1. INVERSE SCATTERING, SEISMICTRAVELTIME TOMOGRAPHY, ANDNEURAL NETWORKSShin-yee Lu and James G. BerrymanInternational Journal of Imaging Systems and Technology, vol 2, 112-118 (1990) Kelompok 6 Muhammad Naufal Hafiyyan 12309031 Muhammad Arief Wicaksono 12309033 Fajar Abdurrofi Nawawi12309054

2. OUTLINEIntroductionInverse Scattering and traveltime tomographyHopfield Nets and OptimizationSeismic Tomography using a Hopfield NetThe Rate of ConvergenceConclusions 3. INTRODUCTION Inverse scattering methods have been shown toinverting line integrals when the scattered field is ofsufficiently high frequency and the scattering issufficiently weak Seismic traveltime tomography uses first arrivaltraveltime data to invert for wave-speed structure. Neural Networks approach eliminates the need forinverting singular or poorly conditioned matrices andtherefore also eliminates the need for the dampingterm often used to regularize such inversions. 4. INVERSE SCATTERING Scattering theory describes the relationship betweenthe physical properties of an actual medium, thephysical properties of a reference medium, and theimpulse response for the actual and reference media. Process of sending in a wave of known characteristics,measuring the scattered waves (i.e., the deviations fromthe incident wave), and then using characteristics ofscattered wave to invert for the structure causing thescattering. For probing some material or region to discover theshape and magnitude of any inhomogenities that mightbe present. 5. INVERSE SCATTERING 6. INVERSE SCATTERING ANDTRAVELTIME TOMOGRAPHY 7. INVERSE SCATTERING ANDTRAVELTIME TOMOGRAPHY 8. HOPFIELD NETS 9. HOPFIELD NETS 10. HOPFIELD NETS 11. HOPFIELD NETS 12. HOPFIELD NETS 13. HOPFIELD NETS 14. SEISMIC TOMOGRAPHY USING AHOPFIELD NETM is an m x n matrixs is slowness vectort is the derived travel timeMs=t 15. SEISMIC TOMOGRAPHY USING AHOPFIELD NET 16. SEISMIC TOMOGRAPHY USING AHOPFIELD NET 17. SEISMIC TOMOGRAPHY USING AHOPFIELD NET Compared results between hopfield net and previous research 18. SEISMIC TOMOGRAPHY USING AHOPFIELD NETHopfield net approach has a sharper contrast around slow anomaly, and the artifact at the top is less pronounced.Hopfield net approach eliminates the need for the damping term often used to regularize singular or poorly conditioned matrices in inversion problems. 19. RATE OF CONVERGENCE The performance of the hopfield net approach is controlled by the gain in the updating rule and the number of minimization iterations (H) applied within each global iteration. 20. RATE OF CONVERGENCE is total gain H is minimization iteration A larger total gain yields faster global For the same , by increasing H, the convergence, but the perfomrance will mean-square traveltime errors may be degrade and the errors diverge quickly if converging, but the model errors will the total gain becomes too largediverge 21. Berryman noted that traveltime tomography reconstructs aslowness model from measured travel time for first arrivals.Therefore,We can define a feasibility constraints Based on the fermats principle, first arrival necessarilyfollowed the path of minimum travel time for the model s.Therefore,any model that violates this equation along any ofthe raypaths is not the feasible model. We can start frominfeasibility and moving toward feasibility boundaries, andusing the feasibility violation number as a performancemeasure. 22. The Feasibility violation number is the number of raysviolating the feasibility constraints for a given model s :whereThe feasibility violation number can be used as astopping criterion in the inversion step. For a model in theinfeasible region, we can move it to the feasibilityboundary by adding s to s : 23. For each global iteration , we compute a new toreconstruct a slowness model. For each minimization,we compute the feasibility violation number andcompare it with previous minimization iteration. We terminate the inversion step when violation numberbegin to deteriorate. 24. Compared with the trial and error results Thisprocedure does not converge as fast as before, but thederived adn H are within the safe range ofconvergence and give a reasonable speed ofconvergence. 25. CONCLUSIONS Three-dimensional inverse scattering theory is quiteclosely related to the traveltime inversion. This Hopfield net reconstruction has fewer artifacts orsmaller errors. Correctly selected gain yield faster convergencewithout degrading the reconstruction. The convergence to the best approximation(minimum norm) is guaranteed. However, the methoddoes not guarantee global convergence for lineartomography. 26. REFERENCE Cheney, M. , and J. H. Rose. 1988. Three-dimensional InverseScattering For The Wave Equation : Weak ScatteringApproximations With Error Estimates. Inverse Problems, 4, 435-477. Hopfield, J. J. 1984. Neurons With Graded Response HaveCollective Computational Properties Like Those Of Two-stateNeurons. Proc. Natl. Acad. Sci. USA, 81, 3088-3092. Hopfield, J. J. , and D. W. Tank. 1985. Neural Computation OfDecisions In Optimization Problems. Biol. Cybernet, 52, 141-152. Jeffrey, W. , and Rosner, R. 1986. Optimization Algorithms:Simulated Annealing And Neural Network Processing. Astrophys.J., 310, 473-481.