11 Reservoir Geophysics

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11 Reservoir Geophysics Introduction Elastic Waves and Rock Properties Seismic Resolution Vertical Resolution LateralResolution Analysis of Amplitude Variation with Offset Reflection and Refraction Reflector Curvature AVO Equations Processing Sequence for AVO Analysis Derivation of AVO Attributes by Prestack AmplitudeInversion Interpretation of AVO Attributes 3-D AVO Analysis Acoustic Impedance Estimation SyntheticSonic Logs Processing Sequence for Acoustic Impedance Estimation Derivation of Acoustic Impedance Attribute3-D Acoustic Impedance Estimation Instantaneous Attributes Vertical Seismic Profiling VSP AcquisitionGeometry Processing of VSP Data VSP-CDP Transform 4-D Seismic Method Processing of 4-D SeismicData Seismic Reservoir Monitoring 4-C Seismic Method Recording of 4-C Seismic Data Gaisers CouplingAnalysis of Geophone Data Processing ofP P Data Rotation of Horizontal Geophone Components Common-

Conversion-Point Binning

Velocity Analysis ofP SData

Dip-Moveout Correction ofP SData

Migration ofP SData Seismic Anisotropy Anisotropic Velocity Analysis Anisotropic Dip-Moveout Correction AnisotropicMigration Effect of Anisotropy on AVO Shear-Wave Splitting in Anisotropic Media Exercises Appendix L:Mathematical Foundation of Elastic Wave Propagation Stress-Strain Relation Elastic Wave Equation Seismic Wave Types Body Waves and Surface Waves Wave Propagation Phenomena Diffraction, Reflection,and Refraction The Zoeppritz Equations Prestack Amplitude Inversion References

11.0 INTRODUCTION

In Chapter 8, we reviewed the two-dimensional (2-D)

and three-dimensional (3-D), post- and prestack migra-tion strategies for imaging the earths interior in depth.In Chapter 9, we learned traveltime inversion techniquesfor estimating a structural model of the earth that isneeded to obtain an accurate image in depth. In Chap-ter 10,structural inversioncase studies for earth model-ing and imaging in depth were presented. By structuralinversion, we define the geometryof the reservoir unit,and the overlying and underlying depositional units.Traveltimes, however, are only one of the two compo-

nents of recorded seismic wavefields; amplitudes are the

other component.In this chapter, we shall turn our attention to in-version of reflection amplitudes to infer petrophysicalproperties within the depositional unit associated withthe reservoir rocks. The petrophysical properties in-clude porosity, permeability, pore pressure, and fluidsaturation. Specifically, we shall discuss prestack am-plitude inversion to derive the amplitude variation withoffset (AVO) attributes (Section 11.2) and poststackamplitude inversion to estimate an acoustic impedance


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1794 Seismic Data Analysis

model of the earth (Section 11.3). The processes of es-timating the acoustic impedance and AVO attributesby way of inversion of amplitudes may be appropri-ately referred to as stratigraphic inversion. Our goalultimately is reservoir characterization based on struc-

tural and stratigraphic inversion of seismic data withcalibration to well data.We appropriately begin this chapter by investigat-

ing the resolution we can achieve from seismic data indefining vertical and lateral variations in the geometryof the reservoir unit. Resolution is the ability to sep-arate two events that are very close together (Section11.1). There are two aspects of seismic resolution: ver-tical (or temporal) and lateral (or spatial). Seismic res-olution becomes especially important in mapping smallstructural features, such as subtle sealing faults, and indelineating thin stratigraphic features that may havelimited areal extent.

Reservoir characterization involves calibration ofthe results of analysis of surface seismic data bothfrom structural and stratigraphic inversion, to welldata. One category of well data includes various typesof logs recorded in the borehole. Logs that are mostrelevant to seismic data are sonic, shear, and density.Another category of well data is a vertical seismic pro-file (VSP) (Section 11.4).

Just as we can seismically characterizea reservoir,we also can seismically monitor its depletion. This isachieved by recording 3-D seismic data over the fieldthat is being developed and produced at appropriatetime intervals and detecting changes in the reservoir

conditions; specifically, changes in petrophysical prop-erties of the reservoir rocks, such as fluid saturation andpore pressure. Specifically, such changes may be relatedto changes in the seismic amplitudes from one 3-D sur-vey to the next. Time-lapse 3-D seismic monitoring ofreservoirs is referred to as the 4-D seismic method (Sec-tion 11.5). The fourth dimension represents the calendartime over which the reservoir is being monitored.

Some reservoirs can be better identified and mon-itored by using shear-wave data. For instance, acousticimpedance contrast at the top-reservoir boundary maybe too small to detect, whereas shear-wave impedance

contrast may be sufficiently large to detect. By record-ing multicomponent data at the ocean bottom, P-waveand S-wave images can be derived. Commonly, fourdata components are recorded the pressure wavefield,and inline, crossline, and vertical components of particlevelocity. Thus, the multicomponent seismic data record-ing and analysis is often referred to as the 4-C seismicmethod (Section 11.6).

This chapter ends with a brief discussion onanisotropy. While exploration seismology at large isbased on the assumption of an isotropic medium, the

earth in reality is anisotropic. This means that elasticproperties of the earth vary from one recording direc-tion to another. Seismic anisotropy often is associatedwith directional variations in velocities. For instance,in a vertically fractured limestone reservoir, velocity in

the fracture direction is lower than velocity in the direc-tion perpendicular to the plane of fracturing (azimuthalanisotropy). Another directional variation of velocitiesinvolves horizontal layering and fracturing of rocks par-allel to the layering. In this case, velocity in the hor-izontal direction is higher than the vertical direction(transverse isotropy). In Section 11.7, we shall reviewseismic anisotropy in relation to velocity analysis, mi-gration, DMO correction, and AVO analysis.

Elastic Waves and Rock Properties

Seismic waves induce elastic deformation along thepropagation path in the subsurface. The term elasticrefers to the type of deformation that vanishes upon re-moval of the stress which has caused it. To study seismicamplitudes and thus investigate their use in explorationseismology, it is imperative that we review wave prop-agation in elastic solids. This gives us the opportunityto appreciate the underlying assumptions in estimatingacoustic impedance and AVO attributes.

A summary of the elastic wave propagation theoryis provided in Appendix L. To facilitate the forthcomingdiscussion on the link between elastic waves and rock

properties, first, we summarize the definitions of elasticwave theory that should always be remembered.

(a) Stress is force per unit area. Imagine a parti-cle represented by an infinitesimally small volumearound a point within a solid body with dimensions(dx, dy,dz) as depicted in Figure L-1. The stressacting upon one of the surfaces, say dy dz, canin general be at some arbitrary direction. It can,however, be decomposed into three components one which is normal to the surface, and two whichare tangential to the surface. The normal compo-

nent of the stress is called thenormal stress

and thetangential components are called the shear stress.A normal stress component istensionalif it is posi-tive and compressionalif it is negative. Fluids can-not support shear stress. In a fluid medium, onlyone independent stress component exists the hy-drostatic pressure.

(b) Strain is deformation measured as the fractionalchange in dimension or volume induced by stress.Strain is a dimensionless quantity. The stress fieldaway from the typical seismic source is so small


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that it does not cause any permanent deforma-tion on rock particles along the propagation path.Hence, the strain induced by seismic waves is verysmall, usually around 106. Consider two points,Pand Q, within a solid body as indicated in Figure

L-2. Subject to a stress field, the solid is deformedin some manner and the particles at points P andQ are displaced to new locations P and Q . Con-sider deformations of specific types illustrated inFigureL-3. The simplest deformation is the exten-sion in one direction as a result of a tensional stress(Figure L-3a). The fractional change in length ina given direction is defined as the principal straincomponent. A positive strain refers to an exten-sionand a negative strain refers to a contraction.Other types of deformation are caused by shearing(FigureL-3b), rotation (Figure L-3c), and a com-bination of the two (Figure L-3d). These angular

deformations are called shear strainssince they re-sult in a shearing of the volume around a pointwithin a solid body (Figure L-3b).

(c) Elastic deformationis a deformation in solid bodiesthat vanishes once the stress is released.

(d) Hookes lawfor elastic deformations states that thestrain at any point is directly proportional to thestresses applied at that point.

(e) Elastic moduliare material constants that describestress-strain relations:(1) Bulk modulusis the ratio of hydrostatic stressto volumetric strain; hence, it is a measure of in-compressibility.

(2) Modulus of rigidity is the ratio of shear stressto shear strain; hence, it is a measure of resistanceto shear stress.(3)Youngs modulusis the ratio of the longitudinalstress to the longitudinal strain associated with acylindrical rod that is subjected to a longitudinalextension in the axial direction. Since strain is adimensionless quantity, Youngs modulus has thedimensions of stress.(4) Poissons ratio is the ratio of the lateral con-traction to longitudinal extension associated witha cylindrical rod that is subjected to a longitudi-

nal extension in the axial direction. Since strain isa dimensionless quantity, Poissons ratio is a purenumber.

(f) Seismic wavesare elastic waves that propagate inthe earth.

(g) P-waves(or equvalently, compressional waves, lon-gitudinal waves, or diltatational waves) are waveswith particle motion in the direction of wave prop-agation.

(h) S-waves (or equivalently, shear waves, transversewaves, or rotational waves) are waves with particle

motion in the direction perpendicular to the direc-tion of wave propagation.

(i) Reflection is the wavefield phenomenon associatedwith the fraction of incident wave energy that isreturned from an interface that separates two layers

with different elastic moduli.(j) Refractionis the the wavefield phenomenon associ-ated with the fraction of incident wave energy thatis transmitted into the next layer.

(k) Diffractionis the wavefield phenomenon associatedwith energy that propagates outward from a sharpdiscontinuity in the subsurface.

In exploration seismology, we are primarily inter-ested in compressional and shear waves that travelthrough the interior of solid layers, and thus are charac-terized as body waves. Whereas in earthquake seismol-ogy, we also make use of Love and Rayleigh waves which

travel along layer boundaries, and thus are character-ized as surface waves.Both body waves and surface waves are different

forms of elastic waves, each associated with a specifictype of particle motion. In the case of compressionalwaves, the particle motion induced by a compressionalstress is in the direction of wave propagation. The com-pressional stress causes a change in the particle dimen-sion or volume. The more the rock resists to the com-pressional stress, the higher the compressional wave ve-locity. In the case of shear waves, the particle motioninduced by a shear stress is in the direction perpendic-ular to the direction of wave propagation. The shear

stress does not cause a change in the particle dimensionor volume; instead, it changes particle shape. The morethe rock resists shear stress, the higher the shear wavevelocity. Under the assumption that both wave typesare elastic, whatever the change induced by the wavemotion the elastic deformation in particle shape, di-mension or volume, vanishes once the wave motion onthe particle vanishes and is propagated onto the neigh-boring particle.

Figure 11.0-1 outlines the interrelationships be-tween the various elastic parameters. Starting withYoungs modulus the ratio of principal stress to prin-cipal strain, and Poissons ratio the ratio of shear

strain to principal strain, Lames constants and are defined. The Lame constant is indeed the modulusof rigidity and the Lame constant = (2/3), where is the bulk modulus (Section L.3). Then, the twowave velocities compressional (P-waves) and shear(S-waves), are derived in terms of the Lame constants,or the bulk modulus and modulus of rigidity, and den-sity.

From the definitions of the P- and S-wave velocitiesin Figure 11.0-1, note that both are inversely propor-tional to density . At first thought, this means that


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FIG. 11.0-1. The relationships between the various elastic parameters for isotropic solids.


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FIG. 11.0-2. Variations in P-wave velocity with various rock types with different densities (Gardner et al., 1974).


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FIG. 11.0-3.Crossplot of P-wave slowness versus S-wave slowness based on laboratory measurements wtih various rocktypes(Pickett, 1963).

the lower the rock density the higher the wave veloc-ity. A good example is halite which has low density (1.8gr/cm3) and high P-wave velocity (4500 m/s). In mostcases, however, the higher the density the higher thevelocity (Figure11.0-2). This is because an increase indensity usually is accompanied by an increase in theability of the rock to resist compressional and shearstresses. So an increase in density usually implies anincrease in bulk modulus and modulus of rigidity. Re-turning to the expressions for the P- and S-wave ve-locities in Figure11.0-1, note that the greater the bulkmodulus or the modulus of rigidity, the higher the veloc-

ity. Based on field and laboratory measurements, Gard-ner et al. (1974) established an empirical relationshipbetween density and P-wave velocity . Known asGardners formula for density, this relationship givenby = c0.25, where c is a constant that depends onthe rock type, is useful to estimate density from veloc-ity when the former is unknown. With the exception ofanhydrites, most rock types sandstones, shales, andcarbonates, tend to obey Gardners equation for den-sity.

In Section 3.0, a brief review of the results of someof the key laboratory experiments on seismic veloci-ties was made. For a given lithologic composition, seis-mic velocities in rocks are influenced by porosity, poreshape, pore pressure, pore fluid saturation, confiningpressure, and temperature. It is generally accepted thatconfining pressure, and thus the depth of burial, has themost profound effect on seismic velocities (Figure 3.0-3). For instance, the P-wave velocities for clastics canvary from 2 km/s at the surface up to 5 km/s and forcarbonates from 3 km/s at the surface up to 6 km/s atdepths greater than 5 km.

Because of the large variations in P-wave veloci-ties caused by all these factors, P-wave velocity alone isnot adequate to infer the lithology, unambiguously. Theambiguity in lithologic identification can be resolved tosome extent if we have the additional knowledge ofS-wave velocities. Here, we examine the ratio of the S-wave velocity to the P-wave velocity, /, which onlydepends on Poissons ratio (Figure 11.0-1). In someinstances, we refer to the inverse ratio /. The higherthe Poissons ratio, the higher the velocity ratio /.


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FIG. 11.0-4.(a) Crossplot ofP-wave velocity versusS-wave velocity derived from full-waveform sonic logs using rock sampleswith different lithologies SS: sandstone, SH: shale and LS: limestone; (b) crossplot of the velocity ratio versus the P-wavevelocity using the same sample points as in (a) (Miller and Stewart, 1999).


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FIG. 11.0-5.Variation of the velocity ratio with respect to (a) shale content and (b) clay content (Miller and Stewart, 1999).


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This relationship is supported by the physical meaningof Poissons ratio the ratio of shear strain to principlestrain. A way to describe the physical meaning of Pois-sons ratio is to consider a metal rod that is subject toan extensional strain. As the rod is stretched, its length

increases while its thickness decreases. Hence, the lessrigid the rock, the higher the Poissons ratio. This is ex-actly what is implied by the expression in Figure11.0-1that relates the modulus of rigidity to Poissons ratio. Unconsolidated sediments or fluid-saturated reservoirrocks have low rigidity, hence high Poissons ratio andhigh velocity ratio/. Here is the first encounter witha direct hydrocarbon indicator the P- toS-wave ve-locity ratio. Ostrander (1984) was the first to publishthe link between a change in Poissons ratio and changein reflection amplitude as a function of offset.

Aside from the direct measurement ofS-wave ve-locities down the borehole, there are three indirect ways

to estimate the S-wave velocities. The first approachis to perform prestack amplitude inversion to estimatethe P- and S-wave reflectivities and thus compute thecorresponding acoustic impedances (Section 11.2). Thesecond approach is to record multicomponent seismicdata and estimate theS-wave velocities from theP-to-Sconverted-wave component (Section 11.6). The thirdapproach is to generate and recordS-waves themselves.

Figure 11.0-3 shows a plot of the S-wave slow-ness (inverse of theS-wave velocity) versus theP-waveslowness (inverse of the P-wave velocity) based on lob-oratory measurements (Pickett, 1963). Figure 11.0-4ashows a plot of the P-wave velocity to S-wave velocity

based on full-waveform sonic logs from a producing oilfield (Miller and Stewart, 1999). The key observationmade from these results is that a lithologic compositionmay be associated with a reasonably distinctive veloc-ity ratio /. The shale and limestone samples fall ona linear trend that corresponds to a velocity ratio of1.9, whereas the dolomite samples have a velocity ratioof 1.8. The sandstone samples have a range of velocityratio of 1.6 to 1.7. Lithologic distinction sometimes ismore successful with a crossplot ofP-wave to S-wavevelocity ratio versus the P-wave velocity itself (Millerand Stewart, 1999). This is illustrated in Figure 11.0-4b which shows the same sample points as in Figure

11.0-4a.Effect of shale and clay content on the velocity ratio/is an important factor in lithologic identification.Field and laboratory data from sandstone cores indicatethat the velocity ratio / increases with increasingshale and clay content as a result of a decrease in S-wave velocity (Figure11.0-5).

Finally, effect of porosity on the velocity ratio /is generally dictated by the pore shape. For limestoneswith their pores in the form of microcracks, the ve-locity ratio increases as the percent porosity increases

(Eastwood and Castagna, 1983). For sandstones withtheir rounded pores, the velocity ratio does not increaseas much with increasing porosity (Miller and Stewart,1999).The difference between the rounded pores and mi-crocracks lies in the fact that it is easier to collapse a

rock with microcracks, hence lower modulus of rigidity.

11.1 SEISMIC RESOLUTION

Resolution relates to how close two points can be, yetstill be distinguished. Two types of resolution are con-sidered vertical and lateral, both of which are con-trolled by signal bandwidth. The yardstick for verticalresolution is the dominant wavelength, which is wavevelocity divided by dominant frequency. Deconvolutiontries to increase the vertical resolution by broadening

the spectrum, thereby compressing the seismic wavelet.The yardstick for lateral resolution is the Fresnel zone,a circular area on a reflector whose size depends onthe depth to the reflector, the velocity above the re-flector and, again, the dominant frequency. Migrationimproves the lateral resolution by decreasing the widthof the Fresnel zone, thus separating features that areblurred in the lateral direction.

Vertical Resolution

For two reflections, one from the top and one from thebottom of a thin layer, there is a limit on how close theycan be, yet still be separable. This limit depends on thethickness of the layer and is the essence of the problemof vertical resolution.

The dominant wavelength of seismic waves is givenby

= v

f, (11 1)

where v is velocity and f is the dominant frequency.Seismic wave velocities in the subsurface range between2000 and 5000 m/s and generally increase in depth. On

the other hand, the dominant frequency of the seismicsignal typically varies between 50 and 20 Hz and de-creases in depth. Therefore, typical seismic wavelengthsrange from 40 to 250 m and generally increase withdepth. Since wavelength determines resolution, deepfeatures must be thicker than the shallow features to beresolvable. A graph of wavelength as a function of veloc-ity for various values of frequency is plotted in Figure11.1-1. The wavelength is easily determined from thisgraph, given the velocity and dominant frequency.


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Table 11-1.Threshold for vertical resolution.

/4 =v/4f

v (m/s) f (Hz) /4 (m)

2000 50 103000 40 184000 30 335000 20 62

The acceptable threshold for vertical resolutiongenerally is a quarter of the dominant wavelength. Thisis subjective and depends on the noise level in thedata. Sometimes the quarter-wavelength criterion is toogenerous, particularly when the reflection coefficient issmall and no reflection event is discernable. Sometimes

the criterion may be too stringent, particularly whenevents do exist and their amplitudes can be picked withease.

Table 11-1 contains the wavelength threshold val-ues for vertical resolution, considering the realistic ve-locity and frequency ranges. For example, a shallow fea-ture with a 2000-m/s velocity and 50-Hz dominant fre-quency potentially can be resolved if it is as thin as 10m. A thinner feature cannot be resolved. Similarly, fora deep feature with a velocity as high as 5000 m/s anddominant frequency as low as 20 Hz, the thickness mustbe at least 62 m for it to be resolvable.

It is now appropriate to ask whether a thin strati-graphic unit must be resolved to be mapped. The an-swer is no. Resolution as defined here and in the geo-physical literature implies that reflections from the topand bottom of a thin bed are seen as separate eventsor wavelet lobes. Using this definition, resolution doesnot consider amplitude effects. The thickness and arealextent of beds below the resolution limit often canbe mapped on the basis of amplitude changes. Thisamplitude-based analysis can be especially precise whenused for mapping gas-generated bright spots in tertiaryrocks. Thus, in many stratigraphic plays, resolution inthe strict sense is not an issue. For these plays, detec-

tion, not resolution, is the problem.Vertical resolution is a concern when discontinu-

ities are inferred along a reflection horizon because offaults. Figure11.1-2shows a series of faults with verticalthrows that are equal to 1, 1/2, 1/4, 1/8, and 1/16thof the dominant wavelength. It is when the throw isequal or greater then one-fourth of the dominant wave-length that the presence of the fault can be inferredeasily. Perhaps a smaller throw can be inferred by us-ing diffractions from faults along the reflection horizon,provided the noise level is low in the data.

FIG. 11.1-1. The relationship between velocity, domi-nant frequency, and wavelength. Here, wavelength = veloc-ity/frequency. (Adapted from Sheriff, 1976; courtesy Amer-ican Association of Petroleum Geologists.)

Clearly the ability to resolve or detect small tar-gets can be increased by increasing the dominant fre-quency of the stacked data. The dominant frequency ofa stacked section from a given area is governed by thephysical properties of the subsurface, processing quality,

and recording parameters. Since we cannot control thesubsurface properties, the high-frequency signal levelcan only be influenced by the effort put into recordingand processing.

The emphasis in recording should be to preservehigh frequencies and suppress noise. The sampling rateand antialiasing filters should be adequate to recordthe desired frequencies. Receiver arrays should be smallenough to prevent the significant loss of high-frequencysignal because of intragroup moveout and statics. How-ever, the arrays should not be too small, since smallarrays are not as effective at suppressing random, high-

frequency noise (wind noise) as large arrays. Finally, thesource effort should be high enough to provide adequatesignal level relative to noise level within the desired fre-quency band. Unless the signal-to-noise ratio of the fielddata is above some minimal level, say 0.25, processingalgorithms have difficulty in recovering the signal. Thesignal has to be detectable before it can be enhanced.

The emphasis in processing should be to preserveand display the high-frequency signal present in theinput data. Filters with good high-frequency responseshould be used for interpolation processes such as NMO


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FIG. 11.1-2.Faults with different amounts of vertical throws expressed in fractions of the dominant wavelength.

removal, datum and statics corrections, and multiplexskew corrections. Extra care should be taken to en-sure that small-scale residual moveout or statics, whichmight cause loss of high-frequency signal during stack-ing, are removed before stack. Nonsurface-consistentalignment programs (often called trim statics programs)sometimes are used for this purpose. Finally, care mustbe taken to ensure that all the high-frequency signal isdisplayed on the final stack. Poststack deconvolution isa useful tool for this purpose.

Lateral Resolution

Lateral resolution refers to how close two reflectingpoints can be situated horizontally, yet be recognizedas two separate points rather than one. Consider thespherical wavefront that impinges on the horizontal pla-nar reflector AA in Figure 11.1-3. This reflector canbe visualized as a continuum of point diffractors. Fora coincident source and receiver at the earths surface(location S), the energy from the subsurface point (0)arrives at t0 = 2z0/v. Now let the incident wavefrontadvance in depth by the amount /4. Energy from sub-surface locationA, orA, will reach the receiver at time

t1 = 2(z0 + /4)/v. The energy from all the pointswithin the reflecting disk with radius OAwill arrivesometime between t0 and t1. The total energy arrivingwithin the time interval (t1 t0), which equals half thedominant period (T/2), interferes constructively. Thereflecting disk AAis called a half-wavelength Fresnelzone (Hilterman, 1982) or the first Fresnel zone (Sheriff,1991). Two reflecting points that fall within this zonegenerally are considered indistinguishable as observedfrom the earths surface.

Since the Fresnel zone depends on wavelength, it

also depends on frequency. For example, if the seismicsignal riding along the wavefront is at a relatively high

frequency, then the Fresnel zone is relatively narrow.

The smaller the Fresnel zones, the easier it is to differen-tiate between two reflecting points. Hence, the Fresnel-

zone width is a measure of lateral resolution. Besidesfrequency, lateral resolution also depends on velocity

and the depth of the reflecting interface the radiusof the wavefront is approximated by (Exercise 11-1)

r=z0

2 . (11

2a)

In terms of dominant frequency f (equation 11-1), the

Fresnel-zone width is

FIG. 11.1-3.Definition of the Fresnel zone AA.


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FIG. 11.1-4. A constant-velocity zero-offset section from an earth model that consists of four reflectors, each with fournonreflecting segments A, B, C, and D. Lateral resolution is governed by the size of the Fresnel zone. The lateral extent ofeach gap is indicated by the solid bars on top. Note that A is hardly recognizable on any of the four horizons; Bcan be inferredon the shallow horizon at 0.5 s; Cis difficult to infer after 2 s, while D is recognizable at all depths. All of these observationsdepend on noise level and how easily the diffractions can be recognized.


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Table 11-2.Threshold for lateral resolution (first Fres-nel zone).

r= (v/2)

t0/f

t0 (s) v (m/s) f (Hz) r (m)

1 2000 50 1412 3000 40 3353 4000 30 6324 5000 20 1118

r=v

2

t0f. (11 2b)

Table 11-2 shows the Fresnel zone radius, wherer = OA in Figure 11.1-3 for a range of frequencyand velocity combinations at various depths t0 = 2z/v.From Table 11-2, note that the shallower the event(and the higher the dominant frequency), the smallerthe Fresnel zone. Since the Fresnel zone generally in-creases with depth, spatial resolution also deteriorateswith depth.

Figure11.1-4shows reflections from four interfaces,each with four nonreflecting segments. The actual sizesof these segments are indicated by the solid bars ontop. On the seismic section, the reflections appear tobe continuous across some of these segments. This isbecause the size of some of the nonreflecting segmentsis much less than the width of the Fresnel zone; theyare beyond the lateral resolution limit.

Spatial resolution is better understood in termsof diffractions. Note that in Figure 11.1-4, the diffrac-tion energy is smeared across the nonreflecting segmentson the deeper reflectors. Since migration is the processthat collapses diffractions, it is reasonable to think thatmigration increases spatial resolution. Remember thatmigration can be achieved by downward continuationof receivers from the surface to the reflecting horizons.As a result of downward continuation, the observationpoints get closer and closer to the reflection points and,therefore, the Fresnel zone gets smaller and smaller. Asmaller Fresnel zone means a higher spatial resolution(equation 11-2).

Migration tends to collapse the Fresnel zone to ap-proximately the dominant wavelength (equation 11-1)(Stolt and Benson, 1986). Therefore, we anticipate thatmigration will not resolve the horizontal limits of someof the nonreflecting segments along the deeper reflectorsin Figure11.1-4. Tables11-1and11-2can be used to es-timate the potential resolution improvement that mayresult from migration. Unless three-dimensional (3-D)migration (Section 7.3) is performed, the actual resolu-tion will be less than that indicated. Two-dimensional

(2-D) migration only shortens the Fresnel zone in thedirection parallel to the line orientation. Resolution inthe perpendicular direction is not affected by 2-D mi-gration.

Figure11.1-5indicates how vertical and lateral res-

olution problems are inter-related. We want to deter-mine the edge of the pinchout. The basis of the pinchoutmodel is a wedge of material represented at a given mid-point location by a two-term reflectivity sequence, oneterm associated with the top and one with the bottomof the wedge. The true thickness of the wedge at vari-ous locations is indicated on top of Figure11.1-5a. Thevelocity within the wedge is 2500 m/s.

We first consider the reflectivity sequence withtwo spikes of equal amplitude and identical polarity.The vertical-incidence seismic response (Figure 11.1-5a) is obtained by convolving the sequences with a

zero-phase wavelet with a 20-Hz dominant frequency.(The zero-phase response simplifies event tracking fromthe top and bottom of the wedge.) Based on this re-sponse, the edge of the wedge can be inferred as leftof location B, where the waveform reduces to a singlewavelet (Figure 11.1-5a). From the resolution thresh-old criterion, the smallest thickness that can be re-solved is (2500 m/s)/(4Hz) = 31.25 m. Figures 11.1-5a, b, and c show the same pinchout modeled us-ing three different zero-phase wavelets with increas-ing dominant frequency (20, 30, and 40 Hz). Sepa-ration between the true location of pinchout A andthe position of the minimum resolvable wedge thick-

ness B decreases with increasing wavelet bandwidth.While the resolution threshold criterion allows us

to say only that the thickness of the wedge is less than31.25 m left of location B, an amplitude-based crite-rion can provide a substantially more accurate loca-tion for the edge of the wedge. Again, refer to Figure11.1-5a and observe the sudden change in amplitudeat location A where the true edge of the pinchout islocated. Hence, the edge still can be reliably detected,even though it may not be resolved, provided the signal-to-noise (signal-to-noise) ratio is favorable. Assumingthat the relative size of the top and bottom reflection

coefficients is known, amplitudes also can be used to es-timate the wedge thickness between locations B and A.

Figures 11.1-5a, b, and c show an apparent lat-eral variation in layer thickness. To see the differencebetween the true thickness and the apparent thickness(peak-to-peak time), refer to Figure11.1-5d. This figureshows the data of Figure11.1-5bwith the actual geome-try of the wedge superimposed on the seismic response.Since the composite wavelet has only one positive peak,the apparent thickness between locations A and B isnearly zero. At location B , the composite wavelet has a


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FIG. 11.1-5.(a) The result of convolving a zero-phase wavelet of 20-Hz dominant frequency with a wedge reflectivity model.The reflection coefficients associated with the top and bottom of the wedge are of equal amplitude and identical polarity. Thetrue edge of the wedge is beneath location A and the true thickness of the wedge is indicated by the numbers on top; (b)same as (a) except the dominant frequency of the wavelet is 30 Hz; (c) same as (a) except the dominant frequency of thewavelet is 40 Hz; (d) same as (b) with the actual geometry of the wedge superimposed on the seismic response; (e) same as(b) except the reflection coefficients from the top and bottom of the wedge have opposite polarity; (f) same as (e) with theactual geometry of the wedge superimposed on the seismic response.

flat top. Immediately to the right of location B , the flattop disappears and the composite wavelet splits. Theflat-top character can be identified as the limit of ver-tical resolution (Ricker, 1953). A short distance to theright of the point at which the composite wavelet firstsplits into two peaks, the apparent thickness becomesequal to the true thickness. This thickness is called thetuning thickness and is equal to peak-to-trough separa-tion (one half the dominant period) of the convolvingwavelet (Kallweit and Wood, 1982). Beyond the pointof tuning thickness, note the apparent thickening of thelayer between locations B and C. To the right of loca-tionC, the apparent and true thicknesses become equal.

Besides apparent thickness, the maximum absoluteamplitude of the composite wavelet along the pinchoutalso changes (Kallweit and Wood, 1982). To the left oflocation A in Figure 11.1-5b, note the single isolatedzero-phase wavelet. Immediately to the right of loca-tion A, the response of the two closely spaced spikeswith identical polarity results in the maximum abso-lute amplitude. This amplitude gradually decreases toa minimum exactly at the tuning thickness. It then in-creases and reaches the amplitude value of the originalsingle wavelet to the right of location C.

Maximum amplitude and apparent thicknesschange in reverse when the reflectivity model consists


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FIG. 11.2-1.A moveout-corrected CMP gather with a re-flection event at 1.25 s that exhibits amplitude variationswith offset. (Courtesy Western Geophysical.)

of reflection coefficients with equal amplitude and op-posite polarity (Figure 11.1-5e). The composite wave-form resulting from this reflectivity model is discussedbyWidess (1973). Two spikes of opposite polarity witha small separation between them act as a derivative op-erator. When applied to a zero-phase wavelet, this op-

erator causes a 90-degree phase shift. This phase shiftcan be seen in Figure 11.1-5e on the wavelets betweenlocations A and B. Widess (1973) observed that thecomposite wavelet within this zone basically retains itsshape while its amplitude changes.

Figure 11.1-5f shows data of Figure 11.1-5e withthe actual geometry of the wedge superimposed on theseismic response. Note that the wedge appears thickerthan it actually is between locations AandB. Also notethe apparent thinning of the layer between locationsB and C. Beyond location C, the apparent and true

thicknesses become equal. Immediately to the right oflocation A in Figure 11.1-5e, the response of the twoclosely spaced spikes with opposite polarity results inthe cancellation of the amplitudes. The largest absoluteamplitude of the composite wavelet gradually increases

to a maximum immediately to the right of location B .It gradually decreases and reaches the amplitude valueof the original single wavelet to the right of location C.

From the above discussion, we see that peak-to-peak time measurements and amplitude informationcan aid in detecting pinchouts that may otherwise beunresolvable. If the size of the reflection coefficients wereknown, then the amplitudes could be used to map thethickness below the resolution limit.

Nevertheless, the reliability of the analysis dependsto some extent on the signal-to-noise ratio. Finally, thedeceptive changes in amplitude and apparent thicknessmust be noted during the mapping of the top and bot-

tom of the pinchout.

11.2 ANALYSIS OF AMPLITUDE

VARIATION WITH OFFSET

In Appendix L, we review the theory of seismic wavepropagation in an elastic continuum. The earths uppercrust that is of interest in seismic prospecting, how-ever, is made up of rock layers with different elasticmoduli. When seismic waves travel down in the earthand encounter layer boundaries with velocity and den-sity contrast, the energy of the incident wave is par-

titioned at each boundary. Specifically, part of the in-cident energy associated with a compressional sourceis mode-converted to a shear wave; then, both thecompressional- and shear-wave energy are partly re-flected from and partly transmitted through each ofthese layer boundaries.

The fraction of the incident energy that is reflecteddepends upon the angle of incidence. Analysis of reflec-tion amplitudes as a function of incidence angle cansometimes be used to detect lateral changes in elasticproperties of reservoir rocks, such as change in Poissonsratio. This may then suggest a change in the ratio ofP-wave velocity to S-wave velocity, which in turn may

imply a change in fluid saturation within the reservoirrocks.

By way of CMP recording geometry, reflectionamplitudes are not measured as a function of angle;instead, they are measured as a function of source-receiver offset. Nevertheless, a range of incidence an-gles is spanned by a range of offsets. Amplitude-versus-offset analysis, therefore, provides the information onamplitude-versus-angle.

Figure 11.2-1 shows a moveout-corrected CMPgather with a strong reflection at 1.25 s. Note the am-


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plitude variations with offset in this case, amplitudesincreasing with offset. By picking the peak amplitudesand plotting them against offset, an amplitude variationwith offset (AVO) curve is derived for a target horizon ateach CMP location. Then, an AVO pattern may emerge,

which can then be used to infer reservoir parameters.The pattern with which amplitudes vary with offsetdepends on the combination of reservoir rock and fluidproperties. Detection of a pattern is dictated primarilyby the signal-to-noise ratio and the range of incidenceangle that is spanned by the offset range of the CMPgather for a target horizon. The shallower the reflec-tor, the wider the range of incidence angle; hence, AVOindicators are best determined for shallow targets.

The following discussion on reflection and refrac-tion of seismic waves is based on flat layer boundaries.Reflection amplitudes also depend upon the dip of thereflecting boundary and its curvature. We can remove

the dip and curvature effects by performing prestacktime migration. The resulting CMP gathers are asso-ciated with reflectors in their migrated positions andreflection amplitudes can then be associated with a lo-cally flat earth model.

Reflection and Refraction

For simplicity, consider a monochromatic compressionalplane wave that impinges at normal incidence upon aflat layer boundary atz = 0. The incident energy is par-titioned between a reflected and transmitted compres-sional plane wave. For this special case, there is onlyone stress component, Pzz, and one displacement com-ponent, w , which is only a function ofz . The equationof wave motion (L-29c) for this special case takes theform

2w

t2 = ( + 2)

2w

z2, (11 3a)

where is density of the medium, and and areLames constants (equations L-19a,b) associated withan isotropic solid. They are directly related to thecompressional-wave velocityby equation (L-35) whichis rewritten below as

=

+ 2

. (11 3b)

A solution to equation (11-3a) can be written as

w0(z, t) = A0 expi

1z it

, (11 4a)

where w0 is the wave function for the incident com-pressional wave, A0 is its amplitude, is the angular

frequency, 1 is the compressional-wave velocity in theupper layer (equation 11-3b), and z-axis is downwardpositive. Similar wave functions can be written for thereflectedplane wave:

w1(z, t) = A1 expi1

z it (11 4b)and for the transmittedplane wave:

w2(z, t) =A2 expi

2z it

, (11 4c)

where2is the compressional-wave velocity in the lowerlayer.

Given the incident wave amplitude A0, we want tocompute the reflected and refracted wave amplitudes,A1 andA2, respectively. Equations (11-4a,b,c) must beaccompanied with boundary conditions at z = 0 thatsatisfy the continuity of displacement and stress. Thecontinuity of displacement condition, w0+ w1 = w2 at

the interface z = 0 gives the relation

A0+ A1= A2. (11 5)

The stress component Pzz can be specialized fromHookes law (equation L-18c) for the present case ofa normal-incident compressional plane wave

Pzz = ( + 2)w

z. (11 6)

The continuity of stress condition at the interface z = 0gives the relation

Pzz0+ Pzz1 = Pzz2. (11 7)Now, differentiate the wave functions of equations (11-4a,b,c) with respect toz, substitute into equation (11-7)and set z = 0 to obtain the following expression:

11A0 11A1= 22A2, (11 8)

where1and 2are the densities of the upper and lowerlayers, respectively.

We now have two equations, (11-5) and (11-8), andtwo unknowns,A1and A2. By combining equations (11-5) and (11-8), we can derive the ratio of the reflectedwave amplitude to the incident wave amplitude, whichis called the reflection coefficientc, associated with thelayer boundary as

c=A1A0

=22 1122+ 11

. (11 9)

Define the product of density and velocity as theseismic impedance, I = . If there is a difference be-tween the seismic impedances of the two layers, then areflection occurs at the interface. If the upper layer hasa higher impedance than the lower layer, the reflectioncoefficient becomes negative causing a phase reversal onthe reflected waveform.


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FIG. 11.2-2.A long, deep-water seismic section that shows internal reflections within the water layer caused by densitycontrast associated with temperature and salinity changes in the water layer. (Data courtesy IFP.)

An impedance contrast at a layer boundary often islargely caused by velocity contrast. Nevertheless, con-ditions exist for which density contrast can be signifi-cant in giving rise to reflections. Figure11.2-2illustratesone such case. The internal reflections within the waterlayer occur because of changes in water temperature

and salinity that causes variations in water density.A typical reflection coefficient for a strong reflector

is about 0.2. The reflection coefficient associated witha hard water bottom is about 0.3. Note that it is theimpedance contrast, and not the density or velocity con-trasts, that gives rise to reflection energy.

In the foregoing discussion, a normal-incident com-pressional plane wave was considered. If the same com-pressional plane wave was incident at an oblique angleto the interface, then derivation of the reflected and

transmitted wave functions gets complicated, and wefind that the reflection coefficient changes with angle ofincidence. Moreover, at non-normal incidence, the in-cident compressional-wave energy is partitioned at theinterface into four components: reflected compressional,reflected shear, transmitted compressional, and trans-

mitted shear waves.For simplicity, consider a 2-D compressional plane

wave that impinges on a layer boundary with an angle ofincidence0as depicted in Figure11.2-3a. The incidentwavefront is denoted by AC. Point A at the layer bound-ary acts as a Huygens secondary source and generatesits own spherical wavefronts associated with compres-sional and shear waves propagating in both upper andlower media with the corresponding velocities. In Fig-ure11.2-3a, only the reflected compressional wavefront


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FIG. 11.2-3.Reflection and refraction of an incident P-wave at a layer boundary. Medium parameters: is density, isP-wave velocity, isS-wave velocity. (a) ReflectedP-wave; (b) reflected S-wave; (c) refractedP-wave; (d) refracted S-wave;(e) raypaths associated with the incident Pwave, and reflected and refracted P- and S-waves. The radius of the circularwavefront associated with Huygens secondary source at A on the layer boundary is CB for the reflected wave, (1/1) forthe reflected S-wave, (2/1)CB for the refracted P-wave, and (2/1) for the refracted S-wave. The relationship betweenthe angles in (e) is given by Snells law (equation 11-10).


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and raypath are shown. By the time the incident wave-front at C reaches the reflecting interface at B, thespherical wavefront associated with the reflected com-pressional wave reaches D, so that AD = CB and thetangential line DB becomes the wavefront for the re-

flected compressional wave. Since the incident and thereflected compressional waves travel with the same ve-locity, the angle of incidence 0 is equal to the angle ofreflection1.

Now consider the reflected shear wave as shown inFigure11.2-3b. By the time the incident wavefront at Creaches the reflecting interface atB, the spherical wave-front associated with the reflected shear wave reachesD, so that AD = (1/1)CB , and the tangential lineDB becomes the wavefront for the reflected shear wave.The angle for the reflected shear wave 1 is no longerequal to the angle of incidence 0.

Huygens principle can also be applied to describe

the refracted wave at the interface. Refer to Figures11.2-3c and 11.2-3d and note that the compressionalincident plane wave at A acts as a Huygens secondarysource and generates its own compressional wavefrontthat travels into the lower medium with velocity2andshear wavefront that travels into the lower medium withvelocity 2. By the time the incident wavefront at C inFigure 11.2-3c reaches the layer boundary at B, thespherical wavefront associated with the refracted com-pressional wave reaches D, so that AD = (2/1)CB ,and the tangential line DB becomes the wavefront forthe refracted compressional wave. Similarly, by the timethe incident wavefront at C in Figure 11.2-3d reaches

the layer boundary at B, the spherical wavefront asso-ciated with the refracted shear wave reaches D, so thatAD= (2/1)CB , and the tangential lineDB becomesthe wavefront for the refracted shear wave.

From the geometry of reflected and refracted ray-paths shown in Figure11.2-3e, Snells law of refractioncan be deducted as

sin01

=sin1

1=

sin22

=sin1

1=

sin22

.

(11 10)

Note that for all four cases in Figure 11.2-3, the hori-zontal distance AB at the interface is the same for the

incident and reflected or the refracted wave. IfACis setto the wavenumber along the propagation path of theincident wave, then AB is the horizontal wavenumberwhich is invariant as a result of reflection or refraction.Actually, Snells law given by equation (11-10) is a di-rect consequence of this physical observation.

If the compressional velocity 1 of the upper layeris less than the compressional velocity 2 of the lowerlayer, then there exists an angle of incidence such thatno refracted compressional energy is transmitted intothe lower layer. Instead, the refracted energy travels

along the interface and is refracted back to the upperlayer with an angle equal to the angle of incidence. Thisangle is called the critical angle of incidence for com-pressional waves and is given by

sinc=1

2. (11 11a)

The critically refracted wave is often called the headwaveand is the basis for refraction statics (Section 3.4).

If the compressional velocity 1 of the upper layeris less than the shear velocity 2of the lower layer, thenthere exists an angle of incidence such that no refractedshear energy is transmitted into the lower layer. Again,the refracted energy travels along the interface and isrefracted back to the upper layer with an angle which iscalled the critical angle ofP-to-Sconversion given by

sinc=12

. (11 11b)

For the general case of non-normal incidence,boundary conditions at the interface involve not onlyprincipal stress and strain components but also theshear stress and strain components. Again, by using therequirement that the stress and displacement must becontinuous at the interface, a set of equations can bederived to compute the amplitudes of the reflected andrefracted P- and S-wave components associated withan incident compressional source (Section L.5):

cos1A1+11

sin1B1

+1

2cos2A2

1

2sin2B2= cos1,

(11 12a)

sin1A1+11

cos1B1

+12

sin2A2+12

cos2B2= sin1,

(11 12b)

cos 21A1 sin 21B1

+21

cos22A2 21

sin22B2= cos 21,

(11 12c)

and

sin21A1 21

21

cos21B1

+222

21

12122

sin22A2+221121

cos22B2= sin 21.

(11 12d)

These are the Zoeppritz equations which can be solvedfor the four unknowns, the reflected compressional-waveamplitude A1, the reflected shear-wave amplitude B1,the refracted compressional-wave amplitude A2, and


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FIG. 11.2-4. Framework for derivation of the Zoeppritz equations. For details see Section L.5.

the refracted shear-wave amplitude B2. Equations (11-12a,b,c,d) have been normalized by the incident-waveamplitudeA0= 1 (Section L.5).

From Snells law (equation 11-10), given the inci-dent angle for the compressional wave and specifyingthe compressional- and shear-wave velocities, the an-gles of reflection and refraction can be computed. Sub-stitution into equation (11-12) yields the required waveamplitudes. These wave amplitudes, of course, dependon the angle of incidence (Figure 11.2-3e).

Figure 11.2-4 outlines the framework for derivingthe Zoeppritz equations based on an earth model that

comprises two layers separated by a horizontal interface.Details are left to Section L.5. Starting with the equa-tions of motion and Hookes law, derive the wave equa-tion for elastic waves in isotropic media in which elasticproperties are invariant in any spatial direction at anygiven location. Then, use the equations of continuitywhich state that the vertical and tangential stress andstress components coincide at layer boundary, plane-wave solutions to the wave equation and Snells law thatrelates propagation angles to wave velocities to derive

the equations for computing the amplitudes of the re-flected and transmitted P- and S-waves.

Refer to Figure 11.2-5 for a specific case of thepartition of energy of an incident compressional-waveamplitude into four components. Note the significantchanges at critical angles of refraction for the compres-sional and shear wave. It is important to keep in mindthat the shape of these curves varies greatly with dif-ferent situations of medium parameters. Also, note thatat normal incidence no P-to-S conversion takes place,and equations (11-12a,b,c,d) reduce to the special casedescribed by equations (11-5) and (11-8).

From a practical standpoint, the angle-dependencyof reflection coefficients implies that the reflection am-plitude associated with a reflecting boundary varieswith source-receiver separation as well as the depth ofthe reflector. For sufficiently deep reflectors and the typ-ical source-receiver separations used in practice, the am-plitude for the reflected compressional wave is nearlyconstant or slowly varying with offset (left of the crit-ical angle on the curve corresponding to the reflectedcompressional-wave energy in Figure 11.2-5). It is this


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FIG. 11.2-5.Partitioning of a unit-amplitude incident P-wave energy into four components reflected and refracted P-and S-waves(Richards, 1961.).

slowly varying portion of the P-to-P reflection curvethat we have to detect from CMP data with limitedoffset range and in the presence of noise. Note also fromFigure11.2-5that the largest P-to-Sconversion occursbeyond the critical angle, corresponding to large source-receiver separations.

Reflection amplitude variations with angle of in-cidence, and therefore with offset, can be modeled us-ing the Zoeppritz equations. For modeling the reflectionamplitudes, you need well-log curves for theP-wave ve-locity, S-wave velocity, and density. Then, using equa-tion (11-13a) compute theP-to-Preflection amplitudesas a function offset. Figure11.2-6shows a modeled CMPgather using well data. A sonic log was first convertedto a blocky form to simplify the modeled amplitudes onthe gather. Then, using the Gardner relation (equation

11-34a), a density profile was derived. Also, using a ra-tio ofP-wave toS-wave velocity, a shear-wave velocityprofile was generated. The objective in this modelingexercise was to see the effect of a change in Poissonsratio at some depth on the amplitude variation withoffset. The Poissons ratio profile shows a change at ap-

proximately 650 ms at which time we observe a markedvariation of amplitudes with offset on the CMP gather.

Reflector Curvature

Define the ratio C Eof the reflection amplitude at nor-mal incidence from a curved boundary to the reflec-tion amplitude from a flat boundary at the same depth


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FIG. 11.2-6. An example of modeling of a CMP gather based on Zoeppritz equations. (Courtesy Hampson-Russell.)

(Hilterman, 1975) as

CE= 1

1 + A1z

, (11 13a)

where z is the depth to the reflector, and A is the re-flector curvature negative for synclines and positivefor anticlines. Note that for a flat reflector, whatever itsdepth,CE= 1. But for a curved reflector, the reflectionamplitude at normal incidence to a synclinal interfaceis greater than the case of a flat reflector, and the re-

flection amplitude at normal incidence to an anticlinalinterface is smaller than the case of a flat reflector. Thephysical basis of equation (11-13a) is that a synclinal in-terface focuses the energy associated with the reflectingwave, whereas an anticlinal interface defocuses it.

The effect of reflector curvature on amplitude vari-ation with offset is quantified as (Shuey et al., 1984)

CE() = 1

1 + A1z/ cos2 , (11 13b)

where is the angle of incidence. Note that equation(11-13b) reduces to equation (11-13a) for the case ofnormal incidence ( = 0). Both equations are for thecase of a 2-D reflecting interface. Equation (11-13b) wasextended byBernitsas (1990) to the case of a 3-D re-flecting interface with curvature in both the inline andcrossline directions.

To understand the effect of reflector curvature onamplitude variation with offset, it is convenient to studythe ratio CE()/CE( = 0) as a function of angle of

incidence (Shuey et al., 1984; Castagna, 1993). Figure11.2-7shows the behavior of this ratio for the cases of asyncline and an anticlne with varying degrees of curva-ture. Note that, for an anticlinal interface, the curvatureeffect defined by the ratio C E()/CE(= 0) decreaseswith angle of incidence or offset (Figure 11.2-7c). Inpractice, this means that the reflection amplitudes froman anticlinal interface at far offsets are lower than thosefrom a flat interface. Also, note that the larger the cur-vature defined by the ratio z/x denoted in Figure11.2-7,


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FIG. 11.2-7. Effect of reflector curvature on angular dependence of reflected wave amplitudes. The incident wave is of

compressional type. (a) A synclinal reflector (x 0), (c) effect of curvature on the angle-dependent reflection amplitudes associated with the reflector as in (a), and (d) the same as in (c) for the reflector as in (b).(Adapted fromShuey et al., 1984.)


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the more prominent the curvature effect. For a synclinalinterface with a mild curvature (0 > z/x > 1), thecurvature effect on amplitudes increases with angle ofincidence, and it decreases with angle of incidence for atight syncline (z/x


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FIG. 11.2-8.P-to-Preflection amplitude as a function of angle of incidence computed by using the exact Zoeppritz equation,and Bortfeld (equation 11-14) and Aki-Richards (equation 11-15) approximations (Smith and Gidlow, 1987).


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reflectivity /, S-wave reflectivity / and frac-tional chnage in density / that we wish to esti-mate from the observedangle-dependent reflection am-plitudes. To use the Aki-Richards equation (11-15) inthe inversion of reflection amplitudes for these elas-

tic parameters, we first need to recast it in successiveranges of angle of incidence. This change of philosophyin arranging the terms in the Aki-Richards equation(11-15) was first introduced by Shuey (1985) and ledto practical developments in AVO analysis. The newarrangement in terms of successive ranges of angle ofincidence is given by

R() =

1

2

+

+

1

2

4

2

2

2

2

2

sin2

+

12

tan2 sin2

. (11 16)

Another practical matter of concern is that theAki-Richards equation (11-15) or any of its modifica-tions that we shall derive in this section describe themodeledreflection amplitudes as a function of angle ofincidence. However, the observed reflection amplitudesare available from CMP data as a function of offset. Aneed then arises either to transform the model equa-tion for the reflection amplitudes from angle to offsetcoordinates (Demirbag and Coruh, 1988) or to actuallytransform the CMP data from offset to angle coordi-

nates. While the first approach is theoretically appeal-ing, the practical schemes are based on the latter ap-proach. We have already discussed such a transforma-tion in Section 6.4 the Radon transform using thelinear moveout equation or its robust variation in theform of slant stacking. Figure 11.2-9 shows Zoeppritzamplitude curves as a function of angle of incidence andoffset (Demirbag and Coruh, 1988).

Based on the theoretical conjecture made earlier byKoefoed (1955) that the elastic property that is mostdirectly related to angular dependence of reflection co-efficient R() is Poissons ratio , Shuey (1985) intro-duced a variable transformation from S-wave velocity to . The relationship between the two variables isgiven by equation (L-49) which we rewrite below as

2 =1

2

1 21

2 (11 17a)

to perform the necessary differentiation

=

1

2

(1 )(1 2). (11 17b)

The compressional-wave velocity is given by equation

(11-3b) and the shear-wave velocity is given by equa-tion (L-47) which is rewritten below as

=

, (11 17c)

where is Lames constant.We also define theP-wave reflection amplitudeRPat normal incidence as

RP = 1

2

+

. (11 18)

Substitute equations (11-17a), (11-17b), and (11-18)into the Aki-Richards equation (11-16) and performsome algebraic simplification to obtain

R() = RP

+ 1

2

2

+

121

+

(1

)2 sin2

+

1

2

tan2 sin2

. (11 19)

Define a new term H

H= /

/ + /, (11 20)

and by way of equation (11-18) note that

= 2 RPH. (11 21a)

Next combine equations (11-20) and (11-21a) to obtain

= 2 RP(1 H). (11 21b)

Finally, substitute equations (11-21a) and (11-21b) intothe second term on the right-hand side of equation (11-19) to obtain

R() = RP+

RPH0+

(1 )2

sin2

+

1

2

tan2 sin2

, (11 22)

where

H0= H 2(1 + H)1 21

. (11 23)

Equation (11-22) is known as Shueys three-termAVO equation. The first term RP is the reflection am-plitude at normal incidence. At intermediate angles(0< < 30 degrees), the third term may be dropped,thus leading to a two-term approximation

R() = RP+ G sin2 , (11 24)

where


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G= RPH0+

(1 )2. (11 25)

Equation (11-24) is known as Shueys two-termAVO equation. In practice, amplitudes picked along

a moveout-corrected event on a CMP gather plottedagainst sin2 can be fitted to a straight line. The slopeof the line gives the AVO gradient attributeand the or-dinate at zero angle gives the AVO intercept attribute.The AVO gradient given by equation (11-25) is directlyrelated to change in Poissons ratio , which in turn,is directly related to fluid saturation in reservoir rocks.The AVO intercept attribute represents the reflectiv-ity RP at normal incidence. Therefore, the AVO in-tercept attribute, in lieu of conventional stack, can beused as input to derive the acoustic impedance attribute(Section 11.3), which is indirectly related to porosity inreservoir rocks.

Shown in Figure 11.2-10a is a portion of a sec-tion derived from 2-D prestack time migration. Theobjective is to identify fluid-saturated reservoir zonesat the apex and the flanks of the structural closure.This image section is derived from the stacking ofcommon-reflection-point (CRP) gathers associated withthe prestack time-migrated data. The CRP gather inFigure11.2-10bshows three events with amplitude vari-ations with offset which are plotted in Figure 11.2-10c.By using Shueys equation (11-24), the AVO gradientand intercept sections are computed from the CRPgathers as shown in Figures11.2-11and11.2-12, respec-tively. Note that the gradient section exhibits a groupof AVO anomalies in the vicinity of the structural apex,possibly indicating fluid-saturated reservoir rocks.

At large angles of incidence beyond 30 degrees, thethird term in equation (11-22) gradually becomes dom-inant. Note that this term is related directly to frac-tional change in P-wave velocity, /. So, not onlythe reflection traveltimes at far offsets (Section 3.1) cor-responding to large angles of incidence, but also the re-flection amplitudes at large angles of incidence makethe biggest contribution to the resolution needed to es-timate the changes in P-wave velocities.

The two-term equation (11-24) can be specialized

for a specific value of Poissons ratio, = 1/3 andH0=1 so that equation (11-25) takes the form

G= RP+9

4, (11 26)

which can be solved for the change in Poissons ratioacross a layer boundary

=4

9

RP+ G

. (11 27)

This is the AVO attribute equation for estimatingchanges in Poissons ratio (Hilterman, 1983). Actually,

as described by equation (11-27), this attribute is thescaled sum of the AVO intercept RPand AVO gradientG attributes.

By recasting the first-order approximation to theZoeppritz equation, Wiggins et al. (1984) and Spratt

et al. (1984) derived a practical expression for S-wavereflectivity. First, drop the term with tan2 in equation(11-16) and rearrange the remaining terms to obtain

R() =

1

2

+

+

1

2

+

8

2

2

1

2

+

sin2

+

2

2

2

1

2

sin2 (11 28)

such that, much like the definition for the P-wave re-

flectivityRPgiven by equation (11-18), an expressionfor the S-wave reflectivityRS

RS= 1

2

+

(11 29)

can be explicitly inserted back into equation (11-28) toget

R() = RP+

RP 8

2

2RS

sin2

+22

2

1

2

sin2 . (11

30)

Set / = 0.5 to make the last term on the right-hand side vanish and obtain (Spratt et al., 1984)

R() =RP+

RP 2 RS

sin2 . (11 31)

This equation is of the form given by equation (11-24)where

G= RP 2 RS, (11 32)

which can be rewritten explicitly in terms of the shear-wave reflectivityRS

RS=1

2RP G. (11 33)

This is the AVO attribute equation for estimating theshear-wave reflectivity. Given the AVO intercept RPand AVO gradient G attributes, simply take half ofthe difference between the two attributes to derive theshear-wave reflectivityRSas described by equation (11-33).

Figure 11.2-13 shows the reflection amplitudes asa function of angle as predicted by equation (11-31) forthree combinations ofRP and RS. Note that equation(11-31) is a good approximation to the P-to-Preflection


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FIG. 11.2-10. (a) A portion of a prestack time-migrated section; (b) portion of a common-reflection-point gather in thevicinity of the structural apex in (a); (c) reflection amplitudes as a function of offset measured along three events indicted bythe red, horizontal trajectories in (b).


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FIG. 11.2-11.The AVO gradient section defined by G in equation (11-24); (b) close-up of (a) in the vicinity of the structuralapex.


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FIG. 11.2-12. (a) The AVO intercept section defined by RP in equation (11-24); (b) close-up of (a) in the vicinity of thestructural apex.


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FIG. 11.2-13. P-to-Preflection amplitude as a function of angle of incidence computed by using the exact Zoeppritz equation(the three curves labeled as A) and the approximate form given by equation (11-31) (the three curves labeled as B ) (Sprattet al., 1984).

amplitudes as predicted by the exact Zoeppritz equationup to nearly 30 degrees of angles of incidence.

Return to the Aki-Richards equation (11-15) andconsider the case ofN-fold CMP data represented in thedomain of angle of incidence. Note that the reflection

amplitudeR() is a linear combination of three elasticparameters P-wave reflectivity /,S-wave reflec-tivity /and fractional change in density /.

Smith and Gidlow (1987)argue in favor of solvingfor only two of the three parameters by making use ofthe empirical relation between density and P-wavevelocity (Gardner et al., 1974):

= k1/4, (11 34a)

where k is a scalar. This relation holds for most water-saturated rocks. Differentiate to get

=1

4

. (11

34b)

Now substitute equation (11-34b) into the originalAki-Richards equation (11-15) and combine the termswith /

R() =

1

2

1 + tan2

+

1

8

1 4

2

2sin2

4

2

2sin2

. (11 35)

Simplify the algebra to obtain the desired two-par-ameter model equation for prestack amplitude inversion(Section L.6)

R()=5

8

1

2

2

2sin2 +

1

2tan2

4

2

2sin2

.

(11 36)Redefine the coefficients ai and bi and write the

discrete form of equation (11-36)

Ri = ai

+ bi

, (11 37)

where

ai =5

8

1

2

2

2sin2 i+

1

2tan2 i (11 38a)

and

bi =

42

2

sin2 i. (11

38b)

Consider the case ofN-fold CMP data representedin the domain of angle of incidence. We have, for eachCMP location and for a specific reflection event asso-ciated with a layer boundary, Nequations of the formas equation (11-37) and two unknowns / and/. We have more equations than unknowns; hence,we encounter yet another example of a generalized lin-ear inversion (GLI) problem (Section J.1). The objec-tive is to determine the two parameters such that the


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FIG. 11.2-14.(a) P-wave and (b) S-wave reflectivity sections derived from the Smith-Gidlow inversion of prestack amplitudes(equation 11-39).


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FIG. 11.2-15. Laboratory measurements of (a) P-wave and (b) S-wave reflectivities at a gas-brine interface in sandstones(Spratt et al., 1984). The horizontal axis in both plots denotes induration which is a measure of rock strength.

FIG. 11.2-16.Portions of (a) P-wave and (b) anS-wave stacked section that show anomalies associated with lithology (coal)and fluid content (gas) (Spratt et al., 1984).


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FIG. 11.2-17. A sketch of crossplot ofP- to S-wave velocity(Castagna et al., 1985;Fatti et al., 1994).

difference between the modeledreflection amplitudesRirepresented by equation (11-37) and the observed re-flection amplitudes Xi is minimum in the least-squaressense (Smith and Gidlow, 1987).

The GLI solution in matrix form is given by (Sec-tion L.6)

Ni a

2i

Ni aibi

Ni aibi

Ni b

2i

=

Ni aiXi

Ni biXi

.

(11 39)This matrix equation is solved for the two parame-

ters P-wave reflectivity / and S-wave reflectiv-ity /. Note from the definitions of the coefficientsai and bi given by equations (11-38a,b) that you haveto choose a value for the ratio / to compute the tworeflectivities.

Shown in Figure11.2-14are theP- andS-wave re-flectivity sections that were derived from prestack am-plitude inversion applied to the CRP gathers associated

with the data shown in Figure11.2-10a. Note the AVOanomalies along the flanks of the structural closure. It isapparent that some of the AVO anomalies in the P-wavereflectivity section stand out more distinctively as com-pared with those in theS-wave reflectivity section. Thismay be related to the fact that changing from brine togas causes a small change in S-wave reflectivity, while itcauses a significant change in P-wave reflectivity. Suchinference is supported by the laboratory measurementsofP- and S-wave reflectivities at a gas-brine interface

in sandstones (Spratt et al., 1984). Figure11.2-15showsP-wave reflectivities for a group of reservoir sandstonesin which the pore fluid changes from brine to gas. Forrocks with low induration (weak rocks), there is a largechange in P-wave reflectivity, while for rocks with highinduration the change becomes less significant (Figure11.2-15a). The change in S-wave reflectivity is negli-gible for rocks with low and high induration (Figure11.2-15b). A demonstrative field data example ofP- andS-wave reflectivity contrasts that arise from a change in


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FIG. 11.2-18. Crossplot ofP- toS-wave velocity measured down a borehole (Fatti et al., 1994).

lithology and fluid saturation is shown in Figure 11.2-16.

Refer to equation (11-17b) and note that the dif-ference between theP-wave andS-wave reflectivities is

related to change in Poissons ratio a direct hy-drocarbon indicator. In fact, Smith and Gidlow (1987)have coined the term pseudo-Poisson reflectivityto de-scribe the difference between the P-wave and S-wavereflectivities

=

. (11 40)

Note from equation (11-17b) that the pseudo-Poisson reflectivity / is not exactly the same aswhat may be referred to as the proper Poisson reflec-tivity /. If we define the ratio

=

(11 41a)

by differentiation, we can derive the difference relationgiven by equation (11-40) and thus show that

=

(/)

/ . (11 41b)

Castagna et al. (1985) defined a straight line inthe plane ofS-wave velocity versus P-wave velocity asshown in Figure11.2-17. This is called the mudrock line

and is represented by the equation

= c0+ c1, (11 42)

where the scalar coefficients c0 and c1 are empirically

determined for various types of rocks. Suggested valuesfor these scalars are c0 = 1360 and c1= 1.16 for water-saturated clastics (Castagna et al., 1985). Gas-bearingsandstones lie above the mudrock line, and carbonateslie below the mudrock line as shown on a crossplot ofP-and S-wave velocities sketched in Figure 11.2-17. Thecrossplot shown in Figure11.2-18is based on log mea-surements at a gas-producing well (Fatti et al., 1994).Note the separation of gas-sandstone cluster from thewater-sandstone and shale clusters in the manner assketched in Figure11.2-17.

A way to quantify the prospectivity of the reservoir

rock of interest is by defining afluid factor attributethatindicates the position of the rock property with respectto the mudrock line. First, apply differentiation to bothsides of equation (11-42) and note that

=c1

. (11 43)

Then, define the fluid factor F

F=

c1

. (11 44)


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FIG. 11.2-19. (a) Pseudo-Poisson and (b) fluid factor attribute sections derived by using equations (11-40) and (11-44),respectively.


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FIG. 11.2-20.(a) The and (b) attribute sections derived by using equations (11-52b) and (11-52c), respectively.


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FIG. 11.2-21.(a) Crossplot ofP-wave impedance versus S-wave impedance, and (b) crossplot of versus AVO attributes.

If F is close to zero, it means that you have water-saturated rock. If you have a gas-saturated sandstone,Fwill be negative at the top and positive at the baseof the reservoir unit (Smith and Gidlow, 1987).

Figure11.2-19shows the pseudo-Poisson and fluid-factor sections that were derived from prestack ampli-tude inversion applied to the CRP gathers associatedwith the data shown in Figure11.2-10a. Note the dis-

tinctive AVO anomalies along the flanks of the struc-tural closure.

The two parameters / and /, estimatedby using the least-squares solution given by equation(11-39), represent fractional changes in P- and S-wavevelocities. As such, they are related to P- and S-wavereflectivities, IP/IP and IS/IS, respectively, whereIP and ISare the P- and S-wave impedances given by

IP = (11 45a)

and

IS=. (11 45b)

From Section L.6, the P- and S-wave reflectivitiesare given by

IPIP

=

+

(11 46a)

and

ISIS

=

+

. (11 46b)

Assuming that the density obeys Gardners relationgiven by equation (11-34a), the P-wave reflectivityIP/IP is simply the fractional change of the P-wavevelocity / scaled by a constant, whereas the S-wavereflectivity IS/ISis a linear combination of the frac-tional changes of the P- and S-wave velocities, /and /, respectively.


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A direct estimation of the P- andS-wave reflectiv-ities given by equations (L-46a,b) can be made by usingan alternative formulation of prestack amplitude inver-sion which is based on transforming the Aki-Richardsequation (11-15) to the new variables IP/IP and

IS

/IS

(Goodway et al., 1998). Solve equation (11-46a)for / and equation (11-46b) for /, and substi-tute into the Aki-Richards equation (11-15). Followingthe algebraic simplification, we obtain (Section L.6)

R() =

1

2

1 + tan2

IPIP

42

2sin2

ISIS

1

2tan2 2

2

2sin2

. (11 47)

Goodway et al. (1998)have implemented a specificform of equation (11-47) to derive the AVO attributesIP/IP and IS/IS. For a specific value of/ = 2

and small angles of incidence for which tan sin ,the third term in equation (11-47) vanishes. Compareequations (11-18) and (11-46a), and equations (11-29)and (11-46b), and note that equation (11-47) with theremaining terms takes the form

R() =

1 + tan2 RP

2sin2

RS. (11 48)

The resulting equation then is solved for the P- andS-wave reflectivities, IP/IP = 2RP and IS/IS =2RS, respectively. Again, consider the case ofN-foldCMP data represented in the domain of angle of inci-dence. In discrete form, equation (11-48) can be rewrit-ten as

Ri= aiRP+biRS, (11 49)

where i is the trace index and the coefficients ai andbiare given by

ai =1

2

1 + tan2 i

(11 50a)

and

bi= 2sin2 i. (11 50b)

The reflection amplitudeRi in equation (11-49) is a lin-ear combination of the two parameters, RP andRS. Asfor the Smith-Gidlow equation (11-37), the two param-eters,RP andRS, can be estimated for a specific eventfrom CMP data using the least-squares solution givenby

N

i a2

i

N

i aibiN

i aibi

Ni b2i

RP

RS

=

N

i aiXiN

i biXi

.

(11 51)

Following the estimation of the P-wave reflectiv-ityRP and the S-wave reflectivity RSusing the least-squares solution given by equation (11-51), the P-wave

impedance IP and the S-wave impedance IS can becomputed by integration.

By using the impedance attributes IP and IS,Goodway et al. (1998)compute two additional AVO at-tributes in terms of Lames constants scaled by density

and . Substitute equation (11-3b) into (11-45a)to get the relation

+ 2= I2P, (11 52a)

and substitute equation (11-17c) into equation (11-45b)to get the relation

= I2S. (11 52b)

Note from equations (11-52a,b) that

= I2P 2 I2

S. (11 52c)

Figure11.2-20shows the and AVO attribute

sections which were derived from prestack amplitudeinversion applied to the CRP gathers associated withthe data shown in Figure 11.2-10a. As in the case ofthe pseudo-Poisson and fluid-factor AVO attribute sec-tions shown in Figure11.2-19, note the distinctive AVOanomalies along the flanks of the structural closure.Goodway et al. (1998) convincingly demonstrates thatseparation of gas-bearing sands from tight sands andcarbonates is much better with the crossplot of theLame attributes (, ) in contrast with the crossplotof the P- and S-wave reflectivities or impedances (Fig-ure11.2-21).

Figure 11.2-22 outlines a summary of the AVO

equations that are described in this section based on thevarious approximations to the Aki-Richards equation(11-15). Start with Shueys arrangement of the termsin the Aki-Richards equation, which itself is an approx-imation to the exact expression for the P-to-Preflectionamplitudes given by the Zoeppritz equations. Then, ap-ply a transformation from S-wave velocity to Poissonsratio, and drop the third term to get a simple expres-sion for theP-to-P reflection amplitude that is a linearfunction of sin2 . This linear approximation yields theAVO intercept and gradient attributes.

Alternatively, you may drop the third term in the

original Shuey arrangement given by equation (11-16)and apply Wiggins rearrangement of the terms. Then,assume the specific case of the P-wave velocity to betwice the S-wave velocity to obtain the equation thatyields, once again, the AVO intercept and gradient at-tributes. The Wiggins AVO intercept and gradient at-tributes, unlike the Shuey counterparts, directly lead toderiving theP-wave and S-wave reflectivity attributes,albeit only under the assumption that the P-wave ve-locity is twice theS-wave velocity.


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FIG. 11.2-23. A raw shot record from the AVO line presented in Section 11.2. (Data courtesy Britannia Gas Ltd.)


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FIG. 11.2-24. The shot record in Figure 11.2-23 after muting for guided waves and t2-scaling for geometric spreadingcorrection. (Data courtesy Britannia Gas Ltd.)


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FIG. 11.2-25.The shot record in Figure11.2-24after spiking deconvolution. (Data courtesy Britannia Gas Ltd.)


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FIG. 11.2-27.Autocorrelogram (a) of the shot record in Figure11.2-23, (b) after muting for guided waves and t2-scaling forgeometric spreading correction, and (c) after spiking deconvolution.


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Returning to the three-term Aki-Richards equa-tion (11-15), you may reduce it to the two-term Smith-Gidlow equation (11-36) by using Gardners relation ofdensity andP-wave velocity. Then, for a specified ratioofP- to S-wave velocity, perform inversion of prestack

amplitudes associated with target events on CMP orCRP gathers to obtain the P- and S-wave reflectivityattributes. In addition, using these attributes to derivetwo more AVO attributes psuedo-Poisson and fluidfactor.

Finally, you may recast the Aki-Richards equation(11-15) in terms ofP- and S-wave reflectivities (equa-tions 11-46a,b) and assume that/= 2 to obtain thetwo-term Goodway et al. equation (11-48). Again, per-form inversion of prestack amplitudes associated withtarget events on CMP or CRP gathers to obtain the P-andS-wave impedance attributes. Subsequently, by us-

ing the relations between impedances and Lames con-stants, you can compute two additional AVO attributes and .

The AVO equations derived in this section are allexpressed in terms of angle of incidence (equations 11-24, 11-31, and 11-36). In practice, the mapping of am-plitudes associated with a reflection event on a CMPgather from offset to angle of incidence needs to be per-formed. Assume that the CMP gather is associated witha horizontally layered earth model so that the eventmoveout on the CMP gather is given by the hyperbolicequation

t2 =t20

+ x2

v2rms

, (11 53)

where t is the two-way traveltime from the source tothe flat reflecting interface back to the receiver, t0 isthe two-way zero-offset time, x is the offset, and vrmsis the rms velocity down to the reflector. The ray pa-rameterp is given by the stepoutdt/dxmeasured alongthe hyperbolic moveout trajectory (Section 6.3). Applydifferentiation to equation (11-53) to get

dt

dx=

1

v2rms

x

t. (11 54)

Since the ray parameter p is also expressed as the ratioof sin /vint, where vint is the interval velocity abovethe reflecting interface, it follows that

sin = vintv2rms

x

t. (11 55)

By using equation (11-55), reflection amplitudes ona CMP gather can be transformed from offset to angledomain, and subsequently used in the AVO equations(11-24), (11-31), (11-36), and (11-48).

Processing Sequence for AVO Analysis

There are three important aspects of a processing se-quence tailored for AVO analysis.

(a) The relative amplitudes of the seismic data must bepreserved throughout the analysis so as to recog-nize amplitude variation with offset. This require-ment often leads to an application of a parsimo-nious sequence of signal processing to avoid distor-tion of amplitudes by undesirable effects of someprocessing algorithms.

(b) The processing sequence must retain the broadestpossible signal band in the data with a flat spec-trum within the passband.

(c) Prestack amplitude inversion to derive the AVOattributes must be applied to common-reflection-

point (CRP) gathers (Section 5.4), not to common-midpoint (CMP) gathers. This is because all theAVO equations described in this section are basedon a locally flat earth model that can be related toCRP raypaths but not to CMP raypaths. Specifi-cally, the CRP gathers are associated with events intheir migrated positions, whereas the CMP gathersare associated with events in their unmigrated posi-tions. The AVO equations described in this sectionare all based on a horizontally layered earth modelthat can be related to CRP raypaths but not toCMP raypaths. Prestack time migration compen-sates for the effect of reflector curvature (equation

11-13b) on amplitude variation with offset so thatthe reflection amplitudes in each of the resultingCRP gathers can be associated with a locally flatearth model.

We shall consider a marine 2-D seismic data setfrom the North Sea recorded over a producing gas field.Data specifications for the line are listed in Table 11-3.

The prestack signal processing that complies withthe a

11 Reservoir Geophysics

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