1642-2511 Statistical simulation of 4D random fields by means of Kotelnikov-Shannon decomposition Symulacja statystyczna 4D pól losowych z użyciem rozkładu Kotelnikova-Shannona Vyzhva Zoya author Vyzhva Andrii author Fedorenko Kateryna author Taras Shevchenko National University of Kyiv Correspondence to: Zoya Vyzhva zoya_vyzhva@ukr.net Correspondence to: Andrii Vyzhva motomustanger@ukr.net Correspondence to: Kateryna Fedorenko slims_mentol@mail.ru 16 11 2016 Vol. 15 (2016) 2016 73 83 Copyright © 2016 2016 <p>This paper researches the real valued random fields, that are homogeneous with respect to time and homogeneous isotropic with respect to spatial variables. An analogue of the Kotelnikov-Shannon theorem for random fields with a bounded spectrum is presented. Models for such random fields by partial sums of series are constructed. Some estimates for the mean square approximation of a random field by its models are obtained. Statistical simulation procedures of realizations of a random field with Gaussian distribution are constructed. The using of these theorems, models and procedures are demonstrated through applications to generate by means of computer adequate realizations of Gaussian random field with some wide-known examples of covariance functions. Spectral analysis of generated noise is considered.</p> 1 Bath M. (1980). Spectral analysis in geophysics. M.: Nedra, 535 p 2 Belyaev Yu. K. (1956). Analitical Random Processes, Theory of Probability and its Applications, Vol. 4, No. 4, pp. 437–444. 3 Chiles J.P., Delfi ner P. (1999). Geostatistics: Modeling Spatial Uncertainty., Inc. New York, Toronto, John Wiley &amp; Sons, 720 p. 4 Demyanov V.V., Savelev E.A. (2010). Geostatistica. M.: Nauka, 327 p. 5 Higgins J. (1996). Sampling Theory in Fourier and Signal Analysis. Clarendon Press. Oxford. New York, 225 p. 6 Olenko A. Ya., Pogany T.K. (2005). A precise upper bound for the error of interpolation of stochastic processes, Theor. Probability and Math. Statist., No. 4, pp. 151–163. 7 Olenko A. Ya. (2005). Bounds Comparison for Approximation Reminder in Sampling Theorem, Bulletin of Kyiv University. Series: Mathematics and Mechanics, No. 13, pp. 41–44. 8 Olenko A. Ya. (2004). Upper bound for interpolation reminder in multidimensional sampling theorem, Bulletin of Kyiv University. Series: Physics and Mathematics, No. 3, pp. 49–54. 9 Olenko A. Ya., Pogany T.K. (2011). Universal truncation error upper bounds in irregular sampling restoration, Applicable Analysis, Vol. 90, No. 3–4, March–April, pp. 595–608. 10 Piranashvili Z.A. (1967). Towards Question about Interpolation of Random Processes, Theory of Probability and its Applications, Vol. 12, No. 4, pp. 708–717. 11 Vyzhva Z.O. (2003). About Approximation of 3-D Random Fields and Statistical Simulation, Random Operator and Stochastic Equation, Vol. 4, No. 3, pp. 255–266. 12 Vyzhva Z.O. (2011). The Statistical Simulation of Random Processes and Fields, Kyiv, Obrii, 388 p. 13 Vyzhva Z.O., Kendzera O.V., Fedorenko K.V., Vyzhva A.S. (2012). The frequency characteristics of under-building-site geology environment determination by using the statistical simulation of seismic noise by the example of Odessa city//Visn. Kyiv University. Geology, № 58, pp. 57–61. 14 Vyzhva Z.O. (2012). The statistical simulation of 2-D seismic noise for frequency characteristics of geology environment determination// Visn. Kyiv University. Geology, №59, pp. 65–67. 15 Vyzhva Z.O. (2013).The statistical simulation of 3-D seismic noise for frequency characteristics of geology environment determination// Visn. Kyiv University. Geology, № 60, pp. 69–73. 16 Vyzhva Z. O., Fedorenko К. (2013). The Statistical Simulation of 3-D Random Fields by Means Kotelnikov-Shannon Decomposition, Theor. Probability and Math. Statist., No. 88, pp. 17–31. 17 Vyzhva Z. O., Fedorenko К. (2016). About Statistical Simulation of 4D Random Fields by Means of Kotelnikov-Shannon Decomposition, Journal of Applied Mathematics and Statistics (in progress). 18 Yadrenko M.Y. (1983). Spectral theory of random fi elds. N.Y.: Optimization Software Inc., 259 p. 19 Yadrenko M.Y., Gamaliy O. (1998). The Statistical Simulation of Homogeneous and Isotropic Three-dimensional Random Fields and Estimate Simulation Error, Theory of Probability and Mathematical Statistics, No. 59, pp. 171–175.