Control Systems and Computers, N6, 2016, Article 10

Upr. sist. maš., 2016, Issue 6 (266), pp. 80-88.

UDC 519.6

Lytvyn Oleg Nikolaevich – Doctor of physical and mathematical sciences, Prof., head. of caf. Higher and Applied Mathematics of the Ukrainian Engineering Pedagogical Academy, Kharkov, 61003, ul. University, 16, E-mail:

Dragun Vladimir Vladimirovich – post-graduate student of the Department of Higher and Applied Mathematics of the Ukrainian Engineering Pedagogical Academy, Kharkov, 61003, ul. University address, 16, E-mail:

The Spatial Distribution of Slowness on the Basis of the First Approximation in the Mine Seismic Tomography

Introduction. Seismic tomography is one of the topical areas of the modern geophysics, based on building the image object by
means of the research paths spread of the seismic signals.
The finding of the spatial distribution of the slowness in the mine seismic tomography by means of the first approximation to the
slowness on the system horizontal plane is suggested.
The purpose of the article: to analyse the literature on the mineral exploration, to explore approximation by Fourier sums discontinuous functions of two variables using discontinuous splines of two variables, approximation of Fourier sums functions of two
variables, to determine the spatial distribution of slowness on the basis of the first approximation in mine seismic tomography, to
carry out computational experiments.

Methods: method of studying and generalization of the advanced experience, analysis of the literature, method of algorithmization.

The results and conclusions. The proposed approach can be viewed as constructing a mathematical model for finding the spatial
distribution slowness of seismic wave propagation of a given three-dimensional region in the earth crust, in the case where a
formula for the slowness approximate is calculated as the Fourier sums at each plane z = zk, k = 1, .., M. At the same time, information
which is used for approximate calculation of these coefficients of Fourier sums on every horizon plane z = zk, is the first time
the arrival of the seismic signal in the observation point from the seismic sources. Coefficients for Fourier sums are calculated using
the projection system along a certain line, located in each of the horizontal planes.

Download full text! (In Ukrainian and Russian)

Keywords: computed tomography, mine tomography.

1. Литвин О.М. Метод знаходження першого наближення для розв’язання задачі шахтної сейсмічної томографії в неоднорідному середовищі. УСиМ. 2016. № 3. С. 71–83.
2. Литвин О.М. Періодичні сплайни і новий метод розв’язання плоскої задачі рентгенівської комп’ютерної томографії. Системний аналіз, управління і інформаційні технології: Вісн. Харків. держ. політех. ун-ту: Зб. наук. праць. 2000. № 125. С. 27–35.
3. Кулик С.І. Математичне моделювання в комп’ютерній томографії з використанням вейвлетів : Дис. канд. фіз.-мат. Наук. спец. 01.05.02 «Математичне моделювання та обчислювальні методи». Харків, 2008. 192 с.
4. Luxbacher K.D. Time-Lapse Passive Seismic Velocity Tomography of Longwall Coal Mines: A Comparison of
Methods: Diss.  Doct. of Philos. in Mining Engin. USA, Blacksburg, Virginia, 2008. 166 p.
5. Чепмен К. Преобразование Радона и сейсмическая томография. Сейсмическая томография. М.: Мир,
1990. С. 34–60.

 Received 25.08.2016