# Trend Analysis

This page provides an online tool to calculate the long-term trend of a time series of data. The output can be coefficients or a graph of the trend.

## How To Use

• Paste daily mean data of a variable in the form to perform analysis. Each data line must contain year, month, day, and value of the variable. Data columns must be separated by one or more space characters. Refer to the CO2 data from Hateruma station as an example.
• The time sequence does not have to be continuous, but should be incremental. No missing value should be used.
• The default value of cutoff frequency (7.3 cycles/year = 50-day cycle) was used by Thoning et al. (1989) to remove short-term variations. You may use a small value (e.g., 0.5) to obtain inter-annual variation.
• The server may refuse a request if too many data are submitted.

## 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.