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

Vladimir Bogdanov; Wladimir Karsten; Valeriy Miroshnichenko; Yuriy Volkov

Open Mathematics (2013)

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

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topVladimir 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>.

@article{VladimirBogdanov2013,

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.},

author = {Vladimir Bogdanov, Wladimir Karsten, Valeriy Miroshnichenko, Yuriy Volkov},

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 - Vladimir Bogdanov

AU - Wladimir Karsten

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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- [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] 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] 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] 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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