Jan, I don't know how much is required to get at the primary spectral components.
This paper applies the technique to the same kind of geophysical time-series data
The multitaper method does reduce the noise background so therefore might lift out some of the weaker spectral lines. However it doesn't do much with respect to the strongest.
This appears to be the most impressive approach I have come across
"Application of stabilized AR-z spectrum in harmonic analysis for geophysics"
Perhaps I should be using MM to recover the time-series? The plain Fourier series works well for me so I don't know what else it will add at this level. For example, I already know that the signal is heavily aliased, but this is due to a physical aliasing not related to sampling aliasing. The spectral leakage caused by having signals of 27.2122, 27.312, 27.554 so closely separated may not be a factor if the time series is long enough. In any case, the physical aliasing separates these as 2.37, 2.715, and 3.91, which are much more widely separated. I think that is what is causing all the confusion in the first place -- the fact that no one is aware that nonlinear physical aliasing is occurring. Maybe get that point across first and then we can use more advanced spectral techniques. I don't think I am missing anything but who knows.