Isometric method: Efficient tool for solving non-linear inverse problems

Studia Geophysica et Geodaetica - Tập 51 - Trang 469-490 - 2007
J. Málek1, B. Růžek2, P. Kolář2
1Institute of Rock Structure and Mechanics, Acad. Sci. Czech Republic, Praha 8, Czech Republic
2Institute of Geophysics, Acad. Sci. Czech Republic, Praha 4, Czech Republic

Tóm tắt

A novel algorithm called Isometric Method (IM) for solving smooth real-valued non-linear inverse problems has been developed. Model and data spaces are represented by using m + 1 corresponding vectors at a time (m is the dimension of model space). Relations among vectors in the data space are set up and then transferred into the model space thus generating a new model. If the problem is truly linear, this new model is the exact solution of the inverse problem. If the problem is non-linear, the whole procedure has to be repeated iteratively. The basic underlying idea of IM is to postulate the distance in the model space in such a way that the model and data spaces are isometric, i.e. distances in both spaces have the same measure. As all model-data vector pairs are used many times in successive iterations, the number of the forward problem computations is minimized. There is no necessity to deal with derivatives. The requirement for the computer memory is low. IM is suitable especially for solving smooth medium non-linear problems when forward modelling is time-consuming and minimizing the number of function evaluations is topical. Applications of IM on synthetic and real geophysical problems are also presented.

Tài liệu tham khảo

Aki K. and Richards P.G., 1980. Quantitative Seismology. Theory and Methods. W.H. Freeman and Company, San Francisco, USA.

Aster R.C., Borchers B. and Thurber C.H., 2005. Parameter Estimation and Inverse Problems. Elsevier, IBSN 0-12-065604-3, USA.

Bäck T. and Schwefel H.P., 1995. Evolution strategies I: Variants and their computational implementation. In: J. Périaux and G. Winter (Eds.), Genetic Algorithms in Engineering and Computer Science, John Wiley & Sons Ltd., New York, 111–126.

Backus G. and Mulcahy M., 1976a. Moment tensors and other phenomenological descriptions of seismic sources. Continuous displacements. Geophys. J. Royal Astron. Soc., 46(2), 341–361.

Backus G. and Mulcahy M., 1976b. Moment tensors and other phenomenological descriptions of seismic sources. Discontinuous displacements. Geophys. J. Royal Astron. Soc., 47(2), 301–329.

Baish S., Bohnhoff M., Ceranna L, Tu Y. and Harjes H.P., 2002. Probing the crust down to 9 km depth: A unique longterm fluid injection experiment at the KTB superdeep drilling hole, Germany. Bull. Seismol. Soc. Amer., 92, 2369–2380.

Billings S.D., Kenneth B.L.N. and Sambridge M.S., 1994. Hypocenter location: Genetic algorithm incorporating problem-specific information. Geophys. J. Int., 118, 698–706.

Calderón-Marcías C., Sen M.K. and Stoffa P.L., 2000. Artificial neural networks for parameter estimation in geophysics. Geophys. Prospect., 48, 21–47.

Fanni A. and Montisci A., 2003. A Neural inverse problem approach for optimal design. Transactions on Magnetics, 39, 1305–1308.

Fogel D.B., 1995. Evolutionary Computation. IEEE Press, New York, ISBN 0-7803-1038-1.

Hoffmeister F. and Bäck T., 1992. Genetic Algorithms and Evolution Strategies: Similarities and Differences. Technical Report No. SYS-1/92, University of Dortmund, Dortmund, Germany, ISSN 0941-4568.

Holland J.H., 1975. Adaptation in Natural and Artificial Systems. University of Michigan Press, Michigan, USA.

Kolář P., 2000. Comparing tests of several non-linear methods on three simple synthetic problems. Acta Montana 15, 84–93.

Kolář P., 2007. How much can we trust some moment tensors or an attempt of seismic moment error estimation. Acta Geodyn. Geomater., 4, No.2(146) 13–20.

Kolínský P. and Brokešová J., 2007. The Western Bohemia uppermost crust shear wave velocities from Love wave dispersion. J. Seismol., 11, 101–120.

Málek J., 1998. Tomographic location of rockbursts using isometric algorithm. Publ. Inst. Geoph. Pol. Acad. Sc., M-22, 167–170.

Málek J., 2005. 3D anisotropic model of seismic velocities in the upper crust. In: P. Alberigo, G. Erbacci and F. Garofano (Eds.), Science and Supercomputing in Europe. CINECA, Bologna, Italy, ISBN 88-86037-15-5, 244–247.

Málek J., Horálek J. and Janský J., 2005. One-dimensional qP-wave velocity model of the Upper Crust for the West Bohemia/Vogtland Earthquake swarm region. Stud. Geophys. Geod., 49, 501–524.

Menke M., 1989. Geophysical Data Analysis: Discrete Inverse Theory. Academic Press Ltd., London, U.K.

Nakayama H., Arakawa M. and Sasaki R., 2001. Optimization with implicitly known objective functions using RBF Networks and Genetic Algorithms. In: V. Kůrková, N.C. Steele, R. Neruda and M. Kárný (Eds.), Artificial Neural Nets and Genetic Algorithms. Proceedings of the International Conference in Prague, Czech Republic, 2001. Springer, Wien, New York, 387–390.

Press W.H., Teukolsky S.A., Vetterling W.T. and Flannery, B.P., 1992. Numerical Recipes in C. The Art of Scientific Computing. Second Edition. Cambridge University Press. Cambridge, U.K.

Price K. and Storn R., 1997. Differential evolution. Dr. Dobb’s Journal, 22(4), 18–24.

Price K., Storn R.M. and Lampinen J.A., 2005. Differential Evolution: A Practical Approach to Global Optimization. Springer, Berlin, Germany, ISBN-10 3-540-20950-6.

Sambridge M.S. and Drijkoningen G., 1992. Genetic algorithm in seismic waveform inversion. Geophys. J. Int., 109, 323–342.

Storn R. and Price K., 1997. Differential Evolution. A simple and efficient heuristic for global optimization over continuous spaces. J. Glob. Optim., 11, 341–359.

Tarantola A., 1987. Inverse Problem Theory. Elsevier Science B.V, Amsterdam, The Netherlands.

Vavryčuk V., Bohnhoff M., Jechumtálová Z., Kolář P. and Šílený J., 2007. Non-double couple mechanisms of induced microearthquakes during the 2000 injection experiment at the KTB site, Germany: a result of tensile faulting or anisotropy of a rock? Tectonophysics., in print.

Zhang Y. and Paulson K.V., 1997. Magnetoteluric inversion using regularized Hopfield neural networks. Geophys. Prospect., 45, 725–743.

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Studia Geophysica et Geodaetica

Tập 51

469-490

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