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

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.

[1]  Peter Craven,et al.  Smoothing noisy data with spline functions , 1978 .

[2]  Zvi Ziegler,et al.  Approximation theory and applications , 1983 .

[3]  G. Wahba,et al.  Some New Mathematical Methods for Variational Objective Analysis Using Splines and Cross Validation , 1980 .

[4]  Latitudinal-Wavenumber Power Spectra of Stratospheric Temperature Fluctuations , 1979 .

[5]  G. Wahba,et al.  Some results on Tchebycheffian spline functions , 1971 .

[6]  G. Wahba,et al.  Design Problems for Optimal Surface Interpolation. , 1979 .

[7]  Donald R. Johnson,et al.  Use of Approximating Polynomials to Estimate Profiles of Wind, Divergence, and Vertical Motion , 1972 .

[8]  W. Freeden,et al.  On spherical spline interpolation and approximation , 1981 .

[9]  S. Eisenstat.  Efficient Implementation of a Class of Preconditioned Conjugate Gradient Methods , 1981 .

[10]  George E. Backus,et al.  Harmonic splines for geomagnetic modelling , 1982 .

[11]  G. Wahba.  Practical Approximate Solutions to Linear Operator Equations When the Data are Noisy , 1977 .

[12]  G. Wahba.  Data-Based Optimal Smoothing of Orthogonal Series Density Estimates , 1981 .

[13]  harald Cramer,et al.  Stationary And Related Stochastic Processes , 1967 .

[14]  G. Wahba,et al.  A Correspondence Between Bayesian Estimation on Stochastic Processes and Smoothing by Splines , 1970 .

[15]  E. Parzen.  An Approach to Time Series Analysis , 1961 .

[16]  G. Wahba.  Spline Interpolation and Smoothing on the Sphere , 1981 .

[17]  G. Wahba,et al.  Generalized Inverses in Reproducing Kernel Spaces: An Approach to Regularization of Linear Operator Equations , 1974 .

[18]  G. Wahba.  Bayesian "Confidence Intervals" for the Cross-validated Smoothing Spline , 1983 .

[19]  G. Wahba.  Improper Priors, Spline Smoothing and the Problem of Guarding Against Model Errors in Regression , 1978 .

[20]  Paul R. Julian,et al.  On Some Properties of Correlation Functions Used in Optimum Interpolation Schemes , 1975 .

[21]  A. Kasahara,et al.  Normal Modes of Ultralong Waves in the Atmosphere , 1976 .

[22]  D. F. Utreras,et al.  Optimal Smoothing of Noisy Data Using Spline Functions , 1981 .