## Method

The trend analysis uses Fast Fourier Transform with Thoning's exponential frequency filter \(H(f)\) (Thoning et al., 1989, J. Geophys. Res., 94(6):8549-8565 ) to remove short-term variations of time series data:
$$
H(f) = \exp(-c \ast { (f/f_{c}) }^p)
$$
where \(f_{c}\) is the cutoff frequency in cycles per year, \(c=ln(2)\) is the normalizing coefficient to make \(H(f)=0.5\) when \(f=f_{c}\) and \(p=4.0\) is used to modulate the decay of frequency.

Fast Fourier Transform requires a sequence of evenly spaced stationary data. Since a time series most likely has trend and gaps of missing data, the equation

$$
y = a + b \ast x + c_{1} \ast \cos(x) + d_{1} \ast \sin(x) + c_{2} \ast \cos(2x) + d_{2} \ast \sin(2x)
$$

is used to remove trend and to fill gaps.

Internally, data of the first and the last years are patched to the beginning and the end respectively to reduce alias effect of end-points.

The final results may be used to estimate variations of long-term and seasonal growth rates of greenhouse gases.