the old and the new in seismic inversion

The old and the new in seismic inversionBrian Russell and Dan HampsonVeritas Hampson-Russell

Introduction

Seismic inversion is a technique that has been in use bygeophysicists for over forty years. Early inversion techniquestransformed the seismic data into P-impedance (the productof density and P-wave velocity), from which we were able tomake predictions about lithology and porosity. However,these predictions were somewhat ambiguous since P-imped-ance is sensitive to lithology, fluid and porosity effects, and itis difficult to separate the influence of each effect. To performa less ambiguous interpretation of our inversion results, we must perform full elastic inversion, in which we estimateP-impedance, S -impedance (the product of density and S-wave velocity) and density. The reason for this can be seenin Figure 1, which plots the P and S-wave velocities as a function of gas saturation. In this figure, it can be noted thatthe P -wave velocity drops dramatically when gas is introduced into the reservoir whereas the S -wave velocity is largely unaffected by the introduction of the gas.

This talk will present both a history of seismic inversion andan overview of the inversion techniques themselves.

December 2006 CSEG RECORDER 5

★ ★ ★ ★ ★ ★

DECEMBER LUNCHEONDATE: December 18, 2006

TIME: 11:30 A.M. Lunch

LOCATION: Telus Convention Centre, Calgary

TICKETS: Contact CSEG office

TELEPHONE: 262-0015 or Fax: 262-7383

JANUARY LUNCHEONJanuary 22, 2007

“DHI / AVO best practices methodology and applications”

William A. FahmySEG / AAPG Fall 2006 Distinguished Lecturer

Brian Russell started his career as an exploration geophysicist with Chevron in 1976, and worked forChevron affiliates in both Calgary and Houston. He then worked for Teknica Resource Development Ltd.and Veritas Seismic Ltd. in Calgary before co-founding Hampson-Russell Software Ltd. in 1987 with DanHampson. Hampson-Russell develops and markets seismic inversion software which is used by oil andgas companies throughout the world. Since 2002, Hampson-Russell has been a fully owned subsidiary ofVeritasDGC. Brian is currently Vice President of Veritas Hampson-Russell and is involved in bothgeophysical research and training. He is also an Adjunct Professor in the Department of Geology andGeophysics at the University of Calgary and is involved with CREWES (Consortium for Research in

Elastic Wave Exploration Seismology) and CHORUS (Consortium for Heavy Oil Research by University Scientists).

Brian was President of the CSEG in 1991, received the CSEG Meritorious Service Award in 1995, the CSEG medal in 1999,and CSEG Honorary Membership in 2001. He served as chairman of The Leading Edge editorial board in 1995, technicalco-chairman of the 1996 SEG annual meeting in Denver, and as President of SEG in 1998. In 1996, Brian and Dan Hampsonwere jointly awarded the SEG Enterprise Award, and in 2005 Brian received Life Membership from SEG.

Brian has presented numerous technical papers at geophysical conferences around the world, including the SEG, EAGE,CSEG and ASEG conferences. His papers have been published in Geophysics, The Leading Edge, Exploration Geophysicsand The Journal of Petroleum Geology. His book “Introduction to seismic inversion methods”, based on course notes froman SEG Continuing Education course, was published by the Society of Exploration Geophysicists in 1988.

Brian holds a B.Sc. in Geophysics from the University of Saskatchewan, a M.Sc. in Geophysics from the Durham University,U.K., and a Ph.D. in Geophysics from the University of Calgary. He is registered as a Professional Geophysicist in Alberta.

F i g u re 1. The effect of gas saturation on P and S-wave velocity.

Continued on Page 7

Seismic interpretation

The seismic reflection method was developed in the first quarterof the twentieth century and was used initially as a tool for iden-tifying structures, such as anticlines, which could act as trappingmechanisms for hydrocarbon reservoirs. Such a structure can beseen in Figure 2, which shows a seismic line recorded over a gas-charged sand. The high on the picked structure corresponds tothe gas sand. The inserted curve is the P -wave sonic log. But thisstructural interpretation is ambiguous when it comes to identi-fying gas sands. By the 1970s, geophysicists had started to realizethat information was contained in the amplitudes of the seismicreflections themselves. This information could be correlated withp o rosity changes, lithology changes, or even fluid changeswithin the subsurface of the earth.

The amplitude anomalies on a seismic section became known as“bright-spots” and were considered to correlate well with gassands. A typical “bright-spot” is highlighted by the rectangle inFigure 1, and is associated with the gas sand. Unfortunately,“bright-spots” were also ambiguous with respect to identifyingfluid anomalies, which lead to the development of the AVO(amplitude variations with offset) technique, to be discussedlater. However, let us first look at how we can “invert” thesection shown in Figure 2.

Post-stack seismic inversion

The seismic traces in the stacked seismic section shown in Figure2 can be modelled as the convolution of the earth’s reflectivityand a bandlimited seismic wavelet, which can be written

(1)

where st is the seismic trace, wt is the seismic wavelet and rt is thereflectivity. The reflectivity, in turn, is related to the acousticimpedance of the earth by

(2)

where rP i is the zero-offset P -wave reflection coefficient at the ithinterface of a stack of N layers and ZP i=ρi VP i is the ith ρ-impedanceof the ith layer, where ρ is density, VP is P-wave velocity and *denotes convolution. Lindseth (1979) showed that if we assumethat the recorded seismic signal is as given in equation (2), wecan invert this equation to recover the P-impedance using therecursive equation given by

(3)

By applying equation (3) to a seismic trace we can eff e c t i v e l ytransform, or invert, the seismic reflection data to P- i m p e d a n c e .H o w e v e r, as also recognized by Lindseth, there are a number ofp roblems with this pro c e d u re. The most severe problem is thatthe re c o rded seismic trace is not the reflectivity given in equation(2) but rather the convolutional model given in equation (1). The effect of the bandlimited wavelet is to remove the lowf requency component of the re f l e c t i v i t y, meaning that it cannot bere c o v e red by the recursive inversion pro c e d u re of equation (3).

After proper processing and scaling of the seismic data, an intuitivea p p roach to recovering the low frequency component is to simplyextract this component from well log data and add it back to theseismic. However, this is a fairly ad-hoc pro c e d u re, and a morerecent approach to inversion is called model-based inversion(Russell and Hampson, 1991). In model-based inversion we startwith a low frequency model of the P-impedance and then perturbthis model until we obtain a good fit between the seismic data and

Continued on Page 8


Luncheon Cont’dThe old and the new in seismic inversion

Continued from Page 5

F i g u re 3. The extracted wavelet from the seismic data of Figure 3, where (a) is the time response and (b) is the frequency re s p o n s e .

F i g u re 2. A seismic section from Alberta, in which the picked even between a timeof 600 and 650 ms re p resents a seismic structure and the rectangle highlights anamplitude anomaly, or “bright spot”, associated with the gas sand.

a synthetic trace computed by applying equations (1) and (2). Bothrecursive and model-based inversion use the assumption that wehave extracted a good estimate of the seismic wavelet. Theextracted wavelet from the stacked section of Figure 2 is shown inF i g u re 3, where the time domain response of the wavelet is shownon the left, and its frequency domain response on the right.

F i g u re 4 shows the inverted sections for the seismic line in Figure2, where the top inversion was done using the recursive techniqueand the bottom inversion was done using model-based inversion.

In Figure 4, notice that although the model-based results looks alittle more geologically reasonable (it has a blockier, lesssmoothed appearance, and less dramatic swings), both inver-sions show a low-impedance zone at the gas sand zone, which isto be expected. However, there are also low impedance zoneselsewhere on both inversions, probably due to shales. Thus, lowimpedance associated with “bright” amplitudes is not an unam-biguous indicator of a gas sand.

Pre-stack simultaneous inversion

The standard seismic data processing flow involves trans-forming a set of CMP gathers into a stacked section. For example,a few of the processed gathers that were used to create thesection shown in Figure 2 are shown in Figure 5, along with a“cut-out” of the stacked section within the zone corresponding tothese gathers.

The assumption behind stacking is that the amplitudes on thegather do not show much variation, so that stacking can beconsidered as simply a noise cancellation technique. However, ifwe look carefully at the gathers in Figure 5 around the zone ofinterest (630 ms) it is clear that the amplitudes show a lot of vari-ation from the near offset on the left to the far offset on the right.In this case, we see an increase in amplitude, often associatedwith an anomaly in which the impedance of the gas layer is lessthan that of the surrounding shales.

The reason behind this can be seen in Figure 6, which shows thatan incident P-wave at an angle θ results in reflected and trans-mitted P and S-waves. This is called mode conversion, and theamplitudes of the reflected and transmitted waves can becomputed using the Zoeppritz equations (Zoeppritz, 1919).

Continued on Page 9

8 CSEG RECORDER December 2006

Luncheon Cont’dThe old and the new in seismic inversionContinued from Page 7

F i g u re 5. Several of the CDP gathers used to create the stacked section of Figure 2a re shown at the top of this diagram, with their location on the final stack shown atthe bottom.

F i g u re 6. Mode conversion of an incident P-wave on an elastic boundary.

F i g u re 4. Inversion of the seismic data shown in Figure 2, where (a) shows re c u r-sive inversion and (b) shows model-based inversion, and the colour bar on the right.



F i g u re 7. Crossplots of (a)l n ( ZD) vs ln(ZP) and (b)

l n ( ZS) versus ln(ZP)w h e re, in both cases, a best

straight line fit has beenadded. The deviations

away from this straightline, ΔLD and ΔLS, are the

d e s i red fluid anomalies.

F i g u re 8. Simultaneous P-impedance inversion ofthe pre-stack data shown

in Figure 5, where theellipse highlights the gas

sand zone.

F i g u re 9. The VP/ VS r a t i ofound by dividing

the inverted P and S-impedance sections fro msimultaneous inversion,

w h e re the ellipsehighlights the gas

sand zone.

(a) (b)

Continued on Page 10


Continued on Page 11

10 CSEG RECORDER December 2006

F i g u re 10. The results of cross-plotting the VP/ VS ratio seen in Figure 6 against the P-impedance seen in Figure 5, where (a) shows the cross-plot, with a red zone high-lighted, and (b) shows the resulting highlighted zone on the seismic section.

(a) (b)

The Zoeppritz equations are a set of four equations in fourunknowns that are difficult to intuitively interpret. However, asshown by Aki and Richards (2002) a linearized version of theseequations can be written for the reflected P-wave in which wedivide the response into three terms. In the original Aki-Richardsequation, the three terms were weighted values of changes in P-wave velocity, S -wave velocity and density. However, their equa-tion was reformulated by Wiggins et al. (1983) as

(4)

w h e re is a linearized approximation to the zero - o ffset P-wave reflection coeff i c i e n t ,

and .

Equation (4) forms the basis for the AVO (amplitude variationswith offset) method, in which the terms A and B, called interc e p tand gradient, are extracted from the seismic data using aweighted stack method and are cros plotted and analyzed forfluid anomalies. We will not discuss AVO in this article, but ratherjump directly to pre-stack inv sion, which can be considered as aquantitative extension of AVO.

The A k i - R i c h a rds equation was re-formulated by Fatti et al. (1994)as a function of zero - o ffset P -wave reflectivity RP 0, zero - o ffset S -wave reflectivity RS 0 and density reflectivity RD in the form

(5)

w h e re

a n d

RP 0 is equivalent to the A term in equation (4), and the other tworeflectivity terms are given by

and . Based on equation (5), a

l e a s t - s q u a res pro c e d u re can be implemented to extract the thre ereflectivity terms from the pre-stack seismic data.

After we have extracted the three reflectivities in equation (5), theycan be inverted using the post-stack inversion method described inthe last section. This is re f e r red to as independent inversion.H o w e v e r, Hampson et al. (2005) developed a new approach thatuses a modification of equation (5) and allows us to invert dire c t l yfor P-impedance, S -impedance, and density. This method isre f e r red to as simultaneous inversion. It was also the goal of thiswork to extend the model-based post-stack impedance inversionmethod described in the last section to perform pre-stack inversion.Although the mathematics of this approach will not be describedh e re (the interested reader is encouraged to read the expandedabstract by Hampson et al., 2005), one of the key assumptions insimultaneous inversion is that we can build linear re l a t i o n s h i p sbetween the logarithms of P -impedance and S -impedance, LP a n dLS, and between LP and the logarithm of the density re f l e c t i v i t y, LD.That is, we are looking for deviations away from this linear fitgiven by ΔLS and ΔLD, as illustrated in Figure 7.

The inverted P -impedance from the simultaneous inversion ofthe pre-stack data in Figure 5 is shown in Figure 8, and displaysless of an impedance drop than the model-based post-stackinversion shown in Figure 4. This is to be expected, since theamplitudes used in the post-stack inversion were increased dueto the AVO effect. As in post-stack inversion, the P -impedance onits own is not a gas sand indicator.

However, when we combine the plot shown in Figure 8 with theratio of the inverted P and S -impedances (which gives the VP/ VSratio since the density terms cancel) shown in Figure 9, the interpretation becomes more clear. Associated with a drop in

P -impedance is a drop in the VP/VS ratio, which is generally anindicator of a gas sand.



To show this more conclusively, Figure 10 show the results of acrossplot of a small portion of the two sections shown in Figures8 and 9. On the crossplot itself, shown on the left of Figure 10,we have highlighted a zone in which both P -impedance andVP/VS ratio are low. This zone is displayed on the seismicsection on the right-hand image in Figure 10. Notice the excel-lent definition of the gas sand zone.

The example we have been considering is a 2D example and themethod discussed in this talk can be applied to 3D datasets togive a spatial image of the reservoir. Such an example, takenf rom the Gulf of Mexico, will beshown in the expanded version ofthis talk given during the CSEGluncheon.

Conclusions

In this talk, we have discussed thehistory of seismic amplitude inver-sion, from its origin as a post-stackprocess to the most recent develop-ments which involves the simulta-neous inversion of pre-stack seismicdata. Although post-stack inversionis a powerful and robust method, itsuffers from the fact that its finalp roduct, P -impedance, does notallow us to discriminate betweenlithology, porosity and fluid effects.This limitation was removed with thedevelopment of both the AVO tech-nique and simultaneous inversion ofp re-stack data. The simultaneousinversion method that we discussedis based on the assumptions thatreflectivity as a function of angle canbe given by the Aki-Richards equa-tion, and that there is a linear rela-tionship between the logarithm ofP-impedance and both S-impedanceand density. We illustrated our inver-sion methods using a gas sandexample from Alberta. R

ReferencesAki, K., and Richards, P.G., 2002, QuantitativeS e i s m o l o g y, 2nd Edition: W.H. Freeman andCompany.

Fatti, J., Smith, G., Vail, P., Strauss, P., and Levitt,P., 1994, Detection of gas in sandstone reservoirsusing AVO analysis: a 3D Seismic Case HistoryUsing the Geostack Technique: Geophysics, 59,1362-1376.

Hampson, D., Russell, B., and Bankhead, B., 2005, Simultaneous inversion of pre-stackseismic data: Ann. Mtg. Abstracts, Society of Exploration Geophysicists.

Lindseth, R. O., 1979, Synthetic sonic logs - A process for stratigraphic interpretation:Geophysics, 44 , no.1, 3-26.

Russell, B. and Hampson, D., 1991, A comparison of post-stack seismic inversionmethods: Ann. Mtg. Abstracts, Society of Exploration Geophysicists, 876-878.

Wiggins, R., Kenny, G.S., and McClure, C.D., 1983, A method for determining anddisplaying the shear-velocity reflectivities of a geologic formation: European patentApplication 0113944.

Zoeppritz, K.,1919, Erdbebenwellen VIIIB, On the reflection and propagation ofseismic waves: Gottinger Nachrichten, I, 66-84.

the old and the new in seismic inversion

Documents