6 th International Conference & Exposition on Petroleum Geophysics Kolkata 006 Summary Simultaneous Inversion of Pre-Stack Seismic Data Brian H. Russell, Daniel P. Hampson, Brad Bankhead Hampson-Russell Softare Ltd, and VeritasDGC. We present a ne approach to the simultaneous pre-stack inversion of PP and, optionally, PS angle gathers for the estimation of P-impedance, S-impedance and density. Our algorithm is based on three assumptions. The first is that the linearized approximation for reflectivity holds. The second is that PP and PS reflectivity as a function of angle can be given by the Aki-Richards equations (Aki and Richards, 00). The third is that there is a linear relationship beteen the logarithm of P-impedance and both S-impedance and density. Given these three assumptions, e sho ho a final estimate of P- impedance, S-impedance and density can be found by perturbing an initial P-impedance model. After a description of the algorithm, e then apply our method to both model and real data sets. Introduction The goal of pre-stack seismic inversion is to obtain reliable estimates of P-ave velocity (V P ), S-ave velocity (V S ), and density (ρ ) from hich to predict the fluid and lithology properties of the subsurface of the earth. This problem has been discussed by several authors. Simmons and Backus (996) invert for linearized P-reflectivity (R P ), S-reflectivity (R S ) and density reflectivity (R D ), here Simmons and Backus (996) also make three other assumptions: that the reflectivity terms given in equations () through (3) can be estimated from the angle dependent reflectivity R PP (θ) by the Aki-Richards linearized approximation (Aki and Richards, 00, Richards and Frasier, 976), that ρ and V P are related by Gardner s relationship (Gardner et al. 974), given by and that V S and V P are related by Castagna s equation (Castagna et al., 985), given by () () (3) (4) V s = (V p - 360) /.6. (5) The authors then use a linearized inversion approach to solve for the reflectivity terms given in equations () through (3). Buland and Omre (003) use a similar approach hich they call Bayesian linearized AVO inversion. Unlike Simmons and Backus (996), their method is parameterized by the three terms and, again using the Aki-Richards approximation. The authors also use the small reflectivity approximation to relate these parameter changes to the original parameter itself. That is, for changes in P-ave velocity they rite here ln represents the natural logarithm. Similar terms are given for changes in both S-ave velocity and density. This logarithmic approximation allos Buland and Omre (003) to invert for velocity and density, rather than reflectivity, as in the case of Simmons and Backus (996). Unlike Simmons and Backus (996), hoever, Buland and Omre (003) do not build in any relationship beteen P and S-ave velocity, and P-ave velocity and density. In the present study, e extend the ork of both Simmons and Backus (003) and Buland and Omre (996), and build a ne approach that allos us to invert directly for P-impedance (Z P =ρv P ), S-impedance (Z P =ρv P ), and density through a small reflectivity approximation similar to that of Buland and Omre (003), and using constraints similar to those used by Simmons and Backus (996). It is also our goal to extend an earlier post-stack impedance inversion method (Russell and Hampson, 99) so that this method can be seen as a generalization to pre-stack inversion. (6) (37)
Post-stack Inversion For P-impedance We ill first revie the principles of model-based post-stack inversion (Russell and Hampson, 99). First, by combining equations () and (6), e can sho that the small reflectivity approximation for the P-ave reflectivity is given by Pre-stack Inversion For P-impedance We can no extend the theory to the pre-stack inversion case. The Aki-Richards equation as reexpressed by Fatti et al. (994) as, () (7) here i represents the interface beteen layers i and i+. If e consider an N sample reflectivity, equation (7) can be ritten in matrix form as RP 0 L LP R 0 P O L P = M 0 0 O M, (8) RPN M O O O LPN here L Pi = ln(z Pi ). Next, if e represent the seismic trace as the convolution of the seismic avelet ith the earth s reflectivity, e can rite the result in matrix form as T T = M TN M 3 0 O 0 0 O L O O O RP R P M R PN, (9) here T i represents the i th sample of the seismic trace and j represents the j th term of an extracted seismic avelet. Combining equations (8) and (9) gives us the forard model hich relates the seismic trace to the logarithm of P-impedance:, (0) here W is the avelet matrix given in equation (9) and D is the derivative matrix given in equation (8). If equation (0) is inverted using a standard matrix inversion technique to give an estmate of L P from a knoledge of T and W, there are to problems. First, the matrix inversion is both costly and potentially unstable. More importantly, a matrix inversion ill not recover the lo frequency component of the impedance. An alternate strategy, and the one adopted in our implementation of equation (0), is to build an initial guess impedance model and then iterate toards a solution using the conjugate gradient method. For a given angle trace T(θ) e can therefore extend the zero offset (or angle) trace given in equation (0) by combining it ith equation () to get () here L S = ln(z S ) and L D = ln(ρ). Note that the avelet is no dependent on angle. Equation () could be used for inversion, except that it ignores the fact that there is a relationship beteen L P and L S and beteen L P and L D. Because e are dealing ith impedance rather than velocity, and have taken logarithms, our relationships are different than those given by Simmons and Backus (996) and are given by and, (3). (4) That is, e are looking for deviations aay from a linear fit in logarithmic space. This is illustrated in Figure. Combining equations () through (4), e get here, (5) Equation (5) can be implemented in matrix form as T( θ) c% ( θ) W( θ) D c% ( θ) W( θ) D c3( θ) W( θ) D LP T( θ) c% ( θ) W( θ) D c% ( θ) W( θ) D c3( θ) W( θ) D = L S M M M M LD T( θ ) c% ( θ ) W( θ ) D c% ( θ ) W( θ ) D c ( θ ) W( θ ) D N N N N N 3 N N (6) (37)
6 th International Conference & Exposition on Petroleum Geophysics Kolkata 006 Fig. : Crossplots of (a) ln(z D ) vs ln(z P ) and (b) ln(z S ) vs ln(z P ) here, in both cases, a best straight line fit has been added. The deviations aay from this straight line, L D and L S, are the desired fluid anomalies. If equation (6) is solved by matrix inversion methods, e again run into the problem that the lo frequency content cannot be resolved. A practical approach is to initialize the solution to, here Z P0 is the initial impedance model, and then to iterate toards a solution using the conjugate gradient method. In the last section ill sho ho to extend the theory in equations (5) and (6) by including PS angle gathers as ell as PP angle gathers. Model And Real Data Examples We ill no apply this method to both a model and real data example. Figure (a) shos the ell log curves for a gas sand on the left (in blue), ith the initial guess curves (in red) set to be extremely smooth so as not to bias the solution. On the right are the model, the input computed gather from the full ell log curves, and the error, hich is almost identical to the input. Figure (b) then shos the same displays after 0 iterations through the conjugate Fig.. The results of inverting a gas sand model, here (a) shos the initial model before inversion, and (b) shos the results after inversion (373)
gradient inversion process. Note that the final estimates of the ell log curves match the initial curves quite ell for the P-impedance, Z P, S-impedance, Z S, and the Poisson s ratio (σ). The density (ρ) shos some overshoot above the gas sand (at 3450 ms), but agrees ith the correct result ithin the gas sand. The results on the right of Figure (b) sho that the error is no very small Next, e ill use Biot-Gassmann substitution to create the equivalent et model for the sand shon in Figure, and again perform inversion. Figure 3(a) shos the ell log curves for the et sand on the left, ith the smooth initial guess curves superimposed in red. On the right are the model from the initial guess, the input modeled gather from the full ell log curves, and the error. Figure 3(b) then shos the same displays after 0 iterations through the conjugate gradient inversion process. As in the gas case, the final estimates of the ell log curves match the initial curves quite ell, especially for the P-impedance, Z P, S-impedance, Z S, and the Poisson s ratio (σ). The density (ρ) shos a much better fit at the et sand (hich is at 3450 ms) than it did at the gas sand in Figure. We ill next look at a real data example, consisting of a shallo Cretaceous gas sand from central Alberta. Figure 4 shos the computed V P /V S ratio from this dataset, here the anomalous gas sand is encircled by the black ellipse. Notice the drop in V P /V S associated ith the gas sand. Fig. 5 then shos a comparison beteen the input gathers over the sand (here a clear AVO Class 3 anomaly is evident), and the computed synthetic gathers using the inverted results. Extension To Converted Wave Data We ill no discuss ho the formulation just derived can be extended to include pre-stack convertedave measurements (PS data) that have been converted to PP time. To do this, e ill use the linearized form of the equation as developed by Aki, Richards, and Frasier (Aki and Richards, 00, Richards and Frasier, 976). It has been shon by Margrave et al. (00) that this equation can be ritten as RPS ( θ, φ) = c4rs + c5rd, (7) here Fig. 3. The results of inverting a et sand model, here (a) shos the initial model before inversion, and (b) shos the results after inversion. (374)
6 th International Conference & Exposition on Petroleum Geophysics Kolkata 006 Fig. 4. The inverted V P /V S ratio for a shallo gas sand from Alberta, here the elliptical region indicates the anomalous region. Fig. 5 The CDP gathers over the gas sand anomaly from Fig.4, here (a) shos the input gathers and (b) shos the synthetic gathers after inversion.. (8) and. The reflectivity terms R S and R D given in equation (7) are identical to the terms those given in equations () and (3). Using the small reflectivity approximation, e can re-rite equation (7) as: Using the relationships beteen S-impedance, density and P-impedance given in equations (3) and (4), equation (8) can be further re-ritten as (9) (375)
Note that equation (9) allos us to express a single PS angle stack as a function of the same three parameters given in equation (5). Also, equation (9) is given at a single angle φ. When e generalize this equation to M angle stacks, e can combine this relationship ith equation (6) and rite the general matrix equation as Equation ](0) gives us a general expression for the simultaneous inversion of N PP angle stacks and M PS angle stacks. Note that e extract a different avelet for each of the PS angle stacks, as as done for each of the PP angle stacks. Conclusions We have presented a ne approach to the simultaneous inversion of pre-stack seismic data hich produces estimates of P-impedance, S-Impedance and density. The method is based on three assumptions: that the linearized approximation for reflectivity holds, that reflectivity as a function of angle can be given by the Aki- Richards equations, and that there is a linear relationship beteen the logarithm of P-impedance and both S-impedance and density. Our approach as shon to ork ell for modelled gas and et sands and also for a real seismic example hich consisted of a shallo Creataceous gas sand from Alberta. Finally, e shoed ho our method could be extended to include PS converted-ave pre-stack data hich had been calibrated to PP time. References Aki, K., and Richards, P.G., 00, Quantitative Seismology, nd Edition: W.H. Freeman and Company. Buland, A. and Omre, H, 003, Bayesian linearized AVO inversion: Geophysics, 68, 85-98. Castagna, J.P., Batzle, M.L., and Eastood, R.L., 985, Relationships beteen compressional-ave and shearave velocities in clastic silicate rocks: Geophysics, 50, 57-58. Fatti, J., Smith, G., Vail, P., Strauss, P., and Levitt, P., 994, Detection of gas in sandstone reservoirs using AVO analysis: a 3D Seismic Case History Using the Geostack Technique: Geophysics, 59, 36-376. Gardner, G.H.F., Gardner, L.W. and Gregory, A.R.,974, Formation velocity and density - The diagnostic basics for stratigraphic traps: Geophysics, 50, 085-095. Margrave, G.F., Steart, R. R. and Larsen, J. A., 00, Joint PP and PS seismic inversion: The Leading Edge, 0, no. 9, 048-05. Richards, P. G. and Frasier, C. W., 976, Scattering of elastic aves from depth-dependent inhomogeneities: Geophysics, 4, 44-458. Russell, B. and Hampson, D., 99, A comparison of post-stack seismic inversion methods: Ann. Mtg. Abstracts, Society of Exploration Geophysicists, 876-878. Simmons, J.L. and Backus, M.M., 996, Waveform-based AVO inversion and AVO prediction-error: Geophysics, 6, 575-588. (376)