Inverse scattering, seismic traveltime tomography, and neural networks
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 Abdurrof‟i Nawawi 12309054
OUTLINEIntroductionInverse Scattering and traveltime tomographyHopfield Nets and OptimizationSeismic Tomography using a Hopfield NetThe Rate of ConvergenceConclusions
INTRODUCTION Inverse scattering methods have been shown to inverting line integrals when the scattered field is of sufficiently high frequency and the scattering is sufficiently weak Seismic traveltime tomography uses first arrival traveltime data to invert for wave-speed structure. Neural Networks approach eliminates the need for inverting singular or poorly conditioned matrices and therefore also eliminates the need for the damping term often used to regularize such inversions.
INVERSE SCATTERING Scattering theory describes the relationship between the physical properties of an actual medium, the physical properties of a reference medium, and the impulse response for the actual and reference media. Process of sending in a wave of known characteristics, measuring the scattered waves (i.e., the deviations from the incident wave), and then using characteristics of scattered wave to invert for the structure causing the scattering. For probing some material or region to discover the shape and magnitude of any inhomogenities that might be present.
SEISMIC TOMOGRAPHY USING AHOPFIELD NET Compared results between hopfield net and previous research
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.
RATE OF CONVERGENCE The performance of the hopfield net approach is controlled by the “gain” λ in the updating rule and the number of minimization iterations (H) applied within each global iteration.
RATE OF CONVERGENCE Ω is „total gain‟ H is minimization iteration A larger total gain yields faster global For the same λ , 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 large diverge
Berryman noted that traveltime tomography reconstructs a slowness model from measured travel time for first arrivals. Therefore,We can define a feasibility constraints Based on the fermat‟s principle, first arrival necessarily followed the path of minimum travel time for the model s. Therefore,any model that violates this equation along any of the raypaths is not the feasible model. We can start from infeasibility and moving toward feasibility boundaries, and using the feasibility violation number as a performance measure.
The Feasibility violation number is the number of rays violating 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 Δs to s :
For each global iteration , we compute a new λ to reconstruct a slowness model. For each minimization, we compute the feasibility violation number and compare it with previous minimization iteration. We terminate the inversion step when violation number begin to deteriorate.
Compared with the trial and error results This procedure does not converge as fast as before, but the derived λ adn H are within the safe range of convergence and give a reasonable speed of convergence.
CONCLUSIONS Three-dimensional inverse scattering theory is quite closely related to the traveltime inversion. This Hopfield net reconstruction has fewer artifacts or smaller errors. Correctly selected gain yield faster convergence without degrading the reconstruction. The convergence to the “best approximation” (minimum norm) is guaranteed. However, the method does not guarantee global convergence for linear tomography.
REFERENCE Cheney, M. , and J. H. Rose. 1988. Three-dimensional Inverse Scattering For The Wave Equation : Weak Scattering Approximations With Error Estimates. Inverse Problems, 4, 435- 477. Hopfield, J. J. 1984. Neurons With Graded Response Have Collective Computational Properties Like Those Of Two-state Neurons. Proc. Natl. Acad. Sci. USA, 81, 3088-3092. Hopfield, J. J. , and D. W. Tank. 1985. Neural Computation Of Decisions 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.
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