Climate change is a topic of great importance, and a lot of work is focused on quantifying significant
variations in the properties of climatological variables, in particular, of surface air temperature (SAT). Although
changes in local seasonal cycles (such as the amplitude and the phase lag to the insolation) have been investigated,
changes in large-scale patterns of faster SAT variability (on a daily time-scale) remain poorly understood.
Here we perform a Hilbert analysis of daily SAT reanalysis data covering the Earth’s surface, and identify
the geographical regions where inter-decadal changes are more pronounced.
A recent study demonstrated that, in a class of networks of oscillators, the optimal network reconstruction from
dynamics is obtained when the similarity analysis is performed not on the original dynamical time series, but on
transformed series obtained by Hilbert transform.  That motivated us to use Hilbert transform to study another
kind of (in a broad sense) “oscillating” series, such as the series of temperature. Actually, we found that Hilbert
analysis of SAT (Surface Air Temperature) time series uncovers meaningful information about climate and is
therefore a promising tool for the study of other climatological variables. 
In this work we analysed a large dataset of SAT series, performing Hilbert transform and further analysis
with the goal of finding signs of climate change during the analysed period. We used the publicly available
ERA-Interim dataset, containing reanalysis data.  In particular, we worked on daily SAT time series, from year
1979 to 2015, in 16380 points arranged over a regular grid on the Earth surface. From each SAT time series we
calculate the anomaly series and also, by using the Hilbert transform, we calculate the instantaneous amplitude
and instantaneous frequency series.
Our first approach is to calculate the relative variation: the difference between the average value on the last
10 years and the average value on the first 10 years, divided by the average value over all the analysed period. We
did this calculations on our transformed series: frequency and amplitude, both with average values and standard
deviation values. Furthermore, to have a comparison with an already known analysis methods, we did these same
calculations on the anomaly series. We plotted these results as maps, where the colour of each site indicates the
value of its relative variation.
Finally, to gain insight in the interpretation of our results over real SAT data, we generated synthetic sinu-
soidal series with various levels of additive noise. By applying Hilbert analysis to the synthetic data, we uncovered
a clear trend between mean amplitude and mean frequency: as the noise level grows, the amplitude increases while
the frequency decreases.
Zappala, D.; Barreiro, M.; Masoller, C. Entropy: international and interdisciplinary journal of entropy and information studies Vol. 18, num. 11, p. 408- DOI: 10.3390/e18110408 Data de publicació: 2016-11-16 Article en revista
The Hilbert transform is a well-known tool of time series analysis that has been widely used to investigate oscillatory signals that resemble a noisy periodic oscillation, because it allows instantaneous phase and frequency to be estimated, which in turn uncovers interesting properties of the underlying process that generates the signal. Here we use this tool to analyze atmospheric data: we consider daily-averaged Surface Air Temperature (SAT) time series recorded over a regular grid of locations covering the Earth’s surface. From each SAT time series, we calculate the instantaneous frequency time series by considering the Hilbert analytic signal. The properties of the obtained frequency data set are investigated by plotting the map of the average frequency and the map of the standard deviation of the frequency fluctuations. The average frequency map reveals well-defined large-scale structures: in the extra-tropics, the average frequency in general corresponds to the expected one-year period of solar forcing, while in the tropics, a different behaviour is found, with particular regions having a faster average frequency. In the standard deviation map, large-scale structures are also found, which tend to be located over regions of strong annual precipitation. Our results demonstrate that Hilbert analysis of SAT time-series uncovers meaningful information, and is therefore a promising tool for the study of other climatological variables.