A Method for Topographical Estimation of Lake Bottoms by B-Spline Surface

H. Bao; K. Fueda

doi:doi:10.11648/j.ajtas.20130204.12

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A Method for Topographical Estimation of Lake Bottoms by B-Spline Surface

H. Bao, K. Fueda

Received: 15 July 2013 Accepted: Published: 10 August 2013

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Abstract

The application of B-spline (Basis spline) surface to the estimation of the lake bottom topography is described.By using the analysis of a bivariate B-spline, the shape of the lake bottom is approximated.According to the validity of the estimation by the bivariate B-spline function the method is applied to the actual data of the lake depth.Surveys over the water area have more difficulties than those on land, and the measurement data are distributed quite irregularly. The locations of the measured data donot exist regularly over the lake.Those locations were distributed along with the wake of the boat on which the sample data were collected. The density of the data is quite high in some small regions and quite low in other wide regions.Based on such irregular data, we tried a statistical estimation.The regularized term with a penalty coefficient makesa proper approximation of the parameters of the B-spline functions. There are many factors, such that the number of knots, the locations of those knots, the number of B-spline functions and the coefficient of penalized term.Appropriate information criterion which has sufficient accuracy and a small amount of computation is applied for determination of the optimal model.

DOI	10.11648/j.ajtas.20130204.12
Published in	American Journal of Theoretical and Applied Statistics (Volume 2, Issue 4, July 2013)
Page(s)	102-109
Creative Commons	This is an Open Access article, distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution and reproduction in any medium or format, provided the original work is properly cited.
Copyright	Copyright © The Author(s), 2024. Published by Science Publishing Group

Keywords

B-spline surface, Cross-validation, Influence function, Generalized cross-validation, Surface model selection, Numerical computation, Topography of lake bottom

References

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[2]	Cox, M.G.(1972). "The numerical evaluation of B-splines", J. Inst. Math. Appl., 10, pp.134-149.
[3]	Cox, M.G.(1975). "An algorythm for spline interpolation", J. Inst. Math. Appl.,15, pp.95-108.
[4]	de Boor, C.(1972). "On calculation with B-splines", J. Approx. Theory, 6, pp.50-62.
[5]	Schoenberg, I. J., Whitney, A.(1953). "On Pólya frequency functions III", Trans. Amer. Math. Soc, Vol. 74. pp. 246-259, pp. 246-259.
[6]	Good, I. J. and Gaskins, R.A.(1971). "Non parametric roughness penalties for probability densities", Biometrika, Vol. 58. pp. 255-277.
[7]	Good, I. J. and Gaskins, R.A.(1980). "Density estimation and bump hunting by the penalized likelihood method exemplified by scattering and meteorite data", Journal of American Standard Association, Vol. 75. pp. 42-56.
[8]	Green, P. J., Silverman, B. W.(1994). "Nonparametric Regression and Generalized Linear Models", Chapman and Hall, London.
[9]	Umeyama, S. (1996). "Discontinuity extraction in regularization using robust statistics", Technical report of IEICE.,PRU95-217 (1996). pp. 9-16.
[10]	Konishi, S., Kitagawa, G. (2008). "Information Criteria and Statistical Modeling", Springer Science+Business Media, LLC.
[11]	Akaike H. (1974). A new look at the statistical identification model. IEEE Transactions on Automatic Control 19(6), 716-723.
[12]	Schwarz G. (1978). Estimating the dimension of a model. Annals of Statistics6(2), 461-464.
[13]	Yuen K.V. (2010). Bayesian methods for structural dynamics and civil engineering. John Wiley and Sons, NJ.
[14]	Hampel, F.R.(1968)."Contributions to the theory of robust estimation", Ph.D. Thesis,University of California, Barkeley.
[15]	Hampel, F.R.(1974)."The influence curve and its role in robust estimation", J. Amer.Statist. Assoc., 62, 1179-1186.

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APA Style

H. Bao, K. Fueda. (2013). A Method for Topographical Estimation of Lake Bottoms by B-Spline Surface. American Journal of Theoretical and Applied Statistics, 2(4), 102-109. https://doi.org/10.11648/j.ajtas.20130204.12

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ACS Style

H. Bao; K. Fueda. A Method for Topographical Estimation of Lake Bottoms by B-Spline Surface. Am. J. Theor. Appl. Stat. 2013, 2(4), 102-109. doi: 10.11648/j.ajtas.20130204.12

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AMA Style

H. Bao, K. Fueda. A Method for Topographical Estimation of Lake Bottoms by B-Spline Surface. Am J Theor Appl Stat. 2013;2(4):102-109. doi: 10.11648/j.ajtas.20130204.12

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@article{10.11648/j.ajtas.20130204.12,
  author = {H. Bao and K. Fueda},
  title = {A Method for Topographical Estimation of Lake Bottoms by B-Spline Surface},
  journal = {American Journal of Theoretical and Applied Statistics},
  volume = {2},
  number = {4},
  pages = {102-109},
  doi = {10.11648/j.ajtas.20130204.12},
  url = {https://doi.org/10.11648/j.ajtas.20130204.12},
  eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.ajtas.20130204.12},
  abstract = {The application of B-spline (Basis spline) surface to the estimation of the lake bottom topography is described.By using the analysis of a bivariate B-spline, the shape of the lake bottom is approximated.According to the validity of the estimation by the bivariate B-spline function the method is applied to the actual data of the lake depth.Surveys over the water area have more difficulties than those on land, and the measurement data are distributed quite irregularly. The locations of the measured data donot exist regularly over the lake.Those locations were distributed along with the wake of the boat on which the sample data were collected. The density of the data is quite high in some small regions and quite low in other wide regions.Based on such irregular data, we tried a statistical estimation.The regularized term with a penalty coefficient makesa proper approximation of the parameters of the B-spline functions. There are many factors, such that the number of knots, the locations of those knots, the number of B-spline functions and the coefficient of penalized term.Appropriate information criterion which has sufficient accuracy and a small amount of computation is applied for determination of the optimal model.},
 year = {2013}
}

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TY  - JOUR
T1  - A Method for Topographical Estimation of Lake Bottoms by B-Spline Surface
AU  - H. Bao
AU  - K. Fueda
Y1  - 2013/08/10
PY  - 2013
N1  - https://doi.org/10.11648/j.ajtas.20130204.12
DO  - 10.11648/j.ajtas.20130204.12
T2  - American Journal of Theoretical and Applied Statistics
JF  - American Journal of Theoretical and Applied Statistics
JO  - American Journal of Theoretical and Applied Statistics
SP  - 102
EP  - 109
PB  - Science Publishing Group
SN  - 2326-9006
UR  - https://doi.org/10.11648/j.ajtas.20130204.12
AB  - The application of B-spline (Basis spline) surface to the estimation of the lake bottom topography is described.By using the analysis of a bivariate B-spline, the shape of the lake bottom is approximated.According to the validity of the estimation by the bivariate B-spline function the method is applied to the actual data of the lake depth.Surveys over the water area have more difficulties than those on land, and the measurement data are distributed quite irregularly. The locations of the measured data donot exist regularly over the lake.Those locations were distributed along with the wake of the boat on which the sample data were collected. The density of the data is quite high in some small regions and quite low in other wide regions.Based on such irregular data, we tried a statistical estimation.The regularized term with a penalty coefficient makesa proper approximation of the parameters of the B-spline functions. There are many factors, such that the number of knots, the locations of those knots, the number of B-spline functions and the coefficient of penalized term.Appropriate information criterion which has sufficient accuracy and a small amount of computation is applied for determination of the optimal model.
VL  - 2
IS  - 4
ER  -

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Author Information

H. Bao

Graduate School of Environmental Science, Okayama University, Okayama, Japan
K. Fueda

Graduate School of Environmental and Life Science, Okayama University, Okayama, Japan

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APA Style

H. Bao, K. Fueda. (2013). A Method for Topographical Estimation of Lake Bottoms by B-Spline Surface. American Journal of Theoretical and Applied Statistics, 2(4), 102-109. https://doi.org/10.11648/j.ajtas.20130204.12

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ACS Style

H. Bao; K. Fueda. A Method for Topographical Estimation of Lake Bottoms by B-Spline Surface. Am. J. Theor. Appl. Stat. 2013, 2(4), 102-109. doi: 10.11648/j.ajtas.20130204.12

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AMA Style

H. Bao, K. Fueda. A Method for Topographical Estimation of Lake Bottoms by B-Spline Surface. Am J Theor Appl Stat. 2013;2(4):102-109. doi: 10.11648/j.ajtas.20130204.12

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@article{10.11648/j.ajtas.20130204.12,
  author = {H. Bao and K. Fueda},
  title = {A Method for Topographical Estimation of Lake Bottoms by B-Spline Surface},
  journal = {American Journal of Theoretical and Applied Statistics},
  volume = {2},
  number = {4},
  pages = {102-109},
  doi = {10.11648/j.ajtas.20130204.12},
  url = {https://doi.org/10.11648/j.ajtas.20130204.12},
  eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.ajtas.20130204.12},
  abstract = {The application of B-spline (Basis spline) surface to the estimation of the lake bottom topography is described.By using the analysis of a bivariate B-spline, the shape of the lake bottom is approximated.According to the validity of the estimation by the bivariate B-spline function the method is applied to the actual data of the lake depth.Surveys over the water area have more difficulties than those on land, and the measurement data are distributed quite irregularly. The locations of the measured data donot exist regularly over the lake.Those locations were distributed along with the wake of the boat on which the sample data were collected. The density of the data is quite high in some small regions and quite low in other wide regions.Based on such irregular data, we tried a statistical estimation.The regularized term with a penalty coefficient makesa proper approximation of the parameters of the B-spline functions. There are many factors, such that the number of knots, the locations of those knots, the number of B-spline functions and the coefficient of penalized term.Appropriate information criterion which has sufficient accuracy and a small amount of computation is applied for determination of the optimal model.},
 year = {2013}
}

Copy | Download

TY  - JOUR
T1  - A Method for Topographical Estimation of Lake Bottoms by B-Spline Surface
AU  - H. Bao
AU  - K. Fueda
Y1  - 2013/08/10
PY  - 2013
N1  - https://doi.org/10.11648/j.ajtas.20130204.12
DO  - 10.11648/j.ajtas.20130204.12
T2  - American Journal of Theoretical and Applied Statistics
JF  - American Journal of Theoretical and Applied Statistics
JO  - American Journal of Theoretical and Applied Statistics
SP  - 102
EP  - 109
PB  - Science Publishing Group
SN  - 2326-9006
UR  - https://doi.org/10.11648/j.ajtas.20130204.12
AB  - The application of B-spline (Basis spline) surface to the estimation of the lake bottom topography is described.By using the analysis of a bivariate B-spline, the shape of the lake bottom is approximated.According to the validity of the estimation by the bivariate B-spline function the method is applied to the actual data of the lake depth.Surveys over the water area have more difficulties than those on land, and the measurement data are distributed quite irregularly. The locations of the measured data donot exist regularly over the lake.Those locations were distributed along with the wake of the boat on which the sample data were collected. The density of the data is quite high in some small regions and quite low in other wide regions.Based on such irregular data, we tried a statistical estimation.The regularized term with a penalty coefficient makesa proper approximation of the parameters of the B-spline functions. There are many factors, such that the number of knots, the locations of those knots, the number of B-spline functions and the coefficient of penalized term.Appropriate information criterion which has sufficient accuracy and a small amount of computation is applied for determination of the optimal model.
VL  - 2
IS  - 4
ER  -

Copy | Download