The document discusses inverse scattering, seismic traveltime tomography, and using neural networks for seismic tomography. It summarizes that inverse scattering can be used to infer underground structures from scattered seismic waves, while traveltime tomography uses arrival times to map wave speed variations. The document proposes using a Hopfield neural network for seismic tomography to eliminate matrix inversion and provide sharper images with fewer artifacts compared to traditional methods. It examines the network's convergence properties and feasibility constraints to guide the inversion process.

1. INVERSE SCATTERING, SEISMIC
TRAVELTIME TOMOGRAPHY, AND
NEURAL NETWORKS
Shin-yee Lu and James G. Berryman
International 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

2. OUTLINE
Introduction
Inverse Scattering and traveltime tomography
Hopfield Nets and Optimization
Seismic Tomography using a Hopfield Net
The Rate of Convergence
Conclusions

3. 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.

4. 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.

5. INVERSE SCATTERING


6. INVERSE SCATTERING AND
TRAVELTIME TOMOGRAPHY


7. INVERSE SCATTERING AND
TRAVELTIME TOMOGRAPHY


8. HOPFIELD NETS


9. HOPFIELD NETS

10. HOPFIELD NETS


11. HOPFIELD NETS


12. HOPFIELD NETS


13. HOPFIELD NETS


14. SEISMIC TOMOGRAPHY USING A
HOPFIELD NET
M is an m x n matrix
s is slowness vector
t is the derived travel time
Ms=t

15. SEISMIC TOMOGRAPHY USING A
HOPFIELD NET


16. SEISMIC TOMOGRAPHY USING A
HOPFIELD NET


17. SEISMIC TOMOGRAPHY USING A
HOPFIELD NET
 Compared results between hopfield net and previous research

18. SEISMIC TOMOGRAPHY USING A
HOPFIELD NET
Hopfield 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” λ in the updating rule and the
number of minimization iterations (H) applied within
each global iteration.

20. 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

21.  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.

22.  The Feasibility violation number is the number of rays
violating the feasibility constraints for a given model s :
where
The feasibility violation number can be used as a
stopping criterion in the inversion step. For a model in the
infeasible region, we can move it to the feasibility
boundary by adding Δs to s :

23.  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.

24.  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.

25. 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.

26. 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.