Introduction
The unprecedented intensification of weather extremes is motivating research
aimed at understanding long-term climatic variations
that can have
profound socio-economic impacts and trigger
complex ecological adaptation mechanisms .
Relative change in the time-averaged Hilbert amplitude.
(a) Amplitude averaged over the first 10 years (January 1979 to
December 1988). (b) Amplitude averaged over the last 10 years
(July 2007 to June 2016). (c) Relative change in the Hilbert amplitude,
Δa/<a>. (d) Relative change in amplitude of the seasonal
cycle computed from the amplitude of the climatology,
Δa(clim)/a(clim). A good qualitative
agreement is seen in the spatial structures in (c) and (d).
Importantly, the structures uncovered by the Hilbert amplitude are well defined
in comparison with those uncovered by the analysis of the climatology
amplitude, which look noisier.
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 ; in addition, reanalysis data
can be unreliable (due to the lack of observational constraints in many
geographical regions), and reanalysis time series are insufficiently long (as
reanalysis starts at the beginning of the satellite era). These challenges
have motivated the use, for climate data analysis, of data-driven approaches
that have been commonly used for investigating observed complex signals in
other fields of science (e.g. neurological, physiological, financial, etc.).
Univariate analysis tools that have been used to analyse SAT time series
include detrended fluctuation analysis, fractional analysis, and wavelet
analysis. Bivariate analysis and the complex network approach has also
allowed us to uncover inter-relations between SAT anomalies in
different regions . In this
approach the seasonal cycle is removed to eliminate the influence of
solar forcing, and the links represent correlations (linear or non-linear) or
statistical similarities between SAT dynamics in different areas
. On the other hand, changes in the SAT
seasonal cycle have also been investigated, and a trend toward reduced cycle
amplitude has been detected in many regions .
However, changes in SAT dynamics over several decades (such as those observed in Fig. 1) have
not yet been investigated at a global scale. In order to fill this gap, we
use Hilbert analysis (described in the Supplement) to
investigate SAT time series with daily resolution (reanalysis covering the
Earth's surface in the period 1979–2016). Our goal is to detect the most
sensitive regions (“hotspots”) where variations in SAT dynamics over the
last decades are more pronounced.
The Hilbert transform (HT) provides, for a real oscillatory time series,
x(t) with t ∈ [1, T], an instantaneous amplitude, a(t), and an
instantaneous frequency, ω(t), for each data point of the time
series and thus allows us to characterise how the amplitude and the frequency
of a signal vary in time. If a signal does not have a sufficiently narrow
frequency band, a(t) and ω(t) will not have a clear physical
meaning . The usual solution is based on band-pass
filtering to isolate a narrow frequency band; however, HT directly applied to
the signal can still yield useful information. An alternative solution is
based on the Hilbert–Huang transform that combines
Hilbert analysis with empirical mode decomposition that decomposes an
arbitrary real time series into components, each having the physical meaning
of a rotation in the complex plane.
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 . As more recent
examples, used Hilbert analysis to characterise the
daily variability of the Seine river flow from 1950 to 2008, uncovering
linkages between river flow variability and global climate oscillations (the
North Atlantic Oscillation and the Madden–Julian Oscillation).
used Hilbert analysis to compute the daily phase
shift between temperature signals recorded at the ground surface and at a
depth of 5 m in two meteorology stations in Taiwan from 1952 to 2008.
Significant reductions in the phase shift from the 1980s to 1990s were found,
which was interpreted to be related to the warming of the Pacific Decadal
Oscillation. applied Hilbert analysis to rainfall
time series in India and found that the multi-scale components of rainfall
series have a similar periodic structure as global climate oscillations (the
Quasi-biennial Oscillation, El Niño Southern Oscillation, etc.).
We have recently applied the Hilbert transform to unfiltered daily SAT reanalysis
. We have shown that the maps of time-averaged
Hilbert frequency, <ω>, and standard deviation,
σω, revealed well-defined large-scale structures which were
consistent with known dynamical processes.
Here we use a(t) and ω(t) to quantify SAT variations. Our hypothesis
is that changes in a(t) and ω(t) can yield information about
variations in SAT dynamics. Specifically, we are interested in addressing the
following questions: which properties of a(t) and ω(t) display
relevant variations? Where are the regions in which these variations are more
pronounced? Which processes can be responsible for these variations? Can these
variations be used as a quantitative measure of regional climate change?
Methods
Hilbert analysis
To apply the Hilbert transform (described in the Supplement) we first pre-process each raw SAT time series, rj(t) (where
j ∈ [1, N] represents the geographical site and t ∈ [1, T] represents
the day): we eliminate the linear trend and normalise to zero mean and unit
variance, obtaining xj(t). The Hilbert transform is then applied to xj(t),
obtaining yj(t) = HT[xj(t)]. From xj(t) and yj(t),
the amplitude aj(t) and the phase φj(t) were calculated as
aj(t) = [xj(t)]2+[yj(t)]2 and
φj(t) = arctan[yj(t)/xj(t)]. Taking into account the signs
of xj(t) and yj(t), the phase is constrained to the interval [-π, π).
Whenever an extreme value of the interval is reached, the phase jumps
to the other end of the interval. By eliminating these sudden jumps (using a
standard library function that appropriately adds ±2π) we
“unwrapped” the phase obtaining a continuous variation in time from which
the frequency time series, ωj(t), was obtained by calculating the
derivative. Since the Hilbert algorithm
gives deviations from the true values of the amplitude, phase, and frequency
near the extremes, in each time series (aj(t), φj(t), and
ωj(t)) we disregarded the initial and final 5% (see the
Supplement). This way, we have time series of length T = 12 328.
Measures used to quantify variations in SAT dynamics
Variations in the Hilbert amplitude were quantified by the relative change,
Δa/<a> = (<a>l - <a>f)/<a>, where <a>f is
the average value of the amplitude during the first 10 years of the time
series (January 1979 to December 1988), and <a>l
during the last 10 years (July 2007 to June 2016). Analogously, we
calculated the relative change in amplitude variance, Δσa2/σa2,
of average frequency, Δω/<ω>,
and of frequency variance, Δσω2/σω2. In the
Supplement we analyse how the spatial structures uncovered depend
on the time intervals used to calculate the relative variations: we compare
with relative variations during the first and final 5 years of the
reanalysis and also during the first half-period and the second half-period
of the reanalysis. While the values of the relative variations vary with the
time interval considered, the spatial maps are remarkably robust as the same
structures are found with the three time intervals considered.
A similar analysis was performed to detect changes directly from the raw SAT
time series, rj(t), by computing the amplitude of the climatology
(or seasonal cycle), cj(t), and the variance of anomaly time
series, zj(t).
Specifically, the amplitude of the climatology was calculated as
aj(clim)(I) = max[cjI(t)] - min[cjI(t)], where
cjI(t) is the climatology series calculated only in the time interval I.
We remark that the climatology amplitude aj(clim)(I) is a
scalar number that depends on the choice of the time interval I. We
calculated the climatology amplitude in the first and last decade, as well as
in the whole series. As before, we used these values to calculate the
relative change Δa(clim)/a(clim). Also, the
variance of the anomaly time series zj(t) was calculated and then used to
find the relative change, Δσz2/σz2.
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 xj(t) we counted the number of
crossings through the mean value, x = 0. As with other quantities, we then
calculated the relative change.
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) Site of coordinates (7.5∘ S,
307.5∘ E) marked with a triangle in
Fig. c. (b) Site of coordinates
(75∘ N, 40∘ E) marked with a circle in
Fig. c.
Significance analysis
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,
Δas/<as>, was calculated. Then, the average over the
100 surrogates, <Δas/<as>>s, and its
standard deviation, σs, were used to define the significance
threshold: the relative change computed from the original data was considered
significant if it was higher than <Δas/<as>>s + 2σs
or lower than <Δas/<as>>s - 2σs. In the colour maps, regions where variations
are not significant are displayed in white. The same test was applied to
frequency variations and the other quantities, except for the climatology for
which a surrogate test is not applicable. In the Supplement various thresholds are considered and, in addition, a
non-parametric significance test is used. Here we present only the maps
obtained with threshold ±2σ because it is a compromise between
uncovering the spatial regions where SAT changes are pronounced and
disregarding the areas where the variations are small.
Results
We analyse the maps of <ω>, <a>,
σω2, and σa2 in the first 10 years and in the last
10 years of the period covered by the reanalysis, as well as the relative
change between the two decades.
Analysis of amplitude variations
Figure a and b
display <a> in the first and in the last 10 years,
respectively, and Fig. c displays the relative
difference (see Sect. for details). In
Fig. c we see an area of large increase (more
than 50 %) in average amplitude located in South America (red spot marked by
a triangle) and an area of large decrease (again, more than 50 %) located
in the Arctic (blue spot marked by a circle). The raw SAT time series in
these regions are displayed in Fig. .
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. a has an increasing amplitude, can be
interpreted in terms of changes in precipitation. In particular, the increase
in the Hilbert amplitude is linked to the decrease in precipitation and to the
lengthening of the dry season (as reported in ). This is due to the fact that when
precipitation decreases, the fraction of solar radiation that is not used for
evaporation is used to heat the ground, which in turn heats the surface air.
This leads to higher extreme temperatures during the dry seasons, as can be
observed in Fig. a. Regarding the blue spot in
the Arctic region where the SAT series shown in
Fig. b has a decreasing amplitude, it can be
interpreted as due to the melting of sea ice. In fact, when ice is present at
the surface of the sea, it acts as an insulator preventing heat exchange
between sea and air. This causes a large amplitude in the SAT cycle. On the other
hand, if the ice melts, the air–sea heat exchange reduces the amplitude of
the cycle. In particular, during winter the air temperature is mitigated by
the sea and tends to have more moderated values. It is important to take into
account that this blue spot is in a region for which the observational
constraints from satellites on the reanalysis are scarce, which decreases the
quality of the reanalysis in the region. Therefore, in order to check whether
the detected changes are robust, we performed the same analysis using the
NCEP-DOE reanalysis dataset. The results, presented in the Supplement, confirm
the presence of the blue spot in the Arctic.
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. ). Since the climatology term
retains the seasonal variation, we expect its amplitude change to give
similar indications as the Hilbert amplitude change. On the other hand, the
anomaly term contains all the rapid variability, so we expect its variance to
give similar results as the variance of the Hilbert amplitude.
Relative change in amplitude fluctuations computed from the variance
of the (a) Hilbert amplitude, Δσa2/σa2;
(b) the anomaly time series, Δσz2/σz2.
Relative change in the time-averaged Hilbert frequency (in units of
oscillations per year). (a) Average in the first 10 years (1979–1988).
(b) Average in the last 10 years (2007–2016).
(c) Relative change in Hilbert frequency, Δω/<ω>.
(d) Relative change in the number of zero crossings of the
normalised SAT time series. In (a) and (b) the colour scale
is adjusted to represent in white the regions where the average frequency is
one oscillation per year. In (c) and (d), a good
qualitative agreement of spatial structures is seen; however, we note that
the
Hilbert frequency detects stronger variations than those measured by the
number of zero crossings.
Figure c and d,
which respectively display the relative change in Hilbert amplitude and in
climatology amplitude, and Fig. a and b, which respectively display the relative
change in Hilbert amplitude variance and in anomaly variance, confirm these expectations.
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. a and b, however, there is a difference in the
eastern Pacific Ocean in the area marked with a circle. In particular, in
Fig. b there is an area with large decrease
in variance (dark blue, around -100 %), while in Fig. a the decrease is less
pronounced (light blue, around -65 %) and extended over a smaller area. In
addition, in Fig. a there is a reddish orange
area that indicates a moderate increase in variance (around 45 %), while in
Fig. b such an area is absent. The reasons underlying
these differences will be discussed later.
Normalised SAT time series and number of zero crossings in the
regions indicated with a circle in Fig. c. In the
red region (2.5∘ N, 245∘ E) (a), the number of
zero crossings increases in the last 10 years with respect to the first 10 years (289 and 202, respectively), while in the blue region (7.5∘ S,
250∘ E) (b), it decreases (128 and 258 in the last and first 10 years).
Analysis of frequency variations
Figure a displays the average frequency <ω>
in the first 10 years, Fig. b
in the last 10 years, and Fig. c displays the
relative change, Δω/<ω>. In
Fig. c we note that in the eastern Pacific
Ocean there are two small areas, enclosed by the circle, of intense increase
(red) and decrease (blue) in frequency. They both represent frequency changes
whose absolute values are larger than 100 % and correspond to the same region
where differences were detected in Fig. .
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. a and b. Therefore, in
the relative change in frequency, in regions corresponding to the initial
position of the ITCZ we see a decrease, while in regions corresponding to the
present position of the ITCZ we see an increase. For the same reason, the two
red areas in the western Pacific Ocean (indicated by two squares) suggest an
expansion of the tropical convective regions. This interpretation is in
agreement with previous works that have related a northward shift of the ITCZ to
an inter-hemispheric temperature gradient, such as the one experienced during the
last decades . Regarding the red areas in
the north Atlantic, the north Pacific, and the south Pacific, they are
consistent with an increase in the occurrence of fronts, which cause large
daily fluctuations of temperature and thus an increase in the Hilbert frequency.
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
(detrended and normalised to zero mean as described in
Sect. ). Then, we count the number of events in the first
10 years and
in the last 10 years and calculate the relative variation.
Figure d displays the map of relative change in
zero crossings. We see that there is a qualitative good agreement with the
spatial structures seen in Fig. c, thus
providing a physical interpretation for the observed variation in the Hilbert
frequency: the areas where the frequency increases (decreases) correspond to
areas where the number of zero crossings increases (decreases). We note that
the relative variations in the Hilbert frequency are more pronounced than those
in the number of crossings, and this specifically holds in the regions where
frequency variations are interpreted in terms of ITCZ migration.
Figure a and b display SAT time
series in the dipole region indicated with the circle in
Fig. c and also indicate (in red) the
zero crossings. We can understand the difference that was detected in this
region between the variance of the Hilbert amplitude
(Fig. a) and the variance of the anomaly
(Fig. b). This difference is explained in the
following terms: in the first decade the seasonal cycle is more irregular
than in the last decade, probably a consequence of an El Niño event
in 1982–1983. The anomaly series contains these slow fluctuations as well as
the rapid ones, and thus its variance is affected by both effects. In
contrast, the Hilbert amplitude is less affected by the slow fluctuations as
its variance captures mainly the rapid fluctuations of SAT.
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 <ω>, <a>, σω2, and σa2, but we
also discuss some relevant differences. In addition, to further understand
the relationship between the statistical properties of SAT and those of the Hilbert
amplitude and frequency, in the Supplement we apply
Hilbert analysis to synthetic data generated by an autoregressive AR(1)
process. We chose an AR(1) process because it is commonly used in the
literature to model climate data. We find that, when increasing the noise
intensity in the synthetic series, the Hilbert amplitude decreases while the
frequency increases, which shows that this trend is also observed in real SAT time series.
Conclusions
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.