Amplitude variation with offsetinterpretation doesn’t have to be ablack box.

AUTHORSRoger Young and Robert LoPiccolo, e-Seis

Even before S.R. Rutherford and R.H.Williams proposed their three-foldclassification of amplitude variation

offset (AVO) types in 1989, the geophysicalcommunity has sought to extract value fromthe information contained in the unstackedseismic gathers. A large number of schemesfor organizing, simplifying or portrayinginformation extracted from the gathers hasbeen proposed. And, while a lot of thesehave merit, one of the unintendedconsequences is that a lot of people think ofthe subject as fundamentally arcane andimpenetrable. Nothing could be furtherfrom the truth.

Basic backgroundModern seismic data are acquired in such away that each point in the subsurface issampled multiple times during each survey.Each sample comes from a differentcombination of seismic sources (shots) andgeophones and, therefore, each sample willhave a different angular relationshipbetween the shot, the sample location andthe geophone (see Figure 1). As thegeophones are moved and different shotsare set off, a vast library of data is developed.

A number of procedures are appliedduring routine processing of the data toensure that the spatial (and temporal)coordinates of each sample are known aswell as possible. The result of this processingis that a catalog of information is assembledfor each X-Y location in the survey. Eachcatalog is known as a common-midpointgather (gather) and is comprised of traces.Each trace represents the reflected energyfrom the series of all time samples beneaththe X-Y location of that gather; differenttraces in the gather result from differentangular relationships between shots andgeophones (see Figure 1 and the left panelof Figure 2).

These traces, in each gather, contain alarge amount of useful information whichmay be made available to the geophysical

interpreter as he seeks to broaden hisunderstanding of the subsurface throughseismic. Unfortunately, there’s no good wayto visualize and map this information in itsraw form. It has to be reduced to somethingmanageable.

AVO analysisReducing the information into somethinguseful is a two-step process. First, theinterpreter must decide what informationhe or she wants to extract. Then, a displaydecision must be made. In the case of thestack, the extracted information is theaverage amplitude for each time sample ofall of the traces in the gather (see Figure 2for a graphic representation of this). It isdisplayed either as a wiggle or with somecolor code designed to reveal themagnitude and direction of the averageamplitude. The big advantage of the stack isthat, while all of the AVO information islost, a lot of the random errors associatedwith each individual sample are eliminated,resulting in a very robust and stable look atthe reflectivity of the subsurface.

AVO analysis is not very much morecomplicated than stacking the data. Figure2 shows a set of traces in a gather. Thesetraces have been selected so that they all arerepresentative of a line of observationsextending vertically from a particular pointon the surface to some pre-selected depth

(in two-way, acoustic travel time). They havebeen adjusted so that correspondingsamples on different traces reference thesame two-way travel time. They are arrangedin order with the trace coming from theshot-receiver pair closest to the gather lyingon the left. Remember that all of the tracesin a gather have the same X-Y location.

Also shown on Figure 2 is a graph. Thisgraph plots the value of the amplitude foreach trace at time t2 on the ordinate (or Y-axis) against sin2� on the abscissa (or X-axis), where � is the angle between the shot(or receiver) and the line of the trace.Higher angles obviously are associated withfarther offsets. For reasons that arediscussed elsewhere, sin2� ransforms therelationship between amplitude and offsetinto a more-linear one. Because therelationship, as graphed, is nearly linear, asimple, least-squares, best-fit line can beused to describe the distribution of theamplitudes (shown on Figure 2 as a grayline going through the data points). Thisline can be completely and uniquelycharacterized by two parameters: the valuewhere it intersects the ordinate (called “P”),and its slope (called “G”). To the extent thatthe line fully describes the distribution ofthe amplitudes, these two parameters (Pand G) fully describe the line and,therefore, the amplitude variation withoffset (AVO).

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AVO analysis demystified

Figure 1. This diagram shows how the angular relationship varies from trace to trace ina common midpoint seismic gather. Each set of colored lines represents the ray paths ofsome of the energy contributing to an individual trace in the seismic gather. (All imagescourtesy of e-Seis)

The terms P and G often go by othernames. The value of P is defined wherethere is no offset; hence, P is often referredto as the “zero-offset” amplitude. A zero-offset would, hypothetically, occur wherethe shot and receiver were at the same placeand the energy went straight down andcame straight back, yielding another namefor P: the “Normal Incident” amplitude. Inaddition to “Slope,” G is often referred to asthe “Gradient” or the “AVO Gradient.”

There are other pieces of information orAVO parameters which may be extracted

from the traces or from the graph of theiramplitudes. Sometimes a subset of thetraces such as those reflected through asmall angle (or from a short offset distance)are stacked independently of the othersand, in this case, referred to as “Nears.”Other groupings, such as those reflectedthrough a large angle or an intermediateangle, are called “Fars” or “Mids.”

This is all of the information one is likelyto see extracted from an “AVO Analysis.”There are other, more esoteric valuessometimes extracted. For example, the sin2

transformation does not make therelationship strictly linear; another term isrequired, and it has petrophysicalsignificance. The problem is that real dataare almost invariably too noisy for that extraterm to have any real value. So, practicallyspeaking, we have the following AVOparameters from which to make ourinterpretations: the Ps and Gs, whichtogether completely define the gathers; thestack, which averages all of the informationfrom the gathers; and the nears, mids andfars, which average subsets of theinformation in the gathers.

Displaying AVO informationThis is where the geophysical communityhas gotten really very inventive. As we’veseen, the Ps and Gs effectively contain all ofthe information from the gathers. It’sinstructive, therefore, to start with the Psand Gs and then see how other productsrelate to them.

Figure 3 is a plot of all of the values of Pand of G from a seismic dataset, cross-plotted against each other; we refer to thisplot as representing AVO space. The left-hand panel of Figure 3 is obviously of littlevalue; it shows that there is some regularityto the data, but little more. The right-handpanel of Figure 3 shows the same data color-coded to highlight the amplitudes of thestacked dataset. The distribution of thecolors should come as no surprise: Anegative amplitude is a trough and a bunchof trace amplitudes distributed so as to yielda strongly negative P with a stronglynegative gradient (G) that will have thehighest average negative amplitudes (large,dark-blue troughs on the stack).Superimposed on the right-hand panel arethe three regions defined by Rutherfordand Williams’ AVO classes. Clearly there’ssomething interesting going on here, butit’s not well-defined by either the stack orthe classes. The stack shows a fairlydiscriminating pattern of amplitudesbetween the classes, but the same range ispresent outside the classes. And the classesfail to cover more than a small portion ofthe possible combinations.

Applying the same approach of lookingat a single parameter from the AVO analysiswithin AVO space to the Ps, Gs, nears andfars yields a similar conclusion. As shown inFigure 4, any single parameter fails to show

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Figure 2. A basic AVO analysis after appropriate geometric corrections the individualtraces for one gather are displayed in the panel on the left; the colors correspond to thecolors in Figure 1. A graph may be constructed at any point in time (e.g. t1, t2 or t3) toanalyze the amplitude variation with offset of the traces at that time; the graph at theright shows one such plot for time t2. The gray line is a least-squares, best-fit to thenatural distribution; values for P and G are extracted from analysis of the best-fit line.Triangle shows the average amplitude of all of the traces at t2.

Figure 3. On the left, a cross-plot shows P versus G for every time sample for everygather in a volume of seismic data. The data have been normalized to constrain P andG to the same range of values. On the right, the same cross-plot has been color-coded toshow the stacked amplitude of each point; light to dark blues are low to high amplitudetroughs, and pink to dark reds are low to high amplitude peaks. The three regionsdenoted 1, 2 and 3 are defined by the three AVO classes of Rutherford and Williams.

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much of the information that’s inherent inthe data. Schemes to combine two of theparameters by multiplying them only makethings worse (see Figure 4, also).

There are techniques, however, whichcan capture a lot of the richness of AVOspace and put it in a form that is useful to aninterpreter. One such approach isillustrated in Figure 5. A couple ofunderlying assumptions and observationsare necessary. The first is that changes inlithology, porosity and fluids affect wherepoints plot in AVO space. The second is thatthe background lithologic trend is shaleand that shale-to-nonshale contrasts will

plot away from the background. When AVO space is divided using this

simple approach, everything begins tomake sense. The power of the Rutherfordand Williams approach is now distributedover the entire AVO space. The richness ofthe information in the gathers is capturedand divided into geologically meaningfuldomains.

In the approach that is illustrated inFigure 5a, AVO space is divided into 10AVO types. An alternate division of theAVO space (Figure 5b) yields informationthat is sensible from a gross lithologyperspective. The assumptions, in this case,

are simple: The background trend is shale,and deviations from the background trendare shale to nonshale contacts. Finally, itcan be seen that there is generally aregular distribution of porosity within AVO space; this distribution is shown inFigure 5c.

There’s no black box in any of this, justuseful information derived from AVOanalysis and displayed in a familiar andintelligible format. The interpreter can nowlook at sections in terms that are useful andintuitive.

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Figure 4. These cross-plots show the distribution of AVOparameters in AVO space. The datasets and cross-plottingparameters are identical to those in Figure 3. In panels a and b,the parameters P and G (which completely define the linearaspects of AVO space) appear as uniformly varying, one-dimensional gradients across AVO space. In panels c and d, thenears and fars show similar one-dimensional gradients, inclined at angles intermediate between those shown by P and G. In panelse and f, two of the AVO parameters are combined and the resultingattribute is displayed in color. In panels a, c and e there is gooddiscrimination by the value of the AVO parameter (orcombination) between the classes of Rutherford and Williams but, as with the stack, there is the same range in values outside the classes.

Figure 5. Using the same datasets as Figures 3 and 4, the AVO spacehas been subdivided into more rational and useful domains. In 5athe entire space has been divided into AVO classes by extending theoriginal classifications around the entire unit circle. In 5b the samespace has been divided into domains that are reflective of lithologicvariations: The shale background (shades of green) is close to theorigin; non-shale rocks, which are softer than shales, are yellows andbrowns (reds and maroons are extreme departures, which may beindicative of compressible fluids); relatively hard non-shale rocksare shades gray and blue. In 5c the entire data set is divided, yetagain, to reflect variations in porosity — low-porosity rocks areobserved to fall in the southeast part of the plot and higher porositiesfall to the northwest. Panels d, e and f show seismic sectionsprepared using the domain divisions illustrated by panels a, b and c.