We study daily surface air temperature (SAT) reanalysis in a grid over the
Earth's surface to identify and quantify changes in SAT dynamics during the
period 1979–2016. By analysing the Hilbert amplitude and frequency we identify
the regions where relative variations are most pronounced (larger than

The unprecedented intensification of weather extremes is motivating research
aimed at understanding long-term climatic variations

Relative change in the time-averaged Hilbert amplitude.

Quantifying variations in surface air temperature (SAT) dynamics over several
decades is a challenging problem because of non-stationarity and the presence
of trends, measurement noise, multiple timescales, memory, and correlations
in the data

The Hilbert transform (HT) provides, for a real oscillatory time series,

Because many natural geophysical time series have a seasonal periodicity,
this has motivated the use of Hilbert analysis to characterise the
time-varying oscillation amplitude and to investigate phase shifts and
phase–amplitude couplings. Applications in various geophysical fields
are discussed in

We have recently applied the Hilbert transform to unfiltered daily SAT reanalysis

Here we use

In the main text we present results from an ERA-Interim daily SAT reanalysis

To apply the Hilbert transform (described in the Supplement) we first pre-process each raw SAT time series,

Variations in the Hilbert amplitude were quantified by the relative change,

A similar analysis was performed to detect changes directly from the raw SAT
time series,

Specifically, the amplitude of the climatology was calculated as

With the goal of relating changes in Hilbert frequency with changes in
the statistical properties of SAT time series, an analysis of the number of
zero crossings was performed: for each

Surface air temperature in two regions where a clear change in the
oscillation amplitude in the last 10 years is observed with respect to the first
10 years.

A statistical significance analysis was performed by surrogating Hilbert
series. For each amplitude time series (i.e. in each grid point) 100 shuffle
surrogates were generated and for each surrogate the relative change,

We analyse the maps of

Figure

In both time series we clearly observe a change in the amplitude of the
oscillations in the last 10 years with respect to the first 10 years,
having a visual confirmation of the changes detected by the Hilbert amplitude.
The red spot in Amazonia, whose SAT series shown in
Fig.

Next, we compare the changes detected by the Hilbert amplitude with those
computed directly from SAT (by decomposing the SAT time series into climatology
and anomaly, as explained in Sect.

Relative change in amplitude fluctuations computed from the variance
of the

Relative change in the time-averaged Hilbert frequency (in units of
oscillations per year).

Figure

The good qualitative agreement seen in the spatial structures in these maps confirms that Hilbert analysis directly applied to unfiltered SAT indeed gives a physically meaningful instantaneous amplitude, with average and variance values that are consistent with those computed from SAT.

In Fig.

Normalised SAT time series and number of zero crossings in the
regions indicated with a circle in Fig.

Figure

These two areas of opposite signs suggest that, between the initial and the
final decade, there is a shift of the inter-tropical convergence zone (ITCZ)
toward the north. The ITCZ involves strong convective activity, which causes
rapid fluctuations of SAT, thus giving high values of instantaneous
frequency, as shown in Fig.

To gain insight into the physical meaning of the changes that are captured by
the Hilbert frequency, we use an alternative approach to estimate frequency
variations: we define “events” as the zero crossings of SAT time series

Figure

Figure

To demonstrate the robustness of our findings, in the Supplement
we compare the results obtained from ERA-Interim with those
obtained from another reanalysis dataset, NCEP-DOE. We find a good
qualitative agreement in the spatial structures in the maps of

We have used Hilbert analysis to quantify the changes in SAT dynamics, on a global scale, that have occurred over the last 3 decades. From the SAT time series with daily resolution we derived the amplitude and the frequency time series and then calculated the relative change (between the first and the last decade) in the average and variance of these series. Large variations in the Hilbert amplitude (more than 50 %) in the Arctic and in Amazonia were respectively interpreted as due to ice melting and precipitation decrease. The analysis of the Hilbert frequency also uncovered areas of large changes. In particular, two areas of opposite changes in the eastern Pacific Ocean and two areas of increase in the western Pacific Ocean suggest a shift towards the north and a widening of the ITCZ. While there is evidence that the ITCZ has moved north–south in the past, to the best of our knowledge our work is the first to confirm this migration in the last decades. Our findings have important implications because, as the ITCZ is the ascending branch of the Hadley cell, its migration affects both the Earth's radiative balance and the release of latent heat that drives the tropical atmospheric circulation. Taken together, these effects have not only local but also far-reaching climatic consequences. Additional analysis provided in the Supplement confirms the robustness of these observations.

As the methodology used here can be applied to many other climatological time series that exhibit well-defined oscillatory behaviour, we believe that our work will stimulate new research to identify and quantify the impacts of climate change directly from observed data.

The Hilbert algorithm used is available in

The authors declare that they have no conflict of interest.

This work was supported in part by the LINC project (EU-FP7-289447), ICREA ACADEMIA (Generalitat de Catalunya, Spain), and by the Spanish MINECO/FEDER (FIS2015-66503-C3-2-P). Dario A. Zappalà also thanks the European Social Fund and the Generalitat de Catalunya for the FI-AGAUR scholarship. Edited by: Sagnik Dey Reviewed by: two anonymous referees