$m$. In an ideal case the total number of degrees of freedom will be close to $n$, meaning that the observations can be fully characterized by those $n$ parameters.

In reality, measurements are susceptible error, and the total number of degrees of freedom in the observed signal (denoted by $d_{s}$), and thus the number of parameters accessible to our retrieval, may be fewer than the number of independent measurements, $n$. Some degrees of freedom,$d_{n}$, can be lost in the noise . The sum of $d_{s}$ and $d_{n}$ must add up to the total number of parameters we are seeking, $n$.

Before calculating the degrees of freedom it is useful to first introduce the averaging kernel, [**A**]{}. The averaging kernel tells us which of the parameters in the state vector have the greatest impact on the retrieval, that is, the sensitivity of the retrieval to a given parameter, given by $${\bf A}=\frac{\partial {\bf \hat{x}}}{\partial {\bf x}}=\frac{\partial {\bf \hat{x}}}{\partial {\bf y}}\frac{\partial {\bf y}}{\partial {\bf x}}={\bf GK}$$ [**A**]{} is an $n\times n$ matrix whose elements are given by $$A_{ij}=\frac{\partial \hat{x}_i}{\partial x_j}$$ If a diagonal element of [**A**]{} is unity, or close to it, then that means for a given change in the true 