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Application of neural networks in nonlinear inverse problems of geophysics. (English. Russian original) Zbl 1450.86004
Comput. Math. Math. Phys. 60, No. 6, 1025-1036 (2020); translation from Zh. Vychisl. Mat. Mat. Fiz. 60, No. 6, 1053-1065 (2020).
Summary: Neural networks (NN) are widely used for solving various problems of geophysical data interpretation and processing. The application of the neural network approximation (NNA) method for solving inverse problems, including inverse multicriteria problems of geophysics that are reduced to a nonlinear operator equation of first kind (respectively, to a system of operator equations) is considered. The NNA method assumes the construction of an approximate inverse operator of the problem using neural network approximation designs (MLP networks) on the basis of a preliminary constructed set of reference solutions to direct and inverse problems. A review of the application of the NNA method for solving nonlinear inverse problems of geophysics is given. Techniques for estimating the practical ambiguity (error) of approximate solutions to inverse multicriteria problems are considered. Results of solving the inverse two-criteria 2D gravimetry problem in combination with magnetometry are presented.
86A22 Inverse problems in geophysics
86-08 Computational methods for problems pertaining to geophysics
Full Text: DOI
[1] A. N. Tikhonov and V. Ya. Arsenin, Solutions of Ill-Posed Problems (Nauka, Moscow, 1979; Winston, Washington, 1977). · Zbl 0354.65028
[2] Lavrent’ev, M. M.; Romanov, V. G.; Shishatskii, S. P., Ill-Posed Problems of Mathematical Physics and Calculus (1980), Moscow: Nauka, Moscow
[3] Strakhov, V. N., “On the integral and functional equations for certain inverse problems of the theory of logarithmic potential and their use for the interpretation of gravitational and magnetic anomalies,” Izv. Akad. Nauk SSSR, Ser. Fiz, Zemli, No., 3, 54-66 (1976)
[4] Gravity Prospecting: Geophysicist Handbook (Nedra, Moscow, 1981) [in Russian].
[5] Novikov, P. S., On the uniqueness of the inverse problem of potential, Dokl. Akad. Nauk SSSR, 18, 165-168 (1938)
[6] Strakhov, V. N., “On the condition of unambiguous calculation of interface between two-dimensional layered media on the basis of gravimetric survey,” Dokl. Akad. Nauk Ukr, SSR, No., 12, 1086-1091 (1975)
[7] Romanov, V. G.; Kabanikhin, S. I., Inverse Problems of Geoelectrics (1991), Moscow: Nauka, Moscow
[8] Dmitriev, V. I., Inverse Problems of Geophysics (2012), Moscow: MAKS, Moscow
[9] M. S. Zhdanov, Theory of Inverse Problems and Regularization in Geophysics (Nauchnyi mir, Moscow, 2007) [in Russian].
[10] Dmitriev, V. I., “On the uniqueness of solution to the 3D inverse electromagnetic sensing problem,” Trudy Fakul’teta Vychisl. Mat. Kibern. Mosck Gos. Univ, Ser. Prikl. Mat. Inform., 57, 5-20 (2018)
[11] Gurvich, I. I.; Boganik, G. N., Seismic Prospecting (1980), Moscow: Nedra, Moscow
[12] Seismic Prospecting: Geophysicist Handbook (Nedra, Moscow, 1980) [in Russian].
[13] Tikhonov, A. N.; Goncharskii, A. V.; Stepanov, V. V.; Yagola, A. G., Numerical Methods of Solving Ill-Posed Problems (1990), Moscow: Nauka, Moscow · Zbl 0712.65042
[14] Shimelevich, M. I.; Obornev, E. A.; Obornev, I. E.; Rodionov, E. A., “The neural network approximation method for solving multidimensional nonlinear inverse problems of geophysics,” Izv, Phys. Solid Earth, 53, 588-597 (2017)
[15] A. A. Nikitin and V. K. Khmelevskii, Joint Interpretation in Geophysical Methods (VNIIgeosistem, Moscow, 2012) [in Russian].
[16] Haber, E.; Oldenburg, D., Joint inversion: A structural approach, Inverse Probl., 13, 63-77 (1997) · Zbl 0878.65047
[17] M. I. Shimelevich, “Improving the stability of geoelectric data inversion using neural network simulation,” Geofiz., No. 4, 49-55 (2013).
[18] A. V. Bukharov, S. I. Kabanikhin, and M. A. Shishlenin, “Combined statement of the inverse electromagnetic sensing problem,” Abstracts of the IV Int. Conference “Functional Spaces, Differential Operators, General Topology, Mathematical Education Issues” (Ross. Univ. Druzhby Narodov, Moscow, 2013), pp. 394-395.
[19] Gallardo, L. A.; Perez, M. A.; Treviño, E. G., A versatile algorithm for joint 3D inversion of gravity and magnetic data, Geophysics., 68, 949-959 (2003)
[20] Gallardo, L. A.; Meju, M. A., Joint two-dimensional DC resistivity and seismic travel time inversion with cross-gradients constraints, J. Geophys. Res., 109, B03311 (2004)
[21] V. V. Spichak, “Modern approaches to complex inversion of geophysical data,” Geofiz., No. 5, 10-19 (2009).
[22] Goncharskii, A. V.; Cherepashchuk, A. M.; Yagola, A. G., Inverse Problems of Astrophysics (1987), Moscow: Znanie, Moscow
[23] Akhieser, N. I., Lectures on Approximation Theory (1965), Moscow: Nauka, Moscow
[24] Shimelevich, M. I.; Obornev, E. A.; Obornev, I. E.; Rodionov, E. A., An algorithm for solving inverse geoelectrics problems based on the neural network approximation, Num. Anal. Appl., 11, 359-371 (2018)
[25] Galushkin, A. I., Synthesis of Multi-Layered Pattern Recognition Systems (1974), Moscow: Energiya, Moscow
[26] Haykin, S., Neural Networks: A Comprehensive Foundation (1999), Upper Saddle River, N.J.: Prentice Hall, Upper Saddle River, N.J. · Zbl 0934.68076
[27] K. V. Vorontsov, Lectures on Artificial Neural Networks, 2009. http://www.ccas.ru/voron/download/NeuralNets.pdf
[28] P. J. Werbos, “Beyond regression: New tools for prediction and analysis in the behavioral sciences,” Ph.D. thesis, Harvard University, Cambridge, MA, 1974.
[29] Raiche, A., A pattern recognition approach to geophysical inversion using neural nets, Geophys. J. Int., 105, 629-648 (1991)
[30] H. Hidalgo, E. Gómez-Treviño, and R. Swiniarski, “Neural network approximation of a inverse functional,” Proc. IEEE Int. Conference on Neural Networks, 1994, Vol. 5, pp. 3387-3392.
[31] Poulton, M.; Sternberg, B.; Glass, C., Neural network pattern recognition of subsurface EM images, J. Appl. Geophys., 29, 21-36 (1992)
[32] Spichak, V. V.; Popova, I. V., “Application of the neural network approach for recovering the parameters of the 3D geoelectric structure,” Izv. Ross. Akad. Nauk, Ser, Fiz. Zemli, 34, 33-39 (1998)
[33] M. Shimelevitch and E. Obornev, “The method of neuron network in inverse problems MTZ,” Abstracts of the 14th Workshop on Electromagnetic Induction in the Earth, Sinaia, Romania, 1998, p. 159.
[34] Shimelevich, M. I.; Obornev, E. A., “An approximation method for solving the inverse MTS problem with the use of neural networks,” Izv, Phys. Solid Earth, 45, 1055-1071 (2009)
[35] Shimelevich, M. I.; Obornev, E. A.; Obornev, I. E.; Rodionov, E. A., “Numerical methods for estimating the degree of practical stability of inverse problems in geoelectrics,” Izv, Phys. Solid Earth, 49, 356-362 (2013)
[36] Shimelevich, M. I.; Obornev, E. A.; Obornev, I. E.; Rodionov, E. A., “A modified neural network method for solving the inverse MT problem,” Izv. Vyssh. Uchebn. Zaved., Geol, Razvedka, No., 3, 46-52 (2013)
[37] S. Dolenko, A. Guzhva, E. Obornev, I. Persiantsev, and M. Shimelevich, “Comparison of adaptive algorithms for significant feature selection in neural network based solution of the inverse problem of electrical prospecting,” Proc. ICANN 2009 (Springer, Berlin, 2009), Part II, pp. 397-405.
[38] Osman, O.; Albora, A. M.; Ucan, O. N., Forward modeling with forced neural networks for gravity anomaly profile, Math Geol., 39, 593-605 (2007) · Zbl 1141.86301
[39] Abedi, M.; Afshar, A.; Ardcstaiii, V. E.; Norouzi, G. H.; Lucas, C., Application of various methods for 2D inverse modeling of residual gravity anomalies, Acta Geophys., 582, 317-336 (2010)
[40] Chen, X.; Du, Y.; Liu, Z., Inversion of density interfaces using the pseudo-backpropagation neural network method, Pure Appl. Geophys., 175, 4427-4447 (2018)
[41] Eshaghzadeh, A.; Hajian, A., 2D inverse modeling of residual gravity anomalies from simple geometric shapes using modular feed-forward neural network, Ann. Geophysics, 61, SE115 (2018)
[42] Al-Garni, M. A., Inversion of residual gravity anomalies using neural network, Arab. J. Geosci., 6, 1509-1516 (2013)
[43] Alimoradi, A.; Angorani, S.; Ebrahimzadeh, M.; Shariat Panahi, M., Magnetic inverse modelling of a dike using the artificial neural network approach, Near Surface Geophys., 9, 339-347 (2011)
[44] Roth, G.; Tarantola, A., Neural networks and inversion of seismic data, J. Geophys. Res., 99, 6753-6768 (1994)
[45] Alfarraj, M.; AlRegib, G., Semisupervised sequence modeling for elastic impedance inversion, Interpretation, 7, SE237-SE249 (2019)
[46] Connolly, P., Elastic impedance, The Leading Edge, 18, 438-452 (1999)
[47] Kailai Xu and E. Darve, “The neural network approach to inverse problems in differential equations,” arXiv:1901.07758 [math.NA], 2019. · Zbl 1442.65411
[48] Bin Liu, Qian Guo, Shucai Li, Benchao Liu, Yuxiao Ren, Yonghao Pang, Lanbo Liu, and Peng Jiang, “Deep learning inversion of electrical resistivity data,” IEEE Trans. Geoscience Remote Sens. 58, 2135-2149 (2019). arXiv:1904.05265, 2019
[49] Shucai Li, Bin Liu, Yuxiao Ren, Yangkang Chen, Senlin Yang, Yunhai Wang, and Peng Jiang, “Deep learning inversion of seismic data,” IEEE Trans. Image Process. 1-15 (2019).
[50] Dolgal’, A. S., Computer Technologies of Processing and Interpretation of Gravimetric, Seismic, and Magnetic Survey in Mountainous Areas (2002), Abakan: FIRMA-MART, Abakan
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