atmospheric state, there is identically the same change in the retrieved state. This suggests that the parameter, $x_j$, is fully characterized by the data. If that diagonal element is less than unity, meaning that the data itself is not of a high enough quality to constrain that parameter, then some fraction of the [*a priori*]{} information must have been used in determining the value of that parameter. If each parameter is fully characterized by the data, that is if, all of the diagonal elements of [**A**]{} are unity, then we would expect to be able to retrieve all $n$ parameters. If the diagonal elements are less than unity, then the sum of the diagonals would be less than $n$. In essence, the diagonal elements of the averaging kernel can be thought of as the degrees of freedom per parameter. If the value of a particular diagonal element is 1, then that parameter is well characterized by the data. If it is much less than 1, then the data contributes little to our knowledge of that parameter. The total degrees of freedom from the signal can be determined by calculating the trace of [**A**]{}. The difference between $n$ and the trace 