Correction of velocity profile by the method of imitation of annealing

М. А. Lazarenko, O. A. Gerasimenko


Reversal of hodograph in 2D formulation, obtained by finite-difference solution of eikonal has been put into practice  by the method of imitation of  “annealing”  in the problem of combinatorial optimization for uneven and interrupted functions. Hodographs of seismic studies of DSS of the territory of Ukraine (profile DOBRE-5) have been used as a target function. Velocity function of the medium was restored by two-dimensional published models of P-waves velocities. Refinements of optimal functioning of the algorithm of “annealing” imitation and of regime of initial function filtration during the process of minimization of the function of losses value are presented in details.


method of imitation of annealing; “eikonal” equation; finite-difference valuation; minimization of function of losses; combinatorial optimization; arrivals of longitudinal waves; solution of direct problem; DSS; function of the losses value; iteration; hodograph


Babich V. M., Buldyrev V. S., 1972. Asymptotic methods in problems of diffraction of short waves. Moscow: Nauka, 455 p. (in Russian).

Bazarov I. P., Gevorkyan E. V., Nikolaev P. N., 1986. Thermodynamics and Statistical Mechanics. The theory of equilibrium systems. Moscow: Publ. MSU, 312 p. (in Russian).

Nolet G., 1990. Seismic tomography. Moscow: Mir, 415 p. (in Russian).

Papadimitru Kh., Stayglits K., 1985. Combinatorial Optimization: Algorithms and complexity. Moscow: Mir, 512 p. (in Russian).

Hutton L., Uerdipton M., Makin J., 1986. Processing of the seismic data. Moscow: Nauka, 285 p. (in Russian).

Ammon C. J., Vidalе J. E., 1993. Tomography without Rays. Bull. Seysmol. Soc. Am. 83(2), 509—528.

Alford R. M., Kelly K. R., Booret D. M., 1974. Accuracy of finite-difference modeling of the acoustic wave equations. Geophysics 39(6), 834—842.

Cao S., Greenhalgh S., 1994. Finite-difference solution of the eiconal equation using an efficient, first arrival, wavefront tracking scheme. Geophysics 59(4), 632—643.

Černý V., 1985. Thermodynamical approach to the traveling salesman problem: An efficient simulation algorithm. J. Optim. Theor. Appl. 45, 41—51.

Goffe W. L., Farrier G. D., Rogers J., 1994. Global optimization of statistical functions with simulated annealing. J. of Econometrics 60, 65—99.

Guerra V., Tapia R. A., 1974. A local procedure for error detection and data smoothing. MRC Technical Summary Report 1452, Mathematics Research Center, University of Wisconsin, Madison.

Kim S. D., 1992. Eiconal solvtrs: first-arrival traveltimes. Geophysics 57(4), 632—643.

Kirkpatrik S., Gelatt C. D., Vecchi M. P., 1983. Optimization by simulated annealing. Science 220, 671—680.

Levievre P. G., Farquharson C. G. Hurich C. A., 2011. Inverision of first-arrival seismic traveltimes without rays, implemented on unstructured grids. Geophys J. Int. 185, 749—763. doi: 10.1111/j.1365-246X.2011.04964.x.

Metropolis N., Rosenbluth F., Rosenbluth V., Teller A., Teller E., 1953. Equation of State Calculation by Fast Computing Machines. J. Chem. Phys. 21, 1087—1092.

Mo L.-W., Harris J. M., 2002. Finite-difference calculations of direct arrival travel times and amplitudes. Geophisics 59(5), 167—176.

Nowack R. L., 1992. Wave fronts and solutions of the eikonal equation. Geophys. J. Int. 110, 55—62. doi: 10.1111/gji.1992.

Podvin P., Lecomte I., 1991. Finite difference computation of traveltime in very contrasted velocity models: a massively parallel approach and its associated tools. Geophys. J. Int. 105, 271—284.

Qian J., Symes W. W., 2002. An adaptive finite-difference method for traveltimes and amplitudes. Geophysics 67(1), 167—176.

Qin F., Luo Y., Olsen K. B., Cai W., Schuster G. T., 2002. Finite-difference solution of the eikonal equation along expanding wavefronts. Geophysics 67(4), 1225—1231.

Starostenko V., Janik T., Yegorova T., Farfuliak L., Czuba W., Środa P., Thybo H., Artemieva I., Sosson M., Volfman Y., Kolomiyets K., Lysynchuk D., Omelchenko V., Gryn D., Guterch A., Komminaho K., Legostaeva O., Tiira T., Tolkunov A., 2015. Seismic model of the crust and upper mantle in the Scythian Platform: the DOBRE-5 profile across the north western Black Sea and the Crimean Peninsula. Geophys. J. Int. 201(1), 406—428.

van Trier J., Symes W. W., 1991. Upwind finite-difference calculation of traveltimes. Geophysics 56(6), 812—821.

Vidale J., 1988, Finite-difference calculation of travel times. Bull. Seysmol. Soc. Am. 78(6), 2062—2076.



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