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Measuring information content from observations for data assimilation: spectral formulations and their implications to observational data compression

The previous singular-value formulations for measuring information content from observations are transformed into spectral forms in the wavenumber space for univariate analyses of uniformly distributed observations. The transformed spectral formulations exhibit the following advantages over their counterpart singular-value formulations: (i) The information contents from densely distributed observations can be calculated very efficiently even if the background and observation space dimensions become both too large to compute by using the singular-value formulations. (ii) The information contents and their asymptotic properties can be analysed explicitly for each wavenumber. (iii) Superobservations can be not only constructed by a truncated spectral expansion of the original observations with zero or minimum loss of information but also explicitly related to the original observations in the physical space. The spectral formulations reveal that (i) uniformly thinning densely distributed observations will always cause a loss of information and (ii) compressing densely distributed observations into properly coarsened super-observations by local averaging may cause no loss of information under certain circumstances