state, ${\bf \hat{x}}$ and its precision ${\bf \hat{S}}$.

Information Content & Degrees of Freedom 
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The information content (Shannon & Weaver 1962) and total number of degrees of freedom are useful quantities that can help diagnose the quality and ability of a spectral data set to contribute to our knowledge of the atmospheric state. The number of degrees of freedom represents how many independent parameters can be retrieved from the spectrum, and the information content is a metric of how much the precision in the retrieved parameters has improved as a result of the observation. In the simplest sense, if there are $m$ independent measurements with no error (eg, fluxes at $m$ different wavelengths), then there will be at most be $m$ independent pieces of information (degrees of freedom) that can be obtained from the observations. If $m$ is fewer than the number of model parameters, $n$, the exact values of $n-m$ parameters cannot be obtained from the observations. We do not discuss those cases in this article, we choose only cases for which $m > n$. For a given forward model, with $n$ parameters, the maximum number of obtainable degrees of freedom will be the smaller of $n$ and 