Waveform Inversion of Amplitude-Versus-Offset Data Usinga Genetic Algorithm

SL2.7

Subhashis Mallick, Western Geophysical

SUMMARY

I cast inversion of amplitude-versus-offset (AVO) data into aframework of Bayesian statistics. Under such a framework, themodel parameters and the physics of the forward problem areused to generate synthetic data. The synthetic data are thenmatched with the observed data to obtain an a-Posteriori Prob-ability Density (PPD) function in the model space. Geneticalgorithm (GA) is a directed random search technique that uti-lizes the natural analogy of the survival of the fittest in directingits search process. GA is not dependent upon the choice of aninitial model and is therefore well suited for the AVO inversion.

The AVO waveform inversion where all or a part of theprestack data are inverted, is nonunique for the direct estimationof the absolute values of velocities, Poisson’s ratios, and densi-ties. However, from simplified approximations to the P-wavereflection coefficient, it can be shown that the normal incidencereflection coefficient the contrast in the acoustic impedance

and the gradient in the reflection coefficient(G) can beestimated from such inversion. From the GA estimated val-ues of and G, and from reliable estimates of velocityand Poisson’s ratio at the start time of the input data, an in-verted model can be obtained. I apply this procedure to ma-rine data and demonstrate that the synthetics computed fromsuch inverted model match the input data to reasonable accu-racy. Comparison of the log data from a nearby well showsthat the GA inversion obtains both the low and the high fre-quency trends (within the bandwidth of seismic resolution) ofthe P-wave acoustic impedance. In addition, GA also obtains anestimate of the Poisson’s ratio, an extremely important param-eter for the direct detection of hydrocarbons from the seismicdata.

INTRODUCTION

Amplitude-versus-offset (AVO) analysis for the direct detec-tion of hydrocarbons from seismic data has been of growing in-terest to exploration seismologists over the past few years. Re-flection records of prestack seismic data contain valuable ampli-tude information that can be related to the subsurface lithol-ogy. With the increasing popularity of AVO, much work hasalso been done on AVO inversion; and the fundamental problemof nonuniqueness associated with such inversion is now well-recognized (Hampson, 1991; Sen and Stoffa, 1992a).

In AVO waveform inversion all or a part of the prestack dataare inverted for travel-times as well as wave-shapes. Such aninversion, like most other geophysical inverse methods, can becast into the framework of Bayesian statistics, where the modelparameters and the physics of the forward problem are used toobtain synthetic data. These synthetic data are then matchedwith the observed data to estimate an a-Posteriori ProbabilityDensity (PPD) in the model-space. For the AVO waveform in-version, the model parameters describe the earth models given

as the P-wave velocities, Poisson’s ratios, densities, and thick-nesses. The physics of the forward problem is expressed by anexact or an approximate elastic wave theory. The synthetic dataare the synthetic seismograms computed for a given model us-ing the physics of the forward problem, and the observed dataare the prestack seismic data. To estimate the PPD, I com-pute an objective fitness function, such as a normalized crosscorrelation giving the degree of similarity between the observedand the synthetic data. This fitness function is a measure ofthe absolute frequency for a given model parameter in its modelspace. The absolute frequencies of many models, when normal-ized by their respective cumulative absolute frequencies, givethe relative frequencies for each model parameter. Providedeach model parameter is accurately sampled in its domain ofdefinition, these relative frequencies are proportional to their re-spective a-Posteriori Probability Density (PPD) functions. Theobjective of an inverse method is to find the global maxima ofthe PPD function for each model parameter to get a maximumlikelihood estimate of the true model.

Genetic Algorithm (GA) is documented in Goldberg (1989).Applications of GA to geophysics are given by Stoffa and Sen(1991), Sen and Stoffa (1992a, b), and Mallick (1992). In prin-ciple, GA consists of the four basic steps: (1) parameter coding,(2) reproduction, (3) crossover, and (4) mutation. A comprehen-sive account of each of the above steps are given in the abovereferences and are not repeated here. I must point out that inaddition to the above steps, an additional operation, namely up-date, introduced by Stoffa and Sen (1991), and Sen and Stoffa(1992a, b) is required for a rapid convergence of GA. Also, toavoid genetic drift, caused by the stochastic errors due to finitepopulation sizes, I implemented a parallel GA in which manyGA are run in parallel and output from each such run are com-bined while estimating the PPD.

EXAMPLE

For the waveform inversion of AVO, I use an angle gather(Todd and Backus, 1985) as the observed data. Each tracein the angle gather represents a plane-wave propagating at aconstant angle-of-incidence. Angle gathers are therefore easilyinterpretable as the variations of the plane-wave reflection am-plitudes as functions of time and angle-of-incidence. Proper dataprocessing, e.g., a prestack migration can take care of any two orthree dimensional effect on the angle-gather data reducing eachmodel in the GA inversion to a horizontally layered earth model;each layer being described by its P-wave velocity, Poisson’s ra-tio, density, and thickness. Finally, I use the reflectivity method(Fuchs and Müller, 1971), modified to compute synthetic anglegathers as the physics of the forward problem.

Figure la shows an angle gather (between a time interval of1800 and 2200 ms) for a marine data set. Frequency analysisshowed that the data contain significant energy between 5 and

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Waveform inversion of AVO data 2

85 Hz frequency. In the following example, I estimated thesource wavelet by trial and error, although procedures of auto-matic estimation of source wavelets can easily be incorporatedinto the algorithm.

Figure lb shows the synthetic angle gather computed us-ing the estimated model obtained from ten parallel GA runs.Figure 2a compares the acoustic impedance estimated from tenGA runs with the log data from a nearby well. Notice the lowfrequency trend of the acoustic impedance is not supplied, butobtained from the GA inversion. Also notice that the GA inver-sion fails to pick some high frequency oscillations of the acousticimpedance as they fall beyond the resolution of the seismic data.Considering that the well is not on the seismic line, the over-all match between the well data and the inverted model is quitesatisfactory. In addition to P-wave acoustic impedance, GA alsogives an estimate of the Poisson’s ratio (Figure 2b), and the P-and S-wave attenuation Qp and Qs (Figure 2c). If I ignore anyattenuation effect, I do not obtain as good a fit between thewell data and the inverted model as the one shown in Figure 2a.However, since the data were deconvolved and compensated forQ at the processing stage, the high Qp and QS shown in Fig-ure 2c may be far from the true ones. Figure 3 compares theGA estimated and the well-log acoustic impedances at differentfrequency bandwidths.

DISCUSSION

The synthetic angle-gather data shown in Figure lb werecomputed using the models estimated from the GA inversion.Figure 4 shows the estimated PPD for the P-wave velocity, Pois-son’s ratio, density, and layer thickness obtained from such run.The PPD distributions in Figure 4 are in a standard wiggle traceformat. Wide PPD distributions in Figure 4 demonstrate thatthe AVO waveform inversion is nonunique for the estimation ofsuch absolute values of model parameters as the P-wave veloc-ities, densities etc. Considering that the model parameters areestimated so poorly in Figure 4, how did the GA-estimated mod-els match so closely to the seismic and well-log data (Figures 1,2a, and 3) ? To answer this question, note that the angle-gatherdata are controlled by the P-wave reflection coefficient at dif-ferent reflecting interfaces. Although the exact formula for thiscoefficient is quite complicated (Aki and Richards, 1980), thereare approximate versions offering physical insights into the pa-rameters playing key roles in the variation of the P-wave reflec-tion coefficient. These approximate formulas relate the P-wavereflection coefficient R to the angle-of-incidence as

R Ro + G (1)

in which Ro is the reflection coefficient at normal incidence andG is a dimensionless gradient term. Equation (1) suggest thatRo and G should be estimated well in the AVO inversion. SinceRo is related to the contrast in the acoustic impedance this quantity should also be well estimated. In addition, since

the arrival time for each reflection event is known, the verticaltravel time (t) for each reflecting interface can also be estimated.Figure 5 shows the PPD for Ro AA, G, and t. These PPD dis-tributions of Figure 5 were computed using the same models asthe ones used to generate the PPD of Figure 4. Tight PPD dis-tributions of Figure 5 demonstrates that in the AVO inversion,Ro, AA, G, and t can be unambiguously estimated. Using reli-able P-velocity, density, and the Poisson’s ratio at the start timeof the input data and the GA-estimated Ro AA, and G, I obtainthe estimated models which were used to compute the syntheticseismograms shown in Figure 1. Note that after the model pa-rameters at the start time are fixed, their relative changes withtime are well-determined.

ACKNOWLEDGEMENTS

I thank Western Geophysical for permission to present thispaper. Many stimulating discussions with Mrinal Sen at theUniversity of Texas at Austin and with Neil Frazer at the Univer-sity of Hawaii at Manoa regarding various aspects of SimulatedAnnealing and Genetic Algorithm were extremely beneficial. Ithank Bill King for assisting me with the data processing for theAVO analyses, Bill Dragoset, Craig Beasley, Frank Levin, Wen-dell Wiggins, Ron Chambers, and Alfonso Gonzalez for manyinteresting discussions.

REFERENCES

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

Fuchs, K., and Müller, G., 1971, Computation of synthetic seis-mograms with reflectivity method and comparison withobservations: Geophys. J. Roy. Astr. Soc., 76, 461-481.

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

Hampson, D., 1991, AVO inversion, theory and practice: TheLeading Edge, 10, No 6, 39-42.

Mallick, S., 1992, Inversion of amplitude-versus-offset data usinggenetic algorithm: 62nd Ann. Internat. Mtg., Soc. Expl.Geophys., Expanded Abstracts, 844-847.

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

Sen, M.K., and Stoffa, P.L., 1992b, Rapid sampling of modelspace using genetic algorithms: examples from seismicwaveform inversion: Geophys. J. Int., 108, 281-292.

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

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, 586-588.

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3 Waveform inversion of AVO data

Figure 1: (a) Angle gather computed from a marine data set. (b) Synthetic angle gathercomputed using a model estimated from ten parallel GA runs.

Figure 2: (a) P-wave acoustic impedance from well data com-pared with the one estimated from ten GA runs. (b) Es-timated Poisson’s ratio from ten GA runs. (c) EstimatedQP and Qs

Figure 3: Comparison of GA estimated acoustic impedancewith well acoustic impedance at different frequency band-widths: (a)unfiltered, (b) 120Hz cutoff, (c) 80Hz cutoff,and (d) 40Hz cutoff.

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Waveform inversion of AVO data 4

PPD Estimates from the Waveform Inversion of Marine Data PPD Estimates from the Waveform Inversion of Marine Data

Figure 4: PPD estimates from 10 parallel GA runs in the wave-form inversion: (a) P-wave velocity, (b) Poisson’s ratio, (c)density, and (d) layer thickness.

Figure 5: PPD estimates from 10 parallel GA runs in the wave-form inversion: (a) normal incidence reflection coefficient(R0), (b) acoustic impedance contrast (AA), (c) gradientin the reflection coefficient (G), and (d) vertical time ineach layer.

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