Q1 : How do you exactly compute filtered fluxes?

Monthly time series are filtered using the curve fitting procedure of Thoning et al. [1] (see full version). This procedure follows three steps:

1. The flux time series are fitted with a function containing a second-order polynomial term and four annual harmonics. The polynome captures the trend in the data while the harmonics capture the seasonal cycle. The fit is based on a least square minimisation of the differences between the function and the raw data.
2. Deviations of the raw data from this fit (residuals) are then filtered in the time domain using a Fourrier Transformatin and a low pass filter of Full Width at Half Maximum of either 3 months or 1 years. We thus obtain two sets of filtered residuals.
3. The filtered residuals are then added back to the polynomial term of the initially fitted function. This results in a so called “smoothed trend” where the seasonal cycle has been removed and "short term" fluctuations partially eliminated (depending on the chosen filter: 3 months or 1 year).

References

1. Thoning, K. W., and P. P. Tans (1989), Atmospheric carbon dioxide at Mauna Loa observatory: 2. Analysis of the NOAA GMCC Data, 1974- 1985, J. Geophys. Res., 94, 8549-8565.