Joint simultaneous inversion of PP and PS angle gathers

Transcription

1 Inversion of and angle gathers Joint simultaneous inversion of and angle gathers Brian H. ussell, aniel. Hampson, Keith Hirshe, and Janusz eron ABTACT e present a new approah to the joint simultaneous inversion of and angle gathers for the estimation of -impedane, -impedane and density. Our algorithm is based on three assumptions. The first is that the linearized approximation for refletivity holds. The seond is that and refletivity as a funtion of angle an be given by the Aki-ihards linearized equations Aki and ihards, 00. The third is that there is a linear relationship between the logarithm of -impedane and both -impedane and density. Given these three assumptions, we show how a final estimate of -impedane, - impedane and density an be found by perturbing an initial -impedane model. After a desription of the algorithm, we then apply our method to both model and real data sets. ITOUCTIO The goal of pre-stak seismi inversion is to obtain reliable estimates of -wave veloity V, -wave veloity V, and density ρ from whih to predit the fluid and lithology properties of the subsurfae of the earth. This problem has been disussed by several authors. immons and Bakus 996 invert for linearized -refletivity, - refletivity and density refletivity, where V V ρ + ρ, and V V ρ + ρ, ρ ρ. 3 immons and Bakus 996 also make three other assumptions: that the refletivity terms given in equations through 3 an be estimated from the angle dependent refletivity by the Aki-ihards linearized approximation Aki and ihards, 00, ihards and Frasier, 976, that ρ and V are related by Gardner s relationship Gardner et al. 97, given by ρ V ρ V, and that V and V are related by Castagna s equation Castagna et al., 985, given by CEE esearh eport Volume 7 005

2 ussell et al. V V 360 /.6. The authors then use a linearized inversion approah to solve for the refletivity terms given in equations through 3. Buland and Omre 003 use a similar approah whih they all Bayesian linearized AVO inversion. Unlike immons and Bakus 996, their method is parameterized by the three terms V / V, V / V, and ρ / ρ, again using the Aki-ihards approximation. The authors also use the small refletivity approximation to relate these parameter hanges to the original parameter itself. That is, for hanges in -wave veloity they write V V lnv 5. 6 where ln represents the natural logarithm. imilar terms are given for hanges in both - wave veloity and density. This logarithmi approximation allows Buland and Omre 003 to invert for veloity and density, rather than refletivity, as in the ase of immons and Bakus 996. In the present study, we present a new approah that allows us to invert diretly for - impedane Z ρv, -impedane Z ρv, and density through a small refletivity approximation similar to that of Buland and Omre 003, and using onstraints similar to those used by immons and Bakus 996. It is also our goal to extend an earlier poststak impedane inversion method ussell and Hampson, 99 so that this method an be seen as a generalization to pre-stak inversion. The input data used to extrat impedane and density onsists of and, optionally, angle gathers, so we refer to the method as joint simultaneous inversion. THEOY e start by reviewing the priniples of model-based post-stak inversion ussell and Hampson, 99. First, by ombining equations and 6, we an show that the small refletivity approximation for the -wave refletivity is given by [ ln Z ln Z ] i ln Z i i+ i, 7 where i represents the interfae between layers i and i+. If we onsider an sample refletivity, equation 7 an be written in matrix form as where L i lnz i. 0 L 0 L 0 0, 8 L CEE esearh eport Volume 7 005

3 Inversion of and angle gathers ext, if we represent the seismi trae as the onvolution of the seismi wavelet with the earth s refletivity, we an write the result in matrix form as T w T w w T 3 0 w w 0 0 w, 9 where T i represents the i th sample of the seismi trae and w j represents the j th term of an extrated seismi wavelet. Combining equations 8 and 9 gives us the forward model whih relates the seismi trae to the logarithm of -impedane: T / L, 0 where is the wavelet matrix given in equation 9 and is the derivative matrix given in equation 8. If equation 0 is inverted using a standard matrix inversion tehnique to give an estmate of L from a knowledge of T and, there are two problems. First, the matrix inversion is both ostly and potentially unstable. More importantly, a matrix inversion will not reover the low frequeny omponent of the impedane. An alternate strategy, and the one adopted in our implementation of equation 0, is to build an initial guess impedane model and then iterate towards a solution using the onjugate gradient method. e an now extend the theory to the pre-stak inversion ase. The Aki-ihards equation was re-expressed by Fatti et al. 99 as + + 3, where + tan, 8γ tan, γ V / V, 3 0.5tan + γ sin, and the three refletivity terms are as given by equations through 3. For a given angle trae T we an therefore extend the zero offset or angle trae given in equation 0 by ombining it with equation to get T / L + / L + 3 L, where L lnz and L lnρ. ote that the wavelet is now dependent on angle. Equation ould be used for inversion, exept that it ignores the fat that there is a relationship between L and L and between L and L. Beause we are dealing with impedane rather than veloity, and have taken logarithms, our relationships are different than those given by immons and Bakus 996 and are given by and ln Z k ln Z + k + L, 3 CEE esearh eport Volume

4 ussell et al. ln Z m ln Z + m + L. That is, we are looking for deviations away from a linear fit in logarithmi spae. This is illustrated in Figure. a b FIG.. Crossplots of a lnz vs lnz and b lnz vs lnz where, in both ases, a best straight line fit has been added. The deviations away from this straight line, L and L, are the desired fluid anomalies. Combining equations through, we get T ~ L + ~ L + 3 L, 5 where ~ / + / k m + 3 and ~ /. Equation 5 an be implemented in matrix form as T ~ ~ 3 L T ~ ~ 3 L L, 6 T ~ ~ 3 If equation 6 is solved by matrix inversion methods, we again run into the problem that the low frequeny ontent annot be resolved. A pratial approah is to initialize the solution to [ ] T L [ L L ln Z ]T, where Z0 is the initial impedane model, and then to iterate towards a solution using the onjugate gradient method. e will now extend the previous derivation to inlude pre-stak onverted-wave measurements gathers that have been onverted to time. e will use the linearized form of the equation was developed by Aki, ihards, and Frasier Aki and ihards, 00, ihards and Frasier, 976. It has been shown by Margrave et al. 00 that this equation an be written as CEE esearh eport Volume 7 005

5 Inversion of and angle gathers CEE esearh eport Volume , +, 7 where [ ] γ γ os os sin tan, [ ] γ γ os os sin tan 5 +, V V / γ, and γ sin sin. The refletivity terms and given in equation 7 are idential to the terms those given in equations and 3. Using the small refletivity approximation, we an therefore re-write equation 7 as: L L T, ext, using the relationships between -impedane, density and -impedane given in equations 3 and, equation 8 an be further re-written as, ~, 5 L L L T where. ~ 5 m k + ote that equation 9 allows us to express a single angle stak as a funtion of the same three parameters given in equation 5. Also, equation 9 is given at a single angle. hen we generalize this equation to M angle staks, we an ombine this relationship with equation 6 and write the general matrix equation as M M M M M M M L L L T T T T ~ ~ ~ ~ Equation 0 gives us a general expression for the simultaneous inversion of angle staks and M angle staks. ote that we extrat a different wavelet for eah of the angle staks, as was done for eah of the angle staks. MOEL EXAMLE e will now apply this method to a model data example. To reate this model, we started with a measured set of well logs -wave, -wave, density, and omputed oisson s ratio for a well in whih the in-situ ase was a partially saturated gas sand. e then used the Biot-Gassmann equations to perform fluid substitution and reate a number of model senarios ranging from the 00% wet ase to the 00% gas ase, in inrements of 0% saturation exept between values of 90% and 00%, where we inserted the 95% ase. e reated the equivalent well log urves for eah ase, as well as a model syntheti, using a wave equation modeling algorithm with a 0 hertz iker wavelet. The model syntheti was then used as the input data for a series of inversions at

6 ussell et al. eah value of. A detailed analysis of the two end members the gas ase and wet ase is shown in Figures and 3. Figure a shows the well log urves for a gas sand on the left in blue, with the initial guess urves in red set to be extremely smooth so as not to bias the solution. On the right we show the model syntheti angle gather omputed from the inversion algorithm, the input omputed angle gather from the full well log urves, and the error, whih is almost idential to the input. The far angle trae for the model and input angle synthetis is 60 o. Figure b then shows the same displays after 0 iterations through the onjugate gradient inversion proess. ote that the final estimates of the well log urves math the initial urves quite well for the -impedane, Z, -impedane, Z, and the oisson s ratio σ. The density ρ shows some overshoot above the gas sand at 350 ms, but agrees with the orret result within the gas sand. The results on the right of Figure b show that the error is now very small. FIG.. The results of inverting a gas sand model, where a shows the initial model before inversion, and b shows the results after inversion. The equivalent wet model for the gas sand shown in Figure is shown in Figure 3. e again performed inversion on this model dataset. Figure 3a shows the well log urves for the wet sand on the left, with the smooth initial guess urves superimposed in red. On the right we show the model syntheti angle gather omputed from the inversion algorithm, the input omputed angle gather from the full well log urves, and the error, 6 CEE esearh eport Volume 7 005

7 Inversion of and angle gathers whih is almost idential to the input. The far angle trae for the model and input angle synthetis is again 60 o. Figure 3b then shows the same displays after 0 iterations through the onjugate gradient inversion proess. As in the gas ase, the final estimates of the well log urves math the initial urves quite well, espeially for the -impedane, Z, -impedane, Z, and the oisson s ratio σ. The density ρ shows a muh better fit at the wet sand whih is at 350 ms than it did at the gas sand in Figure. FIG. 3. The results of inverting a wet sand model, where a shows the initial model before inversion, and b shows the results after inversion. The results shown in Figures and 3 show us that the inversion algorithm is performing very well. However, it should be pointed out that this is noise-free data with an angular aperture of 60 degrees, whih is very optimisti when ompared with real data apertures. Finally, Figure shows the inverted results at eah of the twelve different values of water saturation. Figure a shows the -impedane values in olour, with the olour sale on the left. Figure b shows the V /V ratio results in olour, with the olour sale again on the left. In both Figures, the seismi inversion results at the wells have been dupliated five times for larity, but the original -wave veloity log has been shown only one. CEE esearh eport Volume

8 ussell et al. In Figure, notie that both the -impedane and V /V ratio results are onsistent with the fat that the gas sand is hanging to a wet sand as we move from left to right along the plots. a b FIG.. The results of inverting a variable sand model, from 00% gas on the left to 00% wet on the right where a shows the -impedane results, and b shows the V /V ratio results. ATA EXAMLE e will next look at a real data example, onsisting of a shallow Cretaeous gas sand from entral Alberta. Figure 5 shows the omputed V /V ratio from this dataset, where the anomalous gas sand is enirled by the blak ellipse. otie the drop in V /V assoiated with the gas sand. FIG. 5. The inverted V /V ratio for a shallow gas sand from Alberta, where the ellipse indiates the anomalous region. 8 CEE esearh eport Volume 7 005

9 Inversion of and angle gathers Figure 6 then shows a omparison between the input gathers over the sand where a lear AVO Class 3 anomaly is evident, and the omputed syntheti gathers using the inverted results. a b FIG. 6. The C gathers over the gas sand anomaly from Fig., where a shows the input gathers and b shows the syntheti gathers after inversion. ote that the results given in Figure 6 show that a very good fit to the original real data an be obtained by applying forward modeling to the well logs obtained from the inversion proess. This an be onsidered as an exellent diagnosti, but is not to be interpreted as indiating that our result is absolutely orret, sine there is possible nonuniqueness in the answer. - ATA EXAMLE In this final example, we will show a joint - inversion using a ase study from northeastern Alberta. The workflow for this example involved the following steps:. Correlate the and data to wells. ik the orresponding horizons on both datasets. 3. Use horizon based event mathing to onvert data to time.. Invert and data using simultaneous inversion. Figure 7a shows the orrelated data and Figure 7b shows the orrelated data from our example, where the -wave, -wave, and density logs are shown on the left of eah figure, and the staked data or is shown on the right. The syntheti trae blue to seismi trae red orrelation is shown between the logs and the staked data. CEE esearh eport Volume

10 ussell et al. ote that the ties are quite reasonable in both ases shown. The wavelets used to reate the syntheti were extrated from the seismi data shown. a b FIG. 7. The orrelation results, where a shows the logs on the left and the stak on the right, and b shows the logs on the left and the stak on the right. The blue trae shows the syntheti and the red trae shows the extrated seismi. e then performed two different types on inversion. The first type of inversion, whih we will all independent inversion, involved building independent and - impedane models for the and staks and using a model-based inversion algorithm to independently invert the two datasets. These models were reated by interpolating well log values from eight wells whih interseted the well log survey. This assumes that the stak is a good approximation to zero-offset -refletivity, and that the stak is a good approximation to zero-offset -refletivity. The seond inversion is referred to as joint inversion, and applies the theory disussed in earlier in this paper, speifially in equation 0. otie that we have only two inputs: the stak and the stak. Thus, the problem is under-determined and we therefore deided not to invert for density, only for and -impedane. e assume that the 0 CEE esearh eport Volume 7 005

11 Inversion of and angle gathers stak is equivalent to a angle gather at zero degrees and that the stak is equivalent to a angle stak at 0 degrees. The value of 0 degrees was arrived at by rossorrelating the traes in the stak with model traes at various angles. To hek the validity of the inversion, we performed ross-validation of the results at eah of the eight wells whih interseted the dataset. Cross-validation onsists of leaving eah well in turn out of the model-building proess and then blindly prediting eah well from the inversion results. This results of the ross-validation proess are shown in Figure 8, where 8a shows the independent inversion results and 8b shows the joint inversion results for V /V ratio. a b FIG. 8. The ross-validation of the inversion results at eah of the well loations, where a shows the results of independent inversion, and b shows the results of joint inversion. The parameter being plotting is Vp/Vs ratio. CEE esearh eport Volume 7 005

12 ussell et al. It is lear from Figure 8 that the joint inversion has reated a better visual fit at the wells than independent inversion. To quantify this observation, Figure 9 shows the rossorrelation values at the wells, where Figure 9a orresponds to the independent inversion of Figure 8a and Figure 9b orresponds to the joint inversion of Figure 8b. otie that for the independent inversion Figure 9a, the orrelation oeffiients are entered on a value of 0.5, whereas for the joint inversion Figure 9b the orrelation oeffiients are loser to a value of It an be seen that only two of the wells drop below a 0.5 orrelation for joint inversion, and these are the same two wells with a orrelation lose to zero on the independent inversion. This validates our qualitative observation from Figure 8, that the joint inversion has done a better job than the independent inversion. a b FIG. 9. A display of the ross-orrelation between the known Vp/Vs ratios from the wells and the inverted results of a Figure 8a, the independent inversion, and b Figure 8b, the joint inversion. Finally, the results of applying the two inversion algorithms to the seismi datasets are shown in Figure 0, where the results have been shown for the V /V ratio. Figure 0a shows the result of independent inversion and Figure 0b shows the result of joint inversion. ote that the V /V ratio in olour on the inserted well log mathes the olour on the inversion results and that the sale in shown on the right hand sides of the plots. On the independent inversion results of Figure 0a, note that the math between the well values and the seismi result is not very good, espeially at the top of the setion, between 300 and 350 ms, and the middle part of the setion between 0 and 70 ms. The mismath at the base of the setion is due to the fat that the well log data has not been reorded below about 530 ms. On the joint inversion results of Figure 0b there is good agreement in the two zones just desribed, suggesting that the inversion has done a better job of mathing the well results. Based on the results seen in Figures 8 through 0, it seems lear that the joint inversion algorithm, in whih the and data are oupled in the algorithm, is superior to the independent inversion, in whih there is no oupling between the two datasets. CEE esearh eport Volume 7 005

13 Inversion of and angle gathers a b FIG. 0. The final inverted results for Vp/Vs ratio extrated from the seismi data using a independent inversion and b joint inversion. COCLUIO e have presented a new approah to the joint simultaneous inversion of pre-stak seismi data whih produes estimates of -impedane, -Impedane and density. This method allows us to inorporate both and data into the solution, if we have first alibrated both datasets to time. The method is based on three assumptions: that the linearized approximation for refletivity holds, that refletivity as a funtion of angle an be given by the Aki-ihards equations, and that there is a linear relationship between the logarithm of -impedane and both -impedane and density. Our approah was shown to work well for modelled gas and wet sands, for a real seismi example whih onsisted of a shallow Creataeous gas sand from Alberta, and for a joint - dataset from northeast Alberta. In the first two data examples, pre-stak seismi data was used. In the joint - dataset the prestak data was not available and we applied the method to full staks. Future work will involve the inorporation of pre-stak data. EFEECE Aki, K., and ihards,.g., 00, Quantitative eismology, nd Edition:.H. Freeman and Company. CEE esearh eport Volume

14 ussell et al. Buland, A. and Omre, H, 003, Bayesian linearized AVO inversion: Geophysis, 68, Castagna, J.., Batzle, M.L., and Eastwood,.L., 985, elationships between ompressional-wave and shear-wave veloities in lasti siliate roks: Geophysis, 50, Fatti, J., mith, G., Vail,., trauss,., and Levitt,., 99, etetion of gas in sandstone reservoirs using AVO analysis: a 3 eismi Case History Using the Geostak Tehnique: Geophysis, 59, Gardner, G.H.F., Gardner, L.. and Gregory, A.., 97, Formation veloity and density - The diagnosti basis for stratigraphi traps: Geophysis, 50, Margrave, G.F., tewart,.. and Larsen, J. A., 00, Joint and seismi inversion: The Leading Edge, 0, no. 9, ihards,. G. and Frasier, C.., 976, attering of elasti waves from depth-dependent inhomogeneities: Geophysis,, -58. ussell, B. and Hampson,., 99, A omparison of post-stak seismi inversion methods: Ann. Mtg. Abstrats, oiety of Exploration Geophysiists, immons, J.L. and Bakus, M.M., 996, aveform-based AVO inversion and AVO predition-error: Geophysis, 6, ACKOLEGEMET e wish to thank our olleagues at the CEE rojet and at Hampson-ussell oftware for their support and ideas, as well as the sponsors of the CEE rojet. CEE esearh eport Volume 7 005