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An Algorithm for Solving Inverse Geoelectrics Problems Based on the Neural Network Approximation

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Abstract

A neural network approximation algorithm for solving inverse geoelectrics problems in the class of grid (block) models of media is presented. The algorithm is based on using neural networks for constructing an approximate inverse operator and enables formalized construction of solutions of inverse geoelectrics problem with a total number of sought-for medium parameters of ~ n · 103. The correctness of the problem of constructing neural network inverse operators is considered. A posteriori estimates of the degree of ambiguity of solutions of the resulting inverse problem are calculated. The operation of the algorithm is illustrated by examples of 2D and 3D inversions of synthetic and field geoelectric data obtained by the MTS method.

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References

  1. Berdichevskii, M.N. and Dmitriev, V.I., Inverse Problems of Magnetotellurics: A Modern Formulation, Izv., Phys. Sol. Earth, 2004, vol. 40, no. 4, pp. 276–292.

    Google Scholar 

  2. Van’yan, L.L. and Butkovskaya, A.I., Magnitotelluricheskoe zondirovanie sloistykh sred (Magnetotelluric Sounding of LayeredMedia),Moscow: Nedra, 1980.

    Google Scholar 

  3. Vorontsov, K.V., Lektsii po iskusstvennym neironnym setyam (Lectures on Artificial Neural Networks), 2009; http://www.machinelearning.ru/wiki/images/c/cc/Voron-ML-NeuralNets.pdf.

    Google Scholar 

  4. Galushkin, A.I., Sintez mnogosloinykh sistem raspoznavaniya obrazov (Synthesis of Multilayer Pattern Recognition Systems),Moscow: Energiya, 1974.

    Google Scholar 

  5. Glasko, V.B., Gushchin, G.V., and Starostenko, V.I., On Application of the Tikhonov Method of Regularization to Solution of Nonlinear Systems of Equations, Zh. Vych. Mat. Mat. Fiz., 1976, vol. 16, no. 2, pp. 283–292.

    Google Scholar 

  6. Dmitriev, V.I., Obratnye zadachi geofiziki (Inverse Problems ofGeophysics),Moscow:MAKS Press, 2012.

    Google Scholar 

  7. Dmitriev, V.I. and Kokotushkin, G.A., Al’bom paletok dlya magnitotelluricheskogo zondirovaniya v neodnorodnykh sredakh (Album of Reticulations for Magnetotelluric Sounding in Inhomogeneous Media), Moscow:MGU, 1971.

    Google Scholar 

  8. Dmitrieva, M.V., Numerical Simulation of Physical Processes in Plasma of Tokamak Plants under Influence of ElectromagneticWaves of Alfven FrequencyRange, ExtendedAbstract of Cand. Sci.Diss.,Moscow, 1985.

    Google Scholar 

  9. Ivanov, V.K., On Ill-Posed Problems, Mat. Sb., 1963, vol. 61, no. 2, pp. 211–223.

    MathSciNet  Google Scholar 

  10. Kabanikhin, S.I., Shishlenin, M.A., and Novikov, N.S., Algorithms for Determination of Elastic Parameters Basing on Areal Data Collection Systems (Direct Linear Processing of Seismic Data), Vest. Kiber., 2016, no. 2, pp. 83–91.

    Google Scholar 

  11. Romanov, V.G. and Kabanikhin, S.I., Obratnyye zadachi geoelektriki (Inverse Problems of Geoelectrics), Moscow: Nauka, 1991.

    Google Scholar 

  12. Strakhov, V.N. and Lapina, M.I., Fitting Method for Solving the Inverse Problem of Gravimetry, Dokl. AN SSSR, 1976, vol. 227, no. 2, pp. 344–347.

    Google Scholar 

  13. Spichak, V.V. and Popova, I.V., Application of the Neural Network Approach to the Reconstruction of Three- Dimensional Geoelectric Structure, Izv., Phys. Sol. Earth, 1998, vol. 34, no. 1, pp. 33–39.

    Google Scholar 

  14. Haykin, S., Neural Networks and Learning Machines, 3d ed., Pearson Education, 2009.

    Google Scholar 

  15. Fel’dman, I.S., Okulesskii, B.A., Suleimanov, A.K., Nikolaev, V.I., Kuncherov, V.A., and Chamo, S.S., Electrical Prospecting by the MTS Method as Part of Regional Oil and Gas Exploration in the European Part of Russia, Zap. Gornogo Inst., 2008, vol. 176, pp. 125–131.

    Google Scholar 

  16. Shimelevich, M.I. and Obornev, E.A., An ApproximationMethod for Solving the InverseMTS Problem with the Use of Neural Networks, Izv., Phys. Sol. Earth, 2009, vol. 45, no. 12, pp. 1055–1071.

    Article  Google Scholar 

  17. Shimelevich, M.I., Obornev, E.A., Obornev, I.E., and Rodionov, E.A., NumericalMethods for Estimating the Degree of Practical Stability of Inverse Problems in Geoelectrics, Izv., Phys. Sol. Earth, 2013, vol. 49, no. 3, pp. 356–362.

    Google Scholar 

  18. Shimelevich, M.I., Obornev, E.A., Obornev, I.E., and Rodionov, E.A., The Neural Network Approximation Method for Solving Multidimensional Nonlinear Inverse Problems of Geophysics, Izv., Phys. Sol. Earth, 2017, vol. 53, no. 4, pp. 588–597.

    Article  Google Scholar 

  19. Cybenko, G., Approximation by Superpositions of a Sigmoidal Function, Math. Control, Signals, Sys., 1989, vol. 2, pp. 303–314.

    Article  MathSciNet  MATH  Google Scholar 

  20. Dolenko, S., Guzhva, A., Obornev, E., Persiantsev, I., and Shimelevich, M., Comparison of Adaptive Algorithms for Significant Feature Selection in Neural Network Based Solution of the Inverse Problem of Electrical Prospecting, Proc. ICANN 2009, part 2, Berlin: Springer-Verlag, 2009, pp. 397–405 (Lect. Notes in Computer Science, 5769).

    Book  Google Scholar 

  21. Eisenstat, S., Schultz, M., and Sherman, A., Algorithms and Data Structures for Sparse Symmetric Gaussian Elimination, SIAM J. Sci. Stat. Comput., 1981, vol. 2, no. 2, pp. 225–237.

    Article  MathSciNet  MATH  Google Scholar 

  22. Hidalgo, H., Gymez-Trevico, E., and Swiniarski, R., Neural Network Approximation of a Inverse Functional, Proc. IEEE Int. Conf. on Neural Networks, vol. 5, 1994, pp. 3387–3392.

    Google Scholar 

  23. Kabanikhin, S.I., Definitions and Examples of Inverse and Ill-Posed Problems, J. Inv. Ill-Posed Problems, 2008, vol. 16, pp. 317–357.

    MathSciNet  MATH  Google Scholar 

  24. Lönnblad, L., Peterson, C., and Rögnvalsson, T., Pattern Recognition in High Energy Physics with Artificial Neural Networks—JETNET 2.0, Comput. Phys. Comms., 1992, vol. 70, no. 1, pp. 167–182.

    Google Scholar 

  25. Mackie, R.L., Smith, J.T., and Madden, T.R., Three-Dimensional Electromagnetic Modeling Using Finite Difference Equations: TheMagnetotelluric Example, Radio Sci., 1994, vol. 29, pp. 923–935.

    Article  Google Scholar 

  26. Poulton, M., Sternberg, B., and Glass, C., Neural Network Pattern Recognition of Subsurface EM Images, J. Appl. Geophys., 1992, vol. 29, no. 1, pp. 21–36.

    Article  Google Scholar 

  27. Raiche, A., A Pattern Recognition Approach to Geophysical Inversion Using Neural Nets, Geophys. J. Int., 1991, vol. 105, pp. 629–648.

    Article  Google Scholar 

  28. Shimelevitch, M. and Obornev, E., The Method of Neuron Network in Inverse Problems MTZ, Abstr. of the 14th Workshop on Electromagnetic Induction in the Earth, Sinaia, Romania, 1998.

    Google Scholar 

  29. Werbos, P.J., Beyond Regression: New Tools for Prediction and Analysis in the Behavioral Sciences, PhD thesis, Cambridge,MA: Harvard University, 1974.

    Google Scholar 

  30. Yagola, A., Error Estimation for Ill-Posed Problems with Priori Information, Glob. J. Technol. Optim., 2010, vol. 1, pp. 88–92.

    Google Scholar 

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Authors and Affiliations

  1. Ordzhonikidze Russian State Geological Prospecting University, ul. Miklukho-Maklaya 23, Moscow, 117485, Russia

    M. I. Shimelevich, E. A. Obornev & E. A. Rodionov

  2. Skobeltsyn Institute of Nuclear Physics, LomonosovMoscow State University, Leninskie Gory 1, b. 2, Moscow, 119991, Russia

    I. E. Obornev

Authors
  1. M. I. Shimelevich
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  2. E. A. Obornev
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  3. I. E. Obornev
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  4. E. A. Rodionov
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Correspondence to M. I. Shimelevich.

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Original Russian Text © M.I. Shimelevich, E.A. Obornev, I.E. Obornev, E.A. Rodionov, 2018, published in Sibirskii Zhurnal Vychislitel’noi Matematiki, 2018, Vol. 21, No. 4, pp. 451–468.

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Shimelevich, M.I., Obornev, E.A., Obornev, I.E. et al. An Algorithm for Solving Inverse Geoelectrics Problems Based on the Neural Network Approximation. Numer. Analys. Appl. 11, 359–371 (2018). https://doi.org/10.1134/S1995423918040080

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  • DOI: https://doi.org/10.1134/S1995423918040080

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