The Spectral Density of Hankel Operators with Piecewise Continuous Symbols

In 1966, H. Widom proved an asymptotic formula for the distribution of eigenvalues of the $$N\times N$$N×N truncated Hilbert matrix for large values of N. In this paper, we extend this formula to Hankel matrices with symbols in the class of piece-wise continuous functions on the unit circle. Furthermore, we show that the distribution of the eigenvalues is independent of the choice of truncation (e.g. square or triangular truncation).

Ensemble Data Assimilation with the NCEP Global Forecast System

Real-data experiments with an ensemble data assimilation system using the NCEP Global Forecast System model were performed and compared with the NCEP Global Data Assimilation System (GDAS). All observations in the operational data stream were assimilated for the period 1 January–10 February 2004, except satellite radiances. Because of computational resource limitations, the comparison was done at lower resolution (triangular truncation at wavenumber 62 with 28 levels) than the GDAS real-time NCEP operational runs (triangular truncation at wavenumber 254 with 64 levels). The ensemble data assimilation system outperformed the reduced-resolution version of the NCEP three-dimensional variational data assimilation system (3DVAR), with the biggest improvement in data-sparse regions. Ensemble data assimilation analyses yielded a 24-h improvement in forecast skill in the Southern Hemisphere extratropics relative to the NCEP 3DVAR system (the 48-h forecast from the ensemble data assimilation system was as accurate as the 24-h forecast from the 3DVAR system). Improvements in the data-rich Northern Hemisphere, while still statistically significant, were more modest. It remains to be seen whether the improvements seen in the Southern Hemisphere will be retained when satellite radiances are assimilated. Three different parameterizations of background errors unaccounted for in the data assimilation system (including model error) were tested. Adding scaled random differences between adjacent 6-hourly analyses from the NCEP–NCAR reanalysis to each ensemble member (additive inflation) performed slightly better than the other two methods (multiplicative inflation and relaxation-to-prior).