Linearized AVO inversion with supercritical anglesJonathan E. Downton*, Veritas DGC, and Charles Ursenbach, CREWES, University of Calgary

SummaryContrary to popular belief, a linearized approximation ofthe Zoeppritz equations may be used to accurately estimatethe reflection coefficient for angles of incidence up to andbeyond the critical angle. These supercritical reflectioncoefficients are complex implying a phase variation withoffset in addition to the amplitude variation with offset.This linearized approximation is used as the basis for a newAVO waveform inversion capable of truly incorporatingwide-angle information.

IntroductionSupercritical reflection coefficients arise in situations ofinterest to explorationists. In trying to determine densitythrough AVO, large angles of incidence and offset are usedin the inversion (Downton, 2005). Most AVO inversionapproaches using a linearized approximation require thatonly incidence angles less than the critical angle are used.Since the reliability of the estimates is proportional to therange of angles used in the inversion this limits thereliability of the resulting density estimates.

In addition the critical angle also plays an important rolefor heavy oil plays in northeastern Alberta. Reflections atthe Paleozoic often become critical due to the largevelocity contrast between the overlying clastics andunderlying carbonates. This becomes a problem becausethe large supercritical reflection coefficients obscure theoverlying zone of interest. The traditional way of dealingwith this is to limit the range of angles used in the AVOinversion. This is problematic since the critical angle maybe as low as 25 degrees, again limiting reliability of theestimates of the AVO inversion.

Limiting the angles used in the AVO inversion is due to thecommon and mistaken belief that the Aki and Richards(1980) linearized approximation (equation 5.44) of theZoeppritz equation is only valid for subcritical angles ofincidence. For example, de Nicolao et al. (1993) state thatthe linearized approximation is only valid for precriticalangles. Aki and Richards (1980) state as a precondition forusing the linearized approximation (equation 5.44) that theangles of incidence and transmission must be less than 90degrees, thus precluding the critical angle.

This seems to be supported if the Aki and Richards (1980)equation for the PP reflection coefficient R(p) is written interms of horizontal slowness p

( ) ,412

14121)( 22

22

22

bb

baa

arr

bD

-D

-+

D-= p

pppR

(1)

where r, a, and b are the average density, P-wave and S-wave velocities, and the differentials Dr, Da, and Db are

the change in layer properties for the density andvelocities. As written, the reflection coefficient calculatedby equation (1) must be real for homogenous waves (i.e. pis real). However, supercritical reflection coefficients arecomplex, making use of the complex numbers to containthe phase information.

If however, equation (1) is parameterized in terms of theaverage angle q

,sin4cos2

1sin4121)( 2

2

222

2

2 bb

qab

aa

qrr

qab

qD

-D

+D

÷÷ø

öççè

æ-=R (2)

the reflection coefficient can become complex. This is aconsequence of the fact that q is the average of incidenceand transmitted angles, (qr + qt) / 2. The incidence angle,qr, is always real, but the transmitted angle becomescomplex for angles beyond the critical angle:

1 2 2

1 1

cosh sin , sin 1.2t r ri a ap

q q qa a

- æ ö= - ³ç ÷

è ø (3)

To illustrate this, a simple convolutional model wasgenerated using both the Zoeppritz equation and equation(2) to generate the reflection coefficients. The density, P-wave and S-wave velocity layer information used toconstruct this model are based on a well log from northeastern British Columbia (Downton, 2005). The syntheticis shown with a 10/14-90/110 Hz trapezoidal zero phasefilter. Both synthetics are shown without moveout,geometrical spreading or other losses, highlightingdifferences that arise solely due to the linearizedapproximation. The synthetic data generated usingequation (2) (Figure 1a) closely approximates that using theZoeppritz equation (Figure 1b) as evidenced by thedifference display of the two (Figure 1c). Figures 2a, 2band 2c show amplitude extractions taken from the twomodels at time samples corresponding to interfaces thatinclude supercritical angles. The synthetic modelgenerated using equation (2) closely matches the resultsgenerated using the Zoeppritz equation.

MethodThe above discussion suggests two issues that must be dealtwith if supercritical angles are to be included in the AVOinversion. It is evident that the phase changes as afunction of angle of incidence. This is problematic fortraditional AVO inversion schemes, which work on asample-by-sample basis. These AVO inversions ignore thewavelet and so cannot handle these phase changes. Thissuggests some sort of AVO waveform inversion (Bulandand Omre, 2003; Simmons and Backus, 1996; Downtonand Lines, 2004) should be used.

Secondly, the amplitudes of the synthetic data are verylarge at the far offsets compared to the near offsets. Realseismic data does not behave like this due to geometricalspreading, attenuation, and other losses. To address this,the seismic data is normally preconditioned to correct forthese losses prior to the AVO inversion. However,including angles close to critical make these correctionsunstable. Figure 3 shows the geometrical spreading lossescalculated following Cervany (2001). Note the largechange in scalars close to the critical angle. If the inverseof this is applied as a geometrical correction this willintroduce high frequency noise into the seismic data. It ismore stable to apply this as a forward operation prior to theapplication of the wavelet. This once again is easily doneas part of an AVO waveform inversion.

The approach of Downton and Lines (2004) is easilymodified to incorporate both these considerations. Thebasis of the inversion is Downton and Lines (2004)equation (1)

,1111111

úúú

û

ù

êêê

ë

é

úúú

û

ù

êêê

ë

é=

úúú

û

ù

êêê

ë

é

d

S

P

NNNNNNNrrr

HWNGWNFWN

HWNGWNFWN

d

dMMMM (4)

where rp,rs,rd are the P- and S-wave velocity, and densityreflectivity respectively. These are all vectors whoseelements correspond to different time samples. Likewisethe elements of the data vector dn represent the processedseismic data for the nth offset for the corresponding timesamples. The block matrices describe the physics of theproblem. The matrices F, G, and H are diagonal matricesthat contain weights that describe how the amplitudechanges as a function of offset. These weights are complexand follow from equation (2). Following Claerbout (1992),the block matrix Nn performs NMO. This operator can beconstructed using whatever offset travel time relationshipone desires including the travel times generated from raytracing. In order to invert data at large angles of incidence,it is important to correctly position the event withoutintroducing residual NMO. Implicit in this derivation isthat the velocity is known a priori and that staticcorrections are applied. Lastly, W is a convolution matrixthat contains the source wavelet. In order to insure theoutput of this operation is real, Hermitian symmetry isimposed in the frequency domain using a Hilbert transform.Gain corrections such as geometrical spreading may beimplemented by appropriately scaling each of the diagonalmatrices F, G, and H.

Applying these three operators in series, the block matricesFn,Gn,Hn model the offset dependent reflectivity from thezero offset reflectivity, Nn applies NMO and W convolvesthe offset dependent reflectivity with the source waveletmodeling the band limited seismic data with NMO. Theinversion of equation (4) can be thought of as three separateinversion problems, deconvolution, inverse NMO and AVOinversion. Downton and Lines (2004) solve this usingconjugate gradient and high-resolution constraints.Equation (4) is solved in a similar fashion.

ResultsThe synthetic model shown in Figure (1) was rerun withmoveout, geometrical spreading and other losses. Coherentand random noise was added to the result (Figure 4a) togive a signal-to-noise ratio of 10:1. This was used as theinput into the AVO waveform inversion. Figure 4b showsthe estimated data generated by the AVO inversion whileFigure 4c shows the difference between the estimated andactual data. The AVO inversion was able to estimate theoriginal data quite accurately. Most of the differencearises from the model not being able to estimate the randomand coherent noise although some small theoretical error isevident for two supercritical reflections. Figure 5 shows acomparison between the estimated and reference zero offsetreflectivity. The estimated reflectivity is almost a directoverlay of the ideal zero offset reflectivity. This samemodel was run with signal-to-noise ratios of 4:1, 2:1 and1:1. Density was accurately estimated for signal-to-noiseratios down to 2:1. These results are superior to thatobtained from an equivalent AVO inversion performed on asample-by-sample basis.

Discussion and ConclusionsThis inversion approach is theoretically more accurate andbetter conditioned than traditional AVO inversionperformed on a sample-by-sample basis on a limited anglerange. The method is approaching the accuracy ofreflectivity modeling (Kennett, 1984) if only compressionalwaves and first order reflectivity terms are considered.Because the problem is linearized, and only band limitedreflectivities are being solved for, the algorithm is muchfaster than reflectivity inversion (Sen and Stoffa, 1995).By linearizing the problem we have introduced theadditional constraint that the fractional changes in layerparameters must be small, but this is typically a goodassumption as based on well log data studies.

AcknowledgementsThe lead author would like to thank Veritas for permittinghim to publish this and Dave Gray for many conversationsthat helped to lead to this research. In addition we wouldlike to thank Jan Dewar for many helpful suggestions inproofreading.

REFERENCES

Aki, K., and Richards P.G., 1980, Quantitative seismology: theoryand methods: W.H. Freeman and Co.

Buland, A., and Omre, H., 2003, Bayesian linearized AVOinversion: Geophysics, 68, 185-198.

erveny, V., 2001, Seismic ray theory: Cambridge UniversityPress.

Claerbout, J. F., 1992, Earth Soundings Analysis: Processingversus Inversion, Blackwell Scientific.

Downton, J.E., 2005, Seismic parameter estimation from AVOinversion: Ph.D. Thesis, University of Calgary.

Downton, J., and Lines, L., 2004, Three-term AVO waveforminversion: 74th Ann. Internat. Mtg.: Soc. of Expl.Geophys.

Kennett, B.L.N., 1984, Seismic wave propagation in stratified media: Cambridge University Press.de Nicolao, A., Drufuca, G. and Rocca, F., 1993, Eigenvectors and eigenvalues of linearized elastic inversion: Geophysics, 58, 670-679.Sen, M., and Stoffa P.L., 1995, Global optimization methods in geophysical inversion: Elsevier Science Publishers.Simmons, J.L., Jr. and Backus, M.M., 1996, Waveform-based AVO inversion and AVO prediction error: Geophysics, 61, 1575-1588.

Figure 1: Synthetic model (a) generated with Zoeppritz equation, (b) generated with equation 2 and (c) difference between (a)and (b).

Figure 2: Comparison of extracted amplitude at selected times from Figure 1a and Figure 1b. Figure 2a) is at 172 ms, 2b) at520 ms, and c) at 684 ms. The x-axis is offset in meters.

Figure 3: Geometrical spreading losses shown with a log scale. The critical angle generates a poorly illuminated zoneshown in dark blue.

offset (m)

dept

h (m

)

Geometrical spreading (log value)

0 500 1000 1500 2000 2500

0

200

400

600

800

1000

1200

1400

1600

1800

-9

-8

-7

-6

-5

-4

Figure 4: Input (a) to the AVO waveform inversion with the estimated data (b) and difference between (a) and (b).

Figure 5: Estimate (red) of the P-wave impedance reflectivity (Rp), S-wave impedance reflectivity (Rs) densityreflectivity (Rd) and fluid stack reflectivity (RFl) compared to reference zero offset reflectivity (blue). Note thegood agreement.