generating,processing, and analyzing AVO syn-thetic data, and include the commonlyused methodologies of single-interfacemodeling, single-gather modeling, 2Dstratigraphic modeling, and 2D full-wave elastic-equation modeling. Weillustrate the applications of these meth-ods to P-wave data, converted-wavedata, and elastic rock property inver-sion. In addition, we discuss some com-

monly encountered issues such astuning and the effects of noise contam-ination in AVO processing and analy-sis. Finally, we demonstrate the role ofAVO modeling in calibrating prestackseismic processing and in assisting datainterpretation.

Fundamentals in AVO modeling.Seismic rock properties are directlyresponsible for seismic wave propaga-tion and seismic responses. They may

be catalogued as basic rock seismicproperties (P- and S-wave velocities anddensity, and V

P

/V S

ratio and Poisson’sratio), impedances, modulus rock prop-erties (bulk modulus K, shear modulus µ, Lamé’s constant λ), and anisotropicrock properties.

Often, rock properties and corre-sponding AVO responses can be dis-cerned from well-log data. Todemonstrate this, various rock proper-ties were calculated using the log datafrom the Western Canadian SedimentaryBasin (WCSB) and are displayed in theplots in Figure 1, with the empirical rela-tionships for shale (solid black line),

Practical aspects of AVO modeling

Y ONGYI LI , Paradigm Geophysical, Calgary, Canada J ONATHAN D OWNTON , Veritas, Calgary, Canada,

Y ONG X U , Arcis Corporation, Calgary, Canada

M ARCH 2007 T HE L EADING EDGE 2

Figure 1. Seismic rock properties derived from a set of dipole sonic logs from the Western CanadianSedimentary Basin. Data points of the gas sand, oil-saturated sand, and overlying and underlyingshale are highlighted in red, green, black, and pink squares, respectively. Notice that rock propertiesin different domains have different sensitivity responding to fluid. The contrast between gas sandand overlying shale indicates a class 1 AVO response. These crossplots can be used as templates tointerpret inverted elastic rock properties. The lines in black, blue, and red are the empirical relation-ships for shale, brine-saturated sand, and gas-charged clean sand.

Figure 2. Petrophysical analysis of a well from the Western Canadian Sedimentary Basin. Shalevolume has direct effects on water saturation and porosity, which consequently results in changesin elastic seismic rock properties and seismic responses.



brine-saturated sand (solid blue line), and gas-charged cleansand (solid red line) overlain. Figure 1 demonstrates that thevelocities and impedances do not provide sufficient dis-crimination as to whether the reservoir is gas-charged.

However, the V P /V S ratio, Poisson’s ratio, λρ, and λµ ratio do.Further, as expected, the data points of the oil-saturated sand(green squares) fall on the empirical line for the brine-satu-rated sand. In comparison with the gas sand, the lower imped-ance of the oil-saturated sand (red squares) indicates that ithas higher porosity. The lithology contrast leads one to expecta class 1 AVO anomaly at the top of the gas sand because theimpedance of the gas sand is significantly higher than thatof the shale above it (black squares).

Petrophysical analysis is an important aspect in AVOmodeling because the information can be used to assessreservoir conditions. The relationships between petrophys-ical and seismic rock properties can also be established and

used in reservoir characterization.The petrophysical rock propertiesthat directly relate to seismic rockproperties are volume fractions ofmineralogies, porosity, and water sat-uration (SW). Fluid type, gas/oil ratio(GOR), oil and gas gravity (in API),and brine salinity (in ppm) formanother set of important petrophysi-cal properties. Fluid properties can

be calculated based on derived or in-situ measured pressure (P) and tem-perature (T ). Predicting seismic rockproperties, especially shear-wavevelocity, is important in rock physicsanalysis. The steps to accomplish thisusually include determining in-situconditions, calculating fluid proper-ties, solid properties, and finally cal-culating seismic rock properties. Thelink between seismic rock properties(bulk modulus and shear modulus)and petrophysical rock properties(porosity, fluid type, water saturation,and mineral composition) can be seenin these Gassmann equations:

(1)

(2)

where Kdry = effective bulk modulusof the dry rock; Ksat = effective bulkmodulus of the rock with pore fluid;K0 = bulk modulus of mineral mate-

rial making up the rock; Kfl = effec-tive bulk modulus of the pore fluid; φ = porosity; µdry = effective shearmodulus of the dry rock; and µsat =effective shear modulus of the rockwith pore fluid. In Equation 1, K0, φ,and Kfl are often calculated usingpetrophysical logs. Notice thatEquation 2 indicates that fluid typeand saturation do not affect shearmodulus of rock.

Numerous empirical relationships for seismic rock prop-erties, especially for shear sonic prediction, have been pub-lished. They are useful when either petrophysical propertiesare not available or rock physics theories do not address the

complexity of rock properties. Commonly used empiricalrelationships are the P- and S-velocity linear relation forwater-saturated clastics, also called the mudrock line(Castagna et al., 1985); the velocity-porosity-clay relation(Han, 1986); the critical porosity model for Kdry (Nur et al.,1995); the Greenberg-Castagna relation for mixed lithologies(Greenberg and Castagna, 1992); the velocity relation forclay-sand mixtures (Xu and White, 1996); and the empiri-cal velocity relations for carbonates (e.g., Li and Downton,2000). Often used petrophysical empirical relations are thedensity-velocity relation (Gardner et al., 1974); the sonicand porosity relation (Wyllie et al., 1956); and the resistiv-ity-velocity relation (e.g., Faust, 1953). While the above

296 T HE L EADING EDGE M ARCH 2007



empirical relationships are useful, the empirical relationshipsderived locally are strongly recommended. Further, cali-

bration using available well logs may improve rock prop-erty predictions.

Figure 2 shows a set of petrophysical log curves from aWCSB well. Shale volume (V sh), effective porosity ( φE), and

water saturation (SW) were derivedfrom resistivity, gamma ray (GR),density porosity, neutron porosity,and local petrophysical parameters.The observations from this petro-physical evaluation include:

(1) water saturation is proportional toV sh

(2) gas saturation increases with

decreasing V sh(3) porosity increases with decreas-

ing V sh

Figure 3, based on this petro-physical analysis, shows how the vol-ume of clay influences seismic rockproperties. It is notable that cleansand is most sensitive to gas satura-tion since the distance between theclean sand data points and the empir-ical lines is greatest. The sensitivity ordistance decreases with increasingshale volume. Further, similar to whatis shown in Figure 1, the elastic rockproperties of the oil-saturated sandagain fall on the wet-sand empiricallines. This example shows that petro-physical analysis is useful for under-standing the relationship betweenreservoir quality and seismic rockproperties and, consequently, seismicresponses.

AVO equations and attributes. Toillustrate the relationship betweenrock physical properties and seismicreflections, we will use a two-layerinterface with incident P-waves,

reflected and transmitted P-waves,and converted S-waves as shown inFigure 4. The rock properties in thelayers are P- and S-wave velocities,density, impedances, bulk modulus,shear modulus, Lamé’s parameter,and the attenuation coefficient Q. Theanisotropic properties are P- and S-wave velocity anisotropic parametersε, λ, and δ (Thomsen, 1986). The rockproperties listed above are of greatinterest in prestack seismic inversion.In AVO modeling, they are oftenobtained or derived from well-logdata.

Two main steps of AVO modelingare to generate synthetic gathers,CMPgathers or common-image gath-ers (CIG), by using the exact solutionand then to extract AVO attributes byusing approximate equations. AVOexact solutions are often generated

by using the Zoeppritz equations with ray tracing and aplane-wave assumption, or by using the full wave elasticequation with a finite-difference method.

Aki and Richards (1980) simplified the Zoeppritz equa-tions into a form that can be used to solve for meaningfulreflectivities. The difference between the Zoeppritz exact




solution and the Aki-Richards’ approximation is small whenzero-offset reflectivity << 1. Rearranged forms of the Aki-Richards equation are often used for describing and extract-ing AVO attributes. Commonly used approximate P-waveAVO equations are Shuey (1985) for P-impedance reflectiv-ity Rp and gradient G; Smith and Gidlow (1987) for P- andS-velocity reflectivities RVp and RVs; Fatti et al. (1994) for P-and S-impedance reflectivities RP and RS; and Hilterman andVerm (1995) for P-impedance reflectivity RP and Poisson’sreflectivity R σ. (Table 1)

AVO equations for converted S-wave (e.g., Jin et al.,2000), a vertical transverse isotropic (VTI) medium(Thomsen, 1993), and a horizontal transverse anisotropic(HTI) medium (Ruger, 1996) are listed in Table 2. Using con-verted-wave data, density reflectivity and shear-wave veloc-ity or impedance reflectivity can be calculated. An approachthat solves P- and S-wave reflectivities and density reflec-tivity simultaneously is called simultaneous P-P and P-Sinversion (e.g., Larson, 1999). Including both converted-and P-P wave data in AVO inversion has the potential toobtain a more stable estimation of density reflectivity. Fewefforts have been made to extract AVO attributes in VTImedia because of the difficulties in simplifying anisotropic

AVO equation from usually five to no more than three. Inan HTI medium, Ruger’s two-term equation (for a singleset of fractures) with an elliptical assumption of amplitudeis often used and the attributes to be determined are frac-ture orientation and fracture density.

Commonly used AVO attributes and their relations arelisted in Table 3. Some of these relations are exact and oth-ers approximate. Using Table 3, an AVO attribute can bederived from other AVO attributes by multiplying an oper-ator vector to an attribute vector. For example, fluid factorcan be derived by multiplying operator vector (1, -b/ γ) and

attribute vector (∆V P/V P,∆V S/V S). This yields∆F = (1, -b/ γ)(∆V P/V P, ∆V S/V S) - 1 = ∆V P/V P - (b/ γ) ∆V S/V S, where γ isV P /V S ratio. RP can be calculated from RS by the relationshipRP = b/ γ RS and gradient G can be calculated from RP andRS by G = RP - 2RS. Notice that the fluid indicators, gradi-ent, and fluid factor increase with decreasing V P /V S ratio.

Figure 5 shows the crossplots of AVO attributes calcu-lated from dipole sonic logs. The dashed lines highlight theregional background trends, and the arrows show the direc-tions in which the data points will shift when brine in rockis replaced by hydrocarbons. Figure 5a shows P-wave reflec-tivity RP against S-wave reflectivity. Figure 5b, the crossplotof P-wave reflectivity RP against gradient G, is the one inwhich AVO types for a gas-charged clastic reservoir can bedetermined. In Figure 5d we can see that∆µ/ µ and∆λ/λ havemagnitudes approximately twice RP or RS.

Figure 5c shows the plot of RP against density reflectiv-ity ∆ρ/ρ. This information is useful when density reflectiv-ity is considered robust in discriminating lithologies orreservoirs. In general, the values of density reflectivities aremuch smaller in comparison to RP. To demonstrate this, wederived the equation ∆FP = f (∆V P /V P) - ∆ρ/ρ based onGardner’s relation ρ = dV P

f . Since f has a value of about 1 ⁄ 4,density reflectivity is about one quarter of P-wave velocityreflectivity.

Figure 6 shows a set of synthetic gathers correspondingto class 1–4 AVO responses from the top of a reservoir. Theclass 1 AVO response (higher-impedance gas sand) has a


Figure 3. Influence of shale volume on elastic rock properties. Labels 1–3represent the data points of the gas sand, oil sand, and oil-saturated shalysand. The black, blue, and red lines are the empirical relationships for wetsand, shale, and clean gas sand. The data points of the oil-saturated sandthat fall on the empirical line for wet sand indicate the oil sand and thewet sand have similar rock properties. Based on the distance to the empiri-cal line for wet sand, the sensitivity of the sand responding to gas satura-tion decreases with increasing shale volume.

Figure 4. Reflections and transmissions at an interface. The rock proper-ties above and below the interface determine the energy splitting.

Figure 5. Crossplots of AVO attributes calculated from well logs. Dashedlines indicate regional background trends. Arrows indicate the directionsin which the data points shift when the rocks are saturated by hydrocar-bon. Fluid factor can be obtained using the crossplot of RP and RS by

muting the data points at and near the regional trend. AVO classes can beidentified in the crossplot of RP and gradient. The magnitude of densityreflectivity is much smaller than P-wave reflectivity.



peak at zero-offset that deceases with offset and may changepolarity at far offset. A class 2 AVO response (a near-zeroimpedance contrast) has a weak peak that decreases withoffset with a polarity reversal at far offset, or a weak troughthat increases in amplitude with offset. A class 3 AVO

response (lower-impedance gas sands) has a trough at zerooffset that brightens with offset. Finally, a class 4 AVOresponse (lower-impedance gas sands) has a trough thatdims with offset.

Fluid factor is a hydrocarbon interface indicator. It is theweighted difference between∆V P /V P and∆V S /V S or betweenRP and RS (Table 3). Figure 7 explains how fluid factor andP- and S-wave velocities are related. It shows that the datapoints from the gas sand, with lowered P-wave velocity dueto gas saturation, deviate from the regional backgroundtrend. This results in a trough in fluid factor followed by apeak corresponding to the top and base of the gas zone.Lithologies that do not follow the regional trend produce

false fluid-factor anomalies. For example, reflection from acoal layer can produce a fluid-factor anomaly similar to thatfrom a gas-sand reservoir. A carbonate layer encased bylower-impedance clastic rocks such as shale produces a peakat the top of the carbonate layer followed by a trough at the

base (Figure 7). Ideally, a carbonate line should be used incalculating fluid factor for a porous and hydrocarbon-filledcarbonate reservoir encased by tight carbonates. Further,when multilayer interbedded carbonate and shale are pre-sent, fluid-factor anomalies often become too complex to

interpret. In this circumstance, an understanding of localgeology becomes important to eliminate the false fluid-fac-tor anomalies. In addition, a fluid-factor response mayappear as a single peak or a single trough when seismic rockproperty contrast at the top or base of a reservoir is not strongenough to produce a fluid-factor anomaly.

In the following section we will demonstrate attributeinversion in AVO modeling and analysis by using a syntheticCMP gather. Figure 8 shows the example in which the low-impedance reservoir produces a class 3 AVO anomaly. Wecompared the AVO attributes extracted from the CMP gatherto the attributes directly calculated from the well logs (andconsequently called “reference” AVO attributes). The dif-ferences between the extracted attributes and the referenceattributes are mainly due to offset-dependent tuning, NMOstretch, the angle range used in AVO extraction, and theapproximations introduced in the simplified Zoeppritz equa-tions.

Figure 8a shows angle-limited stacks. The stack with theangle range of 20–30° has a larger amplitude correspond-ing to the top and base of the reservoir. It also shows thatthe full-offset stack differs from the zero-offset stack (RP).This explains why P-reflectivity extracted from CMP gath-ers rather than from a migrated structure (full-offset) stackshould be used as input in impedance inversion. For thistypical example, P-impedance inversion using the full-off-set stack would underestimate the impedance and give anoverly optimistic prediction of porosity.

P- and S-reflectivities extracted using Fatti’s equation are

shown in Figure 8b. The P-reflectivity has a trough corre-sponding to the top of the reservoir since the reservoir haslow P-impedance. However, the S-reflectivity has a peaksince the reservoir has high S-impedance. The fluid factor,as expected, exhibits a trough and a peak corresponding tothe top and base of the reservoir.

The attributes extracted using Smith and Gidlow’s equa-tion are shown in Figure 8c. ∆V P /V P and ∆V S /V S look simi-lar to the RP and RS extracted by Fatti’s equation, telling usthat the velocity-density relations may follow Gardner’srelation.

The gradient extracted using Shuey’s equation is shownin Figure 8d. The negative gradient corresponds to the reflec-tion from the top of the reservoir, indicating an increase inamplitude with incident angle (trough brightening). The

positive gradient corresponds to the base of the reservoir,also indicating an increase in amplitude with incident angle(peak brightening).

Figure 8e shows Poisson’s reflectivity extracted usingHilterman and Verm’s equation. At reservoir level, thePoisson’s reflectivity appears similar to the gradient andfluid factor; that is a trough followed by a peak at the topand base of the reservoir, respectively.

The last panel (Figure 8f) shows λ reflectivity, ∆λ/λ and µ reflectivity, ∆µ/µ extracted using Gray’s equation. Thesereflectivities were scaled by 0.5 because their magnitudesare approximately twice P- or S-reflectivity. Note that ∆λ/λhas a strong trough and peak corresponding to the top and


Figure 6. Class 1–4 AVO responses to gas-charged clastic reservoirs.

Figure 7. Relationship between mudrock line, carbonate line, and fluid factor in P- and S-wave velocity crossplot space. The drop in P-wavevelocity due to gas saturation produces a fluid-factor trough that corre-sponds to the top of the reservoir. In a clastic-carbonate environment, a

fluid-factor peak corresponds to the top of the carbonate.



base of the reservoir.In practice, by examining the extracted AVO attributes

as described above, one may determine which attribute(s)to use in defining a reservoir.

Tuning effects and NMO stretch. Tuning has a significantimpact on seismic resolution and AVO responses. It may

mask or induce an AVO anomaly. In AVO analysis, twotypes of tuning must be considered: tuning due to thicknessof a reservoir and offset-dependent tuning due to amplitudevariation with offset. Both affect the true AVO expressionat an interface. In addition, offset-dependent tuning can befurther affected by the differential traveltime of adjacentreflections. To demonstrate offset-dependent tuning, wederived an equation to calculate the tuning effect on a pri-mary reflection based on offset-dependent Ricker wavelets:

(3)

where A1 and A2 are the zero-offset reflectivities corre-sponding to two adjacent interfaces. Indices p and i repre-sent primary reflection and the interfering reflection,respectively. θ p and θi are the incident angles at the upperand lower interfaces; b is the thickness of the layer, and L0

is the wavelength of dominant frequency. The tuning effectsof a class 1 AVO and a class 3 AVO for a reservoir with V P= 3674 m/s and V S = 2055 m/s were calculated and shownin Figure 9. The lines in red and blue represent the tuning-free AVO responses from the top and base of the reservoir.The AVO responses with ratio b/L0 equal to 1, 1 ⁄ 2, 1 ⁄ 4, and 1 ⁄ 8are shown in Figure 9. A significant change in amplitude


Figure 8. Synthetic CMP gather and commonly used AVO attributes extracted from the gather: (a) angle stacks and full offset stack; (b) P- and S-impedance reflectivity (RP and RS), and fluid factor ∆F ; (c) P- and S-velocity reflectivity (RVp and RVs), and fluid factor ∆F ; (d) gradient G ; (e)

Poisson’s reflectivity, and (f) λ- and µ-reflectivity (R λ and R µ). The subscript r refers to the synthetic traces that were calculated directly from well logs.

Figure 9. Examples of offset (angle) dependent tuning calculated usingEquation 3 and Shuey's two-term AVO equation and Equation 3: (a)class 1 and (b) class 3 AVO responses. The lines in red and blue are AVOresponses without tuning.



can be observed when b/L0 equals 1 ⁄ 4 and 1 ⁄ 8. This illustratesthat AVO analysis performed on reflectivities that sufferfrom tuning will be inaccurate.

To further demonstrate the tuning effect, two syntheticCMP gathers with a class 3 and class 1 AVO anomaly, respec-

tively, were generated from well-log data. The class 3 AVOresulted from an unconsolidated sand reservoir that has anaverage P-wave velocity of 1900 m/s (Figure 10). The fre-quency bandwidth used was 8–14 to 90–100 Hz. Due to thelow impedance of the reservoir, the class 3 AVO anomalycan still be observed when the reservoir thickness is reducedto 5 m. The amplitudes extracted from the top of the reser-voir show a significant tuning effect at a thickness of 1 m.

The class 1 AVO example is from a tight gas-sand reser-voir that has an average velocity of about 4600 m/s (Figure11). The bandwidth used was 10–15 to 100–120 Hz. Whenthe gas sand has a thickness of 18 m, the AVO response hasa peak at near offset that dims with offset and reverses

polarity at far offsets. The AVO response becomes a troughat zero offset and then dims with offset when the thicknesswas reduced to 10 m.

Notice that reducing frequency will result in a similareffect as tuning. Therefore, frequency content in a data setis essential in achieving an accurate AVO analysis.

NMO-corrected gathers may suffer significant differen-tial moveout, which distorts the waveform and bandwidthespecially at far offsets that correspond to large incidentangles. In AVO processing, far-offset traces are often muted

to limit the impact of NMO stretch. This imposes a constraintin the AVO attribute extraction that needs far-offset infor-mation, such as density reflectivity inversion using P-wavedata. Researchers have been attempting to develop meth-ods to combat NMO stretch. Swan (1997) developed amethod by nulling NMO velocity error at a given two-waytraveltime on a CMP gather. In addition, residual moveoutdue to anisotropy may also need to be corrected before AVOattribute extraction.

Effects of noise contamination. In AVO attribute extraction,CMP gathers or common image point gathers obtainedthrough amplitude preserving processing are modeled byAVA equations to solve for AVO attributes. This is often con-ducted using angles of incidence and linear fitting of AVOamplitudes by L2 norm (the least squares) or L1 norm meth-ods. The difference between L2 and L1 norms is that the for-mer minimizes squared deviation while the latter minimizesthe absolute deviation of the data. L1 norm is robust ineliminating coherence noise such as multiples. Ferré et al.(1999) proposed an approach to optimize AVO attributeextraction by using the least median of squares method(Walden, 1991). This method is similar to L1 norm method

but with a progressive converging process to achieve an opti-mized solution.

Signal-to-noise ratio, fold of a CMP gather, and offsetrange all affect the results of an AVO analysis. Offset-depen-dent attributes such as S-reflectivity and gradient are mostinfluenced by these factors. In investigating the effects of

noise, random and/or coherent noise can be added to syn-thetic gathers. A CMP or CIPgather with random and coher-ent noise can also be generated using elastic wave-equationmodeling. Figure 12 shows an example in which randomnoise was added in a CMP gather with a signal-to-noise ratioof 1:0.2. AVO extraction used Fatti’s equation. As expected,noise has little effect on the estimation of P-reflectivity. Theestimated S-reflectivity at reservoir level, however, experi-ences significant distortion.

AVO modeling methodologies. The Zoeppritz equations orthe Aki-Richards equation with ray tracing, and full elasticwave equation with the finite-difference method (FDM) arecommonly used to generate synthetic CMP or CIP gathers.The former has the advantage of being fast and easy for ana-

lyzing primary reflections. Elastic wave-equation modelingcalculates particle displacements in the subsurface andaccounts for direct waves, primary and multiple reflectionwaves, converted waves, head waves, and diffractions. Itovercomes the shortcomings of ray tracing which can breakdown in many cases, such as at edges where the calculatedamplitude is infinite or in shadow zones.

Different AVO modeling methodologies, based on theZoeppritz equations with ray tracing and elastic wave equa-tions, have been developed to take into account issues asso-ciated with data acquisition, processing, and interpretation.The most commonly used methods are single-interface mod-eling, CMP gather modeling, 2D stratigraphic modeling,


Figure 10. Tuning effects on a class 3 AVO anomaly from a low-imped-ance unconsolidated sand reservoir: (left) synthetic gathers with varyingreservoir thickness, and (right) extracted amplitudes at the top of thereservoir. The AVO character is preserved until the thickness reaches to 1m. The CMP gathers were muted at 45°.

Figure 11. Tuning effect of a class 1 AVO anomaly from a gas reservoirwith low porosity and high impedance: (left) synthetic gathers with vary-ing reservoir thickness, and (right) extracted amplitude at the top of thereservoir. The overlying logs on CMP gathers are gamma ray (yellow)and P-impedance (red). The CMP gathers were muted at 45°. The AVOcharacter has a significant change when the thickness reduces to 10 m.



modeling may be used to test strategies in noise attenua-tion. In practice, acoustic wave-equation modeling may beincorporated with elastic wave-equation modeling to iden-tify converted-wave energy.

Finally, we present an example of converted-wave AVO

modeling (Figure 15) which is useful in analyzing multi-component data. In practice, three-component data are sep-arated to P-wave (P-P) data and converted-wave (P-S) datafor AVO processing. Often, converted-wave data have lowerfrequency bandwidth. The P-P data can be incorporatedwith P-S data in AVO attribute inversion. The CMP gathersin Figure 15 were generated using P-P and P-S Zoeppritzequations with ray tracing, where the traveltime of the con-verted-wave gather was corrected to P-wave traveltime.They were processed, and P- and S-reflectivities wereinverted from P-Pgather using Fatti’s equation, and S-waveand density reflectivities were inverted from P-S gatherusing a converted-wave equation (Table 2). In this example,

the frequency bandwidths of 10–14 to 110–120 Hz and 10–14to 55–60 Hz were used in the P-P and P-S AVO modeling,respectively.

Two-dimensional stratigraphic modeling. 2D stratigraphicmodeling takes into account lateral variations which can bedue to structure, reservoir thickness, porosity, lithology,fluid type, and fluid saturation. The velocity and densitymodels may come from seismic interpretation or constructedusing well logs as control points. Using the velocity and den-sity models as input, CMP gathers of a 2D line can be gen-

erated and then processed.Figure 16 shows 2D stratigraphic modeling based on the

wells from a play in the Gulf of Mexico, where the gas zoneshave low P- and S-impedances. It can be seen that P-reflec-tivity section does not provide definitive information on thegas-saturated reservoirs, but the fluid factor does.

A second example is from a WCSB gas play with wellcontrol at four locations (Figure 17). The target zone variesfrom silt sand to gas sand, and then to brine-saturated sand.In this study, we inverted P- and S-reflectivities from thesynthetic CMP gathers, and then performed P- and S-imped-ance inversion and calculated λρ and µρ. Different sensitivityof the inverted elastic rock properties in response to the gas


Figure 14. Single CMP gather AVO modeling for a well in the WesternCanadian Sedimentary Basin: (a) Zoeppritz equation with ray tracing,and (b) elastic wave-equation modeling with finite-difference method. The

former generates primary-only reflections, and the latter generates pri-mary reflections, multiples, and other modes of reflections such as con-verted waves. The elastic modeling shows the distortions in some of the

primary reflections. The interbedded multiples may manifest as real reflec-tions.

Figure 15. P-wave and converted-wave AVO modeling using Zoeppritzequations with ray tracing; the two-way traveltime of the converted-wave

gather was corrected to P-wave time. The attributes shown are the full-offset stacks of the P-P and P-S gathers, inverted P- and S-reflectivities(Rppp and Rpps) from the P-P gather, and inverted S-reflectivity anddensity reflectivity (Rpss and Rps ρ from the P-S gather.

Figure 16. 2D stratigraphic AVO modeling for a play with several low-impedance gas zones in the Gulf of Mexico: (a) P-reflectivity brightens atthe gas well location but does not provide definite information for definingthe reservoirs; and (b) fluid-factor stack shows anomalies with a troughand a peak corresponding to the top and base of each reservoir. The overly-ing logs P-impedance on P-reflectivity, and V P /V S ratio on fluid-factorstack.

Figure 17. Inverted elastic rock property sections from a 2D stratigraphic

AVO modeling for a gas-charged reservoir: (a) P-impedance, (b) S-imped-ance, (c) λρ, and (d) λρ−µρ. P- and S-impedances do not provide defin-ing information on the reservoir. λρ and λρ−µρ, however, show a clearanomaly.




Figure 18. Elastic wave-equation modeling with

finite-difference method for a structurally complex play in MackenzieDelta, Canada: (a) inputP-wave velocity model,(b) a shot gather, (c)structure stack, and (d)

prestack depth-migratedsection. The location of the shot gather is indi-

cated in (c) and (d).

reservoir is evident: The P- and S-impedances are unable to

delineate the reservoir, but the λρ and λρ−µρ attributesshow a clear gas anomaly.

Two-dimensional elastic wave-equation modeling. In struc-turally complex areas, seismic imaging alone can not com-pletely describe a reservoir. Therefore, there is great interestin using 2D or 3D elastic wave-equation modeling to gen-erate synthetic data for AVO analysis. In comparison to ray-tracing methods, elastic wave-equation modeling generatesmore realistic synthetic data. Recent advances in comput-ing power make single-shot gather modeling trivial and 2Dmodeling practical.

Figure 18 shows the application of 2D full wave-elasticmodeling with finite-difference method to a structurally

complex area in the Mackenzie Delta, Canada. The purpose

was to investigate whether the reservoir can be character-ized by AVO attributes extracted from prestack depth migra-tion. In this study, the velocity models and density modelwere constructed from well logs. Figure 18a shows that anunconformity is between 1750 m and 2700 m, and the gasreservoir is under the unconformity and sealed by a fault.Figures 18b–d display a shot gather, the structure stack, andthe prestack depth-migrated section. The results from theAVO analysis are shown in Figure 19. Figures 19a and 19billustrate the conversion of a common image gather fromAVO domain to AVA domain. Figure 19c shows the fluid-factor section based on P- and S-reflectivity extracted fromthe common image point gathers, in which the reservoir was

Figure 19. Commonimage gathers and fluid-

factor section corre-sponding to Figure 18:(a) common image

gather in offset domain,(b) common image

gather in incident angledomain, and (c) fluid- factor section. Fluid- factor section wascalculated based on theP- and S-reflectivitiesextracted from the com-mon image gathers.



clearly defined. This example demonstrates that, for a struc-ture play, an a-priori study using elastic wave-equationmodeling and prestack depth migration helps determinewhether AVO analysis is feasible.

AVO modeling applications. AVO modeling contributes sig-nificantly to prestack data processing and interpretation.Inappropriate processing may be detected by examining

AVO responses in CMP gathers or by analyzing invertedAVO attributes. An understanding of local petrophysics,rock physics, and geology provides constraints for QC inAVO processing and interpretation. Ideally, AVO process-ing and interpretation should be conducted in parallel withAVO modeling. As a result, uncertainty and risk can bereduced.

Data processing. Calibration on CMP gathers and AVO


Figure 20. Seismic CMP gather ties to a synthetic CMP gather: (a) comparison of CMP gathers, and (b) comparison of class 1 AVO responses from the top of the reservoir. The overlying logs are P-impedance (yellow)and gamma ray (black).

Figure 21. Effect of frequency bandwidth in AVO analysis: (a) synthetic gather with the frequency bandwidth of the 2D data ties to a CMP gather from the 3D data, and (b) synthetic gather with the frequency bandwidthof the 3D data ties to a CMP gather from the 3D data. The overlying logsare P-impedance (red) and gamma ray (black).

Figure 22. Comparisonof inverted elastic rock

properties and well-logdata in a carbonate

play: (a) and (b)inverted elastic rock

properties from seismic,where the gas-chargeddolomite porosity zoneis highlighted in blacksquares; and (c) and (d)elastic rock properties

from well logs, where a gas-charged dolomite porosity zone is high-lighted in red squares,and a wet zone is high-lighted in green squares.The overlain empiricallines are for wet sand(blue), shale (black),clean gas sand (red),limestone (black dashedline), and dolomite(green dashed line). Theinverted elastic rock

properties in (a) and (b)indicate that the reser-voir is gas-charged with

porosity similar to the porous dolomite high-lighted in green squaresin (c) and (d).



attributes by AVO modeling is useful in optimizing AVO pro-cessing. It is often implemented by using well logs, syntheticgathers, walkway VSPdata, and known relationships betweenAVO attributes or between seismic rock properties. Calibrationcan be performed at a CMP location, or globally, on a data setto answer such questions as these:

• Have the CMP gathers been properly processed with anamplitude-preserving processing workflow?

• Are phase, tuning, signal-to-noise ratio, and frequency bandwidth influencing the AVO solutions?

Tying synthetic CMP gather(s) to recorded seismic dataoften gives a quick insight into data quality and to type ofAVO response. One may perturb the well logs to representpossible reservoir conditions to examine the variation of seis-mic responses. For example, gas substitution may be per-formed on a wet well to examine the gas effects. Otherparameters often perturbed in AVO modeling are porosity,reservoir thickness, lithology, and frequency and angle to beused for AVO attribute extraction.

Figure 20a shows an example in which a synthetic CMPgather is tied to a recorded CMP gather from a 2D survey. Bycomparing the AVO responses from the top of the gas reser-

voir, a class 1 AVO anomaly with polarity reversal at the faroffsets is confirmed. This is further validated quantitatively

by the amplitudes extracted from both the seismic and the syn-thetic gathers (Figure 20b). In this study, 3D data were initiallyconsidered for AVO processing. Further study, however,revealed that the 3D data have a frequency content below theresolution for delineating the reservoir. Figure 21a shows thata synthetic CMP gather generated using the bandwidth fromthe 2D data ties to the 3D data. It is evident that the frequencycontent in the 3D data is significantly lower than that in the2D data. In Figure 21b, a good well-tie was reached when thefrequency bandwidth of the 3D data was used in the syntheticCMP gather.

Calibration can be applied to other AVO attributes suchas S-reflectivity, gradient, and fluid factor. It can also be applied

by plotting P-reflectivity against S-reflectivity, and P-imped-ance against S-impedance. Figure 22 shows an example usingwell logs to calibrate inverted elastic rock properties in a gas-charged dolomite play. In Figures 22a–b, the inverted elasticrock properties from the zone of interest are highlighted in

black squares. The data points shift toward low λρvalues andlow λ/µ ratios, indicating effects of gas. To calibrate this, dipolesonic logs from the same area were plotted (Figures 22c–d).In this well, the data from the gas-charged dolomite (redsquares) have lower porosity and the data from a brine-satu-rated porous dolomite (green squares) have higher porosity.The comparison between seismic data and well-log data con-firms that the reservoir is gas-charged because it has low val-ues of λρ and λ/µ. Also, it indicates that the reservoir hasporosity similar to the data points highlighted by the greensquares in Figures 22c–d.

Global calibration using AVO modeling is an approach toconduct QC for a prestack seismic data set. For example, AVOwithin a time window or from a specific reflection can be com-pared and calibrated with the results from AVO modeling. Thistype of exercise can tell whether the data were processed byan amplitude-preserving workflow.

Phase analysis is an important aspect in AVO processingQC. Non-zero-phase data result in inaccurate AVO attributesand inverted elastic rock properties. Phase analysis is usuallyaccomplished by crosscorrelation using zero-offset syntheticsand migrated stack. It produces wavelets accompanied byphase information. Well ties using CMP gathers may removesome ambiguity because of additional information from AVOresponses. For example, a typical AVO response or an AVOanomaly can be criteria to tie well to seismic. Also, improvedwavelet(s) may be obtained by using extracted P-wave reflec-tivity (RP) instead of migrated stack because the former areconsidered zero-offset data. Furthermore, AVO attributes suchas fluid factor may be useful in phase analysis. For example,


Figure 23. Elastic wave-equation modeling using

finite-difference method for a dolomite reservoir inthe Wabamun Formationin the Western CanadianSedimentary Basin: (a)with reservoir, (b) with-out reservoir, and (c) thedifference between (a) and(b). The overlying logsare gamma ray (black), P-

impedance (red), and S-impedance (yellow). Inthis play, multiple conta-mination at reservoirlevel is a crucial issue indetermining whether abright spot in amplitudeindicates porosity or theresult of an interbeddedmultiple. The elasticmodeling shows signifi-cant difference in ampli-tudes for the cases withand without the reservoir.



in a clastic environment, a peak followed by a tough in afluid-factor section may indicate that the seismic data havereversed polarity.

Data interpretation. AVO modeling has an important rolein assisting interpretation on CMP gathers, AVO attributes,and inverted elastic rock properties. It helps validate AVOresponses and links seismic expression to known reservoir con-ditions. It is able to integrate petrophysics, rock physics, and

geology with seismic data. As a result, confidence in inter-pretation is increased, and the risk in drilling is reduced.Tuning effects may invoke or mask an AVO anomaly as

described earlier in this paper. AVO modeling using variedfrequency bandwidth or reservoir thickness may yield theanswers. Complex lithologies may produce AVO anomaliesthat can be investigated by AVO modeling as well. For exam-ple, AVO responses from tight sand may manifest as class 1AVO anomalies and show brightening of amplitude in amigrated stack. This type of AVO anomaly may be excluded

by high P- and/or S- impedances. Coal, carbonate, and thelithologies that do not follow the mudrock trend may producefluid-factor anomalies. It is possible to exclude these anom-alies through integrating geologic information. Further, highclay content in rock results in low gas saturation, and this typeof partial gas saturation may be distinguishable because theincrease in clay content results in an increased V P /V S ratio, andconsequently changes the AVO responses.

In data interpretation, synthetic CMPor CIPgathers should be processed through the same data processing sequence used

for field data. The synthetic gathers, AVO attributes andinverted elastic rock properties can thus be directly comparedto their counterparts from seismic data.

As a special case study, Figure 23 shows an example thatuses elastic wave-equation modeling to investigate the inter-ference between primary reflections, multiples, and convertedenergy at the Wabamun dolomite porosity in the WCSB. Thisis a classic case in which multiples were often misinterpreted

as porosity on migrated stack. We were interested in know-ing how multiples and converted energy interfere with theprimary reflections at the reservoir level. AVO modeling wasconducted with and without porosity. Figure 23 shows thatthere is a significant difference in seismic responses betweenthese two cases. Further, the information from this studyhelped optimize noise attenuation and amplitude recovery.

The final example is from a study on a carbonate reser-voir in the WCSB (Figure 24). AVO modeling shows that thegas-charged dolomite reservoir that has an average porosityof about 14% produces a class 3 AVO anomaly. This agreeswith the AVO response observed in the CMPgather at the pro-ducing well. In contrast, a completely different seismicresponse was observed at the tight well locations. In this case,the information obtained from AVO modeling validated theseismic responses.

Conclusions. We have demonstrated the importance andpractical aspects of AVO modeling in prestack seismic pro-cessing and interpretation. We have also presented and dis-


Figure 24. Interpretation of a gas-charged dolomite reservoir with an average porosity of 14% in the Western Canadian Sedimentary Basin: (a) syn-thetic CMP gather, and (b) stack section and CMP gathers at three wells. The gas-charged dolomite porosity produces a class 3 AVO anomaly (peak) atthe base of the reservoir. This was confirmed by AVO modeling.



cussed the AVO modeling methodologies—single-interfacemodeling, single-gather modeling, 2D stratigraphic model-ing, and 2D full wave elastic modeling. Except for the AVOmodeling using Zoeppritz equation with ray tracing, fullwave elastic-wave equation modeling helps AVO analysisin structurally complex areas, and converted-wave AVOmodeling provides useful information on analyzing con-verted-wave data. We have demonstrated that it is essentialto examine seismic rock properties and their contrasts in orderto understand the sensitivity of seismic rock properties

responding to fluid. The information from analyzing petro-physical and seismic rock properties is useful in predictingAVO responses. In AVO modeling, effects of reservoir thick-ness, sensitivity of rock properties responding to fluid andlithology are important and need to be investigated. Specialattention needs to be paid to seismic resolution as it altersAVO responses. Offset-dependent tuning, noise contamina-tion, NMO stretch, limitations in seismic data acquisition,and structural effects need to be studied. Furthermore, AVOmodeling is an exercise in multidisciplinary integration ofpetrophysics, rock physics, seismic, geology, and petroleumengineering. It provides important information on reservoircharacterization and risk reduction in hydrocarbon explo-ration. Finally, AVO modeling of an anisotropic medium,especially an HTI medium, faces challenges due to the dif-ficulties in obtaining accurate anisotropic input.

Suggested reading. Quantitative Seismology by Aki and Richards(Freeman, 1980). “Thin-bed AVO effects” by Bakke and Ursin(Geophysical Prospecting, 1998). “Relationships between com-pressional-wave and shear-wave velocities in clastic silicaterocks” by Castagna et al. (GEOPHYSICS, 1985). “Framework forAVO gradient and intercept interpretation” by Castagna et al.(GEOPHYSICS, 1998). “Seismic modeling” by Carcione et al.(GEOPHYSICS, 2002). “Thin bed AVO: The effect of NMO and NMOcorrection on reflectivity sequences” by Castoro et al. (SEG 1998Expanded Abstracts). “Amplitude responses of thin beds:Sinusoidal approximation versus Ricker approximation” byChung and Lawton (GEOPHYSICS, 1995). “AVO detectability

against tuning and stretching artifacts” by Dong (GEOPHYSICS,1999). “High-resolution AVO analysis before NMO” by Downtonand Lines (SEG 2003 Expanded Abstracts). “Detection of gas insandstone reservoirs using AVO analysis: A3D seismic case his-tory using the Geostack technique” by Fatti et al. (GEOPHYSICS,1994). “A velocity function, including lithologic variation” byFaust (GEOPHYSICS, 1953). “Improving AVO attributes with robustregression and quality control” by Ferré et al. (SEG 1999Expanded

Abstracts). “Formation velocity and density—The diagnostic basis for stratigraphic traps” by Gardner et al. (GEOPHYSICS,1974). “Bridging the gap: Using AVO to detect changes in fun-

damental elastic constants” by Gray et al. (SEG 1999 Expanded Abstracts). “Shear-wave velocity estimation in porous rocks:Theoretical formulation, preliminary verification and applica-tions” by Greenberg and Castagna (Geophysical Prospecting, 1992).Effect of Porosity and Clay Content on Acoustic Properties ofSandstones and Unconsolidated Sediments by Han (PhD dis-sertation, Stanford University, 1986). “Lithology color-codedseismic sections: The calibration of AVO cross-plotting to rockproperties” by Hilterman and Verm (TLE, 1995). “Shear-wavevelocity and density estimation from PS-wave AVO analysis:

Application to an OBS dataset from the North Sea” by Jin et al.(GEOPHYSICS, 2000). “Theoretical reflection seismograms for elas-tic media by Kennett” (Geophysical Prospecting, 1979). AVOInversion by Simultaneous PP and PS Inversion by Larson(Master’s thesis, University of Calgary, 1999). “Application ofamplitude versus offset in carbonate reservoirs: Re-examiningthe potential” by Li and Downton (SEG 2000 Expanded Abstracts).“An empirical method for estimation of anisotropic parametersin clastic rocks” by Li (TLE, 2006). “Amplitude and AVOresponses of a single thin bed” by Liu and Schmitt (GEOPHYSICS,2003). “Critical porosity: The key to relating physical propertiesto porosity in rocks” by Nur et al. (SEG 1995 Expanded Abstracts).Reflection Coefficients and Azimuthal AVO Analysis inAnisotropic Media by Ruger (PhD dissertation, Colorado Schoolof Mines, 1996). “Amplitude-versus-offset variations in gas

sands” by Rutherford and Williams (GEOPHYSICS, 1989). “A sim-plification of the Zoeppritz equations” by Shuey (GEOPHYSICS,1985). “Weighted stacking for rock property estimation anddetection of gas” by Smith and Gidlow (Geophysical Prospecting,1987). “Removal of offset-dependent tuning in AVO analysis”

by Swan (SEG 1997 Expanded Abstracts). “Velocity from ampli-tude variations with offset” by Swan (GEOPHYSICS, 2001). “Weakelastic anisotropy” by Thomsen (GEOPHYSICS, 1986). “Weakanisotropic reflections” by Thomsen (in Offset DependentReflectivity—Theory and Practice of AVO Analysis, Investigationsin Geophysics Vol. 8, SEG, 1993). “Making AVO sections morerobust” by Walden (Geophysical Prospecting, 1991). “Elastic wavevelocities in heterogeneous and porous media” by Wyllie et al.(GEOPHYSICS, 1956). “A physical model for shear-wave velocity

prediction” by Xu and White (Geophysical Prospecting, 1996).T L E

Acknowledgments: Jonathan Downton and Yong Xu were employed byParadigm when this study was done. The authors thank Larry Lines forreviewing that improved the quality of this paper. They also acknowledgeKeith Young, Michael West, Jan Dewar, Paul Hewitt, Bob Somerville,

Huimin Guan, and Luiz Loures for enlightening discussion and assis-tance.

Corresponding author: [email protected]

Practical Aspects of AVO Modeling.li.Downton.xu.Leadingedge

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