of [**A**]{} is the number of degrees of freedom lost to the noise.

The total degrees of freedom, again, tell us how many independent parameters can be determined from the observations. The information content, $H$, tells us quantitatively how well the observations increased our confidence in our estimate of the atmospheric state relative to the [*a priori*]{} knowledge. In a more precise language, the information content of a measurement is the reduction in the entropy of the probability that that an atmospheric state exists given some set of observations, or $$H=entropy(P({\bf x}))-entropy(P({\bf x|y}))$$ The entropy of a Gaussian distribution of width $\sigma$, which the prior and a posterior distributions are assumed to be, can be shown to be proportional to $ln(\sigma)$. Using this fact, and equations (17), (6), and (9), $$H=\frac{1}{2}ln({\bf |\hat{S}^{-1}S_a}|)$$ From this we can see that if the data is good (small error bars), then the elements of ${\bf \hat{S}}$ will be small, resulting in a large $H$. Thus $H$ is a quantitative measure of the reduction in our uncertainty in the retrieved atmospheric state as a result of the observations. The larger the value of $H$, the more useful the observations are in constraining the atmospheric state.

