exists given the observations, [**y**]{}. If we assume Gaussian probability distributions for the observational error and for the [*a priori*]{} information, we can write $$P({\bf y}|{\bf x}) \propto e^{-\frac{1}{2}({\bf y}-{\bf Kx})^{T}{\bf S_{e}^{-1}}({\bf y}-{\bf Kx})}$$ $$P({\bf x}) \propto e^{-\frac{1}{2}({\bf x}-{\bf x_{a}})^{T}{\bf S_{a}^{-1}}({\bf x}-{\bf x_{a}})}$$ where [**S$_{e}$**]{} is the $m \times m$ diagonal error covariance matrix (assuming no correlation between measurements) and [**S$_{a}$**]{} is the $n \times n$ [*a priori*]{} covariance matrix. The [*a priori*]{} covariance matrix represents our prior knowledge of the natural variability of the system and like [**S$_{e}$**]{}, it is assumed to be diagonal. It essentially defines our “trust" region, or how far from the prior state we think the actual state can exist. In general, the prior constraint should be loose enough to allow flexibility in the retrieval but not so loose that the retrieval fails when a measurement contributes no information.

Using Bayes theorem from (4) we can write the posterior probability distribution as a product of (5) and (6) $$P({\bf x}|{\bf y}) \propto e^{-\frac{1}{2}J({\bf x})}$$ where J([**x**]{}) is the cost function and is given by $$\begin{aligned}
J({\bf x})={({\bf y}-{\bf Kx})^{T}{\bf S_{e}^{-1}}({\bf y}-{\bf Kx})} \nonumber \\
+({\bf x}-{\bf x_{a}})^{T}{\bf S_{a}^{-1}}({\bf x}-{\bf x_{a}}) \end{aligned}$$ The first term in 