Firstly, in view of the deviations, it's clearly a bit more than just two frequencies, but the general form is as described. A closer look at freq. spect would be interesting.
explanation: first difference, done twice on daily data. Then triple running mean.
triple running mean : 365 273 204 days.
[technical detail]
freq resp of running mean (sinc fn) has a negative lobe, that badly distorts the result, at tan(x)=pi ; x=1.3371
so each stage is 1.3371 times shorter than the previous one. This removes the neg. lobes and gives a profile similar to gaussian filter but with a true zero at 365, which is better than gaussian for removing a specific fixed frequency.
You say you did the same thing. What did you do for 3RM ?
I also had to 'correct' erratic dates in Cryo Today's data. They are within a day but all over the place with loads of dupes. Esp. leap years but elsewhere too.
[Edit: "all over the place" probably gives false impression. Most intervals are one of two close values but there are a significant number duplicate dates (which thus give infinite rate of change) most but not all of which are related to not correctly processing leap years. There are other irregularities, especially at end of 1980s which is rather messy. None of this too visible if you just look at the time series but needs to be resolved before doing the first difference calculation. Or any other processing, for that matter. Such as filtering with a convolution filter (even a crude running mean) that assumes constant sampling intervals. ]
There are the correct number of data in each year so the best I could do (since they do not answer questions about why their dates are erratic) was to renumber with equally spaced dates.
I would rather have them correct or explain it but in absence of any response it seems a reasonable solution.
I don't think such a result is likely to be the result of my mangling the data but all such questions are fair game.
Does that adequately answer your question ?
Are you now able to get something similar?