Vector Splines on the Sphere, with Application to the Estimation of Vorticity and Divergence from Discrete, Noisy Data

  • Grace Wahba


Vector smoothing splines on the sphere are defined. Theoretical properties are briefly alluded to. An approach to choosing the appropriate Hilbert space norms to use in a specific meteorological application is described and justified via a duality theorem. Numerical procedures for computing the splines as well as the cross validation estimate of two smoothing parameters are given. A Monte Carlo study is described which suggests the accuracy with which upper air vorticity and divergence can be estimated using measured wind vectors from the North American radiosonde network.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. Bates, D., and Wahba, G. (1982) in preparation.Google Scholar
  2. Cramer, H., and Leadbetter (1967). Stationary and related stochastic processes. Chapter V. Wiley, New York.MATHGoogle Scholar
  3. Craven, P. and Wahba, G. (1979). Smoothing noisy data with spline functions: estimating the correct degree of smoothing by the method of generalized cross-validation. Numer. Math., 31, 377.MathSciNetCrossRefMATHGoogle Scholar
  4. Freeden, W. (1981a). On spherical spline interpolation and approximation. Math. Meth. in The Appl. Sci. 3, 551–575.MathSciNetCrossRefMATHGoogle Scholar
  5. Freeden, W. (1981b). On approximation by harmonic splines. Manuscripta Geodaetica, 6, 193–244.MATHGoogle Scholar
  6. Golub, G., Heath, M. and Wahba, G. (1979), Generalized cross-validation as a method for choosing a good ridge parameter, Technometrics 21, 215–223.MathSciNetCrossRefMATHGoogle Scholar
  7. Julian, P.R., and Thiebaux, M. Jean (1975), On some properties of correlation functions used in optimum interpolation schemes. Monthly Weather Review, 103, 7, pp. 505–616.CrossRefGoogle Scholar
  8. Kassahara, A. (1976). Normal modes of ultra-long waves in the atmosphere. Monthly Weather Review 104, 669–690.CrossRefGoogle Scholar
  9. Kimeldorf, G., and Wahba, G. (1970), A correspondence between Bayesian estimation of stochastic processes and smoothing by splines, Ann. Math. Statist., 41, 2.MathSciNetCrossRefGoogle Scholar
  10. Kimeldorf, G., and Wahba, G. (1971), Some results on Tchebycheffian spline functions, J. Math. Anal, and Applic., 33, 1.MathSciNetCrossRefGoogle Scholar
  11. Micchelli, C., and Wahba, G. (1981), Design problems for optimal surface interpolation in “Approximation Theory and Applications: Z. Ziegler, ed. Academic Press.Google Scholar
  12. Nashed, M.Z. and Wahba, G. (1974), Generalized inverses in reproducing kernel spaces: an approach to regularization of linear operator equations, SIAM J. Math. Analysis, 5, 6.MathSciNetCrossRefGoogle Scholar
  13. Parzen, E, 1951), An approach to time series analysis. Ann, Math. Statist. 32, 951–989.MathSciNetCrossRefGoogle Scholar
  14. Schmidt, P.J., and Johnson, D.R. (1972), Use of approximating polynomials to estimate profiles of wind, divergence, and vertical motion. Monthly Weather Review, 100, 5, 249–353.CrossRefGoogle Scholar
  15. Shure, L., Parker, R.L., and Backus, G.E. (1981), Harmonic splines for geomagnetic modelling, to appear, PEPI.Google Scholar
  16. Stanford, J. (1979), Latitudinal-Wavenumber power spectra of stratospheric temperature fluctuations, J. Atmospheric Sciences, 36, 5, pp. 921–931.CrossRefGoogle Scholar
  17. Utreras, F. (1981), Optimal smoothing of noisy data using spline functions, SIAM J. Sci. Stat. Comput. 2, 3, 349–362.MathSciNetCrossRefMATHGoogle Scholar
  18. Wahba, G. (1977a) In invited discussion to Consistent nonparametric regression, C.J. Stone, Ann. Stat, 5, 4, 637–645.Google Scholar
  19. Wahba, G. (1977b), Practical approximate solutions to linear operator equations when the data are noisy, SIAM J. Numerical Analysis, 14, 4.MathSciNetCrossRefGoogle Scholar
  20. Wahba, G. (1978), Improper priors, spline smoothing and the problem of guarding against model errors in regression, J. Roy. Stat. Soc. Ser. B., 40, 3.MathSciNetGoogle Scholar
  21. Wahba, G. (1980), Spline bases, regularization, and generalized cross validation for solving approximation problems with large quantities of noisy data. Proceedings of the International Conference on Approximation Theory in Honor or George Lorenz, Jan. 8–11, 1980, Austin, Texas, Ward Cheney, ed. Academic Press (1980).Google Scholar
  22. Wahba, G. (1981a), Spline interpolation and smoothing on the sphere, SIAf’1 J. Scientific and Statistical Computing, 2, 1.MathSciNetCrossRefGoogle Scholar
  23. Wahba, G. (1981b), Some new techniques for variational objective analysis on the sphere using splines. Hough functions, and sample spectral data. Preprints of the Seventh Conference on Probability and Statistics in the Atmospheric Sciences, American Meteorological Society.Google Scholar
  24. Wahba, G. (1981c), Data-based optimal smoothing of orthogonal series density estimates. Ann. Statist., 9, 1, 146–156.MathSciNetCrossRefMATHGoogle Scholar
  25. Wahba, G. (1981d), Bayesian confidence intervals for the cross validated smoothing spline. University of Wisconsin-Madison Statistics Department Technical Report No. 645, Submitted.Google Scholar
  26. Wahba, G., and Wendelberger, J. (1980), Some new mathematical methods for variational objective analysis using splines and cross-validation. Monthly Weather Review 108, 36–57.CrossRefGoogle Scholar
  27. Wendelberger, J. (1982), Ph.D. thesis, in preparation.Google Scholar

Copyright information

© Birkhäuser Verlag Basel 1982

Authors and Affiliations

  • Grace Wahba
    • 1
  1. 1.Department of StatisticsUniversity of Wisconsin-MadisonMadisonUSA

Personalised recommendations