(K^TS_e^{-1}K+S_a^{-1})^{-1}}$$ As the elements of ${\bf S_a}$ approach $\infty$ or the elements of ${\bf S_e}$ approach 0, then ${\bf G}$ approaches ${\bf K^{-1}}$ which is identically the sensitivity of the state vector to the observations, and thus the retrieval is fully characterized by the data.

If the forward model is linear, then (10) can be solved to obtain the desired state vector. Often, the forward model is non-linear, generally the case in radiative transfer; it is then best to use a numerical iteration scheme to determine the state vector. In the non-linear case the [**Kx**]{} terms in the cost function in (8) are replaced with [**F(x)**]{}. The Levenberg-Marquardt iteration scheme is used to find the minimum of the non-linear cost function. The prescribed scheme is given by $$\begin{aligned}
{\bf x_{k+1}}={\bf x_k}+{\bf [}(1+\gamma){\bf S_a^{-1}+K_k^TS_e^{-1}K_k]^{-1}}\nonumber \\
{\bf\{K_k^TS_e^{-1}[y-F(x_k)]-S_a^{-1}[x_k-x_a]  \}      }\end{aligned}$$ where ${\bf x_{k}}$ and ${\bf x_{k+1}}$ are the state vectors for the $k^{th}$ and $k+1^{st}$ iterations, and ${\bf K_k}$ is the Jacobian matrix calculated at the $k^{th}$ iteration. $\gamma$ is a factor that controls the rate of convergence and is adjusted at each iteration (Press et al. 1995). Equation (13) is iterated until convergence, when $${\bf (x_k-x_{k+1})^T\hat{S}^{-1}(x_k-x_{k+1})}<<n$$ Upon convergence, we obtain the retrieved 