($\chi^2=0$) does not necessarily mean that the true state has been retrieved, because of the degeneracies between some of the parameters. Table 1 compares the true state to the retrieval results along with the retrieval precission. The synthetic retrieval demonstrates the robustness of the retrieval to a poor [*a priori*]{}. The reason for this can be seen by inspecting the elements of the averaging kernal. From Table 1, all but $\kappa_{v1}$ and methane are fairly well characterized by the data ($A_{jj}$ is close to 1). Summing these values gives the total degrees of freedom, and thus the total number of useful retrievable parameters of $\sim 6$.

![image](Figure3.pdf){width="100.00000%"}

[ccccccc]{} $\kappa_{v1}$& 4.00$\times$10$^{-3}$&1.00$\times$10$^{-3}$ & 3.59$\times$10$^{-3}$ &2.76$\times$10$^{-3}$ - 4.68$\times$10$^{-3}$ &0.997 &\
$\kappa_{v2}$ & 4.00$\times$10$^{-3}$ &1.00$\times$10$^{-2}$ & 1.70$\times$10$^{-9}$ & 1.70$\times$10$^{-11}$ - 1.70$\times$10$^{-7}$&0.0\
$\kappa_{IR}$ & 1.00$\times$10$^{-2}$ & 3.16$\times$10$^{-2}$ &8.93$\times$10$^{-3}$ & 7.13$\times$10$^{-3}$ - 1.12$\times$10$^{-2}$&0.998\
$\alpha$ &0.5 & 0.1 & 0.003 &0.00 - 0.022 &0.999\
$f_{H2O}$ & 5.00$\times$10$^{-4}$&1.00$\times$10$^{-6}$ & 4.18$\times$10$^{-4}$&2.58$\times$10$^{-4}$ - 6.76$\times$10$^{-4}$ &0.999\
$f_{CH4}$ & 1.00$\times$10$^{-6}$&1.00$\times$10$^{-4}$ &3.43$\times$10$^{-7}$ & 4.34$\times$10$^{-12}$ - 2.70$\times$10$^{-2}$&0.334\
$f_{CO}$ & 3.00$\times$10$^{-4}$ & 1.00$\times$10$^{-6}$ &1.96$\times$10$^{-4}$ &2.27$\times$10$^{-6}$ - 1.69$\times$10$^{-2}$ &0.896\
$f_{CO2}$ & 1.00$\times$10$^{-7}$&1.00$\times$10$^{-4}$ & 7.70$\times$10$^{-7}$&9.95$\times$10$^{-10}$ - 5.96$\times$10$^{-4}$ &0.768\

Resolution and Signal to Noise Effects on the Degrees of Freedom & Information Content
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The S/N and R 