Prestack seismic amplitude analysis: An integrated overview

Brian H. Russell1

Abstract

In this tutorial, I present an overview of the techniques that are in use for prestack seismic amplitude analysis,current and historical. I show that these techniques can be classified as being based on the computation andanalysis of either some type of seismic reflection coefficient series or seismic impedance. Those techniques thatare based on the seismic reflection coefficient series, or seismic reflectivity for short, are called amplitudevariation with offset methods, and those that are based on the seismic impedance are referred to as prestackamplitude inversion methods. Seismic reflectivity methods include: near and far trace stacking, intercept ver-sus gradient analysis, and the fluid factor analysis. Seismic impedance methods include: independent and si-multaneous P and S-impedance inversion, lambda-mu-rho analysis, Poisson impedance inversion, elasticimpedance, and extended elastic impedance inversion. The objective of this tutorial is thus to make senseof all of these methods and show how they are interrelated. The techniques will be illustrated using a 2D seismicexample over a gas sand reservoir from Alberta. Although I will largely focus on isotropic methods, the last partof the tutorial will extend the analysis to anisotropic reservoirs.

IntroductionThe amplitude variation with offset (AVO) and pre-

stack inversion techniques have grown to include amultitude of subtechniques, each with their own as-sumptions. In this tutorial, I make the assumption thatAVO techniques are based on the computation andanalysis of some type of seismic reflection coefficientseries, or reflectivity for short, and that prestack inver-sion techniques are based on the computation andanalysis of some type of seismic impedance. To under-stand the distinction between impedance and reflectiv-ity, refer to Figure 1, which shows the different ways inwhich geologists and geophysicists look at their data.

As shown on the left side of Figure 1, the goal of allgeoscientists is to understand the subsurface of theearth, which consists of a series of layers of variablestructure, lithology, porosity, and fluid content. To dothis, the geologist generally analyzes borehole welllog measurements of some layer parameter P, whereasthe geophysicist, when using exploration seismology,analyzes changes in this layer parameter at the interfa-ces between successive layers, called the reflectivity(R). The fundamental idea in this tutorial is that thereis a relationship between P and R, and it can be written

RPi ¼Piþ1 − Pi

Piþ1 þ Pi¼ ΔPi

2Pi; (1)

where ΔPi ¼ Piþ1 − Pi; Pi ¼ Piþ1þPi2 and subscript i re-

fers to the ith layer. That is, the reflectivity is found

by dividing the change in the parameter between twosuccessive layers by twice its average value. The sub-script P in the reflectivity RP indicates that the reflec-tivity is associated with parameter P. I will use thisnotation throughout the tutorial. This is, of course,an oversimplification, because the seismic trace alsocontains the source wavelet, amplitude scaling effects,noise contamination due to random noise and coherentnoise such as multiples, static time shifts due to topog-raphy and near-surface weathering, spatial misposition-ing due to structure and subsurface scattering at depth,and so on. A complete discussion of these effects andthe techniques of seismic processing which have beendeveloped to remove or ameliorate them is beyond thescope of this tutorial.

Even if our processing is extremely good, we arealways left with a band-limited reflectivity which canbe modeled using the convolutional model, written inmatrix form as266666666664

W 0 0 · · · 0

..

.W0

. .. ..

.

Wn−1... . .

.0

0 Wn−1. ..

W 0

..

. . .. . .

. ...

0 · · · 0 Wn−1

377777777775

264

R0

..

.

Rm−1

375 ¼

264

S0

..

.

Snþm−2

375;

(2)

1Hampson-Russell, A CGG Company, Calgary, Alberta, Canada. E-mail: brian.russell@cgg.com.Manuscript received by the Editor 10 August 2013; revised manuscript received 1 October 2013; published online 20 March 2014. This paper

appears in Interpretation, Vol. 2, No. 2 (May 2014); p. SC19–SC36, 25 FIGS., 1 TABLE.http://dx.doi.org/10.1190/INT-2013-0122.1. © 2014 Society of Exploration Geophysicists and American Association of Petroleum Geologists. All rights reserved.

t

Special section: Interpreting AVO

Interpretation / May 2014 SC19Interpretation / May 2014 SC19

or S ¼ WR, where R is an m-sample reflectivity (notethat subscript 0 represents zero time),W is the convolu-tional matrix in which each column contains an n-sam-ple wavelet shifted by one sample from the previouscolumn, and S is the nþm − 1 sample result, or seismictrace. This technique of convolution is illustrated picto-rially in Figure 2, where a reflectivity similar to that ofFigure 1 has been convolved with a symmetrical, orzero-phase, wavelet.

We can think of convolution as shifting the waveletso that it is centered at each reflection coefficient, scal-ing the wavelet by the amplitude of each reflection co-efficient, and then summing the result. We say that thisresult is “band limited” because, although the spectrumof the reflectivity contains all frequencies from 0 Hz tothe Nyquist, the seismic wavelet is typically band lim-ited between approximately 10–60 Hz (although someof the new seismic receivers and acquisition techniquesare pushing the low end much lower). It is thus impor-tant to note that in subsequent sections when I refer tothe particular reflectivity model used in an AVO analy-sis, this more precisely refers to the band-limited resultfound from applying equation 2.

A key question is which parameter P we are inter-ested in. For the geologist, the parameter is usuallyone that enables him or her to infer the reservoir prop-erties, such as porosity, water saturation, TOC, etc.However, the geophysicist is usually limited to param-eters that are related to the propagation and reflectionof the seismic signal, such a P-impedance, the productof P-wave velocity and density, and S-impedance, theproduct of S-wave velocity and density. Once theseproperties have been estimated, the geophysicist caninfer the reservoir properties mentioned above, butin this tutorial we will focus more on the estimationof these impedances themselves and the analysis ofthe band-limited reflectivity from which we derivethe impedances.

The geometry of an idealized common midpoint(CMP) seismic gather is shown in Figure 3, in whichtraces recorded at different offsets are grouped by

increasing offset around a CMP point. Notice that eachoffset Xi corresponds to an angle of incidence θi at thereflector. In Figure 3, the single reflector shown willcreate a seismic reflection due to the contrast betweenthe density (ρ), P-wave velocity (VP), and S-wave veloc-ity (VS) in each layer. As we will discuss, the amplitudeof this reflection will change at each offset.

Figure 4 shows a set of real gathers which were ac-quired and processed over a shallow gas field in centralAlberta. This data set will be used to explain many ofthe methods described in this tutorial. In particular, no-tice the amplitudes change as a function of offset on thetrough and peak of the seismic reflection event below620 ms. Our goal in this tutorial is a physical under-standing of this amplitude change, and how it relatesto the gas sand.

Reflectivity methodsNormal incidence reflectivity

Unlike the single incident and reflected raypathsshown in Figure 3, in an elastic earth, we actually

Figure 1. Two different ways of analyzing the subsurfaceof the earth, where the geologist analyzes borehole well logmeasurements of some layer parameter P, whereas the geo-physicist measures changes in the layer parameter, the reflec-tivity R.

Figure 2. Convolution of a reflectivity similar to that inFigure 1 with a zero-phase wavelet, where convolution canbe seen as shifting the wavelet so that it is centered at eachreflection coefficient, scaling the wavelet by the amplitude ofthe reflection coefficient, and then summing the result.

Figure 3. An idealized CMP gather, where the Xi are the off-sets and the θ0 ¼ 0 represent the angles of incidence of eachseismic raypath (note that X0 ¼ 0 and θ0 ¼ 0). The single re-flector shown will produce a seismic reflection caused by thecontrast between the density (ρ), P-wave velocity (VP), and S-wave velocity (VS) in each layer.

SC20 Interpretation / May 2014

observe mode converted reflected and transmittedP- and S-raypaths for an incident P-wave raypath at ar-bitrary incidence angle, as illustrated in Figure 5, whichshows the conversion of an incident P-wave at a non-zero angle of incidence into its reflected and transmit-ted P- and S-components. The P, or compressional,waves are elastic waves that travel by the alternatecompression and rarefaction of the earth’s subsurface,whereas S, or shear, waves are created by side-to-sideshearing of the earth’s subsurface. Shear waves in anisotropic medium with a polarization (particle motion)in a vertical plane are SV-waves, whereas those with apolarization in the horizontal plane are SH-waves.However, there are other directions of polarizationperpendicular to the direction of travel which produceparticle motion that is a combination of SV- and SH-waves. For SV-waves from a horizontal surface, the par-ticle motion is in the plane of Figure 5 but for SH-waves,think of the particle motion as coming out of the planeshown in Figure 5. In HTI-fractured media, like somecarbonates, a vertically traveling S-wave splits intotwo S-waves, one has polarization aligned paralleland the other perpendicular to the direction of the frac-tures with the faster S-wave having a polarization in thedirection parallel to the fractures.

Initially, we will only discuss isotropic materials thathave the same properties in all directions. In this case,SH- and SV-waves have the same velocity, which isslower than the velocity of the P-waves. Intutitively,we can see that a P-wave that strikes a boundary at rightangles will produce only a reflected P-wave, but that aP-wave which strikes a boundary at nonnormal inci-dence (as in Figure 5) will create reflected P- andS-waves. One last observation, which is of fundamentalimportance to prestack seismic analysis, is that becauseP-waves compress the reservoir rocks they are verysensitive to the pore fluid, but because the S-wavesdo not compress the rocks, they are not as sensitiveto the pore fluid.

Zoeppritz (1919) derived the amplitudes of the re-flected and transmitted waves shown in Figure 5 usingthe conservation of stress and displacement across thelayer boundary, giving a forward model with four equa-tions and four unknowns (the amplitudes) which can besolved using matrix inversion. These equations areshown in Appendix A. Although the complete solutionfor an arbitrary angle leads to a complicated set ofequations, it is shown in Appendix A that the normalincidence P-wave reflection coefficient can be deter-mined in a straightforward way from the Zoeppritzequations. This is the quantitative confirmation of theintuitive argument which was given above. Generalizingthe two-layer case given in Appendix A to an arbitrarynumber of layers, we get

RPPið0°Þ ¼ RAIi ¼ρiþ1VPiþ1 − ρiVPi

ρiþ1VPiþ1 þ ρiVPi¼ AIiþ1 − AIi

AIiþ1 þ AIi;

(3)

where AIi is the acoustic impedance, or P-impedance,of the ith layer. As mentioned previously, the reflectioncoefficient RAI is written in such a way as to indicatewhich parameter is associated with the reflectivity; inthis case, the acoustic impedance. To see how equa-tion 3 relates to equation 1, note that we can rewriteequation 3 as

RAI ¼ΔAI2A I

; (4)

where ΔAI ¼ AIiþ1 − AIi and A I ¼ AIiþ1þAIi2 . The sub-

script notation has been dropped in equation 4, so thatRAI can be thought of as them-sample column vector ofreflectivity shown in equation 2. Figure 6 from Russellet al. (2011) shows the CMP stack of the gathers shownin Figure 4, with a picked event at the zero crossingwhich represents the center of the gas sand. The stackis often considered to be a reasonable approximation to

Figure 4. Five CMP gathers from around the well on a seis-mic line recorded over a shallow gas sand in Alberta, with thetime-integrated velocity log overlain at CMP 330. The far-off-set of 647 m corresponds to an angle of approximately 30°(Russell et al., 2011).

Figure 5. Mode conversion of an incident P-wave into re-flected and transmitted P- and SV-waves on the boundary be-tween two elastic layers. P-wave direction of travel andparticle motion is along the raypaths. SV-wave direction oftravel is along the raypaths, but SV-wave particle motion isat right angles to the raypath.

Interpretation / May 2014 SC21

SAI, the convolution of the wavelet with RAI. However,the stack is actually an average over all angles, and sodoes not represent the true normal incidence reflectiv-ity. In the next section, we will discuss a better way toestimate the normal incidence reflectivity.

By a similar analysis, in which we assume that theseismic source creates S-waves and that we recordS-wave reflections, we find that

RSSið0°Þ ¼ RSIi ¼ρiþ1VSiþ1 − ρiþ1VSi

ρiþ1VSiþ1 þ ρiVSi¼ SIiþ1 − SIi

SIiþ1 þ SIi;

(5)

where SI is the shear impedance of each layer. Do notconfuse the reflection coefficient in equation 5 with theconverted wave shear reflectivity which, as shown in

the Appendix A, is equal to zero at normal incidence.Instead, RSI is the true S-wave reflection coefficientwhich would be recorded using a S-wave source andmulticomponent receivers. (Note that there is also anissue of polarity convention when we compare trueS-wave recording with converted S-wave recording[Brown et al., 2002]). As with the acoustic impedancereflectivity, we can rewrite the shear impedance reflec-tivity from equation 5 as

RSI ¼ΔSI2S I

; (6)

where ΔSI ¼ SIiþ1 − SIi and S I ¼ SIiþ1þSIi2 . Again, the re-

flectivity in equation 6 can be thought of as anm-samplevector, and the observed seismic trace could be writtenSAI, the convolution of RSI with the seismic wavelet.

To relate our discussion of P- and S-impedance andreflectivity to the data example shown in Figures 4 and6, Figure 7 shows the P-wave velocity log, S-wave veloc-ity log and density log recorded over the reservoir zoneat CMP location 330. Also shown are logs that havebeen derived from the velocity and density logs and willbe discussed later in this tutorial, such as VP∕VS ratio,Poisson’s ratio, extended elastic impedance (EEI) at56°, λρ, and μρ.

The logs in Figure 7 have all been integrated to timeusing the sonic log traveltimes and correlation with theseismic data. This is shown by the insertion of CMPgather 330 from Figure 4, which shows how the welllogs correlate with the seismic at the gas sand zone, in-dicated by the horizontal lines on the plot. Note thatthis is a shallow gas sand that is characterized bylow P-wave velocity, density, and VP∕VS ratio. Outside

Figure 6. The stack of the gathers in Figure 4, showing a shal-low bright-spot anomaly at 630 ms due to a gas sand. Thepicked horizon starting at 640 ms on the left side of the sectionrepresents the zero crossing from the center of the gas sand.The inserted curve at CDP 330 is the time-integrated P-wavevelocity log from the discovery well (Russell et al., 2011).

Figure 7. The time-integrated P-wave velocity log (VP), S-wave velocity log (VS), and density (ρ) logs measured over the gasreservoir at CMP 330 on Figures 4 and 6, with a number other logs derived from these logs shown in the center of the figureand CMP gather 330 shown on the right of the figure. The time scale is on the left, the depth scale on the right, and the gas sandis indicated by the horizontal lines.

SC22 Interpretation / May 2014

of the gas zone, the S-wave sonic log in Figure 7 hasbeen modeled from the P-wave sonic using the mu-drock line (Castagna et al., 1985). Inside the gas zonethe S-wave log has been modeled using Biot-Gassmannfluid substitution (Mavko et al., 1998). This modelingcould also have been done using the method proposedby Greenberg and Castagna (1992), in which moredetailed lithology parameters are used.

The Aki-Richards approximationIt has been shown (Bortfeld, 1961; Richards and

Frasier, 1976; Aki and Richards, 2002) that, for “small”changes in the P-wave velocity, S-wave velocity, anddensity across a boundary between two elastic media,the P-wave reflection coefficient for an incident P-waveas a function of incident angle can be approximated bythe linearized sum of three terms given by

RPPðθ1Þ ¼ aRVP þ bRVS þ cRD; (7)

where a¼1þ tan2θ, b¼−8ksat sin2θ, c¼1−4ksat sin2θ,RVP ¼ ΔVP

2VP, RVS ¼ ΔVS

2VS, RD ¼ Δρ

2ρ , θ ¼ θ1þθ22 is the average

of the incident, and refracted angles across a layerboundary, VP ¼ VP1þVP2

2 , VS ¼ VS1þVS22 , and ρ ¼ ρ1þρ2

2 arethe average velocity and density values across a layerboundary, ΔVP ¼ VP2 − VP1, ΔVS ¼ VS2 − VS1, andΔρ ¼ ρ2 − ρ1 are the differences of the velocity and den-

sity values across a layer boundary, ksat ¼ ðVS

VPÞ2sat

is the

shear-to-compressional wave velocity ratio for the insitu, or saturated, rocks averaged between layers. SeeFigure 5 for an illustration of a single boundary betweentwo layers, which can be generalized for N layers.

At a more advanced level, it should be noted that thelinearized approximation of Aki and Richards (2002)is based on the minimization of the ray parameterp ¼ sin θ1

VP1, which leads to a definition of the average an-

gle given by θ ¼ tan−1ðΔθ VPΔVP

Þ, where Δθ is the differ-

ence between the incident and refracted angles.Because this expression is almost iden-tical to the average of the incident andrefracted angles, the average value isnormally used instead. Thus, the termson the right side of the linearized AVOequation are always computed usingthe average angle rather than the inci-dent angle. Although there is no firmagreement as to whether the angle onthe left side of equation 7 refers to theincident angle or to the average angle,here I assume that this is the incident an-gle, because the average angle is derivedfrom the incident angle. The same com-ment applies to plots of reflection ampli-tude versus angle, where we normallyassume the angle on the horizontal axisis the incident angle.

Equation 7 is usually referred to as the Aki-Richardsequation. The convention for naming the reflection co-efficients adopted in equation 1 has been continued, sowe can think of the reflectivities in equation 7 as P-wavevelocity, S-wave velocity, and density reflectivities, re-spectively, which are vectors of m-sample length. Bysmall changes, we mean that equation 1 is valid wherethe reflectivity values are in the order of 0.1 or less,although it is still a reasonable assumption for valueslarger than this. Also, again note that the actual mea-sured seismic signal is the convolution of a wavelet withthe reflectivity of equation 7, or SPPðθÞ ¼ WRPPðθÞ, andthat the wavelet is angle-dependent, losing its higherfrequencies as the angle increases. The loss of high

Figure 9. A comparison between the full Zoeppritz equations (see Appendix A)and the two forms of the Aki-Richards equation computed over the top of a gassand, where (a) shows the intercept-gradient-curvature approximation of equa-tion 12 and (b) shows the Fatti et al. (1994) approximation of equation 8. In bothcases, the two-term and three-term sums are shown.

Figure 8. The intuitive derivation of intercept and gradient,where (a) shows the picked amplitudes at time t over a 10-trace angle gather and (b) shows the least-squares fit of thesepicks as a function of sin2θ that could be used to interpret theintercept (RAI) and gradient (RGI).

Interpretation / May 2014 SC23

frequencies is compounded by the effect of NMOstretch. This approximation is also coupled with the lin-earized angle-dependent approximation given by equa-tion 7, which starts to deviate from the full Zoeppritzsolution between 30° and 35°. However, the approxima-tion is quite good out to much larger angles as long as allthree terms are used in the approximation. (In some ap-proximations, the third term is dropped.)

There are several important algebraic rearrange-ments of equation 7 which will be made use of in sub-sequent derivations. The first one that I will discuss isby Fatti et al. (1994), which was based on earlier workby Smith and Gidlow (1987) and Gidlow et al. (1992),and is given by

RPPðθ1Þ ¼ aRAI þ bRSI þ c 0RD; (8)

where the first two weighting terms, a and b, are iden-tical to the weighting terms in equation 7 but the thirdweighting term is given by c 0 ¼ 4ksat sin2 θ − tan2 θ.Equation 8 is used as the basis for extracting the reflec-tivity terms from a CMP gather and for impedanceinversion (Simmons and Backus, 1996; Buland andOmre, 2003; Hampson et al., 2005), both of which willbe discussed later in this tutorial.

A useful way to relate the various reflectivities givenin equations 7 and 8 is to use the logarithmic approxi-mation first derived by Peterson et al. (1955), which canbe written as

RP ≈Δ ln P

2; (9)

where the letter P now refers to any parameter, as inequation 1. Equation 9 follows from equation 1 fromthe calculus formula given by d ln P

dt ¼ 1PdPdt , where we

eliminate the dt term and change the derivative opera-tor d to the difference operator Δ. Using equation 9, wefind that

RAI ≈Δ ln AI

2¼ Δ ln VP þ Δ ln ρ

2≈12

�ΔVP

VPþ Δρ

ρ

�¼ RVP þ RD; (10)

and

RSI ≈Δ ln SI

2¼ Δ ln VS þ Δ ln ρ

2≈12

�ΔVS

VSþ Δρ

ρ

�¼ RVS þ RD: (11)

A third rearrangement of equation 7 (and the mostcommon) is given by

RPPðθ1Þ ¼ a 0RAI þ b 0RGI þ c 0 0RVP; (12)

where RGI¼RVP−8ksatRVS−4ksatRD, a 0 ¼1, b 0 ¼ sin2θ,and c 0 0 ¼ tan2 θ sin2 θ. Equation 12 was initially derivedby Wiggins et al. (1983) and it is the basis of much of the

empirical AVO work performed in our industry. Theterm RGI is called the gradient and is usually writtenas B in the geophysical literature. The RAI term is calledthe intercept and is usually written as A, and the RVPterm is called the curvature and is written as C. How-ever, the A; B; C notation obscures the fact that only RGI(or B) is a new reflectivity compared with the previousmethods, and is itself a weighted sum of the reflectiv-ities given in equation 7. The subscript GI in the gradientreflectivity stands for gradient impedance and antici-pates the EEI concept that will be discussed later inthe tutorial. Equation 12 has the advantage that an es-timate of ksat is not needed in the weighting coefficientsused to extract the three parameters RAI, RGI, and RVP .Although I have written equation 12 to conform with theearlier notation, this obscures the fact that, if we dropthe third term (which is reasonable for angles less thanapproximately 30°) and write the a 0 and b 0 coefficientsexplicitly, we get the usual form of the equation, whichis written

RPPðθ1Þ ¼ RAI þ RGI sin2 θ. (13)

The terms intercept and gradient come from an in-tuitive way of understanding this method, because if wecrossplot the picked amplitudes of an angle gather at agiven time as a function of sin2 θ, and fit a straight line tothe points, the intercept of the plot gives RAI and theslope, or gradient, gives RGI. This is shown in Figure 8,where Figure 8a shows the picks on an angle gather andFigure 8b shows the derivation of the intercept and gra-dient from these picks. I will shortly discuss a morequantitative way of estimating these terms, which canbe generalized to all of the techniques discussed in thissection.

The three-term approximations from equations 8(the intercept-gradient-curvature approximation) and12 (the acoustic impedance-shear impedance-densityapproximation) are shown in Figure 9a and 9b, respec-tively, for the top of a gas sand, where VP ¼ 2250 m∕s,VS ¼ 1125 m∕s, and ρ ¼ 2110 kg∕m3 for the overlyingshale and VP ¼ 2000 m∕s, VS ¼ 1310 m∕s, and ρ ¼1995 kg∕m3 for the underlying gas sand. In Figure 9a,the full three-term form of equation 12 is shown, as wellas the truncated two-term form and the full Zoeppritzcalculation for comparison. For angles up to 30°, allthree curves agree quite well, but after 30°, the two-term truncation (which is often used) becomes lessand less accurate. In Figure 9b, the full three-term formof equation 8 is shown, as well as the truncated two-term form and the full Zoeppritz calculation for com-parison. All three curves agree quite well out to 60°,which suggests that the two-term truncation is quitea reasonable approximation to the full Zoeppritz calcu-lation, but also tells us that the extraction of the thirdterm, the density term, can be very sensitive to noise.Note that in Figure 9, the Zoeppritz calculationswere done using the true angle of incidence and theAki-Richards calculations were done using the average

SC24 Interpretation / May 2014

of the incident and refracted angle. Either angle couldhave been used to annotate the axis, but as noted ear-lier, we have adopted the usual convention and plottedthe amplitudes against incident angle.

Equations 8 and 12 are algebraic reformulations ofequation 7. Another way of reformulating equation 7 in-volves transforming to parameters which are nonli-nearly related to velocity and density. This involvesthe use of differentials as well as algebra. For example,Shuey (1985) transformed equation 12 from depend-ence on VP, VS and ρ to dependence on VP, ρ, and Pois-

son’s ratio ν ¼ V2P−2V

2S

2ðV2P−V

2SÞ. Because the first and third terms

in equation 12 are not dependent on S-wave velocity,only the second term in equation 12 changes in theShuey (1985) formulation, and thus the gradient reflec-tivity term RGI can be written as

RGI ¼ RVP − 2ðRAI þ RVPÞ�1 − 2ν1 − ν

�þ Δν

ð1 − νÞ2 ; (14)

where ν ¼ νiþ1þνi2 , and Δν ¼ νiþ1 − νi. Equation 14 is

written differently from the from the notation usedby Shuey (1985) so that it conforms to the notation usedin this tutorial. It is tempting to reformulate equation 14by introducing a Poisson’s ratio reflectivity Rν ¼ Δν

2ν , butthis would lead to unnecessary complications.

More recently, Gray et al. (1999) reformulated equa-tion 7 for two sets of fundamental constants: λsat, μsat,and ρsat (the first and second Lamé parameters and den-sity, respectively), andKsat, μsat, and ρsat (bulk modulus,shear modulus, and density, respectively), with theshear modulus being identical to the second Laméparameter), where the relationships between theseelastic constants and the saturated compressionalvelocity (VPsat) and shear velocity (VSsat) are given by

VPsat ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiλsat þ 2μ

ρsat

s¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiK sat þ 4

3 μ

ρsat

s; (15)

and

VSsat ¼ffiffiffiffiffiffiffiffiffiffiμ

ρsat;

r(16)

where the subscript sat indicates that the parametersare measured in the in situ, or saturated reservoir.(Adding this subscript is a very important point thatis neglected in many articles, as will shortly be dis-cussed). As with Shuey’s work, this reformulation re-quired the use of algebra as well as differentialsrelating λsat, μsat, and K sat to VPsat, VSsat, and ρsat. Rus-sell et al. (2011) derive a generalized equation based onBiot-Gassmann poroelasticity theory which containsboth of the Gray parameterizations as special cases.This equation is written

RPPðθ1Þ ¼ a 0 0Rf þ b 0 0Rμ þ c 0 0 0RD; (17)

where a 0 0 ¼ ð1 − ksatkdry

Þ sec2 θ2 , b 0 0 ¼ ksat

2kdrysec2 θ − 4ksat sin2 θ,

c 0 0 0 ¼ 1 − sec2 θ2 , kdry ¼ ðVS

VPÞ2dry, Rf ¼ Δf

2f, Rμ ¼ Δμ

2μ , and RD

is as previously defined. The key new parameters inequation 17 are the fluid reflectivity Rf and the squareddry rock shear-to-compressional wave velocity ratiokdry. Although an in-depth discussion of the fluid reflec-tivity is beyond the scope of this tutorial (the interestedreader is referred to Russell et al., 2011), a quick sum-mary is that the fluid term f relates the saturated anddry bulk moduli and first Lamé parameter (that is,K sat ¼ Kdry þ f and also λsat ¼ λdry þ f ) and is writtenexplicitly as

f ¼ α2M; (18)

where α ¼ 1 − Kdry

Kmis the Biot coefficient, M ¼

ðα−ϕKmþ ϕ

Kf lÞ−1 is the fluid modulus,Km is the mineral bulk

modulus, ϕ is the porosity of the reservoir, andKf l is thebulk modulus of the fluid in the pores. Thus, Δf is thechange in the fluid parameter and f is the average fluidparameter between layers. The key point to note is that,in the Gray et al. (1999) formulations, it is the reflectivityassociated with saturated bulk modulus and first Laméparameter that are extracted, whereas in the general re-flectivities Rf and Rμ of equation 17, the fluid and dryparts of reservoir are separated, using the dry rock bulkmodulus ratio kdry as a new parameter. As shown inTable 1 of Russell et al. (2011) (which shows the inverseratio, VP∕VS, squared), the value of kdry allows us to“tune” equation 16 to the two formulations of Gray et al.(1999). If kdry ¼ 3∕4 (i.e., 1/1.333 : : : ) then Rf becomesthe bulk modulus reflectivity given by

RK ¼ ΔK sat

2K sat; (19)

and if kdry ¼ 1∕2, then Rf becomes the lambda reflectiv-ity given by

Rλ ¼Δλsat2λsat

: (20)

For values of kdry less than 0.5, the fluid reflectivityRf is influenced by the type of fluid and the porosity inthe reservoir as they influence Biot’s coefficient α andfluid modulusM , which were defined earlier. For exam-ple, a value of 3/7 (i.e., 1/2.333 : : : ) corresponds to thedry rock velocity ratio squared of a clean, unconsoli-dated sandstone, and the resulting fluid reflectivity willindicate the fluid content and porosity of the reservoiritself.

Because equations 7, 8, 12, and 17 are all linearsums of three reflectivity terms multiplied by weightingterms, they can be used to extract estimates of the threereflectivities by picking and analyzing the amplitudesover all offsets or angles on a seismic gather at eachtime sample t, as shown in Figure 8a. For an N -tracegather, the forward problem can be written as

Interpretation / May 2014 SC25

26664RPPðt; θ1ÞRPPðt; θ2Þ

..

.

RPPðt; θNÞ

37775 ¼

26664a1 b1 c1a2 b2 c2... ..

. ...

aN bN cN

3777524Ra

Rb

Rc

35; (21)

where the term RPPðt; θiÞ represents the picked am-plitude at time t and angle θi, shown as the dots in Fig-ure 8a. Here, ai, bi, and ci are the primed or unprimedcoefficients given in equations 7, 8, 12, and 17, and Ra,Rb, and Rc are the reflectivities associated with these co-efficients. It is not necessary to convert the gathers fromthe offset to angle domain, as we can use the followingrelationship between the sine of the angle and the timeand offset and velocity (Walden, 1991)

sin θ ¼ XV INT

tV 2RMS

; (22)

whereX is the offset, V INT is the interval P-wave velocity,V rms is the root-mean-square (rms) P-wave velocity, and tis the two-way seismic traveltime.

Equation 21 can be solved using the standard least-squares inversion approach to give the three reflectivityterms, which can be written explicitly as

m ¼ ðGTGþ σIÞ−1GTd; (23)

where

d ¼

2666664

RPPðt; θ1ÞRPPðt; θ2Þ

..

.

RPPðt; θNÞ

3777775; G ¼

2666664

a1 b1 c1a2 b2 c2

..

. ... ..

.

aN bN cN

3777775;

m ¼

264Ra

Rb

Rc

375; I ¼

2641 0 0

0 1 0

0 0 1

375

and σ is a prewhitening factor. Once these reflectivityterms have been estimated from the prestack data ateach gather and time sample, they can be used asthe basis for the impedance inversion methods dis-cussed in the next section, or they can be combinedin various ways.

The use of equations 12 and 23 allows us to estimatethe intercept (RAI) and gradient (RGI) and, optionally,the curvature (RVP) in a more quantitative way thanwas shown in Figure 8b. Figure 10, from Russell et al.(2011), shows the product of the intercept and the gra-dient, and clearly shows the gas sand anomaly in red.The reason for this is that, in the type of gas sand shownin Figure 7, called a Class III gas sand, the interceptand gradient are both negative at the top of the gas sandand both positive at the base of the gas sand and thustheir product is positive at the top and base of the gassand, as seen in Figure 10.

Another technique that is commonly used to help lo-cate a gas sand is to crossplot the gradient against theintercept and to identify anomalous outliers on thiscrossplot (Foster et al., 1993). Figure 11 shows sucha crossplot from a 100-ms window centered at thepicked event seen in Figure 10, between CMPs 300and 356. On Figure 11, the red polygonal zone capturesoutliers that have negative intercept and gradient val-ues, and thus correspond to the top of the gas sand;the blue polygon captures outliers that have positive in-tercept and gradient values, and thus correspond to thebase of the gas sand. This is seen in Figure 12, where thecolors on the section correspond to time and CMP val-ues captured by the equivalently colored polygon inFigure 11. The roughly elliptical set of points that areuncolored in Figure 11, and trend at an approximate−45° line on the crossplot, correspond to wet sandsand shales, as shown by Castagna et al. (1998). Thisis obviously an oversimplification, because the clusterof points around the origin in Figure 11 is not confinedto a single linear trend. More recently, Foster et al.(2010) have shown that, although shale upon shale val-ues occupy this approximate −45° line and go throughthe origin, shale upon sand values do not go throughthe origin. Thus, the points seen as the “wet” trendin Figure 11 come from different nongas chargedscenarios.

Although the displays shown in Figures 10–12 clearlyhighlight the gas sand zone, they are qualitative inter-pretations, and tell us nothing about the physical prop-erties of the gas sand. A more quantitative approachinvolves extracting the reflectivities given in equation 8by applying equation 23 with the appropriate weightingcoefficients. That is, we estimate RAI, the acousticimpedance reflectivity, RSI, the shear impedance reflec-tivity, and optionally RD, the density reflectivity. To dothis, we also need an estimate of ksat (the saturated S- toP-wave velocity ratio squared) in the weighting coeffi-cients, usually obtained by smoothing the P-wave veloc-ity estimate from a well log or rms velocity function anddividing this into the smoothed S-wave velocity. The

Figure 10. The intercept times gradient section for the gath-ers shown in Figure 4, where the picked event starting at640 ms on the left of the section represents the transition fromthe top to the base of the gas sand. The amplitudes have beennormalized between −1 and þ1. Note the good definition ofthe gas sand between CMPs 316 and 346 by the product sec-tion, which is positive at the top and base of the sand (Russellet al., 2011).

SC26 Interpretation / May 2014

smoothed S-wave velocity can be estimated from the P-wave velocity using the mudrock equation proposed byCastagna et al. (1985), and given by

VP ¼ dþ eVS; (24)

where d and e are either locally derived scalingcoefficients or are taken directly from the Castagnaet al. (1985) paper, in which case d ¼ 1360 m∕s ande ¼ 1.16. Note that this equation was also used by Cas-tagna et al. (1998) to show that the cluster of points atroughly −45° on the crossplot of Figure 11 correspondsto wet sands and shales.

Another important AVO indicator is the fluid factorΔF , which is the fluid deviation away from the mudrockline defined in equation 24 (Smith and Gidlow, 1987;Fatti et al., 1994). As shown in equation 25, the fluid fac-tor is the weighted difference between the extractedacoustic impedance and shear impedance reflectivities.Because this is a new reflectivity, we will call it RF(notice the use of capital F to differentiate this termfrom the fluid term that came out of equation 17, whichwe denoted by small f ). In our notation, the fluid factorcan be written

RF ¼ RAI − 1.16ksatRSI; (25)

where the value of 1.16 has been used for the coefficiente in equation 24.

Figure 13, from Russell et al. (2011), shows the fluidfactor section computed from the CMP gathers inFigure 4. The color scale has been normalized betweenþ1 and −1, where red shows an increase in the fluidresponse and blue shows a decrease. Note the clearlydefined fluid anomaly between 620 and 640 ms.

For the extraction of the fluid reflectivity in equa-tion 17, we need an estimate of kdry, the dry S, to P-wavevelocity ratio squared, which can be obtained determin-istically by estimating the dry rock ratio using Biot-Gassmann analysis, or empirically by finding theoptimum rotation on a well log crossplot or an invertedf ρ − μρ crossplot (discussed in the next section). Notethat this last approach moves us from reflectivity toimpedance, which will also be discussed in the nextsection. As a final illustration of reflectivity methods,Figure 14 shows the extraction of the fluid modulusfrom the gathers of Figure 4 using a kdry value of1/2.333, or 0.429, which corresponds to a clean, highlyporous sand.

Impedance methodsThe inversion of seismic data to produce an estimate

of acoustic impedance was first proposed by Lindseth(1979). The basic concept was to recursively invertequation 3 to give

AIiþ1 ¼ AIi

�1þ RAIi

1 − RAIi

�: (26)

If our seismic trace consisted of the full-bandwidth,noise-free reflectivity shown in Figure 1, the recursiveapplication of equation 26 would recover the completeimpedance profile shown in Figure 1, assuming that ourestimate of the starting impedance was accurate. How-ever, the fact that the seismic trace is band limited withthe seismic wavelet as shown in equation 2 means thatonly the impedance within the seismic bandwidth canbe recovered. In many cases, this is sufficient andrelative impedance inversion as implemented by equa-tion 26 is commonly applied. It is sometimes preferableto produce a full-bandwidth estimate of the impedancefrom the seismic volume, which involves incorporatingthe missing frequency parts of the impedance fromsome external model. The model can be created byinserting well log impedance profiles at their correct po-sition on a seismic volume and interpolating the imped-

Figure 11. AVO intercept versus gradient crossplot of a zoneof the intercept and gradient sections taken from a 100-mswindow around the event picked in Figure 10, between CMPs300 and 360. The regions captured by the red and blue polygo-nal zones are shown in Figure 12.

Figure 12. The captured red and blue polygonal crossplotzones from Figure 11, where the red zone shows the top ofthe gas sand and the blue zone shows the base of the gas sand.

Interpretation / May 2014 SC27

ances while guiding the structure using picked seismicevents.

Once the model is built, the simplest approach toextracting the low frequencies is to apply a low passfilter to the model and add these low-frequency imped-ances to the recursively inverted traces. However, sev-eral other approaches have been developed over theyears (Oldenburg et al., 1983; Russell and Hampson,1991), the details of which are outside the scope of thispaper. The concept behind all amplitude inversionmeth-ods can be captured in Figure 15, in which we constrainsome type of seismic volume with the information froma model volume using a particular inversion algorithm.

The fundamental question before starting an inver-sion are therefore: What model volume and seismic vol-ume should we use? Traditionally, acoustic impedanceinversion has been done using the stacked seismic vol-ume, but as explained earlier, this is actually incorrect,because the stack is an average over all angles, whereasthe acoustic impedance reflectivity is the normal inci-dence P-wave reflectivity. Thus, a more accurate inputto acoustic impedance inversion would be the band-limited acoustic impedance reflectivity (SAI ¼ WRAI)

estimated from the seismic gathers using equation 23with the coefficients from equation 8 as input. Likewise,the other two outputs from this reflectivity extraction,SSI ¼ WRSI and SD ¼ WRD, can be used as inputs forshear impedance and density inversion. The model vol-umes in Figure 15 would be derived from the P-imped-ance, S-impedance, and density values from the well logmeasurements.

Any inversion algorithm can be used in this process,as long as the model volume used in Figure 15 is theparameter to be inverted for and the seismic volumeis the band-limited reflectivity which has a physical linkto this earth parameter. An alternate approach is to usesimultaneous inversion (Buland and Omre, 2003; Hamp-son et al., 2005) in which equation 8 is still used as thebasis for the band-limited reflectivity extraction, butrock physics constraints are included. In this approach,the extraction of the band-limited reflectivities is notperformed directly and the inversion input consistsof angle gathers, a P-impedance model, and relation-ships between P-impedance, S-impedance, and density.

Once the impedance inversion has been done, thereare several approaches that can be used to interpretthe results. A straightforward method is to divide theP-impedance volume by the S-impedance volume to cre-ate a VP∕VS ratio volume and to crossplot this againstP-impedance using the rock physics template (RPT) ap-proach (Odegaard and Avseth, 2003). Anomalous fluidzones can then be picked on the crossplot and high-lighted on the seismic volume. Figure 16 shows theP-impedance inversion of the gathers in Figure 4 usinga simultaneous inversion approach, and Figure 17shows the VP∕VS ratio that was derived by dividingthe P-impedance inversion of these gathers by the S-impedance inversion. Figure 18 then shows the cross-plot of the two volumes, where a polygonal zone hasbeen identified which has low P-impedance andVP∕VS ratio. Figure 19 shows the region of the seismicstack that corresponds to the polygonal zone inFigure 18. This region correlates very well with theknown gas sand.

Once the gathers have been inverted to P- andS-impedance, the results can be transformed tolambda-rho (λρ) and mu-rho (μρ) volumes (Goodwayet al., 1997), where

Figure 13. The fluid-factor (ΔF) section for the gathersshown in Figure 4, where the picked event starting at640 ms on the left of the section represents the transition fromthe top to the base of the gas sand. As in the intercept timesgradient product section of Figure 8, note the good definitionof the gas sand between CMPs 316 and 346

Figure 14. The fluid reflectivity (Rf ) extraction for the gath-ers shown in Figure 4, where the picked event starting at640 ms on the left of the section represents the transition fromthe top to the base of the gas sand. Note the decrease in thefluid reflectivity at the top of the gas sand, and the good def-inition of the gas sand between CDPs 316 and 346. Figure 15. The basic flow for all seismic inversion methods.

SC28 Interpretation / May 2014

λρ ¼ AI2 − 2SI2; (27)

and

μρ ¼ SI2: (28)

These volumes can also be crossplotted and anoma-lous fluid zones picked on the crossplot highlighted onthe seismic volume, as in the RPT approach. Theoreti-cally, a crossplot of μρ against λρ should show a verticalseparation between the gas sands and the wet sands.That is, gas sands should plot on the low side of theλρ axis. A generalization of the λρ concept was pro-posed by Russell et al. (2003), in which a fluid volumeis created using a more general form of equation 28given by

f ρ ¼ AI2 − k−1drySI2; (29)

where f ρ is the full bandwidth fluid term which is equiv-alent to the fluid term f defined in equation 18 multi-plied by the density. Notice that, when k−1dry is equalto 2, equation 29 reduces to equation 27 and f ρ reducesto λρ. As in the three-term generalized AVO fluid equa-tion (equation 17), this approach uses Biot-Gassmannporoelasticity theory, so the fluid term f ρ in equation 29could be found by inverting the fluid reflectivity Rfin equation 17 and multiplying this by the inverted

density. The advantage of extracting the reflectivityand inverting it is that the inverted results do not needto be squared, as in equation 29. By squaring the imped-ance data, the noise is also squared, which could resultin instability in some data areas. Of course, there arealso potential problems with the nonsquared inversionmethods, most notably the fact that the low-frequencytrend cannot be estimated from the seismic data due tobandwidth limitations and also that the density term isvery sensitive to noise and the limited angle range foundin many surveys. One approach to improving the den-sity estimate is to use simultaneous inversion (Hamp-son et al., 2005), in which rock physics constraintsare built into the inversion algorithm. A final point tonote is that equation 29 also gives us an empiricalway to estimate kdry, by changing this value until thecrossplot of μρ versus f ρ shows the best vertical sepa-ration between the gas sands and the wet sands.

Yet another approach to using the inverted P- andS-impedance results, which also avoids squaring theimpedance, was proposed by Quackenbush et al.

Figure 16. P-impedance estimation from the gathers ofFigure 4 using the simultaneous inversion approach.

Figure 17. The VP∕VS ratio estimation from the gathers ofFigure 4 using the simultaneous inversion approach.

Figure 18. Crossplot of VP∕VS ratio inversion from Figure 17with P-impedance inversion of Figure 16 over a 100 ms extrac-tion window around the gas sand zone between CMPs 300 and360. The polygonal zone highlights the gas sand, which haslow impedance and VP∕VS ratio.

Figure 19. The red zone shows the extent of the polygonalzone in Figure 18 mapped back onto the seismic section. Thewiggle traces show the P-impedance.

Interpretation / May 2014 SC29

(2006) and is called Poisson impedance, or PI. Theequation for PI can be written

PI ¼ AI − cSI; (30)

where c is a derived scale factor which Quackenbushet al. (2006) propose setting to

ffiffiffi2

p. This corresponds

to the square root of the factor 2 used in standardLMR approach, and could therefore be generalizedusing the Biot-Gassmann equations to a value ofk−1∕2dry . Notice that equation 30 defines a rotation of anAI − SI crossplot to find an optimum fluid or lithologydiscrimination.

Elastic impedance (EI)So far, we have discussed several different types of

impedances: acoustic, shear, Poisson, lambda-rho, andmu-rho, which are derived from acoustic and shearimpedance. Connolly (1999) has proposed a new typeof impedance which can be thought of as the imped-ance underlying the angle dependent reflectivity de-rived by Aki and Richards (2002), which he calls EIand defines as

EIðθÞ ¼ VaPV

bSρ

c; (31)

where a ¼ 1 þ tan2 θ, b ¼ −8ksatsin2 θ, and c ¼1 − 4ksatsin2 θ. Note that the ksat term in equation 31is a constant which is averaged over the zone of inter-est. If the coefficients a, b, and c look familiar, it is be-cause we first saw them as the scaling coefficients forthe Aki-Richards equation (equation 7). To see whythese two equations are related, we first take the loga-rithm of equation 31, to get

ln EIðθÞ ¼ a ln VP þ b ln VS þ c ln ρ: (32)Next, we take the differentials of the parameters in

equation 32 to get

Δ ln EIðθÞ ¼ aΔ ln VP þ bΔ ln VS þ cΔ ln ρ: (33)

Finally, we use the logarithmic approximation ofequation 9 (and also a scale factor of 1/2 to keep thereflectivity definitions consistent) to get

REIðθÞ ¼ΔEIðθÞ2E IðθÞ ¼ a

ΔVP

2VPþ b

ΔVS

2VSþ c

Δ ρ

2 ρ; (34)

which gets us back to the original Aki-Richards equa-tion. I have introduced the term REIðθÞ instead ofRPPðθÞ to continue making the reflectivity subscriptconform to the underlying impedance model. Also notethat the EI reflectivity defined in equation 34 has thesame form as the acoustic and shear impedance formsseen in equations 4 and 6, which is the difference of theimpedance across a boundary divided by twice its aver-age. Regardless of what name we apply to the EI reflec-

tivity, in its band-limited form it corresponds to a tracetaken from the seismic angle gather at an angle of θ.That is, SEIðθÞ ¼ WREIðθÞ is the band-limited trace at an-gle θ. Thus, to create the inverted EI volume, we use themethodology suggested in Figure 15, where a constantangle stack is the seismic input, the model volume isderived from equation 34, and your preferred inversionalgorithm is used.

As pointed out by Whitcombe (2002), the limitationof EI is its variable dimensionality, an observationwhich was first made by VerWest et al. (2000). Thatis, as the angle of incidence increases, the value ofEI goes down. To correct for this problem, Whitcombe(2002) introduces the following normalized versionof EI

EIðθÞ ¼ VP0ρ0

��VP

VP0

�a�VS

VS0

�b�ρ

ρ0

�c�; (35)

where VP0, VS0, and ρ are reference constants that arecomputed by averaging the P- and S-wave velocity anddensity over the zone of interest. This scaled version ofthe EI inversion can be computed by using equation 35to create the model volume that is input to the EI inver-sion. Whitcombe et al. (2002) extends this scaling to theconcept of EEI. In EEI, the Aki-Richards equation(equation 12) is modified by dropping the third termand transforming the b coefficient from sin2 θ totan χ, where χ represents a rotation angle in inter-cept-gradient space. The resulting equation is thenmultiplied by cos χ, giving the following form of theEEI reflectivity

REEIðχÞ ¼ RAI cos χ þ RGI sin χ: (36)

The transformation from θ to χ also allows us to re-write the scaled EI model (equation 35) as a function ofχ, giving the following EEI model

EEIðχÞ ¼ VP0ρ0

��VP

VP0

�p�VS

VS0

�q�ρ

ρ0

�r�; (37)

where p¼cos χþsin χ, q ¼ −8ksat sin χ, and r ¼ cos χ− 4ksat sin χ. EEI inversion can be implemented usingthe flowchart in Figure 15 by using equation 37 to createthe model volume and equation 36 to create the seismicvolume. As Whitcombe et al. (2002) show, the interpre-tational advantage of EEI is that at different χ angles itcorrelates to different physical parameters such as bulkmodulus, shear modulus, lambda, VP∕VS ratio, etc. Twospecial angles are 0° and 90°, at which we see fromequation 36 that the EEI equation reduces to RAI andRGI, respectively. Thus, EEI (0°) is identical to acousticimpedance and EEI (90°) can be called gradient imped-ance, or GI, which motivated the subscript to the gra-dient reflectivity when it was introduced earlier.

The optimum χ angle is found by correlation betweenEEI (χ) and a parameter of interest. As an example ofEEI, Figure 20 shows the correlation VP∕VS ratio from

SC30 Interpretation / May 2014

the logs in Figure 7 for our Colony example with EEI (χ)at various χ angles. From Figure 20, we see that the bestcorrelation value is found at an angle of 56°. Referringto Figure 7, note that the VP∕VS ratio and EEI (56°)have been plotted side by side in that figure and thatthe two curves look very similar except for the scalingbetween them.

Figure 21 shows the computation of the EEI reflec-tivity at the computed angle of 56° from the correlationin Figure 20. This is the seismic section that is invertedto give the final EEI result. Finally, Figure 22 shows thecomputation of VP∕VS ratio for our gas sand exampleusing the EEI log model from Figure 7 and seismicmodel from Figure 21 with a rotation angle of 56°.Because EEI does not scale directly to the parameterof interest, the results in Figure 22 have been scaledto VP∕VS ratio values.

Extension to anisotropic mediaSo far in this tutorial, we have assumed only the iso-

tropic case. More recently, AVO and prestack inversionhave been extended to anisotropic reservoirs, usingtechniques such as amplitude versus azimuth (AVAz).In this tutorial, I will focus only on vertically trans-versely isotropic (VTI) and horizontally transverselyisotropic (HTI) media (Thomsen, 1993; Rüger, 1997,1998), but the analysis can also be extended to ortho-rhombicmedia (Tsvankin, 1997; Rüger, 2002). Figure 23,from Rüger (1997), shows schematically the differencebetween VTI and HTI media, where Figure 23a showsthat a VTI medium can be modeled by parallel sheetsin the horizontal, or x1, direction, and an HTI mediumcan be modeled by parallel sheets in the vertical, or x3,direction, and Figure 23b shows the two angles neededin the following equations. For VTI media, we will needonly the incidence angle θ, but in HTI media we will re-quire the incidence the angle θ and the azimuth angle ϕ.

For VTI media, Rüger (1997) expanded previouswork by Thomsen (1993) to show that we could writean extended version of equation 12 as

RVTIðθÞ ¼ RAI þ�RGI þ

Δδ2

�sin2 θ

þ�RVP þ Δε

2

�sin2 θ tan2 θ; (38)

where Δδ and Δε are the change in Thomsen’s δ and εparameters (Thomsen, 1986) across the layer interface.The presence of VTI anisotropy will affect the AVO re-flectivity response and will depend on the magnitudeand sign of the change in the anisotropic parametersbetween layers. However, when the terms in bracketsare extracted using equation 21, these anisotropic termscannot be separated into their isotropic and anisotropiccomponents. Figure 24 shows a set of modeled iso-tropic and anisotropic AVO curves over Class 1, 2,and 3 gas and wet sands, adapted from Blangy(1994). The three Classes were proposed by Rutherfordand Williams (1989), and are separated based on theiracoustic impedance contrast with respect to the sur-rounding shale, where a Class 1 sand has a higherimpedance, a Class 2 sand has a near zero change inimpedance, and a Class 3 sand has a lower impedancewith respect to the shale. Although Rutherford and Wil-liams (1989) focus only on gas sands, Blangy (1994) ex-

Figure 20. The correlation between EEIðχÞ and VP∕VS ratiofrom the logs shown in Figure 7. Notice that the best corre-lation is at an angle of 56°.

Figure 21. The application of the rotation angle of 56° usingequation 36, that is REEIð56°Þ ¼ RAI cosð56°Þ þ RGI sinð56°Þ.This is the seismic section that is inverted to give the EEIresult.

Figure 22. The computation of VP∕VS ratio for our gas sandexample using the EEI log model from Figure 7 and seismicmodel from Figure 21 with a rotation angle of 56°.

Interpretation / May 2014 SC31

tends the models to wet sands to illustrate the effect ofan anisotropic shale above the sand. In each model, thegas sand is isotropic but the shale above the gas sandexhibits VTI anisotropy with δ ¼ 0.15 and ε ¼ 0.30.Table 1 shows the complete set of parameters for themodels.

For the gas case, Figure 24a shows that the effect ofanisotropy is to increase the AVO effect more thanwould be expected. For the wet case, shown in Fig-ure 24b, the effect of anisotropy is to make the wet re-sponse look like a gas response. Although it could beargued that these models are slightly unrealistic dueto the fact that the P-wave velocity does not change be-

tween the wet and gas cases, nonetheless they do showthat the effect of an anisotropic shale over an isotropicsand is a situation that should be modeled before under-taking any drilling program based on the AVO results.

For HTI reservoirs, the reflectivity is now dependenton the average incident angle θ and the azimuth φ (seeFigure 23b), as shown in the following equation fromRüger (1997)

RHTIðθ;ϕÞ ¼ RAI þ ðRGI þ RHTIB cos2 ϕÞsin2 θþ ðRVP þ RHTIC cos2 ϕÞsin2 θ tan2 θ; (39)

where RHTIB ¼ ΔδðVÞ2 þ 8ksat

Δγ2 , RHTIC ¼ ΔδðVÞ

2 sin2 ϕ − ΔεðVÞ2 ,

ΔδðVÞ and ΔεðVÞ are Thomsen’s delta and epsilon param-eters in HTI media, γ is Thomsen’s gamma parameter,

and ksat ¼�VS

VP

�2

sat. Again, I have changed the normal

terminology, in which RHTIB is usually called Bani, to re-present that the fact that this term is an HTI reflectivityassociated with the second, or B, term in the AVO ex-pression.

Although it is theoretically possible to extract RHTIBand RHTIC if we have a full range of azimuths and inci-dence angles, in practice only RHTIB is within the noiserange of typical recorded gathers. Thus, equation 21 canbe modified by dropping the third term in equation 39,but dividing the second term into two parts, as shownhere26664

RPPðθ1;ϕ1ÞRPPðθ2;ϕ1Þ

..

.

RPPðθN;ϕMÞ

37775¼

266641 sin2 θ1 sin2 θ1 cos2ϕ1

1 sin2 θ1 sin2 θ2 cos2ϕ1

..

. ... ..

.

1 sin2 θ1 sin2 θN cos2ϕM

3777524 RAI

RGI

RHTIB

35;

(40)

which can be inverted for RAI, RGI, and RHTIB. As shownby Bakulin et al. (2000), RHTIB, or Bani as they refer to

this term, is proportional to the fracturedensity. Therefore, by extracting theanisotropic gradient in addition to theisotropic intercept and gradient usingthe inverse of equation 40, we can esti-mate the fracture density. Note that inequations 39 and 40, we have used ϕto denote the known azimuth. Anotherunknown is the angle β, which is the an-gle between the chosen zero azimuth di-rection and the direction of the fracturesthemselves. This unknown can be intro-duced into equation 39 by replacingcos2 ϕ with cos2 ðϕ − βÞ, which makesthe equation nonlinear. As shown byJenner (2002), the angle β can also beestimated by reparameterizing the prob-lem. A similar technique was used byGray and Head (2000) to estimate frac-ture direction and density in the Man-

Figure 23. Schematic models for VTI and HTI media, where(a) shows VTI medium can be modeled by parallel sheets inthe horizontal direction, an HTI medium can be modeled byparallel sheets in the vertical direction, and (b) illustrates thatVTI media are defined using only incidence angle θ but HTImedia require incidence angle θ and azimuth angle ϕ (fromRüger, 1997, 1998).

Figure 24. Modeled AVO responses for a shale over (a) a gas sand and (b) a wetsand. The solid line shows the response for an isotropic shale and the dashed lineshows the response for a VTI shale with δ ¼ 0.15 and ε ¼ 0.30 (Adapted fromBlangy, 1994).

SC32 Interpretation / May 2014

derson Field, Wyoming. Figure 25, from Gray and Head(2000), shows the results that were obtained by the au-thors by estimating the fracture density from RHTIB andalso fracture orientation from the angle β. As a finalnote, it should be pointed out that the method just de-scribed has a 90° phase ambiguity in the estimation ofthe azimuth. Recent work (Downton et al., 2011) usingthe Fourier coefficient method shows promise in resolv-ing this phase ambiguity.

ConclusionsIn this tutorial, I have presented an overview of the

historical and current techniques that are in use for pre-stack seismic amplitude analysis, including standardAVO techniques and also prestack seismic inversiontechniques. I have shown that these techniques canbe classified as being based on the computation andanalysis of either some type of seismic reflection coef-ficient series or seismic impedance. Those techniquesthat are based on the computation and analysis of seis-

mic reflection coefficient series, or seismic reflectivityfor short, are called AVO methods, and those that arebased on the computation and analysis of seismic imped-ance are referred to as prestack amplitude inversionmethods. The seismic reflectivity methods that we dis-cussed were intercept versus gradient analysis, P- andS-impedance reflectivity extraction and fluid factoranalysis. The seismic impedance methods that we dis-cussed were P- and S-impedance inversion (the indepen-dent and simultaneous approaches), lambda-mu-rho(LMR) analysis, PI inversion, EI, and EEI inversion.The techniqueswere illustrated using a 2D seismic exam-ple over a gas sand reservoir from Alberta. Although welargely focused on isotropic methods, the last part of thetutorial extended the analysis to anisotropic reservoirs,and we discussed VTI and HTI media.

Of course, it is one thing to classify the methods, butit is another thing entirely to know where and when toapply them. The case study used throughout the mainpart of this tutorial was a clean, isotropic, shallow, high-porosity, highly permeable gas sand from the WesternCanadian Sedimentary Basin (WCSB), and the recordeddata had very high signal-to-noise ratio, so everymethod described did a good job in its delineation.However, as we move to deeper, less-porous sands withlarger percentages of clay content, some of the methodsdescribed here start to perform less well. For example,the intercept/gradient crossplot becomes more difficultto interpret as the resolution of the seismic data is re-duced and the noise content increases. Carbonate res-ervoirs are typically more difficult to interpret usingprestack seismic data than clastic reservoirs, althoughit depends on the type of carbonate. For example,vugular dolomitic type carbonate reservoirs can be in-terpreted in a similar fashion to clastic reservoirs. How-ever, as we move to fractured carbonates, the AVAztechniques described in the previous section becomecrucially important. In general, as the reservoirs be-come more complex, it is my feeling that the prestacktechnique used should also become more complex. Forexample, the inversion techniques described that pro-duce various types of impedance can also reveal subtlefeatures that the reflectivity methods cannot.

As can be seen by the discussion in the previousparagraph, it is impossible to separate the quality ofthe prestack analysis results from the quality of the ac-quisition and processing of the data itself. At the ac-quisition phase, it is important to get the largestangular range possible, as well as a good range of azi-muths if AVAz is the goal. At the processing stage thekey steps to consider are true amplitude preservation,good random and coherent noise elimination, high-frequency signal recovery, and an imaging techniquethat is not only kinematically correct but also pre-serves amplitudes. For very highly structural data,this last step may be unobtainable and the use ofprestack AVO for inversion in very complex areas likesubsalt plays can be problematic. Of course, each ofthe topics just mentioned could lead to its own tutorial.

Figure 25. The estimation of fracture strike and density us-ing the inverse of equation 40 (modified by the inclusion offracture orientation angle β) in the Manderson Field, Wyoming(adapted from Gray and Head, 2000).

Table 1. The parameters for the Type (or Class, whichis the more common terminology) I, II, and III sandsfor the anisotropic study shown in Figure 24 (fromBlangy, 1994).

Material properties VP (m∕s) VS (m∕s) Density δ ε

Overlying shale 3300 1700 2.35 0.15 0.30

Type I gas sand 4200 2700 2.35 0.0 0.0

Type I water sand 4200 2100 2.45 0.0 0.0

Overlying shale 2896 1402 2.25 0.15 0.30

Type II gas sand 3322 2215 2.00 0.0 0.0

Type II water sand 3322 1402 2.25 0.0 0.0

Overlying shale 2307 1108 2.15 0.15 0.30

Type III gas sand 1951 1301 1.95 0.0 0.0

Type III water sand 1951 930 2.20 0.0 0.0

Interpretation / May 2014 SC33

Also, there is no clear consensus on how to achievemany of these processing goals, so the interpreteris encouraged to communicate closely with the proc-essor, and produce the necessary QC plots afterevery step.

Finally, the importance of modeling cannot be over-stressed. There is no point in getting excited about a red“blob” on your AVO results if you have no idea what thatred blob means. It is important to perform many mod-eling steps, including fluid replacement with reasonablerock physics parameters, synthetic modeling with theoption to add events other than just primaries (e.g., con-verted waves and multiples), and realistic wavelet gen-eration. In this tutorial, we have focused on isotropicBiot-Gassmann fluid replacement modeling, but thereare many different techniques available to the inter-preter.

The key thing that I have tried to emphasize in thistutorial is that virtually all of the techniques used inprestack amplitude analysis are derived from a three-(or two-) term linearized approximation to the Zoep-pritz equations, and that they can be classified as ei-ther reflectivity methods or impedance methods Theadvantage of staying in the seismic reflectivity domainis that this is the “natural” domain in which seismicdata is recorded. However, for a geologic analysis ofearth properties, the low-frequency trend of the vari-ous earth parameters is often preferred, and this in-volves the inversion of the seismic reflectivity. Ifinversion is your preferred method of analysis, it isimportant to understand how the low-frequency com-ponent is being incorporated into the final resultand also to understand the possible pitfalls of eachdifferent method. Every method discussed in thistutorial has value, and it is only by comparing themethods on your particular data set, and calibratingthe results against models built from well logs, thatyou will be able to make an informed decision on aparticular method’s usefulness when compared to allthe others.

AcknowledgmentsI want to express my thanks to my longtime col-

league and collaborator Dan Hampson for our fruitfuldiscussions and business and scientific partnershipover the years, as well as my colleague Jon Downton,and a host of industry friends fromwhom I have learnedso much, such as John Castagna, Fred Hilterman, LeonThomsen, George Smith, Maurice Gidlow, Bill Good-way, and David Gray. I also want to thank Fred Hilter-man, Jon Bork, and an anonymous reviewer for theirthorough and thoughtful reviews which helped improvethis tutorial.

Appendix A

The Zoeppritz equations and normal incidenceZoeppritz (1919) showed how to compute the ampli-

tudes of the reflected and transmitted waves shown in

Figure 3 using the conservation of stress and displace-ment across the layer boundary. This gives a forwardmodel with four equations and four unknowns (theamplitudes) which can be arranged in matrix notationas

266666664

− sin θ1 − cos ϕ1 sin θ2 cos ϕ2

cos θ1 − sin ϕ1 cos θ2 − sin ϕ2

sin 2θ1VP1VS1

cos 2ϕ1ρ2V2

S2VP1

ρ1V2S1VP2

sin 2θ2ρ2VS2VP1

ρ1V2S1

cos 2ϕ2

− cos 2ϕ1VS1VP1

sin 2ϕ1ρ2VP2ρ1VP1

cos 2ϕ2 − ρ2VS2ρ1VP1

sin 2ϕ2

377777775

×

26666664

RPP ðθ1ÞRPS ðθ1ÞTPP ðθ1ÞTPS ðθ1Þ

37777775¼

2666664

sin θ1

cos θ1

sin 2θ1

cos 2ϕ1

3777775. (A-1)

Inverting the matrix form of the Zoeppritz equationsin A-1 gives us the exact amplitudes as a function ofangle

26666664

RPP ðθ1ÞRPS ðθ1ÞTPP ðθ1ÞTPS ðθ1Þ

37777775¼

26666664

−sin θ1 −cosϕ1 sin θ2 cosϕ2

cos θ1 −sinϕ1 cos θ2 −sinϕ2

sin 2θ1VP1VS1

cos 2ϕ1ρ2V2

S2VP1

ρ1V2S1VP2

sin 2θ2ρ2VS2VP1

ρ1V 2S1

cos 2ϕ2

−cos 2ϕ1VS1VP1

sin 2ϕ1ρ2VP2ρ1VP1

cos 2ϕ2 − ρ2VS2ρ1VP1

sin 2ϕ2

37777775

−1

×

26666664

sin θ1

cos θ1

sin 2θ1

cos 2ϕ1

37777775. ðA-2Þ

Note that the matrix in equations A-1 and A-2 is asimplification of the general solution given in Aki andRichards (2002) (equation 5-37, page 141) in whichall possible combinations of input P- and SV-planewaves are solved for. Solving equation A-2 will leadto a complicated set of equations, and we discuss inthe tutorial a linearized form of this inversion for theRPP term, called the Aki-Richards approximation.However, the normal incidence case leads to the follow-ing simplification, given as follows

2666664RPPð0°ÞRPSð0°ÞTPPð0°ÞTPSð0°Þ

3777775 ¼

2666664

0 −1 0 1

1 0 1 0

0 VP1VS1

0 ρ2VS2VP1

ρ1V2S1

−1 0 ρ2VP2ρ1VP1

0

3777775

−12666640

1

0

1

377775; (A-3)

with solution

SC34 Interpretation / May 2014

266664RPPð0°ÞRPSð0°ÞTPPð0°ÞTPSð0°Þ

377775

¼

2666666664

0 ρ2VP2ρ2VP2þρ1VP1

0 −ρ1VP1ρ2VP2þρ1VP1

−ρ2VS2ρ2VS2þρ1VS1

0−ρ1V2

S1VP1ðρ2VS2þρ1VS1Þ 0

0 ρ1VP1ρ2VP2þρ1VP1

0 ρ1VP1ρ2VP2þρ1VP1

ρ1VS1ρ2VS2þρ1VS1

0ρ1V2

S1VP1ðρ2VS2þρ1VS1Þ 0

3777777775

2666640

1

0

1

377775:

(A-4)

The normal incidence reflection and transmissioncoefficients are therefore given as

RS0 ¼ TS0 ¼ 0; RP0 ¼ρ2VP2 − ρ1VP1

ρ2VP2 þ ρ1VP1;

TP0 ¼2ρ1VP1

ρ2VP2 þ ρ1VP1¼ 1 − RP0:

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Brian Russell received a B.S. (1975)from the University of Saskatchewan,an M.S. (1978) from Durham Univer-sity, and a Ph.D. (2002) from the Uni-versity of Calgary, all in geophysics.He is vice president, software forHampson-Russell, A CGG Company,a CGG Fellow, and an adjunct profes-sor in the Department of Geoscience

at the University of Calgary. His research interests includerock physics, seismic inversion, and seismic attributeanalysis. He is a past president and honorary member ofSEG and CSEG, a member of EAGE, and received theSEG Cecil Green Enterprise Award in 1996.

SC36 Interpretation / May 2014