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Utilities of the spectral formulations for measuring information content from observations are explored and demonstrated with real radar data. It is shown that the spectral formulations can be used (i) to precisely compute the information contents from one-dimensional radar data uniformly distributed along the radar beam, (ii) to approximately estimate the information contents from two-dimensional radar observations non-uniformly distributed on the conical surface of radar scan and thus (iii) to estimate the information losses caused by super-observations generated by local averaging with a series of successively coarsened resolutions to find an optimally coarsened resolution for radar data compression with zero or near-zero minimal loss of information. The results obtained from the spectral formulations are verified against the results computed accurately but costly from the singular-value formulations. As the background and observation error power spectra can be derived analytically for the above utilities, the spectral formulations are computationally much more efficient and affordable than the singular-value formulations, even and especially when the background space and observation space are both extremely large and too large to be computed by the singular-value formulations.