elastic impedance

It is now commonplace for 3-D data sets to be processedas partial offset volumes to exploit the AVO informationin the data. However, there has been significant asymme-try in the way these volumes could be calibrated andinverted. The amplitudes of near-offset, or intercept, stacksrelate to changes in acoustic impedance and can be tied towell logs using synthetics based on acoustic impedance(AI) or inverted, to some extent, back to AI using poststackinversion algorithms. However, there have been no sim-ple analogous processes for far-offset stacks.

The symmetry can be largely restored using a functionI call elastic impedance (EI). This is a generalization ofacoustic impedance for variable incidence angle. EI pro-vides a consistent and absolute framework to calibrateand invert nonzero-offset seismic data just as AI does forzero-offset data. EI, an approximation derived from a lin-earization of the Zoeppritz equations (Appendix, part 1),is accurate enough for widespread application.

As might be expected, EI is a function of P-wave veloc-ity, S-wave velocity, density, and incidence angle. To relateEI to seismic, the stacked data must be some form of anglestack rather than a constant range of offsets. There are sev-eral ways of constructing suitable data sets by either care-ful mute design or by linear combination of intercept andgradient functions. (Part 2 of the Appendix reviews thesemethods.)

EI was initially developed by BP in the early 1990s tohelp exploration and development in the Atlantic Marginsprovince, west of the Shetlands, where Tertiary reservoirsare typified by class II and class III AVO responses. Figure1 shows a suite of logs from the Foinaven discovery welldrilled in 1992. The 30° elastic-impedance log, EI(30), isbroadly similar in appearance to the acoustic-impedancelog although the absolute numbers are lower; it is a prop-

erty of EI that the level decreases with increasing angle.At this well, the sands are predominantly class III and

so have slightly higher amplitudes at 30° than at normalincidence. This can be more clearly seen in Figure 2 inwhich the EI log has been scaled to have approximatelythe same shale baseline as the AI log. When the sands areclass II, a more dramatic difference is evident between theAI and EI logs.

The seismic data around Foinaven suffer from verystrong peg-leg multiples. Even after demultiple, the sig-nal-to-noise ratio of the near-trace data is often poor, espe-cially from the class II events, whereas the far-offset dataare generally of good quality. EI allows the well data to betied directly to the high-angle seismic which can then becalibrated and inverted without reference to the near off-sets.

Figure 3 shows part of the EI(30) log from anotherFoinaven well overlain on an inverted 30° angle stack. Thedata were inverted using a constrained sparse spike algo-rithm for which the EI log provided the basis for the con-straints and was used to QC the result.

An EI log provides an absolute frame of reference andso can also calibrate the inverted data to any desired rockproperty with which it correlates. In the case of Foinaven,a strong correlation was found between EI(30) and hydro-carbon pore volume, and this relationship was used toestimate the in-place volumes for the field from the inverted30° seismic volume.

Figure 4 shows a section from the inverted 30° volumeused to design the trajectory of the first high-angle devel-opment well. The oil sands correlated closely with theareas of low elastic impedance. The EI volume was usedto design the trajectories of all subsequent developmentwells.

438 THE LEADING EDGE APRIL 1999 APRIL 1999 THE LEADING EDGE 0000

Elastic impedancePATRICK CONNOLLY, BP Amoco, Houston, Texas, U.S.

Figure 1. Comparison of an AI curve with a 30° EI curve for the Foinaven discovery well 204/24a-2.

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The EI formula is an approximation and may not beapplicable in all circumstances; however, the loss of accu-racy is easy to calculate and minimize (Appendix, part 3).In most situations, more general seismic data quality issuesand particularly uncertainty in the estimation of incidenceangle are probably larger than errors in the implied reflec-tivity from the EI values.

Estimating Poisson’s ratio from seismic data hasprompted much comment in the literature and was the sub-ject of a workshop at SEG’s 1998 Annual Meeting. Oneapproach is to invert a 90° angle stack (see part 2 of theAppendix) which, in theory, has amplitudes that areapproximately proportional to changes in Poisson’s ratio.However, the construction of quantitatively accuratePoisson’s ratio stacks is notoriously difficult because of sen-sitivity to residual moveout and bandwidth variations.

EI can provide an optimum compromise. Part 4 of theAppendix shows how one variant of EI has values equalto AI at normal incidence and to (Vp/Vs)2 at 90° (this beingclosely related to Poisson’s ratio) with a smooth transition

between. This allows the user to construct as high an anglestack as is stable and then to calibrate or invert it using theequivalent EI log.

Because of the difficulties and uncertainties of con-structing angle stacks, a method of quality-controlling theresults using available well data is important. EI providesa simple mechanism to produce synthetic seismograms forvariable incidence angle. A Vp term can be factored out ofthe EI expression and the remaining angle-dependentexpression can be used in place of the density log in con-ventional synthetics software (equation 1.3). The Vp log isthen calibrated with a time-depth relationship in the usualway.

Figure 5 shows near- and far-offset ties to a west ofShetlands well. There is much variation of amplitudes


Figure 3. Part of an EI(30) log overlain on the inverted30° angle stack. The log was used to constrain a con-ventional poststack sparse spike inversion and to QCthe result.

Figure 4. A section through the inverted 30° volume, showing the path of the first development well. The locationof oil-bearing sands encountered by the well correlates with the areas of low elastic impedance (yellow).

Figure 2. Detail from Figure 1, but with the EI(30)curve scaled so that the shale baseline isapproximately the same as the AI curve. This showsthe percentage decrease in impedance at the oil-sandinterface is greater than 30° at normal incidence, con-sistent with the class III response of these sands.

with offset in this area, and the two angle stacks are quitedifferent. Despite this, both well ties are of good quality.

A principal benefit of EI within BP has been its valueas a communication and integration tool. EI allows AVOinformation to be displayed in a way that can be under-stood more intuitively by nongeophysical specialists. It iseasily incorporated into petrophysical systems allowingAVO information to be communicated throughout theearth-science community. EI can be used to display rock-property data, either from wireline or core measurements,in a way that can be directly related to far-offset stacks.

Shear-wave data are now recorded routinely in manywells so the calculation of, say, an EI(30) log within anypetrophysical package is straightforward. The combination

of an AI and EI curve is often simpler to relate to the seis-mic response than, say, Vs or Poisson’s ratio logs. An exam-ple is shown in Figure 6 that is a standard BP petrophysicaldisplay from the Gulf of Mexico.

With this type of data established within a petrophys-ical database the EI concept can help with more generalrock-property studies. Figure 7 shows AI/EI crossplots ofdata from 19 Gulf of Mexico wells for shales, brine, andoil sands. By measuring average impedance values wecan quickly estimate the AVO response of various lithol-ogy combinations. This particular data set, for example,shows that the percentage increase in amplitude from 0-30° for a shale/brine sand interface (~18%) is almost exactlythe same as for a shale/oil sand interface (~17%). So, for


Figure 6. Standard petrophysical display for a Gulf of Mexico well (MC619-1). The two right tracks show the AIand EI(30) curves. In this example, the upper sand would be expected to generate little response at normalincidence and a tough-peak pair at far offsets.

Figure 5. Low- and high-angle synthetic ties for a west of Shetlands well. The left side of the display is a conven-tional AI synthetic match to a 10° angle stack. The right side is a 30° EI-based synthetic tied to a 30° angle stack.

these data, AVO gradient would be a poor fluid indicator.However, looking at average values only tells part

of the story. Figure 8 shows simplified, Gaussian fre-quency curves of the AI and EI distributions of the threelithologies.

In each case the normalized standard deviations of theEI(30) data are less than those of the AI data. The area ofoverlap between the oil and brine-sand values at 30° is lessthan half that at normal incidence. I find that many datasets have this characteristic (i.e., that EI values are more


Figure 9. The relationship between oil saturation and AI (left) and EI(30) (right) from core sample measurementsfrom the Foinanven Field.

Figure 8. Gaussian curves equivalent to the histograms of Figure 7 showing the distribution of AI and EI(30) valuesfor the three lithologies. The normalized standard deviations for the shale, brine-sand, and oil-sand AI data are0.15, 0.12, and 0.11. They are 0.10, 0.08, and 0.09 for the EI data.

Figure 7. AI against EI(30) crossplots of data from 10 Gulf of Mexico wells for shales (left), brine sands (center), andoil sands (right). Average AI and EI values can be read from the histograms and from these reflection coefficientsfor any lithology combination at normal and 30° incidence. These data show almost the same percentage increase inamplitude for a shale/brine-sand interface as a shale/oil-sand interface.

uniform than AI values for a given lithology). This impliesthat most forms of amplitude analysis would be lessambiguous at higher incidence angles than lower.

Finally, an example of using EI to display the resultsof core measurement data. The Foinaven Field is the sub-ject of a 4-D, time-lapse seismic experiment and variouscore sample measurements have been made to calibratethe results. Figure 9 shows the relationship between oil sat-uration, AI, and EI(30). Clearly, the far offsets are more sen-sitive to changing saturation than the near ones.

In summary EI is pragmatic technology. It allows atleast first-order AVO effects to be incorporated routinelyinto seismic and rock-property analysis and interpreta-

tion, using no specialist software and with minimal increasein effort. This allows for routine extraction of quantitativeAVO information from large 3-D volumes. LE

Acknowledgments: The examples are the work of many people. I’d espe-cially acknowledge and thank current and former BP Atlantic Marginscolleagues Mike Cooper, Dave Cowper, Robert Hanna, Mike Currie, andDave Lynch and our joint venture partners Shell UK for support andencouragement. Also Ed Meanley, Sue Raikes, Terry Redshaw, StanDavis, and Wayne Wendt for additional help and both Shell and BPAmoco for permission to publish this paper.

Corresponding author: Patrick Connolly, [email protected]


Appendix

1) Derivation. Equation 1.1, a well known linearization ofthe Zoeppritz equations for P-wave reflectivity, is accuratefor small changes of elastic parameters for subcriticalangles.

(1.1)where

and where

and similarly for the other variables (NB. For ease of nota-tion, the “bars” will be omitted from the averaged Vs

2/Vp2

ratios.)We require a function f(t) which has properties analo-

gous to acoustic impedance, such that reflectivity can bederived from the formula given below for any incidenceangle �

Call this function EI (elastic impedance), and use the alter-native log derivation for reflectivity which is accurate forsmall to moderate changes in impedance;

and so,

substituting K for Vs2/Vp

2 and rearranging

but sin2θtan2θ = tan2θ – sin2θ, so

Note that had we used only the first two terms of (1.1),then the above and following expressions differ only bychanging the tan2θ to sin2θ. We substitute again ∆lnx for∆x/x;

now if we make K a constant we can take all terms insidethe ∆s;

and finally we integrate and exponentiate (i.e., remove thedifferential and logarithmic terms on both sides), settingthe integration constant to zero:

(1.2)

An alternative form, with a Vp term factored out, can beused for generating synthetics (see main text).

(1.3)

2) Angle stacks. The ideal angle stack has amplitudes thatrelate to a specific incidence angle over a long time win-dow and has enhanced signal to noise. In other words itshould approximate as closely as possible a band-limitedconstant angle reflectivity sequence. The construction ofan angle stack requires knowledge of the relationshipsbetween offset and incidence angle and between angle

and amplitude. These are both potentially complex areasbut if we restrict ourselves to first-order approximationsthen angle stacks can be calculated with little extra effortbeyond conventional stacking.

These approximations limit applicability to data forwhich the two-term moveout and Dix equations are validand for which amplitudes are proportional to sin2�, (where� is incidence angle). This effectively means layer-cakegeometry, offset less than depth and incidence angle lessthan 30-35°. We are also limiting ourselves to an isotropicmedium. This is probably the most severe limitation;increasing anisotropy will distort raypaths and alter AVObehavior. Correctly balanced, “true” prestack amplitudesare also assumed.

The following expression relates incidence angle to off-set given the above constraints.

(2.1)

where θ = incidence angle,x = offset,t0 = zero offset two way timevi = interval velocityvr = rms velocity

There are essentially two methods of constructing anglestacks; the first uses linear combinations of intercept andgradient, and the second uses averaging between appro-priate muting functions. The former can also be reformu-lated as a weighted stack.

As is well known from Shuey’s classic paper, ampli-tudes are linear with sin2� up to about 30-35°. Therefore,using (2.1) we can estimate sin2� for each sample througha CDP time slice and fit a regression line through theamplitude values up to the maximum appropriate angle.Numerous AVO attribute values can then be constructedfrom this line: the intercept, the gradient, Poisson’s ratiostack (the value at 0=90°) or the value for any intermedi-ate angle (called Finite Angle Stack within BP). This entireprocess can be efficiently programmed to run almost asquickly as a conventional stack.

Perhaps a more intuitive way to look at this process isto rearrange it as a weighted stack. The simple linearregression formula for intercept and gradient formula canbe rearranged as weighting functions as follows, where Xis sin2�, Y is the equivalent amplitude value, and N is stackfold.

A linear combination of these will provide a weightingfunction for any desired angle stack A(�).

Figure A1 shows a typical suite of weighting functions forone time slice to output a range of finite angle stacks from0° (intercept) to 90° (Poisson’s ratio). This more closelyshows the similarity between this process and partial stacking.

An intercept stack is seen to be similar to a near-offsetstack and a midangle stack is similar to a far-offset stack.Higher angle stacks are projections beyond the range ofrecorded data which must exploit the difference betweennear and far offsets. This will inevitably be very sensitiveto residual moveout and phase and bandwidth variationswhich explains some of the difficulty in trying to estimatePoisson’s ratio values from P-wave seismic data.

The second way to produce an angle stack is to designan appropriate muting function. Equation 2.1 can be inte-grated with respect to x to give the average value of sin2θfor a range of offsets stacked between x1 and x2.

(2.2)

Because amplitude is linear with sin2�, the average ampli-tude of the stack will also correspond to this angle.

Either the outer or inner mute can be fixed and thenits pair calculated. For example, for a high-angle stack theouter mute could be a 35° mute calculated from equation2.1 possibly combined with the maximum offset-depend-ent upon acquisition geometry. The inner mute to providesome target angle could then be calculated from equation2.2. In practice a fixed angle can usually only be achievedover some limited window. An example of this process isshown in Figure A2. (Note that the final muting functionsshould be smoothed to some extent.)

This second method of angle stack construction has theadvantage that only conventional stacking software isrequired. Its disadvantages are that it assumes regulargeometry and is only capable of producing stacks at a lim-ited range of angles. The regression approach is far moreflexible.

Of all the errors implicit in the EI/angle-stack approachalmost certainly the largest is the estimate of incidenceangle. Every effort should be made to minimize this andcertainly this should include reestimating the angles at eachvelocity control point when constructing angle stacks.

3) Accuracy. The derivation of the EI formula (Appendix,part 1) requires the Vs/Vp ratio in the exponentials be kept


Figure A1. Finite angle stack weighting functions.

constant for the entire time series (or constant within anysystem in which absolute comparisons are being made).This reduces the accuracy of the derived reflectivity com-pared with that obtained directly from equation 1.1 forwhich the Vs/Vp ratio can be set to be the average acrosseach interface. Again substituting K for Vs

2/Vp2, if ∆K is

the difference between the true local value and the constant value, then from equation 1.1 the error in thereflection coefficient is

(3.1)

This expression can be used to calculate the rms level ofthe error for the entire log. As an example, the error wascalculated for the 204/24a-2 data and is displayed in FigureA3 as signal-to-noise for a range of K values.

This provides a mechanism to obtain the optimum Kvalue for any data set. In this example the signal-to-noiseof 12 should be more than adequate for most purposes.For comparison the signal-to-noise from ignoring AVOand using an AI approximation is 1.3 which would beunacceptable for quantitative analysis. In general errorsintroduced by the EI approximation will probably be muchsmaller than those arising from the estimation of incidenceangle.

It is possible to derive a correction to an EI log suchthat the derived reflectivity is accurate for a locally aver-aged K value rather than the constant value. The correc-tion is, however, recursive, dependent on the overburdenand hence loses the prime advantage of the EI function—that it depends only on instantaneous local properties. IfK is the locally averaged value and K is the constant valuethe following expression is calculated for each sample.

(3.2)

R is the ratio of the correction for the (i-1)th sample to theith sample so the correction works by calculating the run-ning sum of this expression and then multiplying the EIvalues by this factor (Figure A4).

4) High-angle inversions. Both methods for constructingangle stacks (outlined in Appendix, part 2) are based onthe first order AVO equation whereas the EI derivation(Appendix, part 1) is based on the second-order equation.Below 30°, this makes little difference, but if we wish to


Figure A2. A typical stacking velocity function and its Dix interval equivalent. From these and equation 2.1, anouter 35° mute to exclude the nonlinear AVO is derived. Then, from equation 2.2, an inner-trace mute is calculatedto give an average stack angle of 25°.

Figure A3. The ratio of the rms level of the R(30)reflectivity from (1.1) divided by the EI error termfrom (3.1) for a range of K values for the 204/24a-2data. The maximum signal-to-noise is about 12 andcorresponds to a K value of about 0.21.

two-

way

-tim

e (m

s)

two-

way

-tim

e (m

s)

40000

35000

30000

25000

20000

15000

10000

50000

16000

14000

12000

10000

8000

6000

4000

20000

calibrate higher-angle stacks constructed using the regres-sion line projection method then we should use the firstorder EI1 variation which I’ll denote with the subscript. Asnoted in Appendix 1, this is the same as the second-orderversion but with the tangent in the Vp exponential termchanging to a sine.

(4.1)

I’ve already shown that EI(0) = AI and from 4.1 we cannow see that if we let K = 0.25 then EI1(90) = (Vp/Vs)2. Theabsolute levels of EI1(90) will depend on exactly whichvalue for K is being used but relative amplitudes shouldalways be approximately proportional to (Vp/Vs)2. And ofcourse (Vp/Vs)2 can be easily transformed into Poisson’sratio which in appearance is a very similar function.

Figure A5 shows a suite of EI1 functions for the 204/24a-2 well with the gamma and log resistivity curvesfor reference and showing comparisons with (Vp/Vs)2 andPoisson’s ratio curves. The curve values have all been nor-malized. The optimized K value of 0.21 was used for theEI calculations, but the EI1(90) is almost identical to the(Vp/Vs)2 curve.

EI1 curves therefore provide a smooth transitionbetween AI and (Vp/Vs)2 functions. In principle a corre-sponding angle stack could be constructed and invertedfor any desired angle but in practice these will becomeincreasingly unreliable at higher angles. The EI concept,however, allows an optimum balance to be chosen.


Figure A4. An example of the application of the EI correction (3.2) to the 204/24a-2 data. The effect of the correctionis small for this example.

Figure A5. Comparison of a suite of EI1 curves with(Vp/Vs)2 and a Poisson’s ratio curve for the 204/24a-2well. Gamma and log resistivity included for refer-ence. The EI1 curves form a continuum between an AIand a (Vp/Vs)2 curve.

elastic impedance

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