Estimation of Spatial Wavenumber Spectra and Falloff Rate with Unequally Spaced Observations

Abstract

A maximum likelihood estimation procedure is developed for estimating the spatial wavenumber spectrum and falloff slope for a stationary process on a circle when multiple realizations are available at the same unequally spaced locations. The complete model allows for observational error, an arbitrary number of free spectral lines, and the assumed form αk?? for higher wavenumbers, where k is the wavenumber and α and ? are parameters to be estimated. The estimation technique involves iterative, nonlinear optimization; however, since maximum likelihood estimation is used, confidence limits can be obtained for the estimated parameters. A numerical example is given using 500 mb temperature at 15 stations located near 53N, and the estimated falloff rate (?) and 95% confidence limits are 2.03±0.20 when observational error is not included in the model. However, when observational error is included, the slope increases significantly to near 3. This increase is expected since observational error is similar to variance at high wavenumbers. The example demonstrates that care must he taken when interpreting the results of various estimation procedures, since the results are highly dependent on the assumed model. An objective method is used for selecting the appropriate model, and indicates that observational error should be included.