# Application of splines for determining the velocity characteristic of a medium from a vertical seismic survey

Open Mathematics (2013)

• Volume: 11, Issue: 4, page 779-786
• ISSN: 2391-5455

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## Abstract

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A method for solving the inverse kinematic problem of determining the velocity characteristic of a medium from a vertical seismic survey, is proposed. It is based on the combined use of the eikonal equation and spline methods of approximation for multivariable functions. The problem is solved by assuming a horizontally stratified medium; no assumptions about the number of layers and their thickness are made. First, using the data of the first arrival times of the seismic signal from several shotpoints, which are registered by detectors located in the vertical borehole, a spline approximating the function of first arrival time of the signal from source points to any point in the Earth subsurface is constructed. Then with the help of the eikonal equation, the characteristic of the medium around the borehole is determined. Numerical experiments on the model and the real data show high efficiency of the proposed method.

## How to cite

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Vladimir Bogdanov, et al. "Application of splines for determining the velocity characteristic of a medium from a vertical seismic survey." Open Mathematics 11.4 (2013): 779-786. <http://eudml.org/doc/269583>.

abstract = {A method for solving the inverse kinematic problem of determining the velocity characteristic of a medium from a vertical seismic survey, is proposed. It is based on the combined use of the eikonal equation and spline methods of approximation for multivariable functions. The problem is solved by assuming a horizontally stratified medium; no assumptions about the number of layers and their thickness are made. First, using the data of the first arrival times of the seismic signal from several shotpoints, which are registered by detectors located in the vertical borehole, a spline approximating the function of first arrival time of the signal from source points to any point in the Earth subsurface is constructed. Then with the help of the eikonal equation, the characteristic of the medium around the borehole is determined. Numerical experiments on the model and the real data show high efficiency of the proposed method.},
journal = {Open Mathematics},
keywords = {Velocity characteristic of a medium; Seismic survey; Eikonal equation; Traveltime of seismic waves; Vertical borehole; Spline approximation; velocity characteristic of a medium; seismic survey; eikonal equation; traveltime of seismic waves; vertical borehole; spline approximation},
language = {eng},
number = {4},
pages = {779-786},
title = {Application of splines for determining the velocity characteristic of a medium from a vertical seismic survey},
url = {http://eudml.org/doc/269583},
volume = {11},
year = {2013},
}

TY - JOUR
AU - Valeriy Miroshnichenko
AU - Yuriy Volkov
TI - Application of splines for determining the velocity characteristic of a medium from a vertical seismic survey
JO - Open Mathematics
PY - 2013
VL - 11
IS - 4
SP - 779
EP - 786
AB - A method for solving the inverse kinematic problem of determining the velocity characteristic of a medium from a vertical seismic survey, is proposed. It is based on the combined use of the eikonal equation and spline methods of approximation for multivariable functions. The problem is solved by assuming a horizontally stratified medium; no assumptions about the number of layers and their thickness are made. First, using the data of the first arrival times of the seismic signal from several shotpoints, which are registered by detectors located in the vertical borehole, a spline approximating the function of first arrival time of the signal from source points to any point in the Earth subsurface is constructed. Then with the help of the eikonal equation, the characteristic of the medium around the borehole is determined. Numerical experiments on the model and the real data show high efficiency of the proposed method.
LA - eng
KW - Velocity characteristic of a medium; Seismic survey; Eikonal equation; Traveltime of seismic waves; Vertical borehole; Spline approximation; velocity characteristic of a medium; seismic survey; eikonal equation; traveltime of seismic waves; vertical borehole; spline approximation
UR - http://eudml.org/doc/269583
ER -

## References

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1. [1] Aasen J.O., On the reduction of a symmetric matrix to tridiagonal form, Nordisk Tidskr. Informationsbehandling (BIT), 1971, 11, 233–242 Zbl0242.65032
2. [2] Anikonov Yu.E., Bogdanov V.V., Derevtsov E.Yu., Miroshnichenko V.L., Pivovarova N.B., Slavina L.B., Some approaches to a numerical solution for the multidimensional inverse kinematic problem of seismic with inner sources, J. Inverse Ill-Posed Probl., 2009, 17(3), 209–238 http://dx.doi.org/10.1515/JIIP.2009.017 Zbl1162.86003
3. [3] Anikonov Yu.E., Bogdanov V.V., Derevtsov E.Yu., Miroshnichenko V.L., Sapozhnikova N.A., Numerical solution of an inverse kinematic seismic problem with internal sources, Sib. Zh. Ind. Mat., 2006, 9(4), 3–26 (in Russian) Zbl1224.35424
4. [4] Bezhaev A.Yu., Vasilenko V.A., Variational Theory of Splines, Kluwer Academic/Plenum, New York, 2001 Zbl0979.41007
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6. [6] Miroshnichenko V.L., The approximation of functions given by values at scattered points by DMM-splines, In: Vsesibirskie Chteni’ya po Matematike i Mekhanike, 1, Matematika, Tomsk, June 17–20, 1997, Tomsk University, 1997, 211–212 (in Russian)
7. [7] Sayfy A.M., Azzawi K.A., Makky S.M., Seismic inverse problems: determining seismic wave speeds using arrival times, Int. J. Comput. Math., 2006, 83(11), 797–808 http://dx.doi.org/10.1080/00207160601117123 Zbl1119.86002
8. [8] Schaback R., Wendland H., Characterization and construction of radial basic functions, In: Multivariate Approximation and Applications, Cambridge University Press, Cambridge, 2001, 1–24 http://dx.doi.org/10.1017/CBO9780511569616.002 Zbl1035.41013
9. [9] Volkov Yu.S., Miroshnichenko V.L., Constructing a mathematical model of a universal characteristic for a radial-axial hydroturbine, Sib. Zh. Ind. Mat., 1998, 1(1), 77–88 (in Russian) Zbl0916.76064

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