## 4 Discussion
### Model Representation of Atmospheric CO\({}_{2}\)  
We used atmospheric CO\({}_{2}\) concentration observations to corroborate our assimilation results. We used the posterior values from the multi-site, multi-dataset optimization in a global model set-up forced by the global CRUNCEP meteorological driving data, i.e., the terrestrial model simulated global carbon fluxes. The simulated net CO\({}_{2}\) exchange fluxes were then used as input together with a set of background CO\({}_{2}\) fluxes (ocean-atmosphere exchange fluxes, fossil fuel emissions, and land use change emissions) for the global tracer transport model, TM2, to simulate atmospheric CO\({}_{2}\) concentrations for the years, 2010 to 2015. This was essentially the model set-up as described by [19], but here it was only used in a forward modelling sense. Table 7 summarizes the various performance metrics (R\({}^{2}\) and RMSE for the CO\({}_{2}\) time series, and mean error (ME) in the annual growth rates calculated as the method for calculating the annual growth rate in ESRL, [[https://www.esrl.noaa.gov/gmd/ccgg/mbl/crvfit/crvfit.html](https://www.esrl.noaa.gov/gmd/ccgg/mbl/crvfit/crvfit.html)]([https://www.esrl.noaa.gov/gmd/ccgg/mbl/crvfit/crvfit.html](https://www.esrl.noaa.gov/gmd/ccgg/mbl/crvfit/crvfit.html))) for the two stations, Cape Kumukahi (KUM) and South Pole (SPO, see Table A4 for details on the stations). Obviously, the optimized parameter set improved simulated CO\({}_{2}\) concentrations (Figure A3), as shown by the reduction in the average RMSE values in CO\({}_{2}\) concentrations over the whole time period: At station KUM, RMSE for CO\({}_{2}\) was reduced by 10.4%, and at station SPO, it was reduced by 20.6%. Similarly, the ME in the annual growth rates was improved by 5.5% for KUM and 11.3% for SPO, respectively. This demonstrates that the optimized parameters from multi-site assimilation using SMOS soil moisture and JRC-TIP FAPAR observation can provide a more reasonable parameter set for global scale model experiments. We note, however, that the R\({}^{2}\) values slightly worsened in the'sm+fapar (multi)' experiment compared to the 'prior', by about 6% at the station KUM. This was mainly due to a reduction of the amplitude  
\begin{table}
\begin{tabular}{c c c c c c c c} \hline \hline \multirow{2}{*}{**Site**} & \multirow{2}{*}{**Prior**} & **sm** & **sm** & **fapar** & **fapar** & **sm+fapar** & **sm+fapar** \\  & & **(Single)** & **(Multi)** & **(Single)** & **(Multi)** & **(Single)** & **(Multi)** \\ \hline AU-DaS & 1.6232 & 0.87 & 2.26 & 0 & 3.64 & 0.97 & 4.71 \\ CA-Gro & 1.0033 & 5.94 & 18.01 & 5.4 & 18.25 & \(-\)4.55 & \(-\)7.24 \\ CN-Sw2 & 2.6919 & \(-\)7.25 & \(-\)29.11 & \(-\)27.6 & \(-\)32.85 & \(-\)20.79 & \(-\)43.10 \\ DK-Sor & 2.3623 & / & 15.21 & \(-\)8.24 & 19.82 & \(-\)8.24 & \(-\)0.52 \\ ES-LMa & 2.2769 & \(-\)3.31 & 3.80 & \(-\)1.61 & 12.68 & \(-\)12.60 & 16.71 \\ FI-Sod & 1.4224 & / & \(-\)26.36 & \(-\)2.38 & \(-\)29.62 & \(-\)2.38 & 2.14 \\ FR-Pue & 1.6263 & 18.32 & \(-\)14.54 & \(-\)0.76 & \(-\)17.32 & 49.89 & \(-\)0.12 \\ NL-Loo & 2.0675 & \(-\)0.65 & 6.37 & \(-\)0.65 & 6.56 & \(-\)0.46 & \(-\)3.90 \\ US-Kfs & 1.7244 & 55.53 & \(-\)12.35 & 10.96 & 3.93 & 32.82 & \(-\)0.64 \\ US-Wkg & 0.6243 & 0.27 & 1.55 & \(-\)0.02 & 0 & 0.21 & 0.82 \\ \hline \hline \end{tabular}
\end{table}
Table 6: RMSE relative changes (%, positive values mean RMSE was degraded) for net ecosystem productivity (NEP) in comparison with prior simulations, ‘sm’ with soil moisture solely, ‘fapar’ with FAPAR solely, ‘sm+fapar’ with combination of soil moisture and FAPAR data, ‘multi’ denotes multi-site assimilation, and ‘single’ denotes single-site assimilation.  
in the seasonal cycle in atmospheric CO\({}_{2}\) for the optimized case since, in contrast to [19], we did not assimilate atmospheric CO\({}_{2}\) observations in this study.