Comparison between Neighbourhood and Genetic Algorithms on two Analytic Objective Functions and on a 2.5D Synthetic Seismic Inverse Problems

Transcription

1 Comparison between Neighbourhood and Genetic Algorithms on two Analytic Objective Functions and on a 2.5D Synthetic Seismic Inverse Problems A. Sajeva*, M. Aleardi*, A. Mazzotti* and E. Stucchi** * Università di Pisa ** Università di Milano

2 Agenda Stochastic Methods Neighbourhood Algorithm Genetic Algorithm Comparison on Convex Analytic Function Multi-minima Analytic Function 2.5D Seismic Inverse Problem Conclusion

3 Agenda Stochastic Methods Neighbourhood Algorithm Genetic Algorithm Comparison on Convex Analytic Function Multi-minima Analytic Function 2.5D Seismic Inverse Problem Conclusion

4 Stochastic Methods In probability theory, a purely stochastic system is one whose state is non-deterministic so that the subsequent state of the system is determined probabilistically. [M. Kac & J. Logan, in Fluctuation Phenomena, eds. E.W. Montroll & J.L. Lebowitz, North-Holland, Amsterdam, 1976] A geophysical inverse problem consists in obtaining the earth model for which the predicted data best fit the observed ones (Tarantola, 1986). The problem is often non-linear and can be solved using a local linearization method (such as Gauss-Newton, steepest descent or conjugate gradient) or using a Stochastic method such as Simulated Annealing, Genetic Algorithms, and Neighborhood Algorithm.

5 Neighbourhood Algorithm (1) Generate an initial set of ns models uniformly (or otherwise) in model space of dimension nd; (2) Calculate the misfit function for the most recently generated set of ns models and determine the nr models with the lowest misfit of all models generated so far; (3) Generate ns new models by performing a uniform random walk in the Voronoi cell of each of the nr chosen models (i.e. ns/nr samples in each cell); (4) Go to step 2. (Sambridge, 1998,1999a,1999b)

6 Genetic Algorithm Genetic Algorithms (Holland,1975) are based on the mechanics of natural selection and evolution to search through model space of dimension nd. A population of strings (called chromosomes) is evolved toward better solutions during the evolution process. In each generation, the fitness (the error associated to each possible solution) of every individual is evaluated, then multiple individuals are stochastically selected from the current population on the basis of their fitness. model model model Then they are modified (using crossover and mutation operators) to form a new population which is used in the next iteration. The algorithm terminates when either a maximum number of generations has been reached, or the fitness level for the current population is satisfactory. model model

7 Agenda Stochastic Methods Neighbourhood Algorithm Genetic Algorithm Comparison on Convex Analytic Function Multi-minima Analytic Function 2D Seismic Inverse Problem Conclusion

8 Convex function: De Jong m2 m1 m2 m1

9 Inversion Parameters Dim. of model space (nd) [ 2, 5, 10, 15, 20 ] Model ranges ±10 Num. of inversions 10 Convergence Hardware (mtrue-m) < Ɛ esa-core Intel(R) Xeon(R) CPU 48 GB RAM NA num. of models comp. x iter. (ns) 10*nd num. of chosen models (nr) 1 GA num. of individuals x population (num pop) 10*nd SELECTION sel rate 80% sel type stochastic universal sampling pressure sel 2, linear MUTATION mute range 0.2*(model range) mute rate 10%

10 Inversion Results: NA vs GA m2 m1 (nd) (nd)

11 Multi-minima function: Egg-Box m2 m1 m2 m1

12 Inversion Parameters Model ranges ±10 Num. of inversions 5 Convergence (mtrue-m) < Ɛ Hardware esa-core Intel(R) Xeon(R) CPU 48 GB RAM NA dim. of model space (nd) [ 1, 2, 3, 4, 5] num. of initial models ~ 2 ns num. of models comp. x iter (ns) ~ 6 nd num. of chosen models (nr) 1/3 ns GA dim. of model space (nd) [ 2, 5, 10, 15, 20 ] num. pop 200 SELECTION sel rate 80% sel type roulette wheel pressure sel non linear MUTATION mute range 0.2*(model range) mute rate 10%

13 Inversion Results: NA vs GA m2 m1 (nd) (nd)

14 Agenda Stochastic Methods Neighbourhood Algorithm Genetic Algorithm Comparison on Convex Analytic Function Multi-minima Analytic Function 2D Seismic Inverse Problem Conclusion

15 FW-global- True model Inversion 2.5 D Range : 400 m/s Misfit function : L2 Forward Operator : Finite Differences Dimensions of Model Space : 64 Max. Frequency : 1 Hz Cell Dimensions : 288 x 288 m CONVERGENCE CRITERIUM Num. of evaluated models : ~33900 Marmousi ACQUISITION GEOMETRY num. of geophones : 60 (inline ) x 15 (crossline) : 72 m distance btw geophones num. of sources : 15 distance btw sources : 288 m

16 FW-global-Inversion 2.5 D Final model: differences with the true model NA Inversion Parameters num. of models comp. x iter. (ns) 128 num. of chosen models (nr) 32 num. of iterations 264 Data Misfit Evolution

17 FW-global-Inversion 2.5 D Final model: differences with the true model GA Inversion Parameters num. pop 300 num. of generations 140 SELECTION sel rate 80% sel type stochastic universal sampling pressure sel 2 (linear ranking) Data Misfit Evolution MUTATION mute range 0.2*(model range) mute rate 10%

18 Conclusions We compare two different stochastic optimization methods: a Genetic Algorithm and the Neighborhood Algorithm. The tests on the analytic functions (both the convex one and the multiminima one) highlight a better performance of GA with respect to NA in case of high-dimension model spaces. In case of the multi-minima function the inversion parameters for NA must be carefully chosen in order to avoid trapping in a local minimum. The high computational cost of NA for high dimensions is likely due to the computation of the Voronoi cells, whose time is proportional to the square of the number of evaluated models. The synthetic seismic inversion, performed with a 64 dimensions model space, confirms the results obtained with the analytic functions, that is, the better performance of GA for high dimension model spaces (nd >15-20).

19 Aknowledgements These results were obtained within a research project with ENI. We thank ENI for the permission to publish these results.

20 Thank you for your attention. Any questions?