${\bf S_a}$. Once more, the diagonal elements of ${\bf S_a}$ are allowed a large range as we are dealing with a relatively novel type of observations and lack detailed prior information. We also assume that there are no cross correlations between different state parameters (e.g. $f_{CO}$ and $f_{CO_2}$, even though from chemical models we know that such quantities have high correlations). Because the state parameters are logarithmic, the elements of ${\bf S_a}$ are also logarithmic (with the exception of $\alpha$) so we set, somewhat arbitrarily, $\sigma_{\kappa_{v1}}=2$, $\sigma_{\kappa_{v2}}=2$ , $\sigma_{\kappa_{IR}}=2$ , $\sigma_{\alpha}=0.5$ , $\sigma_{f_{H2O}}=6$ , $\sigma_{f_{CH4}}=6$ , $\sigma_{f_{CO}}=6$, and $\sigma_{f_{CO2}}=6$ meaning that the opacities are permitted to span 4 orders of magnitude centered around their [*a priori*]{} value and the mixing ratios are allowed to span 12 orders of magnitude. Such large [*a priori*]{} uncertainties lead to a flat [*a priori*]{} distribution, relative to the data, reducing the current problem to a maximum likelihood estimation (as opposed to Bayesian), with the option of using the priori information if the data is sparse.

The entirety of the forward model can summarized with the Jacobian. Figure 2 shows the columns of the Jacobian evaluated at the true state (response of the flux in 