Inversion of Amplitude-versus-Offset Data Using a Genetic Algorithm SL2.5 Subhashis MaHick, Western Geophysical

SUMMARY

I cast the inversion of amplitude-versus-offset (AVO) data into the framework of Bayesian statistics, where prior informatiox on model parameters and the physics of the forward problen are used to form synthetic data that are matched with the observed data to obtain an r+Posteriori Probability Den.+ (PPD) function in the model space. Genetic Algorithm (GA: uses a directed random search technique to estimate the shape of the PPD. Unlike the classical inversion methods, GA is no1 dependent on the choice of the initial model and is well suitet for the AVO inversion. The inversion results using GA on syn. thetic and real data show that the method works well, ever when the signal-tonoise ratio is low. Comparisons on syn thetic data also indicate that GA inversion obtains more accu. rate results than do the inversion techniques usually practiced in the industry.

INTRODUCTION

Genetic Algorithm (GA) is documented in Goldberg’s 1989 book. Applications of GA to geophysics are given by Frazer et al. (199Q), Stoffa and Sen (1991), Sen and Stoffa (1992a, b), and Smith et al. (1992). In principle, GA consists oi the four basic steps: (1) parameter coding, (2) reproduction, (3) crossover, and (4) mutation.

In parameter coding, each model parameter is coded as an un- signed binary string. In the case of a simple AVO problem consisting of a reflection from a single interface, the model parameters are the P-wave velocity, Poisson’s ratio, density for the medium above the interface, and the contrasts of these quantities across the interface. The coded binary strings for each parameter of a particular model are concatenated to form a chromosome. Each chromosome therefore represents a model. Once a random population of chromosomes is generated, each member of the population is decoded for the actual parameter values. Next, the physics of the forward problem, i.e. the exact ‘ormula for the P-P reflection coefficient (Aki and Richards, 1980, vl, p 144150) is used to obtain the synthetic AVO data br each model (chromosome). These synthetic data are then :ross-correlated with the observed AVO data to obtain a fit- ncs.9 for each chromosome. In reproduction, chromosomes are limply reproduced in number proportional to their fitness. In Erossower, two chromosomes from the reproduced population ue randomly chosen as parcnt.9. With a given probability of :rossover, the gene (bit) contents between the two parents are martially swapped to produce two children. Finally, in mutu- !ion, a single gene in each child is randomly changed, with L specified probability of mutation. Reproduction, crossover, md mutation are the building blocks of GA which take one rcncration of chromosomes into the next, and these three pro :esses are repeated until a generation with many chromosomes itting data with reasonable accuracy are obtained. The fitness

values for every model generated in the course of the run are stored in the model-space. At the end of the GA run, these fitness values are normalized by their respective cumulative fit ness values to get the estimates of the a-Posteriori Probability Density (PPD) functions.

EXAMPLES

Figure la shows a synthetic seismogram computed for a model consisting of a single interface separating two media with P- wave velocities 2350 and 2730 m/s, densities 2.2 and 2.15 g/cm” and Poisson’s ratios 0.4 and 0.95. Thii synthetic seismogram is computed using ANIVEC software developed by Mallick and Frazer (1987, 1988, 1999, 1991). The two arrivals appearing in the seismogram are the P-P and P-W reflections, respec- tively. Figure lb shows the instantaneous amplitudes of the seismogram of Figure la, and Figure lc shows the normaliied P-P instantaneous amplitudes. The AVO response for the P-P reflection in Figure la, b, and c is representative of a class- 1 gas-sand reflection (Rutherford and Williams, 1989). The normalized instantaneous amplitudes of Figure lc are used as observed data in the Genetic Algorithm. Since instantaneous amplitudes do not have polarity information, GA is given the additional information that the contrast in Poisson’s ratio, Au, between the lower and the upper medium is negative ( i.e. the amplitudes are actually decreasing with the angle of incidence). The results of the inversion are shown in Figure Id - f. While the inversion is not able to find the absolute values of model parameters (e.g., the P-wave velocity of layer 1, shown in Fig- ure Id), it uniquely finds I&, the reflection coefficient at normal incidence (Figure le), and Au, the contrast in Poisson’s ratio (Figure If). This result is consistent with the observation by others (e.g., see Hampson, 1991). In the presence of noise as high as 75% of peak signal amplitude (Figures 2a, 2b), GA is still able to find the PPD of & and Au with reasonable ac- curacy (Figures 2c, 2d). Table 1 compares the GA inversion with linearized inversions using the approximate P-P reflection coefficients given by Shuey (1985), and Hilterman (1990), and demonstrates that the GA can give more accurate parameter estimates than the AVO inversion techniques usually practiced in the industry. In addition, note that to perform AVO in- version using Shuey (1985) or Hilterman (199Q), the seismic data must be calibrated such that the amplitudes correspond to the true reflection coefficients. Also, Shuey (1985) requires the knowledge of the P-wave velocities, and the average Pois- son’s ratio, and Hilterman (1990) requires the knowledge of P-wave velocities for the AVO inversion (see Mallick, 1992, for details). The GA on the other hand, does not require any such a priori information, nor does it require that the seismic data be absolutely calibrated. GA works as long as the relative vari- ations of amplitude are preserved.

Figure 3a shows an angle gather computed for marine data (Todd and Backus, 1985). Proper care was taken to preserve

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2 Inversion AVO

relative amplitude information while processing the data set. To enhance signal-to-noise ratio, three adjacent angle gathers were stacked together to produce the seismogram. The data show substantial AVO anomalies in the time interval of 2000 ms and 2200 ms. The amplitudes as functions of incidence angle for events marked 1, 2, and 3 on Figure 3a are shown in Figures 3b, 3c, and 3d, respectively. Although the angle gather was computed to an angle of 32O, the amplitudes could be picked reliably only to 20” for event 1, and to 25O for events 2 and 3. Figure 4 gives the inversion results for these three r+ Section events. Figure 4 shows that GA ia able to estimate the PPD for & and Au as long as the amplitudes can be reliably picked from the observed data. For example, a visual compari- son of amplitudes picked for the events 1,2, and 3 in Figure 3a (see Figures 3b, 3c, and 3d, respectively) indicates that the amplitudes for event 2 are more irregular than those for the events 1 and 3. This irregularity is reflected in the inversion results for Au, shown in Figure 4. Notice that the PPD for Au for event 2 (Figure 4d) has a wider peak than those for the events 1 and 3 (Figures 4b and 4f).

Mallick, S., and Frazer, L.N., 1991, Re&ction/Transmizsion coefficients and azimuthal anisotropy in marine seismic stud- ies, Geophys. J. Int., 105,241-252.

Mallick, S., 1992, A simple approximation to the P-wave relIec- tion coefficient and its implication in the inversionof amplitude versus offset data: Submitted to Geophysics.

Rutherford, S.R., and Williams, R.H., 1989, Amplitude-versus- offset variations in gas sands: Geophysics, 54,68&688.

Sen, M.K., and Stoffa, P.L., 1992a, Rapid sampling of model space using genetic algorithms: examples from seismic wave- form inversion: Geophys. J. Int. (In Press).

Sen, M.K., and Stoffa, P.L., 1992b, Genetic inversion of AVO: The Leading Edge, 11, No. 1,27-29.

Shuey, R.T., 1985, A simpli6cation of the Zoepprite equations: Geophysics, 50,‘6C%614.

ACKNOWLEDGEMENTS

I thank Western Geophysical for permission to present this paper, and Gee-Pacific Corporation for permission to use the ANIVEC software. I also thank Bill Dragoset, Craig Beasley, Frank Levin, Wendell Wiggins, Ron Chambers, Alfonso Gon- sales, Mike Reed, Uwe Albertin, and Dan Wisecup for many interesting discussions and useful suggestions throughout the preparation of this work.

Smith, M.L., Scales, J.A., and Fischer, T.L., 1992, Global search and genetic algorithms: The Leading Edge, 11, NO. 1, 22-26.

Stoffa, P.L., and Sen, M.K., 1991, Nonlinear multiparameter optimization using genetic algorithms: Inversion of plane-wave seismograms: Geophysics, 56, 1794-1810.

REFERENCES

Todd, C.P., and Backus, M.M., 1985, Offset-dependent reflec- tivity in a structural context: 55th Ann. Internat. Mtg., SOC. Expl. Geophys., Expanded Abstracts, 586588.

Aki, K., and Richards, P.G., 1980, Quantitative Seismology, 1, W.H. Freeman and Co.

Fraser, L.N., Basu, A., and Lou, J., 1990, Geophysical inver- ;ion using simulated annealing and genetic algorithms, EOS Trans. Am. Geophys. U., 71, 1477.

Goldberg, D.E., 1989, Genetic algorithms in search optimize tion and machine learning: Addison-Wesley Pub. Co., Inc.

Hampson, D., 1991, AVO inversion, theory and practice: The Leading Edge, 10, No. 6, 3442.

Hilterman, F., 1996, Is AVO the seismic signature of lithology? A case history of ship-shoal-south addition: The Leading Edge, 8, No. 6, 15-22.

Mallick, S., and Fraser, L.N., 1987, Practical aspects of reflec- tivity modeling: Geophysics, 52, 13551364.

Mallick, S., and Frazer, L.N., 1988, Rapid computation of mul- tioffset vertical seismic profile synthetic seismograms for lay- ered media: Geophysics, S&479-491.

Mallick, S., and Frazer, L.N., 1990, Computation of synthetic seismograms for stratified asimuthally anisotropic media: J. Geophys. Res., 95,8513-8526.

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i- t

Fig. 1. (a) Synthetic seismograms for the model consisting 3f two layers with P wave velocities 2350 and 2730 m/s, densi- ties 2.2 and 2.15 g/cmS, and Poisson’s ratios 0.4 and 0.05. (b) Instantaneous amplitudes of the seismogram shown in Figure la. (c) Normalized instantaneous amplitude of the P - P re Bection (event marked by arrows in Figures la and lb). (d) Normalized Posteriori Probability Density (PPD) for the P- wave speed of layer 1. (e) PPD for the reflection coefficient at normal incidence (Ii&). (f) PPD for the contrast in Poisson’s ratio (Au).

Fig. 2. (a) Same as Figure la, but with random noise having a peak amplitude equal to 75% of the peak signal amplitude. [b) Normalized P - P instantaneous amplitude (event, marked by an arrow in Figure 2a). (c) PPD for I&J, (d) PPD for Au.

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4 Inversion AVO

Fig. 3. (a) Angle stack computed from a marine data set. (b) Normalized amplitude 88 a function of offset for the event 1 marked on (a). (c) Normalized amplitude aa a function of offset for the event 2 marked on (a). (d) Normalized amplitude as a function of offset for the event 3 marked on (a).

Fig. 4. (a) PPD for the reflection coefficient at normal in- cidence (RJ), obtained from inverting the amplitude response shown in Figure 3b. (b) PPD for the contrast in Poisson’s ratio (Au), obtained from inverting the amplitude reaponse shown in Figure 3b. (c) Same as (a), but for the amplitude response shown in Figure 3c. (d) Same as (b), but for the amplitude response shown in Figure 3c. (e) Same aa (a), but for the am- plitude response shown in Figure 3d. (f) Same as (b), but for the amplitude response shown in Figure 3d.

Exact Values Noise Free Synthetics Synthetics with 75% Noise

Ra I A0 Ra AU &l AU 0.063 [ -0.35 E&t 1 Err (%) Est 1 Err (%) Est 1 Err (%) Est 1 Err (%)

I I I Shuey (1985) 0.063 0 -0.20 43 0.072 12 -0.16 54

Hilterman (1990) 0.063 0 -0.18 49 0.072 12 -0.15 59

GA 0.063 0 -0.34 2.9 0.060 5 -0.31 11

Table 1. Comparisons of linearized inversions on the synthetic data using Shuey (1985) and Hilterman (1990) with GA inversion. For any quantity A with the exact value A,. and the estimated value Aelt, the error A,,, in the estimate of A is computed using

A,,, = 100 1 A,, - Atrt I / I -4,s I.

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