How to identify the seasonal cycle from a time series?

One might think of the seasonal cycle as the climatological mean of a given variable for a certain time in the year. Hence, we define the seasonal cycle as the average of a variable not on a continuous time axis but on a cyclic one starting from 0 at Jan 1 00:00 and ending one year later after 365.25 days.

However, we can also estimate the seasonal cycle by a series of sine waves, a constant and possibly a linear trend. This comes with the disadvantage that a seasonal cycle of arbitrary shape is only approximated with a finite (and usually small) number of sine waves. In turn, this has several advantages

- There is an analytic and continuous function for the seasonal cycle, that can be evaluated at any time of the year.
- A much smaller amount of parameters is necessary to describe the seasonal cycle. For 4 sine waves, a constant and linear trend this would be 10.
- The seasonal cycle is smooth.

This is an example of sea surface temperature from the ECCO2 data set. The seasonal cycle is nicely estimated and any lagged Autocorrelation at 365.25 days is removed from the time series. The result is totally comparable to the standard approach described above (denoted here by rmean, as a running mean filter is applied afterwards for smoothing)